Upload
independent
View
0
Download
0
Embed Size (px)
Citation preview
Prof. Dr. Uwe Walz
Faculty of Economics and Business Administration
Goethe University Frankfurt am Main
Winter Term 2016/2017
Basic Course
Microeconomics
c⃝2016
Contents
I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
I.1 Economic thinking: Introductory examples . . . . . . . . . . . . . . 6
I.2 Central questions and areas of economics . . . . . . . . . . . . . . . 7
I.3 What is microeconomics about? . . . . . . . . . . . . . . . . . . . . 12
I.4 Some basic (micro-)economic rules . . . . . . . . . . . . . . . . . . 14
I.5 The meaning of economic models . . . . . . . . . . . . . . . . . . . 18
I.6 Overview of the lecture . . . . . . . . . . . . . . . . . . . . . . . . . 19
II Supply and demand: A simple market model . . . . . . . . . . . . . . . . . 22
II.1 Demand and demand curve . . . . . . . . . . . . . . . . . . . . . . 22
II.2 Supply and supply curve . . . . . . . . . . . . . . . . . . . . . . . . 24
II.3 Equilibrium price and adjustment mechanism . . . . . . . . . . . . 24
II.4 Allocation and efficiency . . . . . . . . . . . . . . . . . . . . . . . . 26
III Household decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
III.1 Budget constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
III.2 Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
III.3 Optimal consumption decision . . . . . . . . . . . . . . . . . . . . . 49
III.4 Influence of prices/income on the demand for goods . . . . . . . . . 59
III.5 Overall demand for goods . . . . . . . . . . . . . . . . . . . . . . . 78
III.6 Work-leisure decisions . . . . . . . . . . . . . . . . . . . . . . . . . 85
2
III.7 Intertemporal decisions . . . . . . . . . . . . . . . . . . . . . . . . . 89
III.8 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
IV Production theory and company decisions . . . . . . . . . . . . . . . . . . 103
IV.1 Technology and production . . . . . . . . . . . . . . . . . . . . . . 105
IV.2 Cost minimisation, factor demand and cost functions . . . . . . . . 122
IV.3 Profit maximisation and goods supply of the individual company . . 144
IV.4 Goods supply of all companies in an industry . . . . . . . . . . . . 151
V Market equilibrium with perfect competition . . . . . . . . . . . . . . . . . 157
V.1 Market equilibrium and efficiency . . . . . . . . . . . . . . . . . . . 157
V.2 Interventions in the market equilibrium . . . . . . . . . . . . . . . . 160
VI Market equilibrium with imperfect competition . . . . . . . . . . . . . . . 170
VI.1 Traditional monopoly theory . . . . . . . . . . . . . . . . . . . . . . 171
VI.2 Oligopoly and game theory . . . . . . . . . . . . . . . . . . . . . . . 181
VII Asymmetric information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
VII.1 Asymmetric information and market failure . . . . . . . . . . . . . 194
VII.2 Adverse selection and signals . . . . . . . . . . . . . . . . . . . . . . 196
VII.3 Moral hazard and incentives . . . . . . . . . . . . . . . . . . . . . . 200
VIII Theory of externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
VIII.1 Externalities and the inefficiency of the market mechanism . . . . . 204
VIII.2 Strategies to internalise externalities . . . . . . . . . . . . . . . . . 207
3
Persons
Prof. Dr. Uwe Walz
Professor of Industrial Organisation
Goethe University Frankfurt am Main
Email: [email protected]
Teaching Assistant: Jan Krzyzanowski
4
Notes on the literature
This lecture note summarises the essential content of the course. It is not an alternative
textbook! The overview handed out during the lecture provides some details on standard
works of microeconomics. The following book is urgently recommended:
• Pindyck, R. S. and Rubinfeld, D.L. (2013), Microeconomics, Prentice Hall, New
Jersey, 8th edition
The lecture notes contain details on the relevant chapters in Pindyck/Rubinfeld (2009),
abbreviated to PR. The concluding exam contains only questions and tasks that have been
covered in the course. However, we reserve the right to address a few questions whose basics
are covered in the course, but which require additional reference to PR. In addition, the
textbook by Mankiw is an excellent accompaniment and offers a fundamental introduction
to economics:
• Mankiw, G. (2011), Principles of Economics, International Edition, South-Western,
6th revised edition
5
I Introduction
Literature for preparation and follow-up:
Mankiw, chapters 1 and 2
I.1 Economic thinking: Introductory examples
The essential objective of economics in general and microeconomics in particular is to
impart and to train economic thinking. The following introductory examples provide an
initial insight into the manner in which economists think and analyse problems.
Example 1: Deciding on a degree course and a subject
If we examine the decision on the time of starting a degree course and the choice of subject,
we observe that young people typically embark on their studies immediately after finishing
school or at the latest after completing an apprenticeship. The subjects favoured in this
case are economics and law.
Older people who decide upon a degree course, on the other hand, (e.g. mature students
over 50) typically choose subjects such as philosophy, history or comparative cultural
sciences.
How can these observations be explained (economically)?
Example 2: Travelling by train or car?
You would like to visit friends in Hanoi and are considering whether to make the journey
by car or train. You have the following information about the costs:
• Train journey:
– You purchased a Bahncard for 230 monetary units in September.
– The cost of the train journey to Hanoi (one way), with Bahncard, including
suburban transit: 110 monetary units.
6
• Journey with your own car:
– The outward journey is 400 km.
– Annual insurance and tax costs are 900 monetary units.
– Annual mileage: 12,000 km
– Capital costs: 1200 monetary units
– Fuel consumption: 9 litres/100 km, fuel price: 1.50 monetary units per litre
– Tyres and repairs: 1200 monetary units/year
What is the cheapest means of transport for your trip to Hanoi?
Example 3: Why is the food served in airplanes so bad?
A common complaint of air travellers is that the food on board an airplane is relatively
poor. Any restaurant offering a similar quality of food would probably lose most of its
customers and go bankrupt. Due not least to the fact that the poor quality of food tends
to deter rather than positively encourage air travellers, some airlines (low-cost carriers)
have de facto completely suspended the provision of meals on board.
Why is this so? Think about the costs and benefits of providing meals on an airplane!
I.2 Central questions and areas of economics
I.2.1 Areas
Microeconomics as a subarea of economics
Economics is the science of deciding about scarce resources. Modern societies, oriented to
the market economy, function by means of the interaction of millions of decision makers
such as private households and companies. The discipline of economics observes all deci-
sions and processes in an economy such as Vietnam, the USA or also that of privinces.
It is not limited to the analysis of companies, as is typically the case in the discipline of
7
business administration. Rather, the behaviour of households is also observed, while the
interplay between companies and households in different markets is analysed.
Business administration versus economics
The differentiation between business administration and economics is often difficult and
indistinct. The discipline of business administration concentrates on the analysis of compa-
nies and their relationship to their environment (capital markets, product markets, taxes,
etc.). The discipline of economics analyses the decision of households and companies to
the same degree, and the actions of the state are also considered. All of the processes in an
economy are examined, while the view of company activities is less detailed. However, the
distinction is often blurred, especially in the field of microeconomics; an essential difference
is that in business administration, assigned economists address more specifically the de-
tails of the activities of companies (organisation, personnel, finance, accounting, marketing
etc.), while microeconomics abstracts more strongly from details, observes the company
as a whole, and concentrates on its actions with competitors in the market.
I.2.2 Central questions of economics
How do market economies work and what role does the state play in this? Why do market
economies work quite well with regard to efficiency and economic growth? An initial answer
can be found in Adam Smith (1776): the market and price mechanism as an invisible hand.
Outline of the basic idea: Households and companies react to price changes by adjusting
their behaviour with regard to demand and supply. The market price signals scarcity in an
economy: if something is relatively scarce (e.g. crude oil), it becomes more expensive and
is less demanded (e.g. use of alternative fuel). In addition, an incentive to supply more
crude oil arises (e.g., by means of increased exploration). Prices coordinate the behaviour of
individuals, i.e., demand and supply correspond at the balanced market price. Ultimately,
the pricing and market mechanism means that self-serving individual behaviour leads to
the best results for society. To quote from Adam Smith:
It is not from the benevolence of the butcher, the brewer, or the baker that we
expect our dinner, but from their regard to their own interest. [...] By pursu-
ing his own interest he frequently promotes that of the society more effectually
8
than he really intends to promote it.
Adam Smith (1776), „An Inquiry into the Nature and Causes of the Wealth of
Nations“
In other words it is not the altruistic attitude of the individual that leads to the production
of goods, but rather the legitimate pursuit of profit by the producers in competition that
leads to an increase in the welfare of all.
The subjects in economics are very varied. Examples of specific questions include:
• What determines the demand behaviour of households?
• Which products are offered at which prices by companies to the market?
• How do taxes work?
• Why does inflation exist?
• Can share prices be predicted?
• What effect does demographic change have on security in old age?
• Which causes are responsible for the fact that the economy in some countries grows
much more quickly than in others?
• Why do countries trade with each other?
• Why do multinational companies exist?
• When do currencies appreciate in value, when do they devalue?
I.2.3 The different areas of economics
There are different possibilities for dividing the subject. One can first classify according
to fields: e.g., labour market economics, international economics, monetary economics.
Furthermore, the different methodologies can be distinguished (classical categorisation in
9
Germany): economic theory, economic policy and finance. This course follows the division
according to the relevant approach: microeconomics, macroeconomics, economic policy.
In the following, we outline briefly the three areas, before addressing microeconomics in
more detail.
Microeconomics
Microeconomics is generally concerned with the analysis of individual economic decisions
by households and companies. Furthermore, the interaction between households and com-
panies in markets -especially in product, capital and labour markets-are observed. Some
of the following questions typically arise:
• What effect do oil price increases have on the automobile market?
• What price will be determined in the context of an (internet) auction? What is an
optimal bidding strategy?
• What effects do minimum wages have on unemployment?
• How should managers be rewarded?
• What impact does a prohibition on drugs have on crime?
• What strategy should a competitor of Facebook adopt in order to prevail in com-
petition, considering the highly installed basis of Facebook? What are the main
problems of entering the market as experienced by GooglePlus, for example?
• What incentive problems arise if the liability of the individual for his actions is
suspended or greatly reduced, for example, in the course of bailing out banks and
states?
Macroeconomics
Macroeconomics examines general economic phenomena, i.e., economic factors that affect
the entire economy. Typical questions in macroeconomics include:
• What effect does an oil price increase have in the prices in all markets, i.e. on the
level of prices and inflation?
10
• What are the consequences of national debt?
• How valuable is the independence of the central bank?
• What determines the growth rate of an economy as a whole?
Please note that micro and macroeconomics are very closely connected. Behind all macroe-
conomic developments are many individual decisions that are analysed more closely by
microeconomics. Therefore, modern macroeconomics places great value on a strong mi-
crofoundation of macroeconomic models and analyses. On the other side, macroeconomic
developments naturally also have repercussions for individual economic entities and mar-
kets. For this reason, a good microeconomist should always bear the overall economy in
mind.
Economic Policy
The discipline of economic policy addresses the ways in which the state influences economic
events and the alternatives. Some examples of questions include:
• What consequences do the actions of the state have (positive analysis)?
• How should the state act (normative analysis)?
• What effect does regulation have on the labour market (e.g. employment protection
legislation)?
• What consequence does an expansive monetary policy have on inflation and employ-
ment?
As a subject and a course, economic policy uses the methodical content of micro and
macroeconomics in order to analyse specific questions. Against this background the tradi-
tional role of the subject of economic policy is receding increasingly into the background
in Germany universities. Economic policy questions are divided into micro and macroe-
conomics and the typical economic curriculum is divided into microeconomics, macroeco-
nomics and econometrics (see, for example, the structures of our PhD programme in the
Faculty of Economics and Business Administration at the University of Frankfurt).
We now come to a more precise definition of microeconomics.
11
I.3 What is microeconomics about?
I.3.1 More precise portrayal of the field
As already explained, microeconomics analyses the behaviour of individual economic enti-
ties and their interaction on markets. The following simple diagram illustrates the funda-
mental mechanism of supply and demand-the decisive factors of microeconomics.
Main graphic of the lecture
6
-
D
D’S
S’
Price p
Quantity y
The diagram shows a demand curve DD’ and a supply curve SS’ in an arbitrary market.
The curves show the different price-quantity combinations (p-y) of the demanders and
suppliers. Here, the connection between price and quantity is very simple: at high prices,
demanders tend to buy a smaller quantity of goods, while suppliers have hardly any in-
centive at low prices to offer a large quantity of goods. The intersection of both curves
gives the market equilibrium, at which the quantities supplied and demanded correspond.
Essentially, product and factor markets (i.e., labour and capital markets) are considered.
The determining factors of demand DD’ on the product market can be traced back to
optimal household behaviour, while on the labour and capital markets, companies emerge
as demanders of labour and demanders of capital (in the form of borrowed or equity cap-
ital). The determining factors of supply SS’ on the product market can be attributed to
optimal company behaviour, while households offer their labour on the labour market and
12
their savings as capital on the capital market. Microeconomic theory provides very useful
instruments to explain the market economy processes described. The individual elements
(chapters in this course) will develop slowly but surely to form an overall picture. Starting
from the market we examine and derive the individual supply and demand curves before
ultimately bringing them together again. It therefore makes sense to refer constantly to the
diagram above so as not to lose sight of the big picture. Ultimately, the goal of the analysis
is the overall market equilibrium, even when, for example, a very detailed examination of
the household aspect is being made.
I.3.2 Microeconomics as a decision-making theory
Microeconomics is essentially a decision-making theory. The core of the analysis is always
how individuals decide under economic scarcity. Scarcity in this context means that the re-
spective quantity of time, money, natural resources, etc. is limited and the economic entity
must economise with the quantity available. In order to be able to analyse this decision-
making behaviour theoretically, some fundamental assumptions must be made about the
behaviour of individuals. The central - neoclassical - assumption of the course corresponds
essentially with the artificial construction of the homo economicus. This is the ideal type
of the rational individual who acts in a self-interested manner. In its widest sense it can
also include altruism if it is in the interest of the individual concerned to improved the
situation of others also (such as one’s own children or relatives, but certainly also other
people). Even though the concept of the homo economicus has been the subject of some
criticism, it is nevertheless a good starting point for analysis and for ordering thoughts
and arguments. Examples of decision making under the assumption of the homo economi-
cus include the demand for goods, the choice of degree course, participation in general
democratic elections, the avoidance of environmental pollution, decisions on inheritance,
etc.
An essential differentiation between decision-making situations results from the number of
persons involved in the decision-making process. First, there is a wide range of decision-
making situations with a large number of market participants. This generally leads to
non-strategic behaviour among the participants. Each participant decides independently
of the others. Market prices and the behaviour of others are taken as given (i.e., cannot
13
be influenced). An example of non-strategic behaviour is the bread market: many small
bakeries offer bread and many households demand bread. Quite a different situation occurs
in decisions made in small groups, where strategic behaviour can frequently be observed.
In such a case, one’s own decision has a significant influence on the decisions of others and
vice versa. Examples of strategic decision-making behaviour include fuel stations situated
opposite each other on an arterial road (if one company reduces its prices, this has a large
impact on the other fuel station, which will/must react) or the behaviour of countries (e.g.
USA and Vietnam) in foreign trade (if one country increases its import duties, this has
consequences for the other country reaction might possible be anticipated). This type of
strategic interaction is addressed in the context of Game Theory (Chapter 6.2).
A further important point in microeconomic analysis is individuals’ current state of knowl-
edge or information. Do they know everything, i.e., are they fully informed, or are some
things unknown, for example in the future (such as the development of currency rates or
the weather)? Are all economic agents informed to the same degree or are there asymme-
tries of information (some of the market participants are better informed than the rest)?
We shall also address this aspect later in the course.
Last but not least, the question as to whether the economic entities bear the costs of or
benefit from their actions. If this is not the case, these are known as external factors.
An immediate example is provided by environmental pollution must be answered. For
instance, if we fly away on holiday, this leads to the emission of pollutants, the costs of
which - in the absence of relevant taxes - the individual must not bear. These are known as
negative external factors. We shall address the case of external factors in the final section
of the course.
I.4 Some basic (micro-)economic rules
We now introduce seven basic rules of economics that are important from the perspective
of microeconomics, cf. Mankiw (2011). The basic rules 1-4 refer to the decision-making
behaviour of individuals, while the basic rules 5-7 are concerned with the interaction of
people (via markets).
14
Basic rule 1: Nothing is for free
„There is no such a thing as a free lunch!“ (Originating from corresponding signs in restau-
rants in the American Midwest)
In exchange for anything he wishes to have, the individual must give something up. This
leads to permanent conflict of interests that can be solved by means of a cost-benefit
consideration. Examples include:
• Additional hours invested in studying BMIK costs leisure time (or an hour studying
BMGT)
• An additional bottle of wine implies less consumption of other goods
These considerations do not only apply to individuals, but also to societies as a whole:
• Additional expenditure on coal subsidies is not available for other uses (e.g. educa-
tion)
• Important conflict of interest: efficiency/growth and redistribution; for example:
more social welfare, higher income taxes, less exertion/investment, lower growth
• Tax reductions or spending programmes that are financed by national debt must be
repaid in future, or at the very least the interest burden must be serviced, which in
turn must be financed later by means of higher taxes or lower spending.
Basic rule 2: Opportunity costs must be considered
Opportunity costs must be sacrificed in order to be able to consume a unit of product X.
An initial example: a family firm is thinking about continuing the retail business. The
following data apply: the premises used belongs to the family; expected turnover: 250,000;
costs of material, power, personnel: 170,000
Argument: the company is profitable, which is why the business will be continued. How-
ever, the decision taken on this basis is wrong, as the opportunity costs have not been
considered. The family’s own premises could be let for 40,000. The salary of the shop
owner in alternative employment could amount to 50,000, for example. Therefore, a more
15
precise analysis, taking consideration of the opportunity costs, would suggest that aban-
doning the business would make more sense.
Second example: studying in Germany is free of charge! This statement is not correct, as
it does not take into account the income foregone during the period of study.
Basic rule 3: Principle of marginality
Individuals think in terms of limits: should I study for an additional semester or not?
Should I drink another beer or not?
An example from goods production:
Fixed costs: 200; variable costs per unit: 4; production: 100 units; average costs: 6
An additional order provides a yield of 5 per unit. Should the order be accepted? Answer:
yes, the marginal costs (additional costs) are lower than the yield
Further example: last-minute trips or the sale of cars (high fixed costs!)
Basic rule 4: People react to incentives
Changes to the framework conditions and/or the price change people’s behaviour. This
realisation is also important to companies, for example, when it comes to paying remuner-
ating staff (higher salary changes work motivation), and to economic policy, e.g. in the
health system. Zero-excess contribution in health insurance can lead to an excessive use
of services.
Further examples: higher repair costs are incurred in the case of full comprehensive insur-
ance, as driving behaviour is often more risky, which leads to a higher accident probability
and more frequent repairs. Similarly, the introduction of compulsory seatbelt wear can
have undesired consequences: does this change driving attitudes negatively? It is possible
that driving behaviour might become more risky if the consequences of an accident are
less severe?
Basic rule 5: In most cases, free markets lead to good results
In certain circumstances, which we shall explain in more detail in the lecture, free markets
lead to efficient results, i.e., to the avoidance of resource wastage. Think back to the
invisible hand: the pricing system and market mechanism coordinate the behaviour of a
16
large number of individuals. There is also empirical evidence of the benefits of a market
economy: the collapse of centralised planned economies in the last 15 years; growth is a
phenomenon of the last 150 years, i.e., since the existence of modern market economies.
Basic rule 6: State action can possibly improve the market results
Market failure can occur, for example, due to the influence of external effects (e.g. envi-
ronmental pollution) or when individual companies have control of the market (monopoly
etc.). The intervention of the state in the market can then possibly make sense, in order
to increase efficiency. The state can become active, for example, with an environmental
policy → introducing a market for environmental goods or charging environmental taxes.
Another example is the regulation of banks that systematically take risks (i.e. because of
the linkage of the banking system, their bankruptcy leads to very high costs in another part
of the banking system). This regulation, for example by means of capital requirements,
can increase wealth. The dismantling of market control by means of regulating monopolies
takes place in the context of competition policy.
Basic rule 7: Foreign trade is good for all
Countries are simultaneously competitors and partners, since trade between countries is
not a zero-sum game: everyone can profit, even if a country is less productive in all areas of
manufacturing than the other country. Countries specialise in goods that they can produce
relatively better (concept of comparative advantage). Here is a small example in figures:
• Country A (B) needs 1 (2) labour unit(s) to produce one unit of product and 4 (20)
labour units to produce one unit of product .
• Each country has 220 labour units available and it is assumed that the demanders
want to consume the same amount of each product.
• Self-sufficiency: Country A produces/consumes 44 units of each product and Country
B produces/consumes 10 units of each product.
• Specialisation through trade: Country A specialises completely in production (55
units), Country B produces 60 units of product and 5 units of product through
exchange, each country is in a better position, as the overall production is greater.
17
Because the countries’ resources are limited, specialisation means that all of them can
improve their situation. This can also been seen in the real world: countries that are
integrated in the global economy tend to have higher economic growth. The same principle
of specialisation also applies to other economic groups such as the members of a family.
I.5 The meaning of economic models
Modern industrial societies engaged in the market economy are characterised by the fact
that millions of economic entities act within them on a large number of markets (on labour,
intermediate goods, financial and goods markets). This leads to a high degree of complexity
and hampers the analysis of economic problems. Models have the goal of illustrating the
essential factors of the question in order to make analysis easier.
In the process, assumptions play a central role, as they allow us to concentrate on
decisive contexts and connections. Models must be unrealistic, as certain aspects of reality
must be ignored in order to gain knowledge. What is decisive here is the separation
between essential and non-essential aspects. An exact image of reality does not provide
any knowledge, as illustrated by the example of a map: a scale of 1:1 makes no sense; the
size (scale) of the map depends on the intended use. The more detailed the question, i.e.,
the more the specific structure of a company, for example, is affected, the less general the
formulation of the assumptions.
The model language in modern economic sciences is mathematics. Mathematics makes
things easier, although only a maximum of 10 percent of students in the basic degree
course believe so. A great advantage of formulating models mathematically is the fact
that assumptions must be named specifically. This avoids inconsistencies of argumentation,
which, upon more exact analysis, constantly arise in political talk shows, for example. The
mathematical abilities required for the foundation degree course are limited to differential
calculus (derivatives) and simple algebra. These instruments will be dealt with again in
the course before they are needed.
Generally, thinking in models promotes economic understanding. A two-track approach to
economic problems makes sense (also in the exam!): technical (graphical or analytic) and
economically intuitive.
18
I.6 Overview of the lecture
The lecture is divided into two areas. In the first part the basic model of perfect competition
with perfect information is observed. In conclusion, some assumptions of this market model
are suspended or modified.
The basic model makes the following fundamental assumptions:
• There are many demanders and also many suppliers. Example: stock markets, agri-
cultural markets (such as wheat or meat markets).
• All suppliers and all demanders know the prices and quality of all of the goods.
In the context of this approach the behaviour of households and companies is consid-
ered very comprehensively. Companies and households are suppliers and/or demanders in
different markets:
• On goods markets:
– for end products: companies are suppliers, households are demanders
– for intermediate products: companies are both suppliers and demanders
• On labour markets: companies are demanders of labour, households are suppliers of
labour
• On capital markets: companies are demanders of capital, households are suppliers of
capital
Accordingly, the first part of the lecture will deal initially with the theory of the house-
hold (analysis of the behaviour of private households) and the theory of the firm (anal-
ysis of the behaviour of companies). These two groups meet on each of the markets and
their interaction describes the market conditions and the market equilibrium.
Using the specific example of the market for leisure bicycles:
In the theory of the household:
What determines when and which bicycle is purchased and at which price?
19
In the theory of the company:
How many bicycles will be offered under which circumstances and at which prices?
In the market equilibrium:
Which price will prevail in the market? What factors change this market equilibrium (such
as environmental tax, household incomes, aluminium prices, etc.)?
Theory of the household
The preferences, goods and factors prices are given. The households decide in favour of
that which makes the best of their situation or (to phrase it technically) which maximises
their utility at a given income. The following decisions must be made:
• Optimal distribution of income across different goods
• Optimal distribution of income across different years (periods), i.e. how much should
be consumed, how much saved?
• Optimal provision of labour: how much do I want to work at a given wage rate?
Consideration must be taken of working burden and consumption capacity!
• How should one act in the event of insecurity? What risks are one prepared to take
(e.g.with investments), how much insurance is demanded?
Theory of the firm
Analysis of production and sales decisions by companies with given technologies and pre-
scribed goods and factor prices:
1. Production of each output unit at minimum costs. For this purpose the cost-
minimum demand for production factors (capital, labour and intermediate goods)
is determined.
2. Optimal sales quantity: how many product units should be sold at which sales price
in order to maximise profit?
20
Market equilibrium with perfect competition
The decisive assumption of perfect competition refers to the number of market partici-
pants: many companies, many households, each individual economic entity is relatively
small. Behaviour is passive, strategic considerations do not play any role. To analyse the
equilibrium, the following questions must be asked:
1. What does the market equilibrium look like?
2. How do taxes work? What effect do exogenous changes to price and quantity have on
the equilibrium? Example: impact of a depreciation of the dollar on the PC market
in Vietnam.
3. Is the market solution efficient?
In the second part of the lecture we depart somewhat from the basic model and various
assumptions will be suspended. In detail this means that the market equilibrium, among
other things, will be derived when there is only a small number of market participants
on one of the market sides. This is the case, for example, on an oligopolisticautomobile
market with only a few suppliers. It is also interesting to consider the possibility that
information is distributed unevenly (asymmetrically). A classic example of this can be
found in the used car and insurance markets. We will also examine the aforementioned
external effects, e.g., with regard to environmental pollution. The extensions to the basic
model appear again in the following overview:
Imperfect competition
There are only a few market participants in this model, for example the international oil
markets. The agents then act strategically. Game theory is an appropriate method of
examination.
Asymmetric information
Two economic entities that are connected to each other typically have different informa-
tion. Examples include employer/employee, insurer/insured, or used car salesman/used
car buyer. We shall examine the economic consequences that result.
Theory of external effects
In many economic activities the agents are not credited with the full benefits or costs of
21
their actions. The most important example of this effect is environmental pollution. The
consequence of external effects is that the market system is no longer efficient. We consider
approaches to solving this problem.
II Supply and demand: A simple market model
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 2
Supply and demand are the two terms that are used most frequently by economists. The
interaction of supply and demand characterises the essential working mechanism of market
economies. The following brief introduction is set against this background, using a simple
market model. The details of this market model are part of the microeconomics course in
the basic degree course.
Definition 1 A market consists of demanders and suppliers of a product or service.
Markets can take various different guises. With regard to the degree of organisation we are
presented with very differently structured markets: e.g., securities markets (Xetra trading)
or the sale of food and drinks at a swimming lake in summer. Furthermore, markets can
be classified according to the number of participants:
• Polypoly: one supplier, many demanders, e.g., stock markets
• Oligopoly: few suppliers, many demanders, e.g., airline industry
• Monopoly: one supplier, many demanders, e.g., local water utilities
II.1 Demand and demand curve
Typically, demanders differ not least in their willingness to pay for a certain product.
The overall demand function will now be derived using the example of a package holiday
market. We take a market for a trip with the following product description: 1 week in
Cuba, 1st week of November, 4-star hotel.
There exist 10 demanders with the following willingness to pay:
22
Demander Max. willingness to pay
A 2000 monetary units
B 1900 monetary units
C 1850 monetary units
D 1700 monetary units
E 1680 monetary units
F 1600 monetary units
G 1400 monetary units
H 1200 monetary units
I 1150 monetary units
J 900 monetary units
Graphically, the overall demand curve looks as follows:
6
-
1 2 3 4 5 6 7 8 9 10 Quantity
Price
2000
19001850170016801600
1400
12001150
900
.........
..................
...........................
.........
..................
.........
If there is a large number of demanders, the demand function becomes a downward-sloping
curve. The lower the price, the higher the overall demand. This correlation applies empir-
ically in the vast majority of cases. It is known as the law of demand. Further determining
factors of demand as well as the price of the product are personal preferences (how much
does one want to travel to Cuba?), income (can one even afford a holiday, or perhaps
even a more expensive one?) and the price of similar goods (how expensive are alternative
holiday offers?).
23
II.2 Supply and supply curve
The situation is mirrored on the supply side. The suppliers (tour companies) offer more
holidays, the higher the price. There are 4 suppliers with the following price-quantity
behaviour:
Supplier Min. price Individual quantity
A 900 2
B 1400 3
C 1600 1
D 2000 4
We now turn to the question of what determines supply behaviour. As well as the achiev-
able price, the input prices must also be considered. For example, suppliers must cover
the costs of hotels, taxes, airline fuel, airplane leasing rates, etc. The technology used is
also decisive: how well does the combination of the different component services work?
As in the determination of overall demand, the overall supply is the sum of the quantities
supplied by all individual suppliers. Below is the diagram of the overall supply curve:
6
-
Price
2000
900
2 Quantity
.................
10
.........................
5 6
.........
.................................
14001600
II.3 Equilibrium price and adjustment mechanism
What is the balanced market price and how many people will take the package holiday to
Cuba? The price mechanism brings supply and demand in line. In this example we can
distinguish between two cases:
24
• Case 1: p = 1900 → supply > demand → price sinks
• Case 2: p = 1200 → supply < demand →price rises
At the balanced price p = 1600, supply = demand. Here the market is in equilibrium.
The illustration below shows this adjustment process:
6
-
1 2 3 4 5 6 7 8 9 10 Quantity
Price
2000
19001850170016801600
1400
12001150
900
.................................................................................Excess of supply
.................................................................................Excess of demand
Total demand
Case 1
Total supply
Case 2
Effect of exogenous changes:
Exogenous changes such as increased oil prices leads to shifts of the supply or demand
curves. Then a new balanced price emerges. Hereafter we refer to changes to the exogenous
parameters of the supply and demand curves (e.g., wage increases, tax changes, income
changes, etc.) and the analysis of their effect on supply, demand and equilibrium quantity
and price as comparative-static analysis. Example of a demand curve shift : due to terrorist
attacks, demanders are less willing to travel, with the result that demand sinks. To be
more precise, this means that demanders demand fewer holidays at an unchanged price,
because they judge the risk to be too high. Graphically, this leads to a downward shift of
the demand curve. Now fewer demanders want to travel at the same price, or the same
number of demanders will only travel if the price decreases.
Note on the diagram: for reasons of simplicity the curves are drawn as straight lines.
25
6
-
Price
Number of trips
1200
1600
Supply
DemandDemandafter attacks
At the previous balanced price p = 1600, the demand is lower than the supply. The
price therefore decreases until a new equilibrium is found. In the case presented, the new
balanced price is reached at p = 1200.
Prospect of further approaches in the course
Until now the maximum willingness to pay of households and the minimum price demanded
by companies have been taken as given. Later in the course we analyse determining factors
of these variables. In the context of the theories of the household and the company a rough
explanation will be given of what lies behind the supply and demand functions. Technology
can be a determining factor for an offer, while the prices of alternative goods, for example,
could have an effect on the demand for a product.
II.4 Allocation and efficiency
Until now the price mechanism has been shown as the only allocation mechanism: whoever
pays the most gets the product or service. The question now is whether this mechanism
makes sense. How does the price mechanism compare to other allocation mechanisms, e.g.,
whoever comes first gets the scarce product (firstcome, firstserve).
26
Example: Housing market in the centre of Ho Chi Minh City
6
-
Supply
Demand
Rent
Apartments
p∗
Price mechanism: Whoever can/wishes to pay less than the balanced price p∗ will not
get an apartment. But: is that reasonable/fair? To answer this question we shall look at
possible alternative allocation mechanisms.
First we address allocation according to the principle of chance (lottery). One possible
consequence might be that people who are not terribly interested in having a home in
the city center might receive an apartment, irrespective of income. The occurrence of a
secondary market is then very likely. Demanders who receive an apartment that they do
not require would sell it on to the highest bidder.
A waiting list (first come, first serve) would also be conceivable. Then, however, the
demand with the most urgent need (e.g., demanders who do not own a car) might not
be satisfied. As a consequence, other indirect allocation mechanisms, especially secondary
markets, will emerge.
Another possibility is the determination by the state of a rent limit. However, this would
lead to a reduction in the supply of apartments, as it may no longer be worthwhile for
house owners to rent out apartments. The quality of the apartments might also decline.
This situation, in which apartments are insufficiently renovated, can be observed in many
27
cases where upper rent limits exist. It is also very likely that some demanders can no
longer be served.
A further mechanism might be conducted by means of income redistribution – for
example by direct payments (more on that later). This would not affect the distribution
of apartments by means of the price mechanism.
Conclusion: The price mechanism is efficient, i.e., it is the most sensible and cheapest
method to distribute scarce resources (rental apartments in this example). Not only do
other allocation mechanisms lead to an inefficient use of resources, but they also often lead
to the creation of (secondary) markets as a result. People often benefit, however, in a
very arbitrary manner (e.g., those who first receive information about the availability of
apartments because they have perhaps lived nearby for a long time win, while individuals
who only just move to a city have higher burdens to bear, i.e., they must search longer or
pay higher prices in a secondary market).
Example: Air travel
Due to rebooking options held by business class customers it makes sense for airlines – and
they actually do – to overbook flights. There are usually enough seats, but in rare cases
more passengers show up than seats are available. Possible allocation rules:
1. First come, first serve (in the USA until 1975 and currently in the EU): Whoever
comes first has prior claim to a seat. The other passengers receive a fixed compen-
sation payment.
2. Auction (usual among US airlines): the flight personnel asks who would be prepared
to forego a seat on the current plane and take the next one in return for, e.g., 100
US dollars. If there are fewer volunteers than overbooked seats, the price, i.e., the
compensation offer by the flight personnel, rises.
Which is the most sensible allocation mechanism? A market is obviously created by the
second allocation mechanism. Those passengers who urgently need to fly (for example
if they have to attend an urgent business appointment, or the passenger must go to the
funeral of a close relative) can fly, while a student with a lot of time on his hands but
a tight budget is happy to wait for the next flight in exchange for 100 US dollars. The
28
example of the funeral also demonstrates clearly that it is not always the person with the
most money who is willing to pay the highest price, but that quite other aspects can play
a key role.
29
III Household decisions
Introductory examples
1. Following an extremely poor harvest in Spain and France and the accompanying
scarcity in the wine supply and higher purchase prices, a large wine wholesaler con-
siders raising his prices. However the fact that demand collapsed a few years ago in
a similar situation causes him to hesitate. His doubts are further strengthened by
the fact that, due to the weak economy, customers are in any case slow to buy his
relatively expensive wines.
2. The Finance Minister of the Federal Republic of Germany wishes to increase the
tax on tobacco by 2 Euro in the medium term in order to finance a shortfall in the
budget. However, in the course of this discussion critics repeatedly argue that such
a drastic tax increase could lead to a lower consumption of cigarettes and therefore
lead to lower, if not even negative tax income.
In both cases the reaction of the demanders (behaviour of consumers) plays a key role: in
the first case from the perspective of company policy, in the second case from the perspec-
tive of the state’s income. Below we outline the standard theory of microeconomics (or
indeed economics and business sciences in general), with whose help the demand behaviour
can be illustrated and possibly predicted.
Overview
The theory of the household is generally concerned with household decisions. The fun-
damental problem of the household is that it wants to consume more goods (cars, house,
leisure time, eating, etc.) than is possible. This is due, in the widest sense, to income
limitations. The different facets of this problem will be analysed below:
1. Demand and consumer decisions −→ Demand for goods
2. Work and leisure decisions −→ Labour supply
3. Savings decisions (consumption today vs. consumption tomorrow) −→ Capital sup-
ply
30
4. Decisions regarding security and insecurity
The result of each optimisation problem is the optimal demand for goods and the optimal
supply of factors (capital and labour). Thus, we have derived one market side in each of
our three core markets (goods, labour and capital market): the demand for goods on the
goods market and the supply of labour or capital on the labour or capital market.
Fundamental structure of the problem
Objective function (preferences): To consume as much as possible at a given deployment
of labour, or to work as little as possible at a given level of consumption.
Restriction: Consumer goods cost money, one needs income
Further approaches in the next two chapters
The basic idea of the theory of the household is that the individual always demand what
is best for him, provided he can afford it. We therefore assume rational behaviour of the
part of the households.
Open questions:
a) How can one illustrate “being able to afford something”?
b) What is “the best” for households?
Initial answers:
to a) The budget restriction describes the possible alternatives at given prices.
to b) Preferences that are depicted by a utility function indicate how high the optimal
consumer demand is.
31
III.1 Budget constraints
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 3
Fundamental question: Which bundle of goods can a household buy (at maximum) with
a given income?
For the sake of simplicity we make the following assumptions in our considerations. First,
income is taken as given. Households can consume two goods, which may also be services.
We shall follow a one-period approach, i.e., everything is spent in one period and there are
therefore no savings decisions. We are using a model of perfect competition and we assume
perfect information: the household knows the product quality and prices of all companies.
In this basic model the goods prices are given: the household is the price taker, i.e., he
cannot influence the goods price by means of his demand behaviour.
While assumptions such as the two-product approach merely serve the sake of simplicity,
some of the other assumptions will be suspended successively later, as illustrated by the
following examples:
• Introduction of uncertainty :
A family has built a house near a river that is subject to flooding at large intervals,
which would cause great deal of damage to the house. How much is this family
prepared to spend on water damage insurance?
• Introduction of savings:
The entire income is not spent today, but instead part of it is saved for future
consumption multi-period approach. Example: a business student expects to have
a much higher income after his degree than during his years as a student. Against
this background he considers financing a holiday with a loan and then repaying this
after completing his degree.
• Introduction of labour supply decisions :
We do not assume a given income, but ask how this income arises from labour supply
decisions. How much is the individual prepared to work at different wage levels? The
business student considers whether it might make more sense to take a part-time job
32
instead of a loan, and earn additional money. That said, he knows that he will then
have to either restrict his leisure time or neglect his studies.
• Introduction of asymmetric information:
We suspend the assumption of symmetric information between all market partici-
pants. One of the market participants has better information than others, for exam-
ple about the product quality of a car: What price am I willing to pay for a used car
if I know that the seller is better informed about the quality of the car than I am?
Is a low price an incentive to buy?
III.1.1 Some definitions
• Two goods X and Y . Example: apartment and visit to a restaurant
• Bundle of goods (x, y): quantities of good X and Y . Example: size of the apartment
in sqm and number of restaurant visits
• Goods prices px und py. Example: price of the apartments per square meter or menu
prices in the restaurant
• Income I: the budget of the household/consumer. Example: monthly income
• Budget restriction: The household cannot spend more than its income. Formally:
pxx+ pyy ≤ I
Because each consumed goods provides utility (cf. non-satiation approach in the following
chapter 3.2) and there is no incentive to save in the one-period model, this restriction
becomes a budget constraint:
pxx+ pyy = I
i.e.the budget restriction has an equal sign, which implies that the household does not
leave any money unused.
Diagram of the budget line:
33
6
-
y
xIpx
Ipy
Budget constraintSlope: −px
py
The budget constraint indicates the bundle of goods (x, y) that the household can afford
at a maximum. All combinations below this are also possible, but they leave elements of
the income unused. It will become clear later in the course that such combinations do not
maximise utility. In order to display the budget constraint graphically, we solve the above
equation for y:
y =I
py− px
pyx
Accordingly the slope in the budget constraint is dy/dx = −px/py and indicates how
many units of Y the consumer must forego in order to be able to consume an additional
(marginal) unit of X. Specifically in this example, the slope indicates how many restaurant
visits the household must forego if it wishes to rent an apartment that is larger by one
unit (e.g.one square meter). A steeper budget constraint (i.e., a budget constraint with
a higher absolute slope) implies that the household must forego more square meters of
apartment in order to fund more visits to the restaurant. The y-axis shows the maximum
apartment size (I/py = ymax), while the x-axis shows the maximum number of restaurant
visits (I/px = xmax). Naturally, these extreme cases will never occur. Who on earth goes
to a restaurant one hundred times and then sleep on the streets.
34
III.1.2 Changes to the budget line
Changes to the budget line can occur due to a) a change in income or b) the change to
one or both prices.
a) In the case of a change in income a parallel shift of the budget line occurs:
6
-
y
xI1px
I1py
I0px
I0py
I1
I0
The income change does not alter the slope of the budget line. An outward parallel shift
takes place, i.e., at given prices the maximum possible consumption level of the product
in question increases, for example from I0/px to I1/px.
b) In the case of changes to a price a rotation of the budget line occurs:
6
-
y
xI1p1x
I1p0y
I1p0x
35
If, e.g., px (p1x < p0x) decreases, the maximum consumption level of good X (xmax = I/px ↑)
increases. Due to the reduced price of the apartment, as a good, the income is sufficient
for a larger maximum apartment size (if only the good ’apartment’ is demanded).
c) A change to both prices represents a change in income. If both prices decrease to
the same extent in percentage terms, this will in turn create a parallel shift in the budget
line, as this represents a real increase in income. To illustrate this briefly, we assume that
the household has an income of 100 monetary units and the prices of each good are 10
monetary units each. In this case the household can purchase a maximum of 10 units of
each good. Now if the price of both goods sinks to 5 monetary units, i.e. half the original
price, the household can purchase a maximum of 20 units of each product, if it still has its
income of 100 monetary units. We get the same result if the original prices remain at 10
and income increases to 200. For this reason we consider the price reduction to represent
a real increase in income, as the household has more purchasing power.
III.1.3 Rationality assumption
Independently of the form of preferences of a household, the following should apply: if the
preferences of a rational household do not change, and if the budget line is exactly the
same in two situations, the household will make the same decision. Here are two examples:
1. Loss of a CD (cost: 15 monetary units) after leaving the shop or loss of 15 monetary
units before entering the shop
2. Non-purchase or sale of a share
In principle, the same decision is made in each of the cases, as the same budget constraint
applies in each case.
Summary
1. The budget line shows the maximum bundle of goods that the household can
afford at a given price. While bundles of goods below the budget line are realisable
in principle, they do not maximise utility in a one-period model.
36
2. Economic interpretation of the slope of the budget line: How many units of a good
must the consumer forego in order to be able to consume an additional (marginal)
unit of the other good?
3. Changes to the relative goods prices change the slope of the budget line. Changes
to income (and proportional changes to both prices) lead to a parallel shift.
III.2 Preferences
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 3
Fundamental question: What is “the best” from the perspective of the household?
The household chooses a so-called consumption bundle form a variety of goods. For rea-
sons of simplicity we shall restrict ourselves here to two consumer goods, X (apartment in
sqm) and Y (number of restaurant visits). The above question in this context is: Which
combinations of goods (x, y) are better or worse than, or equally as good as other combi-
nations?
We first discuss the modeling of the consumer preferences (utility function). Then in chap-
ter III.3 the consumer preferences will be combined with the budget line. The maximum
utility choice of consumption bundle (apartment size or restaurant visits) at a given bud-
get restriction will then produce the optimal consumer behaviour and thus the household’s
demand for goods.
III.2.1 Preference order
We shall examine three consumer goods combinations, named in the following consumption
bundles:
• Consumption bundle A (xA, yA): 160 sqm apartment and 1 restaurant visit (per
month)
• Consumption bundle B (xB, yB): 120 sqm apartment and 2 restaurant visits
37
• Consumption bundle C (xC , yC): 100 sqm apartment and 4 restaurant visits
Only a ranking hierarchy is made: A is preferred over B (A ≻ B), B is preferred over
A (A ≺ B), or both are ranked equally (A ∼ B). It should be noted that preferences are
subjective, i.e., can differ potentially from consumer to consumer. Below we shall posit
some properties of the preference order of a rational consumer.
Axioms of consumer theory
i) Completeness:
The consumer has an estimate of all potential consumption bundles. He therefore
has, e.g., an exact estimate of the consumption bundle of a 160 sqm apartment
and going out to eat once a month as opposed to 120 sqm and going out to eat
twice. In other words, he prefers the first or the second consumption bundle or he is
indifferent. This possibility excludes the possibility that the consumer cannot decide.
This property can be problematic in extreme cases.
ii) Transitivity:
A ≻ B and B ≻ C =⇒ A ≻ C
Transitivity ensures that the consumer preferences are consistent and therefore ratio-
nal. There is a similarity to size: if Lisa is taller than Anna, who in turn is taller than
Sarah, then Lisa is also taller than Sarah. Comparative relationships do not always
comply with the assumption of transitivity. For instance, just because Schalke 04
beats Borussia Dortmund and loses against Bayern Munich, it is by no means self-
evident that Dortmund will lose against Bayern (perhaps this season, but certainly
not always). However, for our comparison of consumption bundles the assumption
of transitivity is appropriate in most cases.
iii) Non-satiation:
A ≻ B, if xA > xB and yA ≥ yB
If consumption bundle A comprises a 150 sqm apartment and 2 restaurant visits,
and bundle B 140 sqm and 2 restaurant visits, then A will be preferred over B. Note:
at least the not-worse assumption A ⪰ B is unproblematic. A simple reason is that
the larger apartment can be rented at a higher price. This leaves enough money to
spare that can be used for other things.
38
III.2.2 Indifference curves
Definition: The indifference curve is the connecting line of all consumption bundles, which
the household values as equal.
The indifference curve can be derived with the help of the properties of the preference
order:
6
-
Restaurant visits
Apartment size
....................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....
.............................
............................
............................
............................
.............................
............................
............................
............................
.............................
............................
............................
............................
.............................
............................
..
•
•
•
•
A
S
D
QBetter than A
Worse
than A
We start with the analysis in point A. Due to the property of non-satiation, A ≻ S and
A ≺ Q. Due to transitivity and completeness, a consumption bundle D exists between S
and Q, for which A = D applies. Frequent repetition of this consideration with variation
of S and Q produces the indifference curve:
6
-xA xD
yD
yA
Restaurant visits
Apartment size
........................................................................................................................................................................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ...........................................................................................
............. ............. ............. ............. ............. ..............................................................................
•
•
A
D
“Better”-Quantity
“Worse”-Quantity
All points on the indifference curve are ranked equally.
Important: Indifference curves cannot intersect.
Therefore, the following case cannot occur:
39
6
-
y
x
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................................................................................................
•
•
• W
ZX
............. ............. ............. .............
Proof by contradiction: the consequence of non-satiation and definition of the indifference
curve is X ≺ W ;W = Z =⇒ X ≺ Z =⇒ contradiction. As X and Z lie on one indifference
curve, X ≺ Z cannot apply.
We shall now examine a number of indifference curves. The following basic principle
applies: the further the indifference curve is situated away from the origin, the higher the
welfare of the consumer.
6
-
y
x
.........................................................................................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................................................................................... �higher preference
The marginal rate of substitution
Technical definition: Slope of the tangents on the indifference curve.
Diagram:
6
-
y
x
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................
•
•∆y
∆x
MRS = −∆y∆x
........
........
........
........ ........ ........ ........
40
Economic interpretation: The marginal rate of substitution (MRS) indicates the will-
ingness to exchange for at the same level of preference.
The marginal rate of substitution is a very important concept, which we will use in different
situations below. Interpretation based on an example: The MRS answers the question as
to how many addition restaurant visits must be offered to the household so that it will
accept an apartment that is smaller by one unit (indifference). During the course we shall
observe only very small (marginal) changes: MRS = − dydx
. This means that the change ∆x
is around zero. It is impossible to bear this in mind, especially when we want to illustrate
the marginal rate of substitution with a concrete, specified indifference curve. Moreover
it should also be noted that we give the MRS a negative sign, i.e.the MRS always has a
positive value. The marginal rate of substitution thus provides the amount (!) of the slope
of the indifference curve at a certain point.
Different types of preference orders/indifference curves
a) Perfect substitute:
6
- x
y
..........................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
The consumption of a good is not always necessary. X can be replaced completely
by Y and vice versa. Here is an example: cola or coffee consumption during exam
preparations. Caffeine is the only thing that keeps you awake. With a cola-coffee
caffeine-content ratio of 1:2, coffee can be replaced by twice the amount of cola.
Later in the course we will define the case of the perfect substitute in such a way
that the indifference curves intersect the axes. A closer definition would be that the
41
indifference curve could also be represented by a line, so that there is a constant
marginal rate of substitution.
b) Limiting preferences (Leontief preferences)
6
- x
y
............................................................................................................................................................................................................................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................
•
•
•
This specification infringes the aforementioned assumption of non-satiation. How-
ever, certain goods only make sense in a certain combination, e.g.ingredients for a
meal or cars and tyres.
c) Imperfect substitute:
Both goods are necessary to reach a certain preference level. The apartment/restaurant
example above represents the case of an imperfect substitute. We shall address this
case later in the course.
Convexity of the indifference curve
6
- x
y
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
42
Convex indifference curves imply a decreasing marginal rate of substitution: This
sounds very technical, but it has a directly relevant economic implication. The more that
is consumed of a good (e.g. X), the lower is the willingness to exchange Y for X. If,
for example, you live in a huge apartment and no longer have enough money to visit a
restaurant, you would be more willing to give up 5 square metres of your apartment for an
additional restaurant visit than you would if you lived in a small room of only 10 square
meters. Put generally, this means: the lower the (relative) consumption level of a good,
the more important it becomes for the consumer. An indirect consequence is a preference
for a wide variety of products.
III.2.3 Utility functions
The utility concept provides the opportunity to describe the preferences of the consumer
formally. Our attention was previously concentrated on the ordinal utility theory:
consumption bundle X is better than Y , but Y is not 3 times as good as X. This last
quantitative statement can only be made in the cardinal utility theory, which prevailed
in the 19th century. According to this theory, each of the consumption bundles is allocated
a quantitative utility level u:
u = u(x, y) z.B. u(xA, yA) = 5 =⇒ welfare indicator
However, for the modern theory of the household, ordinal utility theory says that only
the design of the indifference map is important. For this theory we need only the afore-
mentioned properties (completeness, transitivity and non-satiation) and thus much weaker
assumptions than those of the cardinal utility theory, which requires from households pre-
cise information about the utility values of individual consumption bundles.
Ordinal utility function
Alternative consumption bundles are given values, ensuring that equally preferred con-
sumption bundles receive the same values, while preferred consumption bundles receive
higher values. With a preference relation of (xA, yA) ≻ (xB, yB), u(xA, yA) > u(xB, yB) ap-
plies. u(.) is the utility function that allocates values to each of the consumption bundles.
The exact amount of these values and the exact distance from other values is irrelevant;
43
only the design of the indifference map is important. An arbitrary number of utility func-
tions can be allocated to a preference order (indifference map).
Note that the ranking of preference orders remains the same in monotonic transformation.
The following applies for monotonic transformation:
uA > uB
f(uA) > f(uB)
We observe two monotonic transformations of the utility function u(x, y):
f1(u) = u2
f2(u) = u+ 5
u(xA, yA) = 5 > u(xB, yB) = 3
f1(uA) = 25 > f1(uB) = 9
f2(uA) = 10 > f2(uD) = 8
To reiterate: The utility function u = u(x, y) describes only the ranking. Consumer
behaviour is not dependent on the absolute utility level, but only on the order of the
utility levels. To put it another way (based on the above example): it is relevant for
our results whether we show the household behaviour on the basis of the utility function
u1 = xy or by means of the utility function u2 = x2y2, which results from the monotonic
transformation from u1; in both cases we receive the same prediction about the demand
behaviour of the household.
Marginal utility and marginal rate of substitution
As explained above, marginal analysis is essential to economic analysis. We now want to
answer the question as to how large the additional utility is with the additional consump-
tion of good X or good Y . An answer is provided by the marginal utility function:
MUx =∂u(x, y)
∂xund MUy =
∂u(x, y)
∂y
y or x express that a partial variation of x or y is observed, whereby the consumption level
of the other good in each case remains unchanged. It is suggested below that the marginal
utility (MU) of each good diminishes with increasing x or y. The economic reason for
44
this is due to increasing satiation. For example, the additional utility from the 1st pair of
shoes is greater than from the 100th pair. Furthermore, the non-satiation assumption is
made, i.e. MUx and MUy are always positive. This means that an additional unit of a
good always increases utility.
Diagram of marginal utility
6
- x
MUx
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
The marginal rate of substitution can be calculated from the utility function u(x, y). A
curve with the same utility (indifference curve) is defined by u(x, y) = u with a fixed utility
level u. The total differentiation of this function and solving the equation for − dydx
gives us
the MRS:
du = 0 =∂u
∂xdx+
∂u
∂ydy
and thus
MRS = −dy
dx=
∂u/∂x
∂u/∂y=
MUx
MUy
.
Properties of the MRS:
The marginal rate of substitution shows the relationship of the marginal utility of each of
the goods. A diminishing marginal utility of the good X implies a diminishing marginal
rate of substitution! The MRS is invariant with regard to the monotonic transformation
of u. The larger the MRS, i.e., the steeper the indifference curve, the stronger is the
preference for X. This correlation is illustrated in the next diagram:
45
6
-
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................
•
•∆yA
1A
........
........
........
........ ........ ........ ........
................ ........ ........•• B
∆yB 1
x
y
At the point A(B) the household is willing to give up a unit of good X if it is compensated
with ∆yA(∆yB). As ∆yA > ∆yB, the household has a stronger preference for good X at
point S than at point B.
Excursus: Partial and total differential
As we shall be using the concept of total differentiation hereafter, we will reiterate the
concept briefly (you will remember it from OMAT:)).
The differential of a function with only one variable is dy = f ′(x)dx. Accordingly, a
function with many independent variables has many (partial) differentials, which indi-
cate the approximate change of the function value when the relevant independent variable
changes by dx (e.g. dx = 1, i.e., by one unit). Here is an example based on function:
Partial differential with regard to X: ∂f(x,y)∂x
dx
Partial differential with regard to Y : ∂f(x,y)∂y
dy
The total differential indicates how much the function value changes (approximately)
when all independent variables change. It is given as the sum of the partial differentials.
The total differential for u = f(x, y) is therefore produced as
du =∂f(x, y)
∂xdx+
∂f(x, y)
∂ydy.
End of excursus
46
Examples of utility functions/preference orders
a) Cobb-Douglas utility function:
This is a very frequently used utility function in cases of imperfect substitutes,
the most important specific utility function in the further course of the lecture. The
utility function, marginal utility function, 2nd derivative and MRS are:
u = xαyβ (0 < α, β < 1)
MUx = αxα−1yβ
∂2u
∂x2= α(α− 1)xα−2yβ < 0 for α < 1.
MRS =∂u/∂x
∂u/∂y=
α
β
y
x
The expression Cobb-Douglas is often used only for the case α+ β = 1. However we
shall also use it to describe α + β = 1. For goods bundles with an identical (x− y)
ratio, the MRS is constant in Cobb-Douglas preferences. This property implies that
the MRS is constant along a path through the origin, as shown in the diagram:
6
-
y
x
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
•
•
•..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
the same slope at all
three points
b) Additive separable utility function:
Utility function for the case of a perfect substitute. Specific example: u =√y+√x.
In this form of utility function the marginal utility of good X (MUx) is independent of
the consumption level of the other good (Y ). The calculation of the marginal utility
of good X produces: MUx = 0, 5x−0,5. MUy is also independent of x: MUy = 0, 5y0,5.
47
c) Leontief preferences
A limiting utility function that reproduces the Leontief preferences described above
is, for example, the minimum function u = min(2x, 3y).
Summary
1. The preference order of a rational consumer shows the ranking of all possible
consumption bundles. Note should be taken of the following axioms of consumer
theory: completeness, transitivity and non-satiation.
2. Indifference curves indicate all consumption bundles that lead to a fixed level of
consumption by the consumer. Indifference map is the term for a number of indiffer-
ence curves, which each lead to a different level of consumption. It is important that
these indifference curves can never intersect. The marginal rate of substitution
(absolute slope of the indifference curve) is the exchange ratio of two goods where
the utility remains the same.
3. Possible types of preference orders or indifference curves are perfect substitute,
limiting goods relationships (Leontief preferences) and imperfect substitute. The
latter is the normal situation, which leads to convex indifference curves, i.e., to a
diminishing marginal rate of substitution.
4. Preference orders can be illustrated formally by means of utility functions. The
ordinal utility theory forms the core of the modern theory of the household: only
rankings are described, the absolute utility level is not important. Monotonic trans-
formations of the ordinal utility function do not change the ranking of the consump-
tion bundle.
5. Key to the theory of the household is the concept of marginal utility. This is
generally positive and decreases with the increasing consumption of a good. Formally,
the marginal rate of substitution is produced by the marginal utility ratio of the
goods.
48
6. Types of utility function: additive separate utility functions indicate perfect sub-
stitutes, minimum functions are examples of Leontief preferences. Imperfect substi-
tutes are often described with Cobb-Douglas utility functions.
III.3 Optimal consumption decision
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 3
III.3.1 General optimisation problem
We shall now combine the preferences of the household (the "best", Chapter 3.2) with the
budget line (what the household can “afford”, Chapter 3.1).Therefore we are once again
attempting to provide an answer to the question: what is the best consumption bundle
that the household can afford? The optimisation problem of the consumer at a give income
and given prices is then:
max u(x, y)
s.t. I = pxx+ pyy
We call the optimal decision by the consumer the optimal consumption plan. This means
the consumption of the goods bundle (x, y), which maximises the utility to the household
under consideration of budget restrictions.
Graphical derivative of the optimal consumption plan
The income I and the goods prices px and py are given. The consumer’s objective is to
reach the highest possible indifference curve (highest utility level!). Graphically, the tan-
gential point of budget line and indifference curve provides the solution to the optimisation
problem:
49
6
- x
y
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
U1
U2
U3
•
U1 < U2 < U3
x∗
y∗........ ........ ........ ........ ........ ........ ........ ................................................................
The consumption bundle with maximum utility is reached at (x∗, y∗). Every other con-
sumer good bundle either cannot be reached (i.e.the household cannot afford it with its
given income) or would lead to a lower utility level. In general it is the case that, ideally,
budget lines and indifference curves should not intersect, because otherwise an even higher
indifference curve could be reached. In the case of convex indifference curves and imper-
fect substitution there is a tangential solution (internal optimum). Otherwise there is the
possibility of corner solutions. Here is an example:
6
-
y∗
x
y
............................................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................
•
The concept of imperfect substitutes or the convexity of the indifference curve is always
assumed below. In other words, we shall concentrate essentially on internal solutions.
Further below we shall also briefly address corner solutions and go into detail in the
context of the exercise.
50
Formal derivative of the solution
Problem: Maximise the utility to the household at a given income and goods prices by
choosing the optimal consumption bundle!
A concrete application could take the following form: choose the consumption bundle
with the maximum utility, comprising apartment size and restaurant visits, when 2000 is
available per month, the square metre price is 12 and an average restaurant meal costs 20:
max u(x, y)
s.t. 12x+ 20y = 2000
Excursus: The Lagrange approach - a cookbook recipe
In order to solve an optimisation problem under external constraints, we shall usually
revert to the Lagrange approach in future. This approach will not be explained extensively
(mathematically) in the microeconomics course, but rather conveyed merely as a kind of
cookbook recipe, with the help of which the relevant applications can be carried out. In the
Lagrange approach the objective function (e.g.utility function) is subject to one or more
constraints. For the sake of simplicity we shall concentrate here on only one constraint
(e.g.budget restriction) and two variables (x1 and x2, e.g.the consumption levels of two
goods).
The constraint g(x1, x2) ≡ 0 is multiplied with a factor, the Lagrange multiplier, and
added to the (objective) function whose extreme values are examined.The solution to the
problem can be divided into three different steps:
1. From the (objective) function y = f(x1, x2) the Lagrange function becomes
L = f(x1, x2) + λg(x1, x2)
2. The extreme values of this function must then be examined. A necessary condition
for the existence of an extreme value of the Lagrange function is that the first partial
derivatives with respect to x1, x2 and λ are equal to zero:
51
∂L∂x1
!= 0
∂L∂x2
!= 0
∂L∂λ
!= 0
3. This produces an equation system with 3 equations and 3 unknowns. The solution
of this equation system is provided by the points at which the extreme values of the
objective function lie, subject to constraints.
End of excursus
We shall now solve the above utility maximisation problem, under consideration of the
budget restriction, by means of the Lagrange approach. The relevant Lagrange function
is:
L(x, y, λ) = u(x, y) + λ(I − pxx− pyy) (1)
with λ as the Lagrange multiplier, which can be interpreted economically as the marginal
utility of the last monetary unit used. Based on this interpretation we assume a positive
Lagrange parameter. Against this background, the Lagrange function should be formulated
in the above manner, thus: the constraints are reformulated so that spending can be
deducted from income. This is all multiplied with the Lagrange parameter and added to
the objective function. An obviously equivalent formula is:
L(x, y, λ) = u(x, y)− λ(pxx+ pyy − I).
Both formulas mean that the optimisation solutions lead to positive solutions for all three
endogenous variables (x∗ > 0, y∗ > 0 and λ∗ > 0).
The first order conditions are:
∂L∂x
=∂u
∂x− λpx
!= 0 (2)
∂L∂y
=∂u
∂y− λpy
!= 0 (3)
∂L∂λ
= I − pxx− pyy!= 0 (4)
52
In convex preferences the second order conditions are always fulfilled (due to diminishing
marginal utility). For this reason, the extreme values discovered are always a maximum.
The interpretation of (2) and (3) can be made by means of a cost-benefit comparison.
Ideally, marginal utility (e.g. ∂u/∂x) should correspond with marginal costs (λpx). From
the three equations above the three unknowns (x, y and λ) can now be found. We get the
demand functions:
x = x(I, px, py)
y = y(I, px, py)
Accordingly, the amount of each demand for goods depends on the household income,
on the goods price and on the price of alternative goods. We shall derive these demand
functions later for specific utility functions.
Correct step-by-step approach
a) Set up the Lagrange function
b) Derive the optimisation conditions
c) Solve the first two optimisation conditions for λ and equate
d) Solve for x and enter into the third optimisation condition (budget restriction)
e) Solution for y produces y = y(I, px, py)
f) Same procedure for x
(2) and (3) produce the essential property of the optimal consumption plan:
∂u/∂x
∂u/∂y=
pxpy
(5)
Interpretation of this optimisation condition
With optimal utility the ratio of the marginal utility of both goods is equal to the price
ratio. With ∂u/∂x∂u/∂y
> pxpy
it would be advantageous to buy more of X and less of Y .
53
The marginal utility of X would fall as a result (diminishing marginal utility!), while the
marginal utility of Y would increase: (∂u/∂x ↓ and ∂u/∂y ↑). This process would continue
until equation (5) once again applies.
Example of perfect substitutes
Before we turn our attention to the derivative of demand functions in the case of Cobb-
Douglas utility functions, we shall examine a brief utility maximisation problem for the
case of perfect substitutes, in which corner solutions are used:
We take up the above example again. Student M is an avowed caffeine consumer. He uses
his entire drinks budget for caffeinated drinks. The only measure of utility is the caffeine
content. He considers coffee and Coca-Cola, whereby coffee contains twice as much caffeine
as Coca-Cola. A liter of cola costs 2 monetary units, a cup of coffee (0.2 litres) 0.5 monetary
units. What will M do? What effect would an increase of the coffee price to 0.8 monetary
units per cup have? Think about the solution and also try to create an initial diagram to
solve the problem!
III.3.2 Cobb-Douglas utility function and demand functions
Below the demand functions will be derived as an example of a Cobb-Douglas utility
function. This will happen in two steps: first the general derivative will be presented,
which allows a good interpretation. This is followed by a small numerical example, to
clarify the problem further.
1. General derivative
The utility function u = xαyβ with 0 < α, β < 1 includes, under consideration of the
budget restriction, the optimisation problem:
maxx,y
u = xαyβ
s.t. I = px · x+ py · y
Lagrange approach:
L = xαyβ + λ(I − px · x− py · y) (6)
54
First order conditions:
∂L∂x
= αxα−1yβ − λpx = 0 (7)
∂L∂y
= β · xαyβ−1 − λpy = 0 (8)
∂L∂λ
= I − px · x− py · y = 0 (9)
(7) and (8) produceαxα−1 · yβ
β · xαyβ−1=
α
β
y
x=
pxpy
(10)
and thus
y =β
α
pxpy· x (11)
respectively
x =α
β
pypx· y (12)
Using (11) in the budget equation produces
I = px · x+ pyβ
α· pxpy· x
respectively
x =α
α+ β
I
px(13)
The same for y produces
y =β
α+ β
I
py(14)
For α + β = 1 the optimal distribution of consumption spending in accordance with the
exponents given in the utility function:
px · x = α · I
py · y = β · I
The two coefficients α and β therefore indicate the spending share of both goods in relation
to overall income. The more the good is appreciated, i.e.the larger the coefficient in the
utility function, the more is spent on that good.
55
Further note: This result is independent of the special form of the Cobb-Douglas utility
function. A monotonic transformation of the utility function, for example by taking a
logarithm:
U = α lnx+ β ln y
produces a completely identical solution.Each of you should carry out this calculation
yourself in order to check that you have understood what you have read.
2. A small numerical example
The utility function is u = x0,5y0,5. Income is 100 monetary units, the price of good X is
1 monetary unit and of good Y 2 monetary units.
How many units of good X will the household demand?
Result: it will demand 50 units. Recalculate it to check your knowledge or use it in the
above formula!
III.3.3 An application of the consumption decision model
We shall now turn to an application from the field of social policy. The main objective of
social policy is to improve the living conditions of the “poor”. For the sake of simplicity
we shall concentrate on two possible political instruments: allocation of a council flat
or a direct income transfer. We will observe the effects of both possibilities based
on a specific household. The economic question is: Which of the two instruments works
best? Specifically this means, which of the political measure will achieve the socio-political
objective at the lowest cost (efficient social policy)? Or to put it another way: Which
instrument can better achieve the objective with the same use of funds?
We shall follow our proven example, in which solely the apartment and the restaurant
visits provide utility to the household. As before, x denotes the size of the apartment and
y the number of restaurant visits. We assume that the target group has an original income
of I0 = 500 and that prices are px = 20 or py = 5.
56
6
-x∗ x
y∗
x
y
Optimal consumption bundle of hh:
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
I0
I1
U0
UW
U1
︸ ︷︷ ︸x− x∗
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
In the above diagram, (x∗, y∗) denotes the original optimal consumption bundle without
social policy. x indicates the apartment size targeted by social policy for low-earning
households. Let us assume that the government is supporting the construction of social
housing (more precisely: the construction of an x sqm sized council flat) with (x−x∗) ·px.
The needy household can move into a council flat if it is in possession of a social housing
eligibility certificate (Wohnberechtigungsschein or WBS: the cheap flats in the newspaper!).
The proprietor will, however, still demand (x∗) · px from the household, and only the
difference will be paid by the state. We now come to a utility comparison: U0 denotes the
achievable indifference curve (utility level) without social policy, while UW represents the
achievable indifference curve (utility level) upon the allocation and occupation of a council
flat. The utility level of the poor household has therefore increased; the social policy has
improved the living conditions of the household.
We must still examine whether a direct income transfer into the amount of the social
housing construction funding (x−x∗) ·px, i.e.the same amount of money) would produce a
better or worse living standard for the household. With the income transfer the household
budget increases to exactly I1 (same budget line as in the case of the social housing!).
Now, however, the household is free to choose its own optimal consumption bundle and
is not obliged to rent an x sqm sized apartment. According to its preferences, which are
57
expressed by the indifference curve, it can now reach the utility level U1, which is higher
than UW and U0. The direct income transfer can therefore be called the more efficient
instrument of social policy.
Economic intuition:
In social housing the household „consumes“ more apartment (sqm) and fewer restaurant
visits (intersection (x, y) of UW and I1) than with the direct transfer of income (tangential
point of U1 and I1). In terms of its optimal consumption plan it consumes too much of good
X, as it can only take the x sqm sized council flat (fixed apartment size!) and therefore
achieves a lower utility level than with the income transfer. The state intervenes in the free
choice of action of the individual and distorts the consumption decision of the household.
The economic policy implication of this very simple model: If the objective of social policy
is to achieve the highest possible welfare (best living standard) for poor households at the
given costs of the redistribution programme, direct income transfers should be prioritized
over social housing!
Summary
1. The optimal consumption plan of a consumer indicates the utility maximising con-
sumption bundle under consideration of the budget restriction. Formally, the optimal
consumption decision can be derived by maximising the utility function subject to
constraints. Graphically, the optimal consumption bundle can be determined by
means of the tangential point of the budget lines with the highest possible indifference
curve.
2. The Lagrange approach allows us to solve general maximisation problems subject
to constraints. The demand functions of consumers for each good can be derived
with this approach. The demand for a good depends on income, goods price and the
price of the other goods. In the consumer’s optimal utility, the marginal utility ratio
represents the relative price of the goods.
3. The example of the the Cobb-Douglas utility function illustrates the formal
derivative of the optimal consumption decision in the case of imperfect substitutes.
58
For α+ β = 1 the optimal distribution of consumption spending in accordance with
the exponents given in the utility function. The two coefficients therefore indicate
the spending share of both goods in relation to overall income. The more the good
is appreciated, i.e. the larger the coefficient in the utility function, the more is spent
on that good.
4. There are many possible applications of the theory of the household. It is clear
from the social housing example how microeconomic theory can be used to recom-
mend policy.
III.4 Influence of prices/income on the demand for goods
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 4
In this chapter we shall address the question of how changes to price and income influence
the optimal consumer decision that results from the relevant demand functions and budget
restrictions. Methodically, we will use the concept of comparative statics.
Definition: Comparative statics describes the effects of changes to parameters and/or
exogenous variables on endogenous variables.
This exercise is relevant against the background of a whole range of applications. It is best
to imagine the situation from the perspective of a company that asks itself how changes to
price and income will affect the demand for its product and what the impact will be. The
immediate question that arises is, what reductions in demand must the company accept
if it raises the price (e.g. by 10 percent). Or: the competitor (who supplies a substitutive
good) has reduced his price by 15 percent. By how much must our company reduce its
price in order for demand to remain at least constant? Income changes and their effect
on demand are relevant for capacity planning in the business cycle, for example, but even
more so for long-term objectives. In principle we have already done this on the previous
pages (at least implicitly), but now we want to observe and analyse the consequences of
price and income changes in more detail.
59
III.4.1 Income changes and demand
The goods prices remain constant and only the income I is changed. We want to find out
how the optimal consumption bundle (x∗, y∗) adjusts to the new income situation. For the
analysis we shall use the so-called income-consumption curve.
Definition: The income-consumption curve is the connecting line of all optimal consump-
tion plans at varying income and constant goods prices.
We begin with the graphical analysis of the optimal household decision at varying income
(see the illustration below). The diagram shows that a varying income leads to a parallel
shifting of the budget lines. The demand functions x = x(px, py, I) bzw. y = y(px, py, I)
apply.
6
-
y
x
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
•
•
•
•
Income-consumption curve
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
..........................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................
Definition: The Engel curve displays the effects of an income variation on the individual
demand for goods, i.e. the relationship between the goods demand x or y and income I.
The Engel curve is shown graphically in a (x, I)-diagram or (y, I)-diagram:
6
-
I
x
......................................................................................................................................................................................
.................................
.........................................
........................ x(I)Engel curve
6
-
I
y
............................................................................................................................................................................................................................................................................................................... y(I)Engel curve
60
Two general types of goods can be distinguished, depending on the path of the Engel
curves: inferior and normal goods.
1) Inferior goods:
The Engel curve has a negative slope, as a formula (e.g. for X): ∂x/∂I < 0. In other
words, at a higher income the good is in less demand.
6
-
I
x
...............................................................................................................................................................................................................................................................................................................
X inferior good:
x(I)
6
-
y
x
..................................................................................................................................................................
..................................................................................................................................................................
..................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................
••
•
Example for income-consumption curve with Y as inferior good
The prototype of an inferior good is a good for which there is a clearly preferred but more
expensive substitute (higher quality). Here are some examples:
1. Buying meat: the demand for meat as a good can be satisfied by pork with a large
proportion of fat. If, however, income increases, meat with a lower amount of fat
61
is often purchased instead; consumers with a high income generally tend to demand
more expensive, but lower-fat meat (e.g. steak).
2. Buying a car: small car, ... , sports car
3. Watches: Swatch, ... , Rolex
The broader the definition of a good, the less likely it is to be an inferior good. Thus, fatty
pork is typically an inferior good, but meat in general is less so, while food in general even
less (never).
2) Normal goods:
The Engel curve has a positive slope (see diagram). Formally this means (e.g. for X):
∂x/∂I > 0. With an increase in income, therefore, there is more demand for the good.
This is the case with most goods.
6
-
I
x
........................................................................................................................................................................................................................................................................................................
X normal good:
x(I)
In addition, normal goods can differ according to the extent to which demand changes
when income changes:
Luxury goods, necessary goods and homothetic preferences.
a) Homothetic preferences
The demand for goods and income increase to the same extent: if income increases by
e.g. 10%, demand for good X also rises by 10%. That also means that the distribution of
income across goods always remains the same, even if the income varies. An example of a
62
consumption bundle: always 30 percent of expenses on food, 40% for the apartment and
30% for the car, irrespective of whether the monthly income is 1,000, 2,000, ... or 10,000.
The Cobb-Douglas utility function illustrates homothetic preferences. The corresponding
demand function (derivation, see p. 54f) is:
x =α
α+ β
I
px
Numeric example: α = 0.15 β = 0.6 px = 2 I = 10 −→ x = 1
A doubling of income I (I = 20) leads to a doubling of demand: x = 2.
Technical definition of homothetic preferences
If (x∗0, y
∗0) is the optimal consumption bundle at I0
−→ at I1 = t · I0 : (x1, y1) = (t · x∗0, t · y∗0)
Economic interpretation of this formula
The distribution of income across different goods is independent of the amount of income.
b) Luxury goods
Disproportionately high change in demand when income changes; examples include holi-
days, caviar etc.
c) Necessary goods
Disproportionately low change in demand when income changes; examples include food,
water supply, heating.
For all three types of normal goods we observe a positive ascending Engel curve, but the
path of the Engel curve differs between the three types of normal goods. While the Engel
curve for homothetic preferences is a straight line, it takes a concave (convex) form in the
case of luxury goods (necessary goods).
III.4.2 Price changes and demand at a given household income
Basic question: What effect do price changes (e.g. growth in px) have on the optimal
consumption plan / demand quantity of households at a given household income?
63
This is a very important question for market research undertaken by a company (see our
argumentation above). Examples of applications for the analysis of price changes are taxes,
administered prices, exogenous price changes (e.g. oil price changes) etc. The formula for
the problem is taken from the demand function (see above):
x = x(px, py, I)
y = y(px, py, I).
Ordinary goods and Giffen goods
With regard to changes in demand as a reaction to price changes, goods can be divided
into two types: ordinary goods and Giffen goods. An ordinary good is characterised by in-
creasing (decreasing) demand for a good, when the price of this good decreases (increases),
i.e. ∂x∂px
< 0. In the opposite case we speak of a Giffen, i.e. when ∂x∂px
> 0. Most goods
are ordinary ones, while Giffen goods are an exception which can occur under certain
circumstances, which we shall discuss in more detail below.
The price-consumption line
The connection of optimal consumption plans for alternative prices of good X:
6
-
y
x
Ipy
Ip0x
Ip1x
Ip1x
Ip1x
............................................................................................................................................
...........................................................................................................................
•
• ••
price-consumption line
The diagram (price-consumption line) presents the optimal consumption plans at varying
prices for good X. If we transfer all of the points on this line to a (x, px) diagram, we get
the (inverse) demand curve. It represents quantities demanded at varying prices:
64
6
-
p
x
............................................................................................................................................................................................................................................................................................................................................................................ x(p)
Income and substitution effect
We know from our previous considerations that an increase in price changes not only
the price relationship between goods (substitution effect), but also restricts real income
(income effect). It should be noted that the nominal income remains unchanged. But at
a constant nominal income, a reduction in prices means that the household can buy more,
i.e. the household has a higher real income. It often makes sense to divide the overall effect
of a price change into these two partial effects. Let us first look at a sample application:
Example: Environmental tax
At the start of Germany’s first “red-green” federal government it decided to introduce an
environmental tax in a number of stages, in order to raise the price of petrol. However, the
price of petrol increased severely as a result of this (and other factors) and there were very
strong protests from the population. This led to the decision that at least those employees
who drove to work by car would receive compensation in order to balance out the effect
on their income. This prompted many commentators to question the effectiveness of the
measure.
65
General analysis: Let us now examine in detail the effects of a change in price:
I/p1x I/p0x
6
-
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................
y
x
..........................................................................................................................................................................................................
........................................................................................................................................................................................................................................
•A•D
A −→ D: Total effect
The starting optimum is given in point A. An increase in the price of good X then leads
to the new optimal consumption bundle D. The basic idea is that the overall effect is
produced by two partial effects:
1. Substitution effect
A price change changes the relative price of both goods and makes a now relatively
more expensive good relatively less attractive.
2. Income effect
The increase in the price of a good reduces the purchasing power of the income of a
household. This becomes clearest when we imagine an increase in both goods’ prices.
In this case, at a constant nominal income, a real reduction in income occurs. This
effect already occurs when only one good’s price increases, albeit the real reduction
in income is then also lower.
3. Total effect
Sum of substitution effect and income effect
The following applies when the price of good X increases:
The substitution effect (SE) is always clear: if the price of good X increases, the demand
(x ↓) decreases (and vice versa).
66
The income effect (IE) is ambiguous: if the price of good X increases, there is a real
reduction in income. As a result the type of good must be defined precisely: for normal
goods the income effect has the same direction as the substitution effect SE (demand
for X ↓ when price of good x increases), for inferior goods, the income effect acts in the
opposite direction to that of the SE ((demand for x ↑).
Analysis for normal goods / increase in the price of good X
6
-
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
y
x
..................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................
•A
• B
•D
A −→ B: SEB −→ D: IE
SE: Compensated change in demand: the income is changed so that the household can
just about buy the old consumption bundle at a new price condition. The new (no-
tional) budget line then goes through A and displays the new slope (−→ presentation
in accordance with Slutsky). Accordingly, an increase in price leads to a notional
increase in the nominal income.
IE: Reversion to the old nominal income. Graphically, this means a parallel shift of the
notional budget line.
TE: The total effect on the X demand is negative (dx/dpx < 0), i.e. when the price of
good X increases, demand for this good decreases.
67
Quantitative determination of both effects based on an example
The original prices be p0x = 10 and py = 20, the nominal income is I0 = 200. We assume
the utility function u = x0,6y0,4. The corresponding demand quantities x0 = 12 and y0 = 4
can be calculated by means of the Lagrange approach.
Now: Price increase of good X to p1x = 20. The total effect on the X-demand can
be calculated by means of TE=x(p1x, py, I0) − x(p0x, py, I
0). The new demand quantity is
x1 = 6, the total effect comprises a reduction in demand of 6.
Calculation of the substitution effect:
• Compensated income (income that allows the purchase of the old consumption bundle
at new prices): I1 = p1xx(p0x, py, I
0) + pyy(p0x, py, I
0) = 320
• The substitution effect on the X-demand is given by:
x(p1x, py, I1)−x(p0x, py, I0) = 0, 6 ·320/20−0, 6 ·200/10 = −2, 4. In the above diagram
this is the movement from A to B.
Calculation of the income effect:
x(p1x, py, I0)− x(p1x, py, I
1) = 6− 9, 6 = −3, 6
Accordingly, we calculate the X-demand at new prices for the respective income levels and
their difference. In the diagram this is the movement from B to D.
The Law of Demand
If the income effect is positive (dx/dI > 0), i.e. if it is a normal good, then an increase
(decrease) in the price of the good and the associated decrease (increase) in real income
leads clearly to a reduction (increase) in demand −→ falling demand curve.
Note
In many textbooks (e.g. Pindyck/Rubinfeld) the substitution effect (SE) does not refer to
the original consumption bundle, but rather to the original utility level (−→ presentation in
accordance with Hicks); our presentation corresponds with the so-called Slutsky analysis.
68
Analysis for inferior goods / increase in the price of good X
6
-
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
y
x
..................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
•A
• B
•D
A −→ B: SEB −→ D: IE
If the price of the inferior good X increases, demand for this good will decrease (from
consumption bundle A to B). As already mentioned, the direction of this SE is always
clear, independent of the distinction between goods. The IE on the other hand, is not
clear. In the case of an increase in the price of the inferior good X, the income effect leads
to an increase in the demand for good X (from consumption bundle B to D). The total
effect on the demand for the inferior good in the event of a price increase (TE : SE+IE) is
therefore generally undefined for inferior goods. If the SE is overcompensated by the IE,
this is a so-called Giffen good. The demand curve has a positive slope, i.e. an increasing
price would increase demand. This is certainly an exception, but cannot be excluded
completely. One example is given by experiences of increases to the price of essential
foodstuffs in developing countries. If the price of bread increases (bread is typically an
inferior good), it can happen that the demand for bread grows rather than declines if
the income effect is very strong (bread takes up a large share of the household budgets).
Another example, which we shall discuss later (see section III.6), is the demand for leisure
time. Is leisure time a Giffen good, i.e. does the demand for leisure time increase (and,
correspondingly, is less work supplied) if the price of leisure time (wages) increases?
69
Summary of the individual effects using the example of good X
X is a normal good X is an inferior good
px ↑ SE: x ↓; IE: x ↓, TE: ↓ SE: x ↓, IE: x ↑, TE: ?
py ↑ SE: x ↑, IE: x ↓, TE: ? SE: x ↑, IE: x ↑, TE: x ↑
The table provides an overview of the different directions of income effects and substitution
effects on the demand for good X when the price of X or Y is increased. Of course, the
effects would be precisely the opposite if the prices of both goods were to sink. Note that
substitution effects and income effects also occur for good X when the price of good Y
changes!
Cross-price elasticities
Cross-price elasticities, i.e. the effects of changes to the price of a good on the demand for
another good, lead to the definition of substitutes and complements.
Good X is a substitute for good Y , when:
∂x
∂py> 0
i.e. the rise in price of good Y leads to good Y being substituted by good X, as good X
has now become relatively cheaper. Example: If the price of travelling by car increases
due to a higher petrol price, some motorists will switch to travelling by train, even though
the railway price remains unchanged.
Good X is a complement to good Y , when:
∂x
∂py< 0.
i.e. the increase in price of good Y leads to lower demand for good X. Example:
chairs/tables; PCs/printers; butter/bread. We find complementary relationships between
goods particularly in many new industries. Take, for example, the iPad and apps; the new
70
Kindle Fire and the books supplied by Amazon or smartphones and mobile phone tariffs.
This presents interesting questions for the price decisions of companies, if they supply both
goods. Why, for instance, is the Kindle Fire sold below its manufacturing costs? We shall
return to this question briefly at a later stage.
Concluding example: Impact of the reduction in coffee price
(Coffee and tea are imperfect substitutes and normal goods)
Good Substitution effect Income effect Overall effect
Coffee Coffee has become
cheaper, so the con-
sumer buys more
coffee
The purchasing power
of the income in-
creases, he buys more
coffee
Income effect and sub-
stitution effect work in
the same direction
Tea Tea has become rel-
atively expensive, so
the consumer buys
less tea
The purchasing power
of the income in-
creases, he buys more
tea
Income effect and sub-
stitution effect work in
opposite directions
The concept of consumer surplus
How much does a consumer gain from a transaction? We shall try to answer the question
with the help of the concept of consumer surplus.
Definition: Consumer surplus is the monetary equivalent of the utility gain from a trans-
action.
71
This can be presented graphically by means of the (inverse) demand curve:
x1 x2 x3
p3
p2
p1
x
p
6
-
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........ ........ ........ ........ ........ ................................................................................................................................................
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........................................................................................................
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ................................................
•C
O
•B
The points on the demand curve indicate the reservation price (maximum price) that the
consumer is prepared to pay for an additional (marginal) unit of the good. The relevant
price represents the marginal utility of the additional unit of the good (cf. previous analysis
of the optimal consumption decision). Example in the diagram: p1 for an additional unit at
x1, p2 at x2, p3 at x3 and so on. The area beneath the demand curve is known as the gross
consumer surplus, e.g. with a consumption of x3 : Ox3CB. Subtracting the spending of
the consumer (p3Ox3C) produces the (net) consumer surplus (p3CB). To interpret the
consumer surplus: a household would be willing to pay a maximum of Ox3CB to receive
x3 quantity of goods. However, due to the existence of only one market price (p3) the
household only pays Op3Cx3. The difference then produces the (net) consumer surplus,
which denotes the monetary equivalent of the utility gain of the household. The sum of all
consumers’ surpluses gives the aggregate consumer surplus. This is an important measure
for analysing the welfare effects of data changes (e.g. policy analysis). We now come to a
specific application.
72
Change to the consumer surplus in the case of exogenous price shocks
Specific situation: an exogenous shock (e.g. oil price rise) leads to an increase in the
market price. The diagram shows a price increase from p0 to p1 dar:
p0
p1
x
p
6
-
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................................................................................................................................................................
......................................................................................................................................................................................................................V
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
.
..
..
..
.............
This results in a reduction of the consumer surplus to the amount of V.
III.4.3 Buying and selling
In the previous chapter it was assumed that the households under examination have an
exogenously given income. This chapter, in contrast, analyses the effects of price changes
on households’ demand for goods when these households have an initial endowment of
goods. In other words, the households possess a certain amount of those goods that they
demand. As before, the goods’ prices are given exogenously.
An immediate example is a farm that demands goods: agricultural products and cars. But
the farm already owns an initial endowment of agricultural products, which generally even
exceeds its consumption demands. The question now is what is the effect of an increase
in the price of agricultural goods? This price change obviously also influences the total
income of the farm, as it counts such products among its initial endowment and can thus
be exchanged for (more) money. This is exactly what we want to observe more closely
below. Further examples include: What effect does a wage increase have in my labour
supply, given that my time endowment has become more valuable (I can „ sell my time “
for higher wage rates on the labour market, cf. section III.6)? Or: what is the effect of an
interest rate rise on my savings, if I already have initial assets (cf. also section III.7)?
73
Gross and net demand
Let us discuss these cases in general. In doing so we assume that the household demands
two goods (1 and 2). Of these goods, the household possesses an initial endowment amount-
ing to ω1 (quantity of good 1) and ω2 (quantity of good 2). We now differentiate between
the gross gross and the net demand of the household. The gross demand (x1, x2) denotes
the quantity that the household optimally consumes of goods 1 and 2. The net demand
is produced by the difference between the gross demand and the initial endowment, i.e.
(x1− ω1, x2 − ω2). The net demand therefore indicates how many units of goods 1 and 2
the household purchases above and beyond its initial endowment, i.e. on the market.
Budget restriction
As the income of the household consists entirely of the initial endowment, the budget
restriction is given with
p1x1 + p2x2 = p1ω1 + p2ω2
whereby p1 or p2 represent the prices for the goods 1, respectively 2. The initial endowment
bundle (ω1, ω2) lies on the budget line, as the household can always afford to consume these
quantities. In the diagram (x1, x2) the budget line has a slope of −p1p2
.
Change to the initial endowment
A change to the initial endowment from (ω1, ω2) to (ω′1, ω
′2) leads to an outward shift of
the budget line when p1ω′1+p2ω
′2 > p1ω1+p2ω2 and to an inward shift when p1ω
′1+p2ω
′2 <
p1ω1+ p2ω2. In the case of p1ω′1+ p2ω
′2 = p1ω1+ p2ω2 the budget line does not change, but
the new initial endowment bundle lies at a different point on the budget line.
Price changes and budget constraints
We shall now examine what influence a price change has on the optimal consumption
decision of the household at a given initial endowment. As the household budget consists
of the initial endowment bundle as valued at market prices, a price change also has an
effect on the value of the initial endowment of the household. In this case, therefore, price
changes imply an automatic change in income. If the price of a certain good changes, the
slope of the budget line also changes. However, because the household can still consume
74
its entire initial endowment independently of prices, the initial endowment bundle must
also lie on the new budget line, i.e. the budget line must rotate around the point of the
initial endowment (ω1, ω2).
Price changes and demand effects
In the previous chapter (i.e. at a constant monetary income) it was shown that, in the event
of a change to the price of a certain good, the overall effect of the change in demand can
be divided into two effects: the substitution effect and the income effect. Here, however,
because there is no constant monetary income in the case of a price change, there is an
additional effect: the endowment income effect. The endowment income effect comprises
those parts of the demand change that are caused by the change in income (i.e. the initial
endowment bundle, valued at market prices) when prices change. The change in demand is
therefore composed of the sum of three partial effects: substitution effect, ordinary income
effect and endowment income effect.
Graphical analysis
As in the previous chapter at a constant monetary income, the overall effect of a change
in demand for a certain good will here also be broken down into partial effects, using a
graphical analysis. We observe a price reduction for good 1, where good 1 is a normal
good.
66
�
x2
x1-
7
/
e
ee
e
e
A B C D
AA
O
75
In the diagram the initial endowment bundle is marked AA. Prior to the price change the
household consumes bundle O, while its original gross demand for good 1 is illustrated at
point A. The change to the price leads to a rotation of the budget line at the point of
the initial endowment. After the price change the household consumes C units of good
1. The overall effect of the demand change with regard to good 1 therefore passes from
A to C. According to Slutsky this overall effect can be divided into three partial effects.
The substitution effect is the movement from A to B. In the process, the new budget line
makes a parallel shift to the original optimal consumption bundle, in order to compensate
the household notionally for the changed prices so that it can afford the original optimal
consumption bundle at the changed prices. The ordinary income effect is shown by the
movement from B to D. Here it is assumed that the household has a constant monetary
income. Finally, the endowment income effect is the movement from D to C.
Summary
1. In this chapter we examined the effects of price and income changes on the
optimal consumption decision. The changes to the demand for each good were de-
termined by means of comparative statics.
2. Income-dependent optimal consumption bundles are represented in the (x, y)-diagram
by income curves, and individual demand for goods in the (x, I)-diagram by Engel
curves.
3. In general, goods can be distinguished according to the demand reaction to varia-
tions in income. For normal goods demand increases in income, whereas demand
decreases in income for inferior goods.
4. Furthermore, normal goods can be distinguished according the extent to which de-
mand changes in relation to income changes. Luxury goods experience a dispro-
portionately high expansion of demand when income increases, necessary goods
a disproportionately low one. In the case of homothetic preferences, which can
be described for example by a Cobb-Douglas utility function, demand and income
increase at the same ratio.
76
5. With regard to changes in demand for a good in reaction to price changes to that
good, goods can be divided into two types: ordinary goods and Giffen goods.
An ordinary good exists when the demand for a good increases (decreases) when the
price of this good decreases (increases). A Giffen good exists when the demand for a
good increases (decreases) when the price of this good increases (decreases).
6. The effects of price changes on the demand for goods can be described by a price-
consumption line (in the (x, y)-diagram), from which the individual demand func-
tions can be derived (illustration in the (x, p)-diagram).
7. The total demand effects of price changes (at a constant monetary income) can be
divided analytically into substitution effects and income effects. The substitu-
tion effect is derived by using the new relative prices and the compensated income.
According to the Slutsky definition, the compensated income is understood to be
that income which allows the consumer to purchase the previous consumption bun-
dle at new prices. The income effect then arises by means of the reduction of the
income to the actual nominal income.
8. It is important to note that substitution effects and income effects also occur to the
good concerned when the price of an alternative good changes. These cross-price
elasticities depend on whether the goods are substitutes or complements.
9. Finally, the effects of price changes on household utility can be quantified by means
of the concept of consumer surplus.
10. If consumers have an initial endowment of goods, the overall demand effects of
price changes can be divided into three partial effects: substitution effect, ordinary
income effect and endowment income effect. The endowment income effect is the
influence of a price change on the demand for a good that arises from a change in
income (i.e. the change to the initial endowment bundle as valued at market prices).
77
III.5 Overall demand for goods
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 4
III.5.1 Derivation of the overall demand function
Until now we have been concerned with the analysis of individual demand curves (e.g.
x(px)) i.e. with the demand of the individual household for the goods in question. We
shall now turn to the market demand or overall demand the description of which is an
important objective when analysing market exchanges.
Approach: Horizontal addition of the individual demand quantities
The individual demand quantities are determined for a certain price and then compiled.
This is done for many (all) prices, thus producing the overall demand curve.
Graphical analysis
4 8
8
16
x
px6
-
..............................................................................................................................................................................................................................................................................................................
........
........
........
........
........
........
........
........
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........
H1
A market with 2 consumers:
4x
px6
-
.......................................................................................................................................................
H2
4 12x
px6
-
⇒
......................................................................................................................................................................................................................................................................................................................................................................................................
........
........
........
........
........
........
........
........
Overall demand
Formula
x1 = 8− 0, 5p
x2 = 4− 0, 5p
78
Important: Always solve for the quantity! The overall demand function must be defined in
sections. To do so we determine the prohibitive prices of the households: for H 1 (x1 = 0):
p = 16; for H 2 (x2 = 0): p = 8.
=⇒ Overall demand function:
xtot = x1 + x2 =
0 for p > 16
8− 0, 5p for 16 ≥ p ≥ 8
12− p for 0 ≤ p < 8
III.5.2 Price elasticity of demand
From a business point of view it is very important to know the reaction in demand to
your own price changes. In this context, the price elasticity of demand proves to be a very
helpful instrument.
Definition: The price elasticity of demand is the ratio between the percentage change of
demand and the percentage change of the price.
Formally, in marginal notation, it is described by the point-price elasticity:
ϵ = −%x
%p= −dx/x
dp/p= −dx
dp
p
x
As the demand function generally has a negative slope (dx/dp < 0), the price elasticity of
demand is usually defined as negative =⇒ ϵ is a positive number.
Some examples of actual elasticities
The following overview presents the results of empirical studies:
Good/service Demand elasticity
Peas 2.8
Electricity 1.2
Beer 1.19
Cinema 0.87
Flights 0.77
Shoes 0.7
Theatre/opera 0.7
79
What, for example, does ϵ = 2.8? mean? When the price changes by one percent, the
demanded quantity changes by 2.8 percent. The greater ϵ is, the more elastic the demand.
Different demand functions and price elasticity
We differentiate between different (extreme) cases of elasticity in demand functions. These
cases will play an important role later (such as in the analysis of tax shifting).
1. Isoelastic demand: ϵ = constant
The demand function is a hyperbola:
p = ax−1/b
This is an isoelastic demand function, i.e. the elasticity is independent of the quantity
consumed. The price elasticity of demand is calculated thus
x =(pa
)−b
to
ϵ = −(− b
a
(pa
)−b−1)
p
(p/a)−b=
b
a
(pa
)−1
· p = b
The greater b, the more price elastic the demand function, and the stronger the reaction
of demand to price changes.
A small numerical example will illustrate the effects of b: b = 1, b = 2, a = 1
x \ p 1 2 3 4
x(b = 1) 1 12
13
14
x(b = 2) 1 14
19
116
Explanation: If the price of a good rises, the demand will decrease (normal good). However,
the more elastic the demand, the more pronounced the reduction in demand.
80
2. Perfectly elastic demand: ϵ→∞
-
6
p
x
p
.............................................................................................................................................................................................................................................................................................x(p)
Each marginal change to the price (away from p) leads to an infinitely large change in
demand. It will result either in x −→ 0 (at p ↑) or in x −→ ∞ (at p ↓). Approximate
examples for such cases are goods that have very close substitutes, such as an internet or
telephone product from Supplier Z or a bunch of carrots at the local market from Farmer
Y or the demand for standardised pistons for cars.
3. Perfectly inelastic demand: ϵ = 0
-
6
x
p
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
....p(x)
Price changes have no effect on demand. Approximate examples are essential necessary
goods such as important medication as well as the demand for hard drugs (e.g. heroin
etc).
81
4. Linear demand function:
x = A− bp (15)
The elasticity formula is:
ϵ = −dx
dp
p
x(16)
While dxdp
= −b is constant, px=
(xp
)−1
=(
Ap− b
)−1
is not. The demand elasticity is
therefore not constant along a linear demand function: the elasticity differs depending
on how high the original price is, from which a price increase or decrease is to be made.
Inserting this into (16), gives:
ϵ =+bA−bp
p
=+bp
A− bp
We can derive some properties of demand elasticity with linear demand functions:
1. p −→ 0 =⇒ ϵ −→ 0: demand does not react to a change in price
2. p −→ Ab
(saturation price)=⇒ ϵ −→ ∞: quantity reduces to zero already with
marginal price increases
3. At which p is ϵ = 1?
• ϵ = bpA−bp
= 1 =⇒ A− 2bp = 0
• p = A2b
• x = A− bp = A− bA2b
= A2
4. ϵ increases monotonically in p.
82
We get the following diagram:
A/2 A
A/(2b)
A/b
x
p
-
6
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ................................................................................
ε =∞ε > 1
ε = 1
ε < 1
ε = 0
...................................................................................................................................................................................................................................................................................................................................................................................................................
This diagram illustrates that elasticity of demand rises with low quantities and higher
prices. We shall see later that a company that has price-setting power will never choose
the area of the demand curve in which demand elasticity is smaller than one. Why?
Economic intuition provides the answer. Just think what effect a price increase would
have on turnover and costs if ϵ is smaller than one.
III.5.3 Income elasticity
The income elasticity
e =dx
dE
E
x
describes the ratio between a percentage change of demand to a one-percentage change in
income. The different types of goods can be differentiated by income elasticity:
e = 1 =⇒ homothetic preferences
e > 1 =⇒ luxury good
e < 1 =⇒ necessary good
e < 0 =⇒ inferior good
Income elasticity is incredibly helpful in terms of the reaction of demand in the business
cycle (in which income is subject to temporary fluctuation), or the long-term development
of demand (assuming that the income develops positively in the long term). It also explains
83
why the shares of utility companies (e.g. electricity) have little fluctuation in the business
cycle. The following table, which contains the results of empirical studies, provides an
answer:
Good/service Income elasticity
Car 2.46
Furniture 1.48
Restaurant meal 1.4
Cigarettes 0.64
Electricity 0.2
Margarine -0.2
Pork -0.2
Public transport -0.36
Electricity has low income elasticity, i.e. the demand for electricity hardly fluctuates in
the business cycle, so the same applies to the share prices of utility companies in normal
business cycles.
Summary
1. The aggregation of the individual demand functions of all households leads to the
overall demand function in the economy under examination. Graphically, the
overall demand function can be derived by adding horizontally the individual demand
functions in the (x, p)-diagram. Formally, the demand functions of the households
must be solved for the quantity and added. The overall demand function is defined
in sections. The intervals can be determined by means of the relevant prohibitive
prices of the households.
2. In keeping with the analysis of the individual demand for goods, the price and
income elasticity of the overall demand can be determined. The properties of
different demand functions become clear from the derivation of the implied elastici-
ties.
84
3. Isoelastic demand functions lead to price elasticity of demand, which is inde-
pendent of the quantity consumed. With perfectly elastic demand functions,
price changes lead to infinitely large changes in demand. With imperfectly elastic
demand functions, on the other hand, demand quantity is dependent on price.
4. In the frequently used case of linear demand functions, the price elasticity of
demand is dependent on the price level in each case. Generally, price elasticity
increases monotonically with the price.
5. In accordance with the income elasticity of overall demand a differentiation can
be made, along the lines of the analysis of individual demand for goods, between
homothetic preferences, luxury goods, necessary goods and inferior goods.
III.6 Work-leisure decisions
Until now we have observed the choice between two goods, with the income taken as
given. In reality income is not given, but rather depends greatly on our labour income.
This applies to most people. In that respect it is interesting to ask how labour income is
earned. Perhaps an even more important reason to analyse earned income is that the labour
market forms the key social problem in many countries. In principle the labour market
is just as much a market as the market for bananas or houses, i.e. supply and demand
mechanisms apply, which lead to a market equilibrium. The labour market is where labour
supply (households) and labour demand (companies, analysis in chapter IV.2.2) meet. In
this section we shall examine the labour supply decisions of the households.
Labour supply: How much is one willing to work at which price (wage rate)?
The starting point of our analysis is the idea that labour supply results from the choice
between leisure time and work time, which provides consumption possibilities. In partic-
ular, the labour supply decision is relevant to the question of whether overtime should be
worked or how long a working life should be. For self-employed people the question is often
whether or not to accept an additional contract, while single parents often have to decide
about whether to work full-time or part-time. Every labour supply decision is a decision
85
on time allocation: With a given time budget the household must choose between labour
time (L) and free time available for leisure activities (F ).
In the work-leisure decision, two restrictions must be taken into consideration. First, the
budget restriction (income = consumer spending), which we have already addressed:
wL+M = pC (17)
For reasons of simplicity there is only one consumption good (C) at price (p). Non-earned
income (e.g. from assets) is denoted with M while w represents the nominal wage. In
addition, logically, a time restriction must also be observed:
Z = L+ F, (18)
whereby Z indicates the maximum time budget (24 hours).
The budget and time restrictions together ((17) in (18)) produce the expanded budget
restriction:
L = Z − F = 24− F
w(24− F ) +M = pC (19)
Utility function
We assume that the utility of the households is influenced positively by both consumption
and by leisure time. We can establish the following general utility function:
U = U(C,F ) with ∂U/∂C > 0 and ∂U/∂F > 0,
as well as negative second derivatives
The equivalence to the usual household optimisation approach for two goods is obvious:
the structure of the constrained optimisation problem is identical.
86
Graphical analysis
-
6
L∗
C∗
(M + 24w)/p
F
C
.......................................................................................................................................................................................................................................
•........ ........ ........ ........ ........ ................................................................................
Indifference curve
w(24− F ) +M = p · C
24 +M/w
..............................................................................................................................................................................................................................................................................................................
It should be noted here that the consumption of free leisure time is limited to 24. In other
words, part of the budget line is notional (when M > 0) and cannot be reached by the
household.
Analytical solution
Max U = U(C,F )
s.t. : w(Z − F ) +M = pC
Lagrangian:
L = U(C,F ) + λ(M + w(24− F )− p · C) (20)
First order conditions:
∂L∂C
=∂U
∂C− λp = 0
∂L∂F
=∂U
∂F− λ · w = 0
∂L∂λ
= M + w(24− F )− p · C = 0
=⇒ ∂U/∂F
∂U/∂C=
w
p(21)
87
We now come to an interpretation of the optimality condition (21): The marginal rate of
the substitution between leisure time (F ) and consumption (C) represents the real wagewp. The real wage is the nominal wage that has been adjusted for inflation and is thus
defined in units of purchasing power. In terms of the concept of opportunity costs, the
wage rate can be interpreted as the price of leisure time, cf. Chapter I.4.
We can briefly consider the implications of a wage rate increase on the labour supply: in
the context of the substitution effect this leads to a reduction in demanded leisure time
and thus to an increase in the labour supply by the household. This is determined by the
fact that the opportunity costs of leisure time increase (alternative: working for a higher
wage). This leads to a lower demand for leisure time, and thus to a higher labour supply.
Furthermore, there are now 2 reasons for an income effect. We have already observed
the first in the Slutsky division, the wage rate increase leads to an “ordinary” income
effect. Leisure time becomes more expensive and real income decreases as a result (at a
given nominal income). Because leisure time is a normal good, leisure time is demanded
less due to the ordinary income effect, and more labour is supplied by the household. But
now there is also a second effect: an endowment income effect. The change in price
not only changes the relative value of leisure time, or consumption, but also influences
the income level. A higher wage rate means that the household’s time endowment gains
in value. This higher (nominal) income leads to an increased demand for leisure time (a
lower supply of labour). As the substitution effect and the ordinary income effect point
in the same direction (more labour supply), but the endowment income effect indicates a
lower supply of labour, the overall effect of a wage rate increase is a priori indeterminate.
Summary
1. The labour supply of households is derived from the decision between leisure time
and consumption possibilities that result from additional earned income. It is a
decision on time allocation: work time versus leisure time.
2. The household’s optimisation problem is now subject not only to the aforementioned
budget restriction, but also to a time restriction. In the work-leisure decision, the
88
arguments of the utility function encompass leisure time and consumption. Other-
wise the optimisation problem represents the consumption decision in the two-goods
model at a given price.
3. Ideally the marginal rate of substitution for leisure time and consumption represents
the real wage. The latter is the nominal wage rate in units of purchasing power,
which can also be interpreted, according to the concept of opportunity costs, as the
price of leisure time.
4. Similar to the price effect analysis in the context of consumption decisions, the effects
of wage rate changes on labour supply can be analysed. A wage increase leads to
a negative income effect and to a positive substitution effect on the labour supply.
The overall effect is therefore ex ante indeterminate.
III.7 Intertemporal decisions
Until now we have analysed household decisions between two alternatives at a point in
time: Consumption of X ←→ consumption of Y or leisure time ←→ consumption
Now we want to address the question as to how households distribute their income across
time (periods). The simplest approach is a two-period model that can illustrate the con-
sumer decision to „consume today“ (C0) versus „to consume tomorrow“ (C1). We label
the income in each period M0 and M1. Applications include consumption this year as
against consumption next year, or consumption during wage-earning years versus that in
retirement.
Derivation of the budget restriction
We shall first examine the easiest possible case (intertemporal budget restriction without
interest rate), before then turning to the more realistic case with a positive interest rate.
1. Interest rate =0 (“cushion“)
Budget restriction: C0 + p1 · C1 = M0 +M1
89
p1 denotes the relative price of the consumer good in period 1 in relation to the price in
period 0. Below we shall ignore inflation (p1 = 1), the budget restriction becomes:
C0 + C1 = M0 +M1
M0
M1
C0
C1
-
6
•........ ........ ........ ........ ........ ........ ........ ................................................................
Slope: −1
Endowment point
..............................................................................................................................................................................................................................................................................................................
Implicit assumption: no debt restriction, C0 > M0 is possible.
2. Positive interest (realistic case)
Budget restriction:
Present value of consumer spending = Present value of income
Restriction in period 0:
C0 + S0 = M0 (22)
with S0, savings in period 0, which will be consumed in period 1.
Restriction in period 1 with interest rate r:
C1 = S0(1 + r) +M1 (23)
=⇒ C1
1 + r− M1
1 + r= S0 (23’)
(23’) in (22) gives:
C0 +C1
1 + r= M0 +
M1
1 + r(24)
90
The present value (or capital value, right-hand side of the equation) of a payment flow in
the period is the only correct conversion of the payments into their present monetary value.
This allows payment flows in different periods to be compared. The series of payments
with the highest present value (PV ) should always be preferred. With unrestricted debt
and borrowing possibilities at an interest rate r a payment flow with a higher PV can
always deliver more consumption in every period.
How is the present value of income distributed across periods?
Basic idea: Households have a preference for present consumption, i.e. a positive time
preference rate ρ. A possible cause of a positive time preference rate could be uncertainty
about consumption possibilities in the future. Moreover, there is a positive probability of
dying. The time preference rate differs individually: someone who rides a motorbike on
the motorway at a speed of 220 obviously has a high time preference rate.
Utility function:
U = u(C0) +u(C1)
1 + ρ
Whereby u′(Ci) > 0 and u“(Ci) < 0 (convex preferences!). The larger ρ, the stronger
the preference for present consumption.
Graphical analysis:
M0 C0
C1
M1
C0
C1
-
6
.........................................................................................................................................................................................
•
•
•
........ ........ ........ ................................................................................................
........ ........ ........ ........ ........ ........ ........ ................................................................
AI
AII
E
..............................................................................................................................................................................................................................................................................................................
If the endowment point is located to the northwest of the optimal point E (e.g. AI), the
household takes a debtor position, i.e. it borrows money in period 0 (C0 > M0). In the
opposite case (e.g. AII) the household becomes a creditor in period 0 (C0 < M0). The
91
slope of the budget line is −(1 + r), cf. budget restriction (24). If the interest rate rises,
the budget line becomes steeper, and more consumption is transferred to the future: C1 ↑
respectively C0 ↓.
Analytical solution:
The optimisation problem:
max U(C0, C1)
s.t. C0 +C1
1 + r= M0 +
M1
1 + r
Solution with the help of the Lagrange function:
L = u(C0) +u(C1)
1 + ρ+ λ
(M0 +
M1
1 + r− C0 −
C1
1 + r
)(25)
First order conditions:
∂L∂C0
= u′(C0)− λ = 0 (26)
∂L∂C1
=u
′(C1)
1 + ρ− λ
1
1 + r= 0 (27)
∂L∂λ
= M0 +M1
1 + r− C0 −
C1
1 + r= 0 (28)
Interpretation of equation (26): more consumption in period 0 (C0)
- Advantage of an additional unit C0: marginal utility in period 0 (u′(C0))
- Disadvantage: lost consumption in period 1 measured by λ
Interpretation of equation (27): more consumption in period 1 (C1)
- Advantage of the additional consumption in 1: marginal utility u′(C1), discounted
with (1 + ρ).
- Disadvantage of the consumption: lost consumption possibility in 0 (u′(C0) = λ),
discounted with (1 + r), because one must only forego 11+r
units of C0 in t = 0, in
order to be able to consume one unit C1 in t = 1.
92
Optimality requires that (26) and (27) apply:
u′(C0)
u′(C1)=
1 + r
1 + ρ
This equation determines the distribution of the present value of the income across the
consumption levels in both periods. The marginal rate of substitution ideally represents
just the relative price ratio. The (gross) price of consumption in period 0 is (1 + r) (lost
interest), while the (gross) price of consumption in period 1 is (1 + ρ) (relinquishment of
present consumption). We can distinguish the different cases:
a) r = ρ =⇒ C0 = C1: The level of consumption in the period is constant
b) r > ρ =⇒ 1+r1+ρ
> 1 =⇒ u′(C0) > u
′(C1)
=⇒ C1 > C0 : The level of consumption rises over time.
• Economic reasoning: the utility of saving is relatively high in comparism to the costs
of saving (relinquishment of consumption today), due to the high interest on savings
in relation to the low time preference rate (ρ).
c) r < ρ =⇒ 1+r1+ρ
< 1
=⇒ u′(C0) < u
′(C1) =⇒ C1 < C0
Economic reasoning along the above lines.
If we now assume a specific utility function, the consumption path can also be determined
explicitly. u(C0) = lnC0 and u(C1) = lnC2. This produces, optimally:
C1
C0
=1 + r
1 + ρ
If we insert this into equation (28) and solve it accordingly, we get:
C∗0 =
1 + ρ
2 + ρ
(M0 +
M1
1 + r
)and
C∗1 =
1 + r
2 + ρ
(M0 +
M1
1 + r
)
93
For the specific ρ = r = 0 this then produces the intuitive result: C0 = C1 = 0, 5(M0+M1),
i.e. the household distributes half of its consumption on the sum of the incomes in both
periods.
Important implication: Life cycle hypothesis of consumption
In contrast to this is the traditional consumption hypothesis: consumption is only
one function of the (present) periodic income, i.e. C0 = f(M0) and C1 = f(M1). Future
expected income thus does not influence present consumer behaviour at all. The traditional
consumption hypothesis is the basis for Keynesian business cycle policy (“mass purchasing
power“). An increase in income today (e.g. by tax cuts) increases consumption today.
This also applies when the tax reductions of today will be fully compensated tomorrow by
tax increases.
However, our intertemporal decision model implies: consumption is a function of
lifetime income. Compare here equation (28): M0 +M1
1+r= C0 +
C1
1+rbei r = ρ =⇒
C0 = C1. With many periods and increases of a single periodic income, the lifetime income
increases only slightly and there is only a very small consumption effect.
Numeric example
We shall observe a household with a lifetime of 40 periods, the interest rate is r = 0.
Assume that the present income rises by 10$, e.g. due to a temporary tax reduction. The
effect on the lifetime income is then 1/40 of the ten-percent increase of the present income
(0, 25% ↑).
Furthermore, in the case of debt-financed tax reductions, higher taxes must be raised
later in order to repay the national debt. In this case there is absolutely no effect on
present consumption. Example: The state reduces taxes by 100 today at unchanged
state expenditure. The lower tax revenue thus increases the national debt by 100. At
an interest rate of 10$ the state will have to repay 110 in the next period. If it finances
this with a tax increase, the lifetime income of the private household has not changed:
+100−110/1, 1 = 0. This effect is known as Ricardo-Barro Neutrality in the literature.
Empirical studies show that the life cycle hypothesis of consumption is relevant to the
behaviour of households.
94
Summary
1. Intertemporal modelling approaches of the theory of the household are con-
cerned with how households convert income into consumption at different points in
time. It is noted that households can act as both takers and providers of credit
on capital markets. This possibility means that income can be distributed (at will)
across periods, in order to achieve optimal utility. This is important, as the marginal
utility of consumption decreases with the amount of consumption.
2. Accordingly, it is not the actual income in each period that is decisive for consumer
decisions, but rather the capital value of the income from all observed periods.
3. In a two-period model, the relationship between capital market interest and
time preference rates determines the relationship between present and future
consumption. A high (positive) time preference leads to a relatively high present
consumption, while a higher interest rate makes saving more attractive and increases
future consumption. These connections also apply in more realistic multi-period
models.
4. The most important implication of intertemporal models is the importance of the
lifetime income for the consumption decisions of households. This is a strong con-
trast to Keynesian approaches, which base the consumption decision on the current
(available) income.
5. The difference between the two approaches is by no means merely a theoretical con-
flict, as the economic policy recommendations diverge greatly. If the lifetime income
is emphasized, then debt-financed tax reductions, for example, do not produce the
desired effect of increasing mass purchasing power, because the households anticipate
future tax payments (Ricardo-Barro Neutrality).
95
III.8 Uncertainty
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 5
Until now we have assumed that households have complete information about future de-
velopments. Now we expand the previous model to include uncertainty, or risk. Future
income and its associated consumption possibilities can therefore no longer be predicted
with certainty. However, households form expectations about future events. This implies
the establishment of assumptions on probability distribution. Simple examples:
1. When choosing a course of study we form an opinion (expectation) about the pro-
fessional opportunities after completing our degree and the probability of success,
without knowing either with certainty.
2. When choosing a job we have expectations about future career chances.
3. When we go to the cinema, we have certain expectations about the movie.
The question thus arises as to how a household behaves in a world with uncertainty and
what the implications of this are. A decision must be made about how high the consump-
tion of a good should be in different environmental circumstances. Examples include:
1. Revenue/consumption possibilities when stocks perform well versus poorly.
2. Consumption possibilities in conditions of employment/unemployment or illness/health.
3. Remaining consumption possibilities in the event of a car accident/non-accident.
Generally, we are talking about the determination of consumption quantities depending
on circumstances and the resulting utility.
Illustration of the basic idea: The insurance example
Assuming a household owns a house to the value of 500.000 monetary units. With proba-
bility q there will be a fire and a resulting loss in value of 400.000 monetary units. With
96
the opposite event probability 1− q nothing will happen. What insurance premium is the
household prepared to pay?
Economic consequence of the insurance: an uncertain payment is transformed into a certain
payment. There are two possible states of nature that lead to two alternative „consumption
quantities“. With probability q the house will burn down:
C0 = 100.000 =⇒ U(C0)
With probability 1− q nothing happens:
C1 = 500.000 =⇒ U(C1)
The household values the alternatives according to the expected utility they will provide.
The objective function of the household is the expected utility EU . This can be ex-
pressed more formal as follows
EU = qU(C0) + (1− q)U(C1)
This target figure is also known as a von Neumann-Morgenstern utility function.
The valuation of both states/consumption levels does not depend on the expected value
of the payout E(C) = q · C0 + (1 − q) · C1. It is very important to distinguish between
the utility of the expected value, U(E(C)), and the expected utility, EU . Reason:
a decision based on E(C) always implies risk neutrality of the household. The von
Neumann-Morgenstern utility function, however, allows different attitudes towards risk.
Below we look at these risk attitudes (risk neutral, risk averse and risk loving).
a) Risk neutrality U(Ci) = Ci , i.e. U ′′ = 0 .
-
6
Ci
U
............................
.............................
..............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
•
•
•
Risk neutrality:
97
The household is indifferent between consuming the expected value E(C0, C1) (no uncer-
tainty) and consumption in an uncertain situation → U(E(C)) = EU .
b) Risk aversion: U′(Ci) > 0, U
′′(Ci) < 0, concave utility function, i.e.
U(E(C)) > EU = qU(C0) + (1− q)U(C1)
C0 E C1
E U
U(E(.))
Ci
U
-
6
.......................................................................................................................................................................................................................................................................................................................................................................................
.................................
.....................................
............................................
........................................................
................
...........................................................................................................................................................................................................................................................................................................................................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ............................................................................................................................
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ....
•
•
Risk aversion:
The utility of the expected value of consumption is determined as the value of the utility
function at position E; the expected utility is determined by the weighted sum of the utilities
C0 and C1 and is located vertically above E on the connecting line between U(C0) and
U(C1). Accordingly, with risk aversion, it holds that U(E(C)) > EU and the household
thus prefers the certain alternative. In our example the expected value of consumption
with a damage probability of q = 0, 1 is exactly E(C) = 460.000 monetary units. This
implies an expected loss of 40.000 monetary units. The household is then willing to pay
a premium that is even higher than 40.000 monetary units, to be sure of having 500.000
monetary units. To put it another way, this means that the household prefers a safe
consumption of C < 460.000 monetary unitsto an uncertain situation. This is because the
utility from the safe 460.000 monetary units is already higher than the expected utility
that would emerge from the utility in the event of damage and the utility in the event of
non-damage.
98
c) Risk loving: Convex utility function: U′′(C) > 0, i.e.
U(E(C)) < EU = qU(C0) + (1− q)U(C1)
C0 E C1
U(C0)
U(E)E U
U(C1)
Ci
U
...........................................................
..............................................
......................................
.................................
...............................
..................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................................................................
........ ........ ........ ........ ........ ........ ........ .....................
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ................................................................................................................................
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........................................................................................
•
•
Risk loving:
A further example: Lottery
The utility function of a household is U(C) = C12 . There is the opportunity to take part
in the following lottery:
with q = 0, 5 C0 = 0 1− q = 0, 5 C1 = 1.000.000
The premium for participation is P = 160.000 monetary units. Will the household take
part?
Decision situation: Initially there is a negative payout −P and later with probability q
there will be no payout (C0 = 0) and with probability (1− q) there will be a payout of C1.
We can evaluate the lottery using the von Neumann-Morgenstern utility function and the
utility equivalent of the premium:
−P
q 1− q
C0 C1
.....................................................................................................................................................................................................................
.....................................................................................................................................................................................................................︸ ︷︷ ︸EU − 1 · U(P )
99
Expected payout:
EC = 0, 5 · 0 + 0, 5 · 1000.000 = 500.000
Expected utility:
EU = 0, 5 · 0 + 0, 5 · (1.000.000)1/2 = 500
The utility that would result from the consumption of the premium if it did not have to
be paid would be:
U(P ) =√160000 = 400
Because U(P ) < EU(lottery), participation in the lottery increases utility! For a premium
with a utility level greater or equal to 500 (√P = 500 → P = 5002), participation no
longer makes sense for the household.
Yet another example: Decision about a course of study or training
A high-school graduate has two alternatives: Course U (e.g. Business Economics at a
university) and Course B (e.g. vocational training).
Course U
With q = 0, 6 : success (very good grades) −→ good job, lifetime income: y = 4.000.000
1− q = 0, 4: failure (poor grades) −→ bad job: y = 640.000
Course B
Certain success: y = 1.000.000
The utility function of the high-school graduate is U(y) =√y. Which course should the
high-school graduate chose?
Uncertain case U
EUU = 0, 6 ·√4.000.000 + 0, 4 ·
√640.000
= 1200 + 320 = 1520
Certain case B
EUB = 1 ·√1.000.000 = 1000
EUU > EUB =⇒ university course is ex-ante optimal.
100
Application: The advantage of diversification/risk spreading
The above insurance example has shown that a risk adverse household can increase its
utility by means of insurance. This need for insurance also plays a large role in investment
decisions on capital markets. This context will now be explained by using a very simple
model. Assume that there are two possible state of nature (Z1, Z2):
• Z1: Drastic increase in mineral oil tax (probability: q)
• Z2: No increase in mineral oil tax (probability: 1− q)
Furthermore, only two companies exist:
• Company A: automobile company, loses profit in Z1:
company value (Z1A): 100; company value (Z2
A): 1000
• Company B: Bicycle manufacturer, loses profit in Z2:
company value (Z1B): 1000; company value (Z2
B): 100
The company values are perfectly negatively correlated. The purchase of the shares of only
one company leads to a risky investment strategy. The purchase of both shares means lower
risk (no risk at all here!). Diversification improves the investment result for a risk adverse
investor!
Further possible applications: financial investments (stocks/government bonds), choice of
profession, restaurant visit (especially to a new restaurant), decision to marry, choice of
residence.
Note: A key prerequisite for diversification is that the risk factors are not strongly corre-
lated (cf. current financial crisis).
Summary
1. The introduction of uncertainty or risk, to the basic model of the theory of the
household suspends the assumption of complete information. Normally, future in-
come flows are uncertain by nature. However, households form expectations about
future events and their probability distribution.
101
2. The insurance example illustrates the basic concepts of modeling uncertainty. In-
surance transforms uncertain payment flows into safe ones. A household is willing
to pay a premium for this service, as long as it is risk adverse. Von-Neumann-
Morgenstern utility functions can determine how high these premiums have to
be in order to provide the household with the same or greater utility.
3. In order to determine gains in utility, a distinction must be made between two con-
cepts. The expected utility, put simply, describes the average value of two utility
levels, which represent two possible future income situations. The utility of the
expected value on the other hand, specifies the utility of the average future income.
4. With the analytical instruments of the insurance example we can also address other
applications such as the lottery, the choice between a university degree and voca-
tional training, or the investment decision of an investor.
102
IV Production theory and company decisions
In this part of the lecture we are concerned with the behavior of companies. The general
topic is the determination of the factor demand of companies and the optimal supply of
goods. By factor demand we mean the demand for production factors such as labor, capital
and intermediate goods. We therefore derive the counterparts of the goods demand of the
household and its labor and capital supply functions: the goods supply as well as the labor
and capital demands of the company.
The optimal supply of goods is influenced by the cost structure of the companies, the
prevailing market structure and the demand situation. Here, and below, the term goods
includes both material goods and services.
The rough structure of the optimization problem of companies can be illustrated by the
following diagram:
Inputs −−−−−−−−−→Input prices Company/Technology −−−−−−−−−−→Output prices Output
Companies demand input (production factors) on the factor markets that are necessary
for the production of the output (goods supply). The production conditions (technology)
can be illustrated by means of production functions. The assumed objectives of operations
are profit maximization, respectively cost minimization. The relevant input prices and the
technology used determine the optimal sales price determination of the companies.
The company behavior can be derived in a number of steps:
1. Description of the technology (input-output ratios)
2. Cost-minimal factor input, derivation of cost curves
3. Derivation of the goods supply and the factor demand of a company
4. Goods supply and factor demand for many companies
In the process, the entire decision making process will be divided into many partial deci-
sions. The following simple example will briefly illustrate our future approach.
103
Introductory example: Automobile production
A car can be produced with 400 working hours and 2 robots or with 200 working hours
and 3 robots. These contexts will be referred to below as technology. The production of
10 cars requires ten times the deployment of technology than to produce one car.
If a working hour costs 40 monetary units and 1 robot 5,000 monetary units, it is obviously
cheaper to choose the second option, i.e. to produce capital-intensively. The minimal cost
of a car would then be 23,000 monetary units and that of ten cars 230,000 monetary units.
If working hours cost only 20 monetary units per hour, the first technology option would
be cheaper. The minimal costs of one car would then be 18,000 monetary units (10 cars:
180,000 monetary units).
−→ In many aspects it involves a similar approach and method as in the theory of the
household.
Overview: Commonalities and differences between the theory of the company
and the theory of the household
Households Companies
Restriction Budget restriction Technology (production
function)
Objective function Utility function (prefer-
ences)
Cost/profit function
Behaviour hypotheses Utility maximisation Profit maximisation/ cost
minimisation
Method of analysis Maximisation subject to
constraints
Maximisation subject to
constraints
Result Goods demand/factor sup-
ply
Goods supply/factor de-
mand
Essential differences Utility function of house-
holds is ordinal
Profit and cost function is
cardinal (valued in mone-
tary units)
104
IV.1 Technology and production
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 6
IV.1.1 Essential terms
Technology describes input-output ratios. Questions arising from this include: How
many inputs (work, capital, intermediate goods) are needed to produce a certain quantity
of output (goods or services)? Or: What is the maximum amount of output that can be
produced with a certain input quantity?
Inputs are factors of production such as work, machines and primary products.
Outputs are the sales goods, which can also include services.
The input-output ratio is described by a production function:
y = f(K,L)
y denotes the output (e.g. cars, television production), K the capital input and L the work
input. In other words, the production function indicates how much can be produced (e.g.
10 cars) with a certain input of capital (e.g. 10 machines) and workers (e.g. 5 workers).
Note: There are different kinds of "capital": physical capital (machines), human capital
(employee training) and money capital (short-term loans). Below the term capital generally
refers to physical capital. In principle, however, it is also possible to take account of human
capital.
An example of a specific production function:
y = K1/2L1/2
The following table presents the output levels depending on different input combinations:
105
K\L 1 4 9 16
1 1 2 3 4
4 2 4 6 8
9 3 6 9 12
16 4 8 12 16
IV.1.2 Short-term production function and the law of diminishing marginal
returns
In general we distinguish between long-term and short-term production functions. Long
term denotes the shortest period of time in which all production factors can be changed.
Short term means that one or more production factors cannot be changed. The following
examples will illustrate these differences.
1. In an industrial company the number of working hours (overtime) can be changed at
short notice, but no new machines can be procured and no new factory can be built.
2. In a restaurant - for example in a holiday resort - more waiters can be employed in the
short term in order to improve the speed of service (and to reduce the average duration of
a guest’s visit), but it is not possible to build a second kitchen in the short term.
We call the production factor that can be changed in the short term (e.g. working hours)
the variable production factor. The production factor that cannot be changed at short
notice (capital input in the widest sense) is known as the fixed production factor. The
following applies (empirically) with the fixed input of one or more factors:
Law of diminishing marginal returns
With an additional input of one unit of the variable production factor, and no change in
the other production factors, the additional output sinks with the increased input of the
variable production factor.
In our restaurant example this means: if only one waiter is employed initially, a second
waiter will naturally have a noticeable positive effect; however, if more and more additional
106
waiters are employed, they will end up standing around and the additional return from an
extra waiter tends towards zero.
=⇒ Diminishing marginal returns are the essential characteristic of the short-term pro-
duction function.
Illustration based on the specific production function y = K1/2L1/2
Assume that the production factor capital (K) can only be changed in the long term, i.e.
K = const., e.g. K = 1. We always need more labour input (L), in order to produce
one additional output unit (see also the relevant values in the first column of the above
table for K = 1). Alternative formulation: with an additional input of one unit of L the
result will always be less additional output as shown in the examples above (overtime, or
additional waiters). As a formula, we can calculate the marginal return (or also the partial
marginal productivity) of the factor of work to analyse this problem:
∂y(L, K)
∂L=
1
2L−1/2K1/2
This decreases because the 2nd order derivative is negative (diminishing marginal returns).
The following diagram presents the production function and the marginal return function.
y
L-
6
.......
.......
.......
.......
.......
.......
.................................................................................................................................................................................................................................................
..................................
.......................................
..............................................
......................................................
.................................................................
................................................. y(L, K)
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ∂y/∂L
107
Relation between average return (average product) and marginal return (marginal
product)
Average return (y/L): is produced graphically by the slope of a connecting line from the
origin and the point to be examined on the production function.
Marginal return ( ∂y∂L
): is produced graphically by the slope of the tangents at a point of
the production function.
Graphical representation (based on a production function according to the law of dimin-
ishing returns):
-
6
L
y
•
•
•
........................................................................
.............................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................................................................................................
................................................y(K, L)
-
6
•
y/L∂y/∂L
L...........................................................................................................................................................................
................................................................................................................................................................................................................................................................................................∂y/∂L
............................................................................................................
.................................
..............................................
........................................................................................................................y/L
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
......
As long as ∂y/∂L > y/L applies, average productivity increases. As soon as ∂y/∂L < y/L
holds, y/L decreases. Economic intuition: In order for y/L to increase, an additional work
unit must provide a greater increase in output than the previous average. In general, the
∂y/∂L-curve intersects with the y/L-curve at its maximum. For ∂y/∂L = 0 the output y
is maximal (per definition).
The production function according to the law of diminishing returns is the
most important example of short-term production functions.
108
The essential features of a production function according to the law of diminishing returns
are as follows: one production factor is kept constant (e.g. capital) and the other pro-
duction factor (e.g. work) varies. It is therefore a short-term production function. The
path of this production function, already shown above, is characterised by the fact that
for initially low output quantities (y) there is an increasing marginal return of the factor
of work. For large output quantities (y) the marginal return sinks with additional work
input.
A numerical example of a production function according to the law of diminishing returns:
y = 5L2 − 1
100L3
This function can result, for example, from
y = 5L2K − 1
100L3K2 for K = 1
The average productivity is then:
y
L= 5L− 1
100L2
The following holds for marginal productivity:
∂y
∂L= 10L− 3
100L2
The slope of the marginal productivity function is determined by
∂2y
∂L2= 10− 6
100L
The maximum of the marginal productivity function (LG) is given by
∂2y/∂L2 = 0 =⇒ LG = 166, 6
and finally the maximum of the average productivity function with
∂(y/L)
∂L= 5− 1
50L = 0 LD = 250
109
Illustration by means of a table:
L y yL
∂y∂L
∂2y∂L2
0 0 0 0 10
10 490 49 97 9,4
30 4230 141 273 8,2
50 11250 225 425 7,0
75 23909 318,8 581,3 5,5
100 40000 400 700 4,0
125 58593 468,8 781,3 2,5
150 78750 525 825 1,0
166,6 92592 555, 5 833,3 0,0
175 99531,3 568,8 831,3 -0,5
200 120000 600 800 -2,0
250 156250 625 625 -5,0
300 180000 600 300 -8,0
333,3 185185 555,6 0 -10
110
The diagram based on the table:
185185
L
y
-
6
...................................................
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................y(L)
166.7 250 333.3
833.3625
-
6
y/L
∂y/∂L∂2y/∂2L
L....................................................................................................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
...
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
...............................................................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................∂y/∂L
.................................................................................................................................................................
...............................
....................................
...............................................
.........................................................................................................................................................................................y/L
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................∂2y/∂2L
Note: In the exam, depending on the question, it must be possible to present the basic path
of a production function according to the law of diminishing returns without a table. It is
always helpful to calculate each of the maximums and zero points, in order to characterise
the production function.
IV.1.3 Long-term production functions
Until now only one production factor could be changed, while the other remained fixed.
Now we shall turn to the possibility that all production factors are variable. This means,
for example, that both the number of working hours and the capital resources can be
changed.
111
Central questions:
1. What effect does a change in the input of both production factors have on the output?
Restaurant example: To what extent does the number of guests change if the space
and the personnel are doubled?
2. With which combination of both production factors can the same output be gener-
ated? Restaurant example: What are the savings in personnel when a deep-frying
machine is used (with the same number of guests)?
The matter is illustrated with the help of so-called isoquants.
Definition: An isoquant indicates input combinations that lead to the same output (similar
to the indifference curve).
Example: y = K0,5L0,5 ⇐⇒ y2 = KL ⇐⇒ K = y2/L
-
6
L
K
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
y1
y2
y3
y3 > y2 > y1
Properties of the long-term production function
The slope of the isoquant (−dK/dL) at one point indicates the marginal rate of tech-
nical substitution (MRTS) The economic interpretation of the MRTS is directly anal-
ogous to the marginal rate of substitution (MRS) of the indifference curve in the context
112
of the theory of the household: How much additional capital (K) must be spent in order
to compensate for the reduction of the work input (L) by one unit?
MRTS = −∆K
∆Lor MRTSK/L = −∆K
∆L ∆L→0= −dK
dL
Interpretation in case of marginal changes: How much must K be altered in the case of a
marginal change of L in order to keep y constant?
The marginal rate of technical substitution decreases with additional work input, i.e. the
slope of the isoquant becomes flatter when L increases.
Economic intuition: The higher the work input at the beginning, the less additional capital
must be used in the event of a marginal reduction of L.
Illustration using the example of a Cobb-Douglas production function:
y = K1/2L1/2
MRTS = −dK
dL=
∂y/∂L
∂y/∂K=
12K1/2L−1/2
12K−1/2L1/2
=K
L
∂MRTS
∂L= −K
L2< 0
The calculation of the MRTS is carried out by means of the total differential. Compare
this with the derivation of the marginal rate of substitution in the chapter on the theory
of the household.
Different types of isoquants
Different types of isoquants depict different production processes. A classification of these
processes can be made according to the degree to which the production factors are substi-
tutes or complements.
113
-
6
L
K
.......................................................................................................................................................
...................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................................................................
Perfect substitutes:
e.g.: y = K + L
-
6
L
K
..................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................
............................................................................................................................................
•
•
•
Perfect complements:
e.g.: y = min(
KaK
, LaL
)aK and aL: Input coefficients
-
6
L
K
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
Imperfect substitutes:
e.g.: y = KαLβ
114
Explanation and examples:
1. Perfect substitutes: neither of the two production factors is absolutely necessary.
Stock exchange: only floor trading (work) or fully electronic trading system (capital);
the comment from the theory of the household also applies here: this can, but must
not necessarily, be accompanied by a constant MRTS, the key issue is merely that
the isoquants intersect the axes.
2. Perfect complements: both production factors must be provided, the factor with the
lower input level decides the production quantity.
Example: (1 PC - 1 typist) or (1 bicycle frame - 2 wheel rims)
3. Imperfect substitutes: if one production factor is particularly expensive, it can be
replaced, at least partially, by the cheaper factor.
Example: Capital and work input in automobile manufacturing
IV.1.4 Isoquants, short-term production function and marginal productivity
In order to illustrate the relation between short and long-term production functions and the
associated forms of presentation, these will in the following be analysed again in context.
Basic idea:
(1) The isoquants reflect the long-term production function (all factors are variable).
(2) If we now keep one factor constant (e.g. capital) and vary only one factor (e.g. work),
we can derive the short-term production function.
(3) The (partial) marginal productivity is given by the slope of the short-term production
function.
These steps will be shown graphically below.
Starting point: Factors are (imperfect) substitutes.
115
-
6
L0 L1 L2 L3
K
K
L
.................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
• • • •
y0y1
y2
y3
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
Example:
y = AKαLβ , A > 0 .
It holds that y1 − y0 = y2 − y1 = y3 − y2
-
6
L0 L1 L2 L3
y0
y1
y2
y3
y
L.................................................................................................................................................................................................................................
.................
..................
....................
......................
.........................
.............................
......................................
.............................................................................
........ ........ ........ ........
........ ........ ........ ........ ........ ........ ........
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ...
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........
•
The representation refers to the case of constant capital input (K = K) in the illustration
above. The horizontal intersection through the isoquant map gives us the short-term
production function, shown in the lower illustration. Now the partial marginal productivity
can, in turn, be found by deriving the short-term production function for L. The partial
marginal productivity is often referred to as marginal product or marginal return.
The marginal productivity of the factor L decreases, i.e. the higher the work input,
the lower the output quantity increase produced by the last (marginal) worker. This
is produced graphically by the concave path of the short-term production function: the
result is a disproportionately low increase in outputs relative to input (L) at constant K.
In the upper part of the diagram, L, due to diminishing marginal returns, must increase
116
ever more in order to achieve a constant growth of y. This is demonstrated in the following
diagram, which shows the marginal productivity curve derived from the slope of the short-
term production function:
L0L1 L2 L3
-
6
∂y/∂L
L
y1−y0
y2−y1
y3−y2
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
Total factor variation (−→ long-term production function): Economies of Scale
A core question for the composition of an industry and for strategic company decisions
such as the decision on company growth by expanding production or by buying up other
companies (takeover or merger) is whether production is efficient for relatively large or
small quantities. Relative in this context means relative to the size of the market. The
answer to this question determines whether there are many or few companies in an industry.
From a technological viewpoint the type of economies of scale are of key significance.
Definition: Economies of scale show how the output changes if all production factors are
changed to the same extent.
The question of economies of scale plays a large role when a decision must be made on
whether an additional production unit should be set up, given an identical combination
of factors. For example, when an automobile company wishes to set up a new car manu-
facturing plant, the following question is raised: How many more cars can be produced if
both the work input and the capital input are doubled?
Previously: What happens when the input of a single factor is increased and the other
remains constant? −→ partial factor variation
Now: What happens when all factors change proportionately? −→ total factor variation
117
Three cases can be distinguished when all inputs are changed by the factor λ (−→ three
types of economies of scale):
1. Constant economies of scale: Increase in all inputs by 10 percent leads to an
increase in output by 10 percent (the size of the company is irrelevant)
2. Decreasing economies of scale: Increase in all inputs by 10 percent leads to an
increase in output of less than 10 percent (small is beautiful)
3. Increasing economies of scale: Increase in all inputs by 10 percent leads to an
increase in output by more than 10 percent (big is beautiful)
re 1) Proportional output change: constant economies of scale
y(λ · L, λ ·K) = λ · y(L,K);
Example: y = L1/2K1/2
with that: y(λL, λK) = (λL)1/2(λK)1/2
= λ1/2L1/2λ1/2K1/2 = λL1/2K1/2 = λy
Doubling of the input factors implies doubling the output.
re 2) Disproportionately low output change: decreasing economies of scale
y(λ · L, λ ·K) < λ · y(L,K);
Example: y = L1/4K1/4
with that: y(λL, λK) = (λL)1/4(λK)1/4
= λ1/2L1/4K1/4 = λ1/2y
Doubling of the input factors implies a disproportionately low increase in output (here to
the factor of√2). This usually results from organisational frictions with the increasing
size of the company. These might include bureaucratic conditions in large corporations, a
lack of checks and controls, communicative deficits, etc.
118
re 3) Disproportionately high output change: increasing economies of scale
y(λ · L, λ ·K) > λ · y(L,K);
Example: y = LK1/2
with that: y(λL, λK) = (λL)(λK)1/2
= λ3/2LK1/2 = λ3/2y
Doubling of the input factors implies a disproportionately high increase in output (here by
a factor of 2 ·√2 > 2). This often results from specialisation advantages: in large units the
individual can become more specialised and is therefore more productive. For example,
doctors in large (university) hospitals are much more specialised than doctors in (small)
local clinics. Therefore, the large hospital is more efficient.
−→ Increasing (decreasing) economies of scale produce an advantage (disadvantage) in
terms of costs, depending on the size of the company.
The type of economies of scale can also be illustrated by the isoquant map:
1. With constant economies of scale the distance between isoquants is the same when
output is doubled, tripled, quadrupled, etc.
2. With increasing economies of scale, the isoquants shift closer together.
3. With decreasing economies of scale, the isoquants shift further apart.
Below is a diagram of the general connection between economies of scale and the shape of
the isoquants, using the example y = KαLβ.
-
6
L
K
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
y = 10
y = 20
y = 30
To 1: constant economies of scale: α + β = 1
119
-
6
L
K
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
y = 10y = 20y = 30
To 2: increasing economies of scale: α+ β > 1
-
6
L
K
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................
y = 10
y = 20
y = 30
To 3: decreasing economies of scale: α + β < 1
Summary
1. The production technology of a company indicates the relation between produc-
tion factors and output quantities. As a formula, these technologies can be described
by means of production functions.
2. An important distinction must be made between short-term and long-term produc-
tion functions. Long-term production functions are used when all production
factors can be input with variable quantities. In the short term, however, some pro-
duction factors are often fixed. This is the case, for example, when machines (i.e.
capital) are used that cannot be procured or sold at just any time.
3. The most important characteristic of short-term production functions is the law
of diminishing returns. If only one factor can be varied, the optimal input rela-
tionship of the production factor diminishes in general. The result is the decreasing
efficiency of the factor whose input is expanded.
120
4. We can revert to the relevant marginal and average return functions to describe
the properties of a production function. The former intersects the latter at its max-
imum. The maximum production level is reached by definition at a marginal return
of zero.
5. An important short-term production function is the so-called production function
according to the law of diminishing returns. Here, the marginal return of the
variable production factors first increases with increasing output quantities and then
decreases.
6. Long-term production factors can be described by means of isoquants. Similar to
the indifference curve concept of the theory of the household, an isoquant shows the
different factor input quantities that lead to a fixed level of output.
7. The marginal rate of technical substitution formally describes the slope of the
isoquant. The economic significance is: By how much must the input of the alter-
native production factor be expanded if a (marginal) unit of a production factor is
to be foregone, without reducing the production quantity. If the production fac-
tors are imperfect substitutes, there will be a decreasing marginal rate of technical
substitution.
8. Long-term production factors and the corresponding isoquants can be distinguished
according to the degree to which the production factors are substitutes or comple-
ments: perfect substitutes, perfect complements and imperfect substitutes.
9. If both (all) production factors vary, this is known as total factor variation. The
resulting production effects differ greatly, depending on the production function, and
can be noted by means of the concept of economies of scale. These show the percent-
age increase in production in relation to the percentage increase in all production
factors.
10. A distinction is made between constant, increasing and decreasing economies of
scale. These technological production contexts are decisive for the market structure
(many small versus few large companies) and for strategic company decisions.
121
IV.2 Cost minimisation, factor demand and cost functions
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 7
Until now the discussion of the company sectors was influenced solely by technological
considerations. These form the basis of company decisions. However, in order to be able
to choose between different alternatives (e.g. point on the isoquants), information about
the costs of the production factors is an essential decision criterion. Now, based on the
factor prices, we can translate the production theory into the cost theory.
Previously: Connection between input and output −→ technological restriction
Here: Connection between costs and output −→ input market restriction
Schematically, the subject matter of the cost theory can be illustrated as follows:
Output −→ Input (input markets) −→ costs
Objective: Analysis of the connection between output and costs, specifically the minimal
cost connection
The question, therefore, is how a company can produce at minimal cost with a given
production technology and at given factor prices (interest and wages). In this context we
are looking for the minimal cost demand for work and capital: If a certain output is to be
produced at minimal cost, with how much work and how much capital must this be done?
There are two possible approaches or possible solutions for the company decision problems:
1. Cost minimisation (2 steps)
A) Cost-minimal factor inputs at a given output =⇒ cost function
Formula:
minL,K
C s.t. y = y(K,L)
B) Choice of the profit-maximising output,
Formula:
maxy
P = py − C(y)
122
2. Profit-maximal choice of factor input (direct)
Formula:
maxL,K
P (L,K) = py(K,L)− wL− rK
Below we shall choose the first option. The second produces the same result. We shall first
come to the first step: the determination of the cost-minimising factor input quantities
and thus the (minimal) costs that result from the production of a certain output quantity.
Accordingly, the cost function of the company will be derived. The second step will
be presented in Section IV.3 In the discipline of business studies the two steps are divided
roughly into production and sales.
Procedures in this section:
1. Derivation and discussion of cost functions (IV.2.1)
2. Derivation and discussion of (cost-minimising) factor demand (IV.2.2)
3. Implications and properties of different cost functions (IV.2.3)
IV.2.1 Cost functions
In determining cost functions two cases can be distinguished:
1.) Production with one production factor, for example when the input of the other
factor is constant (analogous to production theory!)
=⇒ Short-term cost function
2.) Production with many (variable) production factors
=⇒ Determination of the optimal input combination
=⇒ Long-term cost function
1.) Cost function with one input
We shall now determine the cost function based on the production function, the cost
equation and the cost function. It is important to distinguish between the cost equation
and the cost function. The only (variable) production factor is the work input L.
123
Production function:
y = f(L) (29)
=⇒ Output as a function of the work input L
Cost equation:
C = C(w,L) w = wage rate (30)
=⇒ Production costs as a function of the input: When the work quantity L is input, then
how high are the costs?
Cost function:
C = C(w, y)
=⇒ Costs as a function of the output, not the input (factor input).
Both times the factor costs naturally influence the production costs. The determination
of the cost function is done by combining (29) and (30).
We shall illustrate what we have just said with an example. Assume that the production
function of a company is y = Lα. The factor demand function dependent on the output
level is then L = f−1(y) = y1/α and the cost function is C = w · y1/α. In this example
we have assumed implicitly that the company accepts the wage rate as a given. This is
especially true in the case of small companies.
The graphical derivation of the cost function is done in two steps: first the production
function is inverted (mirrored at the 45-line). Then the ordinates are “multiplied“ by the
wage rate w:
124
-
6f(L)
f−1(y)
L, y.......................................................................................................................................................................................................................................................................
................................................
..................................
..............................................................................................................................................................................................................................................................................................................................................................................................................
......
.......................................................................................................................................................................................................................................................................................................................
.................................
........................................
...................................................
.......................................
.......
.......
.......
.......
.......
......................................................
f(L)
f−1(y)
45-
6
y
C
...............................
.............................
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................C(w, y)
We now make the connection between the cost function and the economies of scale of
the production function. If the cost function is C = wy1/α, as in the above example, the
following statement can be made:
1. Constant economies of scale (α = 1) −→ linear cost function
2. Increasing economies of scale (α > 1) −→ concave cost function
3. Decreasing economies of scale (α < 1) −→ convex cost function
At this point these considerations serve to provide a very simple illustration of the con-
nection between technology and cost functions. We can do this at this point only because
we are observing a production function with one production factor here. In other words,
partial and total factor variations are identical. A more detailed observation of the con-
nection between economies of scale and cost functions can be found below. But it shall
become apparent that the general connections just discussed are generally valid.
The diagram above presents the case of a convex cost function. Production then increases
only at a disproportionately low rate with the factor input L. For this reason, the costs C
increase at a disproportionately high rate with the output quantity y.
2.) Cost function with many production factors
The objective of the company is to maximise profit, which can be achieved by means of the
optimal combination of factors. Cost minimisation is therefore the prerequisite for profit
125
maximisation. If goods are not produced at the minimum cost, the maximum profit can
never be reached. In the case of many production factors the cost-minimising production
of a given output level by a suitable combination of production factors produces:
Production function: y = y(K,L) (31)
Cost equation: C = w · L+ r ·K (32)
Graphical solution:
-
6K
L
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
y0
y1
y2•
•
•
................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................................................... Expansion path
C0
C1
C2y0 , y1 , y2 : Isoquants from (31)y0 < y1 < y2
C0 , C1 , C2 : Isocost lines from (32)C0 < C1 < C2
MCC0
MCC1
MCC2
Analogous to the theory of the household, the cost-minimising factor combination is given
at the tangential point between isoquants and the isocost line. We call the optimality
points minimum cost combinations (MCC). The connection of all MCCs represents the
expansion path. All other points on the isoquant lead to higher costs and are therefore
not optimal.
126
Effects of short-term and long-term production changes:
-
6K
L
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................................................................
•
•
•
................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................• • •
.............................
.............................
.............................
Long-Run Expansion Path(K and L are variable)
Short-Run Expansion Path(only L variable)
The diagram shows that the company is forced in the short term to produce with a factor
input relationship that is generally not cost-optimal. In the long term both production
factors can be varied, so that the optimal input relationship arises.
Analytical derivation of the cost function:
The analytical approach allows a more general treatment of the optimisation problem.
However, the same principle applies as with the graphical solution.
The optimisation problem of the company is:
minL,K
C = w · L+ r ·K (33)
s.t.: y0 = y(L,K)
This means that costs necessary to achieve a certain prescribed production level y0 should
be minimised by means of the input quantities of the production factors L and K. We
now apply the Lagrange method of minimising subject to constraints.
Lagrange function:
L = w · L+ r ·K − λ(y(L,K)− y0) (34)
127
First order optimality conditions:
∂L∂L
= 0 −→ w − λ∂y
∂L= 0 (35)
∂L∂K
= 0 −→ r − λ∂y
∂K= 0 (36)
∂L∂λ
= 0 −→ y(L,K) = y0 (37)
(35) and (36) produce:w
r=
∂y/∂L
∂y/∂K= MRTS (38)
This optimality condition indicates the cost-minimising input combination (minimum cost
combination). At given factor prices the optimal factor input relationship can be read
from condition (38).
(37) and (38) produce
L∗ = L(y, w, r) (39)
K∗ = K(y, w, r) (40)
These equations show the cost-minimising factor demand quantities depending on the
factor price and the output.
Placing (39) and (40) in the cost equation (33) gives the cost function:
C = f(y, w, r)
Accordingly, production costs generally depend on the output quantity and the factor
prices. The individual factor input quantities do not seem to be an argument of the cost
function, as each target production level leads to an optimal factor input relationship.
Summary of the steps for deriving the cost function:
1. Form the Lagrange approach
2. Derive the optimality conditions
3. Derive the conditions for the minimum cost combination
4. Solve this condition for one factor (e.g. L)
128
5. Place in the production function (produces L∗ = L(w, r, y))
6. Follow steps 4 and 5 for the other factor (produces K∗ = K(w, r, y))
7. Place L∗ and K∗ in the cost equation −→ cost function
A concrete example:
The production function of a company is y = K0,5L0,5. The wage rate is 10, the interest
rate is 2. This produces the following Lagrange function:
L = 10L+ 2K − λ(K0,5L0,5 − y)
The first order conditions are thus:
10− λ0, 5K0,5L−0,5 = 0 (41)
2− λ0, 5K−0,5L0,5 = 0 (42)
K0,5L0,5 − y = 0 (43)
The two equations (41) and (42) produce the MCC:
10
2=
K
L
If we solve this for K we get K = 5L. That means that, optimally, five times as much
capital is inputted than work. Placing this in (43) produces:
L∗ =√2/10y
The same procedure produces the following for K:
K∗ =√5y
Placing both equations in the cost equation C = 10L+ 2K produces:
C =√80y
This illustrates the general case that a production function with constant economies of
scale leads to a linear cost function.
129
Excursus: Cost function for Cobb-Douglas production function
In this excursus the cost function is derived for the general Cobb-Douglas production
function and general factor prices. In the process we see that increasing (decreasing)
economies of scale lead to sublinear (superlinear) cost paths.
Production function:
y = KαLβ
Optimisation problem:
minL,K
C = wL+ rK
s.t. y = KαLβ
Lagrange function:
L = w · L+ r ·K − λ(KαLβ − y) (44)
First order optimality conditions:
w − λβKαLβ−1 = 0 (45)
r − λαKα−1Lβ = 0 (46)
KαLβ − y = 0 (47)
(45) and (46) produce:w
r=
βK
αL(48)
(48) (solved for K) results in:
K =wα
rβ· L (49)
(49) in (47) produces: (w
r
α
βL
)α
· Lβ = y
L = y1
α+β ·(w
r
α
β
)− αα+β
(50)
= y1
α+β ·(r
w
β
α
) αα+β
130
Reformulation of (49) (solved for L) produces:
L =r
w
β
α·K (49’)
(49’) in (47):
(r
w
β
αK
)β
·Kα = y (51)
K = y1
α+β
(w
r
α
β
) βα+β
Placing (50) and (51) in the cost equation:
C = r ·K + w · L = y1
α+β
[r
αα+βw
βα+β
(α
β
) βα+β
+ wβ
α+β rα
α+β
(β
α
) αα+β
]
=
[(α
β
) βα+β
+
(β
α
) αα+β
]w
βα+β r
αα+β y
1α+β
For the special case α + β = 1 (constant economies of scale):
C =
[(α
1− α
)1−α
+
(1− α
α
)α]w1−αrαy
Thus the general result, derived above, is that constant economies of scale lead to a linear
cost function (all factors except for y are constant!).
End of excursus
IV.2.2 (Cost-minimising) factor demand curves
The minimum cost combination applies for given factor prices and output levels. What
happens now in the event of parameter changes, especially with changes to the factor
prices?
An answer is provided by conditional factor demand curves.
Definition: The conditional factor demand shows the cost-minimising factor demand for
alternative factor prices and outputs.
131
The conditional factor demand curves (39) and (40) were derived generally above. Here
we want to concentrate on the influence of the factor prices on the cost-minimising factor
demands.
Graphical derivation using the example of an interest rate increase:
-
6
L
K
...................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
•
•
•
A3
A2
A1
-r ↑
y = yC3 C2 C1
The graphical analysis was carried out for a given wage rate w, a given output level y and
a change to the interest rate r. We are observing specifically the case of an interest rate
increase (r ↑). This diagram is constructed as follows: The tangential point for a certain
factor price relationship wr
(slope of the cost equation) is sought on the isoquant. If the
wage rate is fixed (w = w) and if the interest rate rises (r ↑) this produces:
r1 > r2 > r3 : A1/A2/A3
From this it follows that: the higher the interest rate r, the more capital will be substituted
by work. The economic intuition is: If interest rates rise, work becomes relatively cheaper.
It is then worthwhile inputting the relatively cheaper factor of work. The factor demand
curves can be derived from the above diagram. We shall now present the capital demand
curve as an example:
132
-
6r
K
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................
Factor demand curves K = f(r, w, y)
Some implications of the condition for the optimum input mix:
As shown above, the optimum is:∂y/∂L
∂y/∂K=
w
r
Due to the assumption of diminishing marginal returns (technically: negative second
derivation) the following connection applies:
(wr
)↑=⇒ L
K↓
If, accordingly, the relative price of the factor of work increases, work will be substituted by
capital. We call this phenomenon factor substitution. The adjustment process follows
from the assumptions about the production function. If the work input L is reduced,
the marginal productivity of work increases (∂y/∂L ↑). If more capital K is inputted, the
marginal productivity of the capital sinks (∂y/∂K ↓). This is repeated until the optimality
condition is fulfilled.
Examples of applications on the topic of factor endowment, factor prices and
factor demand
a) Wage level and employment
If the wage rate w rises, for example due to wage negotiations, work becomes relatively
more expensive. The input relationship between work and capital then sinks with the
optimising behaviour of the company, i.e.(LK
)↓ and there is therefore less employment.
133
Economic intuition: A cost-minimising company owner adapts his production to the new
factor price relationship. The relatively expensive factor is replaced by the relatively cheap
factor. In the example this would mean that work is replaced by capital.
b) Factor mix and factor endowment
Why is agriculture conducted in a much more work-intensive manner in India than in
Germany?
Possible answers include the following: Because Germany is wealthier and can afford more
machines, or because machines are relatively cheaper in Germany.
The second answer is the correct one: Work is relatively widely available in India, which
leads to a lower wage rate in relation to the interest rate than in Germany (w/r)I < (w/r)D,
with a worldwide identical interest rate and wI < wD. The following diagram illustrates
this issue.
-
6
L
K
...................................................................................................................................................................................................................................................................................................................................
...........................................
......
.......
..................................
.........
..........................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................................................................
•
•
D
I
α β
tanα = wD
r
tan β = wI
r
The optimal factor input relationship KL
is much higher in Germany than in India. Pro-
duction occurs at point D in Germany, and at point I in India.
IV.2.3 Cost concepts and cost curves
Our ultimate objective is to determine the supply behaviour of the company. As we shall
soon see, the cost situation is the essential determining factor for the goods supply of
companies. Against this background it is very important to take a closer look at the cost
trends (and their various versions, e.g. short-term and long-term costs). Therefore, we will
134
now turn to the analysis of different cost concepts and figure out the connection between
these concepts.
We differentiate three essential pairs of terms:
1. Fixed costs versus variable costs
In contrast to variable costs, fixed costs are independent of the production level y.
2. Average costs versus marginal costs
This distinction will be addressed in detail below.
3. Short-term versus long-term costs
A differentiation that is very similar to fixed costs / variable costs.
As before we will assume constant factor prices and examine the connection between costs
and output. Ultimately the production level is the decisive variable for the company.
1. Fixed costs and variable costs
Fixed costs (FC) are independent of the output quantity. Precisely what fixed costs are
depends on the period of observation. Wages or buildings costs are fixed in the short term,
but in the medium and long term they are variable.
Variable costs (VC) change with the output quantity.
Total costs (TC) are the sum of fixed costs and variable costs TC = V C + FC.
Some important properties:
Important properties of fixed and variable costs can be illustrated well by the average fixed
costs (FC/y = AFC) and the average variable costs (V C/y = AV C). The AFC sink with
the output level:(
∂(F/y)∂y
< 0).
In relation to the variable costs it is often assumed that the AVC first decreases then
increases. This assumption can be motivated by a production function according to the
law of diminishing returns. Intuitively, with high capacity utilisation (high production
135
level) the situation arises that the fixed factor constrains the productivity of the variable
factor. This usually leads to increasing AVC.
If, for example, a machine is the fixed production factor, an increase in the machines
running time often leads to an exponential consumption of electricity and lubrication
(growth in variable costs!), if the running time is already very high. The management of a
company can also be interpreted as a fixed production factor. Controls and communication
within the company become ever more difficult with increasing size (production), which
can lead to a growth in variable costs. The economic rule of thumb „small is beautiful“
should be seen in this context.
Purely technically, the typical course of average variable costs, as just outlined, means
the following: For small production quantities y there are increasing economies of scale
(related to the variable inputs). For large production quantities y on the other hand, there
are decreasing economies of scale. The following diagram shows the paths of the AFC and
AVC. Furthermore, the implied average total costs (ATC) are also shown, which also have
a U-shaped path.
-
6
y
ATCAVCAFC
ATC
AVC
AFC
U-shaped path of the average total costs-curve (ATC):
...................................................................................................................................................................................................................................................................................................................................................................................................................
...................................................................................................................................................................................................................................................................................
In anticipation of the later analysis of the market structure, we point to the fact that
136
U-shaped cost curves represent a constraint on the maximum company size.
2. Connection between average and marginal costs
The analysis of the average and marginal costs is directly analogous to the above discussion
of marginal and average productivity. We begin first with a short discussion of marginal
costs. These are defined as additional costs that result from the production of an additional
(marginal) output unit:
MC =∆C(y)
∆y
∣∣∣∣∆−→0
=dC(y)
dy= C
′(y)
Marginal costs are decisive economic factors of company calculations. If a company can
accept an additional order, what are the relevant costs for the decision? The calculation
only makes sense on the basis of marginal costs. The economic reason for this is that only
marginal costs (i.e. the costs of an additional unit) can really be attributed to this order.
A concrete example:
The cost function of a company is C = 100 + y1/2. In the starting situation y = 100 Out-
puteinheiten produziert. output units are produced. Now an additional order is received
for an additional unit at a price of 0, 08 monetary units. Should the company accept the
order?
The answer is yes, as the additional costs (marginal costs = 0, 5y−0,5 = 0, 5(100)−0,5 =
0, 05) are smaller than the price. A calculation of the total additional profits shows
that they are positive. If the decision had been made based on average variable costs
(AVC=y−1/2 = (100)−0,5 = 0, 1) the order would have been rejected erroneously.
Essential connections
1. For sinking (rising) ATC and AVC the marginal costs (MC) are smaller (greater)
than ATC or AVC.
This is the reason: When the average decreases, the additional costs that result from
the production of an additional unit must be smaller than the previous average, and
vice versa.
137
2. The MC curve intersects the AVC curve at its minimum.
Formal evidence:
CV (y)
y−→ min =⇒ ∂(CV (y)/y)
∂y= 0
∂(CV (y)/y)
∂y=
∂CV /∂y
y− CV
y2= 0 =⇒ ∂CV /∂y =
CV
yq.e.d.
Quotientrule!
3. The MC curve intersects the ATC curve at its minimum
Argumentation as above. Formal evidence should be carried out by all (exam?!).
Graphical presentation:
-
6p
y
ATC
C ′(y)
y
•
•
Numeric example:
C =1
100y3 − y2 + 50y + 720
ATC =1
100y2 − y + 50 +
720
y
V C =1
100y3 − y2 + 50y
AV C =1
100y2 − y + 50
FC = 720
AFC = 720/y
MC =3
100y2 − 2y + 50
138
The marginal costs (MC) are independent of fixed costs (derivation of a constant = 0).
Calculation of the minimum of the average variable costs:
AV Cmin: ∂AV C∂y
= 0 ⇒ 150y − 1 = 0 ⇒ y1 = 50
Calculation of the minimum of the average total costs:
ATCmin: ∂ATC∂y
= 0 ⇒ 150y − 1− 720
y2= 0 ⇒ y2 = 60
y ATC AVC MC AFC
0 ∞ 50 50 ∞
10 113 41 33 72
20 70 34 22 36
30 53 29 17 24
40 44 26 18 18
50 39,4 25 25 14,4
60 38 26 38 12
80 43 34 82 9
100 57,2 50 150 7,2
3. Short-term and long-term cost curves
a) Long-term versus short-term average costs:
How are short-term and long-term average cost curves connected?
Restricted in terms of production (see discussion above), a factor is fixed in the short term
and therefore cannot be used in the optimal quantity. In the long term this factor can be
changed in order to achieve the optimum input factor mix. The long-term cost curve arises
from the short-term cost curve by the optimum choice of the fixed factor. In other words
there are many short-term cost curves at alternative input quantities of the fixed factor.
139
Example from automobile manufacture:
The working hours are variable in the short term (at least within certain limits), while the
machine equipment, in contrast, cannot be adjusted in the short term. That means that
there is a short-term cost function for each alternative machine plant (e.g. five production
lines and ten production lines). In the long term the machine plant is also variable, and
the number of production lines is chosen that is most cost-minimising for the target output
level.
In the graphical solution we seek the optimum level of fixed costs at which just the relevant
output level y is produced at minimum cost. Precisely at this point the long-term average
cost curve (ATCL) intersects the short-term average cost curve (ATCS).
-
6
yy
ATCL/S
.......
......
.......
......
.......
......
....
ATCS
ATCL•
Intuition on the diagram:
To the left of the marked tangential point the capital stock (relative to y) is too high, i.e.
the fixed costs are higher than in the cost-minimised factor combination. To the right of
this tangential point the capital stock (relative to y) is too low, i.e. the fixed costs are
too low (decreasing economies of scale). It should be noted, however, that the tangential
point generally does not lie at the minimum of the short-term total average cost curve.
This is only the case when there are constant economies of scale over the entire course
of all production levels; in this case the long-term total average cost curve is a horizontal
line, the ATCS and the ATCL each intersect at the minimum of the ATCS. Otherwise,
the above argumentation applies.
140
If we repeat this process for different capital stock levels we get the long-term average cost
curve enveloping all short-term average cost curves. Essential characteristic: its path is
less sloped, as the fixed factor can be adjusted optimally:
-
6
ATCL/S
For many K:
ATCS′
y
ATCL
ATCL as the envelope of all ATCS
b) Long-term versus short-term marginal costs:
The same applies to the relationship between long-term and short-term marginal costs:
the long-term MC curve is much flatter than the short-term.
Intuition as above: If we want to vary output in the short term, we can only change the
short-term factor (e.g. overtime). This is more expensive than achieving the adjustment
to output changes by changing short-term and long-term variable factors.
4. Long-term (average) costs and market structure
The path of the long-term average cost curve is decisive for the number of companies
in an industry (market structure). There are only very few companies in so-called
concentrated industries
141
The main categories of market structure:
1. Polypoly: very many small companies in one industry (perfect competition)
Example: agriculture
2. Oligopoly: many large companies in one industry
Example: automobile industry
3. Monopoly: only one very large company
Example: German railway
Alternative average cost paths:
We shall first take a closer look at two different average cost paths (assumption: falling
ATC over a very large area of the production quantity y):
-
6ATC
y
ATC1
ATC2
ATC1: natural monopoly
ATC2: few companies
Due to the continuously sinking average costs in the first case, the average costs of a large
company are smaller than the average costs of two medium-sized companies. This leads
to predatory competition, from which only one producer (monopolist) remains.
If the average costs rise at some time in the future, and if this point lies at a large output
quantity relative to the overall quantity, there is a concentration of fewer companies, but
no monopolisation tendency (2nd case). That is then an oligopoly.
142
There is little or no company concentration with the following average cost paths:
-
6ATC
y
ATC3
ATC4
ATC5
y0 (very small)................................................
In the three cases presented (increasing, U-shaped with minimum at small y and constant
average costs) a market structure with many small companies emerges, who do not have
any market power (polypoly).
Summary
1. Company decisions can be optimised when the target production quantity is manu-
factured at minimum cost.
2. The path of the cost function depends decisively on the characteristics of the
production function. Rising (falling) economies of scale lead to concave (convex)
cost curve paths.
3. Analogous to production theory a distinction is made between short term (only
one production factor is variable) and long-term (all factors can be varied) cost
functions.
4. The minimum cost combination of the production factors can be derived from
the cost-minimising calculation. This then produces the factor demand functions
and the cost function, depending on the production quantity and the factor prices.
143
5. The conditional factor demand functions indicate the production factor de-
mand when the relative factor prices change. The rearrangement of the factor input
relationship can also be called factor substitution.
6. The marginal costs lie below (above) the average costs, when the average costs
fall (rise). Therefore the marginal cost curve intersects the average cost curve at the
latters minimum. The same applies with average variable costs.
7. The long-term average cost curve can be interpreted as an enveloping function
of all short-term cost curves when choosing the optimum input quantity of the fixed
input factor.
8. The shape of the average cost curve has significant implications for the market
structure in an industry, as it determines the number of companies in a market
(monopoly, oligopoly, polypoly).
IV.3 Profit maximisation and goods supply of the individual com-
pany
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 8
IV.3.1 Perfect competition and profit maximisation
We shall now observe the goods supply of a single company in perfect competition.1 Until
now the decisions of the company were subject to a technological restriction (produc-
tion function) or an economic restriction (production function + factor prices + cost
minimisation =⇒ cost function). Now the market restriction on the goods market is
added to the mix. The additional restriction is the goods demand. The question is how
much can be sold at what price.
As already shown in the introduction, from an overall market perspective there is a de-
creasing demand curve, which we show again here:1Perfect competition generally means many small companies. We shall come to a more exact definition
of the term later.
144
-
6
......................................................................................................................................................................................................................................................................................................................................................................................................................
p
xN
Here, however, we are interested in the demand from the perspective of a single company.
The demand functions from the viewpoint of a company generally differ from the overall
demand. The so-called monopoly case (only one company) is a special case. In this
instance, individual and overall demand is identical. Below we shall concentrate on the
case of perfect competition.
Essential characteristics of a market with perfect competition:
1. Companies are relatively small compared to the total market
2. Companies produce a standardised good
3. Perfect information on the demand and supply side
4. No preference on the part of the buyer for a specific company: Absence of personal,
spatial, time-related preferences
Examples of markets in which perfect competition reigns include the wheat market, bread
market or shoe market (at least for standard shoes). In its pure form the conditions of
perfect competition are doubtlessly never fulfilled, but the model often represents a good
approximation of market conditions.
From the assumptions made it follows that companies are price takers, i.e. they take the
market price that has been set. The companies assume that the market price will not be
changed by their own supply. Furthermore they assume that they can sell any number of
145
goods at a given market price. This is rational behaviour, as we assume that the single
company is very small. The individual demand function therefore looks like this:
-
6
........................................................................................................................................................................................................................................................................................................................................................................................................
p
xiN
Behavioural assumptions: Profit maximisation
First we must define what we actually mean by profits. Profits are the residuum from
revenue minus costs: Profits = revenue costs.
The cost concept encompasses not only direct spending by the company such as wages for
employees, capital costs for interest on loans (borrowed capital) and rent for buildings, but
also opportunity costs such as company owner salary (what could the company owner
earn at the same time elsewhere?), lost rental revenue from buildings used by the company
and lost interest revenue from investments not made. The consideration of opportunity
costs is a very important and often decisive economic concept!
Economic profit = balance sheet profit: The balance sheet profit contains a number
of distortions (accruals and deferrals), including not considering interest on equity capital
as a cost!
Assertion:
In markets with perfect competition, companies make no profits.
Basic idea:
If income exceeded the costs of all input factors including opportunity costs, this would
lead to a market entry by other companies. Entry by new companies leads to more
supply and therefore to a sinking price, until income corresponds with the costs. This is
completely compatible with positive balance sheet profits that display the interest on equity
capital. Accordingly, for example, dividend distributions by public listed companies are
146
not, for the most part, included in the calculated economic profit of the company.2 Profit
maximisation in this model is based on the simplified assumption that managers act in
the interests of the investors. This assumption is restricted in part in more complicated
models (main degree course).
IV.3.2 Supply decision of a single company with perfect competition
The company maximises the profit function
P = py − C(y)
whereby p denotes the given price (assumption of perfect competition). C(y) is the cost
function that was described above. The question now is which output level y will the
company choose in order to maximise profit. The necessary maximisation condition ∂P∂y
= 0
leads to
p− ∂C(y)
∂y= 0
p = C′(y)
This means that, optimally, the (given) price must correspond with the marginal costs of
production: price = marginal costs. Formulated more generally, the marginal revenue
corresponds with the marginal costs: marginal revenue = marginal costs. If the price
were higher than the marginal costs, an additional unit would bring additional profit.
Optimally, the company would then expand production even further. Conversely, a price
below the marginal costs would mean that an additional production unit would reduce the
profit.
Sufficient condition for the maximum:
∂2P
(∂y)2< 0 −→ −C ′′
< 0 C′′> 0 convex costs
Accordingly, the second profit maximisation condition with perfect competition requires
increasing marginal costs; if they were to fall, the average costs would also fall. In this2„Nestlé makes a profit of 5 billion“ With equity capital of 50 billion and 10 percent interest this means
a profit, as defined here, of zero.
147
case there is an incentive to continue expanding the production quantity even further, as
this increases profit. This results in the concentration tendencies described above, which
breach the assumptions of perfect competition.
Illustration of the optimality conditions:
y1 y2
p
C ′(y)
y-
6
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......• •
C ′(y)
The necessary condition p = C′(y) is fulfilled in y1 and y2. The sufficient condition,
however, is fulfilled only in y2. The production level y1 produces a local loss maximum, as
the marginal costs for all y < y1 are higher than the price (unit revenue). On the other
hand, the profit maximum is reached in y2. From y1 additional production is profitable, as
(p > C′). The rising line of the marginal cost curve (more exactly: from p = C
′ ≥ AV C)
becomes the supply curve of the individual company, as otherwise losses are threatened
(price < average costs). The concluding diagram illustrates this connection once more:
-
6p
y
AV C
C ′(y)
yiA
•
•.....................................................................................................................................................................................
.....................................................................................................................................................................................
.....................................................................................................................................................................................
148
Long-term versus short-term supply curve of the single company
Until now we have been concerned with the derivation of the optimal supply (exact condi-
tion) of the company, i.e. how much is produced to give maximum profit at a given price?
We did not ask whether production leads in total to (positive) profits. If this were not the
case, it would naturally make sense to choose y∗ = 0 and not to supply any goods.
The question that should be asked is when (from which point) is makes sense to engage
in short-term production, or what is the minimum price that must be attained so that
an additional order makes sense in the short term. Furthermore we wish to address the
question of when (from which point) is makes sense to engage in long-term production,
or under which price limit would it be better to close the company altogether. We shall
proceed in two steps. In the first step we will distinguish between short term and long
term with regard to whether the average variable or the average total costs are covered.
In the process we shall ignore the fact that all factors can be varied in the long term, and
that therefore the long-term marginal and average cost curve is flatter than the short-term
marginal and average curve. We will consider that in the second step.
1st step: AVC vs. ATC
In the short term the following must apply:
p ≥ AV C : operating minimum p = AV C
Only if the price covers at least the AVC is it worthwhile to produce at all.
In the long term, on the other hand, the fixed costs must also be covered. The following
must therefore apply:
p ≥ ATC : operating minimum p = ATC
In other words, in the long term the price must be greater than or equal to the average
total costs (p ≥ ATCL), as all factors can be inputted variably in the long term.
2nd step: The slope of the short-term and long-term cost curves
149
As shown above, the marginal cost curve is flatter in the long term than in the short
term. This means that the long-term supply curve is more elastic than the short-term
supply curve. We use the term elastic to describe the property of the supply function
as to how much supply changes in the event of price changes (demand changes). The
economic reasoning for the elasticity properties is the same as above: If a factor is fixed in
the short term (bottlenecks can arise), the marginal costs rise faster than if all factors can
be adjusted optimally (no bottlenecks). The following diagram illustrates the long-term
supply curve of an individual company:
-
6p
y
ATCL
MCL
•
long-term supply curve
The relevant part of the long-term supply function is the line of the MCL after the inter-
section with ATCL.
The concluding diagram compares the long term with the short term.
-
6p
y
...................................................................................................................................................................................................................................................................................................................
................................
................................
................................
................................
................................
................................
................................
................................
................................
.........................
short-term supply curve
long-term supply curve
An important reminder: The supply function always corresponds with the marginal cost
curve (follows from the profit maximisation rule p = MC).
150
Summary
1. The profit maximisation of a single company with perfect competition occurs under
the decisive assumption that the company is small enough to consider the sales price
to be set, i.e. it cannot be influenced. The result is the necessary profit maximisation
condition price equals marginal costs.
2. Profit denotes the economic profit. In contrast to the balance sheet profit, op-
portunity costs, such as the alternative interest on equity capital, are taken into
account. This means that the model implication of zero profits can also be justified
empirically. Zero profits occur because in the event of positive profits, companies
can join in at all times and undercut the prevailing market price.
3. The sufficient profit maximisation condition with perfect competition implies rising
marginal costs or convex total cost paths. If this condition is breached, either the
company suffers losses or the market becomes concentrated, which contradicts the
assumption of perfect competition. The decisive difference between the long and the
short term is that in the long term fixed costs must also be covered in order to be
able to produce profitably. The long-term supply function is flatter (more elastic),
because the short-term fixed production factor can also be adjusted.
IV.4 Goods supply of all companies in an industry
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 8
Until now we have only looked at the goods supply of a company. However, as we are
interested ultimately in the analysis of the overall market (in which all companies supply),
it is necessary to summarise (aggregate) the goods supply of all companies.
Definition: The total supply in a market (in one industry) is the sum of the supplies of
the individual companies:
SM(p) =N∑i
ySi (p)
151
SM(p) : industry-wide supply, ySi (p) : supply of the ith company
Graphical derivation of the total supply function:
-
6
y1(p)
p1
p
p
y
............. ............. ............. ............. ............. ...............................................................................................................................................
.........................................................................................................................................................................................................................yS1
-
6
y2(p)
p2
pp
y
..............................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ...............................................................................................................................................
yS2
-
6
y1(p) + y2(p)
p2
p1
p
p
y
............................................................................................................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ...........................................................................................................................................
..........
⇒
-
6
p∗
p
y............................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ..............................................................................
Combined with demand function: =⇒ equilibrium price
Aggregate supply curve
Aggregate demand curve
x∗
Procedure:
1. Determination of the individual supply quantities at a given price
2. Addition to the total supply quantity
3. Repeat for alternative prices
4. Draw the connecting line −→ total supply function
Short-term total supply function
In the short term, not all factors are freely variable. For this reason the individual supply
function is ySi (p) = MCi for p ≥ AV Cs. Furthermore, the number of companies is
fixed.
152
Analytical derivation of the total supply function in the short term
yS1 =
0 for p < p1
50 + p for p ≥ p1
yS2 =
0 for p < p2
70 + 2p for p ≥ p2
p1 > p2
=⇒ total supply function
yg =
0 for p < p2
70 + 2p for p1 > p ≥ p2
120 + 3p for p ≥ p1
Long-term total supply function
The long-term equilibrium is characterised by the fact that all factors are variable and are
inputted in a cost-minimising manner. There is free market entry. As long as p > ATCL,
there is an incentive to enter the market. The individual long-term supply curve is the
long-term marginal cost curve, which runs above the ATCL-curve.
-
6p
y
...........................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................
•
MCL
yS(L)1
ATCL
2 companies: =⇒ long-term supply curve
-
6p
y
...........................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................
•
yS(L)2
-
6p
y
............................
............................
............................
............................
............................
............................
............................
... yg(L)
Industry equilibrium
Where there is free entry to the market, companies will continue to enter until the entry of
an additional company would lead to losses. For very many companies, on the other hand,
153
our zero profit condition applies with equilibrium. The diagram illustrates the situation
in the case of identical companies:
p∗
p
y-
6
.......................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
..............................
...................
......................................................................................................................................................................................................................................................................................................................................................................................................................
ySE(1) ySE(2)yAE(3)ySE(4)
ySE(5)
demand
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............
p∗: minimum of the long-term average total cost curve
In the example, four companies will supply on the market under observation.
Reason:
If only three companies are active in the market, the equilibrium price (produced by
intersection of demand curve and total supply function ySE(3)) is so high that a fourth
company can also enter. In the new equilibrium, the resulting equilibrium price is greater
than p∗. That means that a fourth company can cover its costs. This is no longer the case
if a fifth company enters. The resulting market price (from the intersection of demand
function and ySE(5)) lies below p∗. The companies can then no longer cover their long-
term average total costs. Therefore, a fifth company will not enter the market, in view of
the expected losses. Naturally, with perfect competition it continues to be the case that
the industry comprises many more than four companies and therefore zero profits still
represent a good approximation (would be the case here with p∗).
Reasons that inhibit free market entry:
1. Statutory licensing and market entry restrictions (e.g. taxi licenses)
2. The existence of special knowledge (patents, know-how, etc.)
3. Talent (e.g. entrepreneurial talent)
154
4. Network effects: the advantage of large networks with increasing economies of scale
(e.g. railway network)
5. Fixed natural factor (raw materials, land)
In these cases economic rent can emerge (higher-than-normal profits, positive(!) profits).
By this we mean an income that goes beyond the competitive remuneration (and thus
beyond the opportunity costs). In dynamic industries positive profits typically occur,
which provides an incentive to develop new knowledge and new technologies.
Profit and producer surplus
In market analysis the so-called producer surplus (in principle, variable profit) a measure
for the profit situation of the company. Graphically, the producer surplus can be shown
as follows.
y
A
p
-
6
y
p
Producer-surplus
.....................................................................................................................................................................................................................................................................................................................................................................
............................
.............................
............................
............................
............................
.............................
............................
............................
............................
.............................
............................
............................
............................
.............................
..
O
B
Supply function
The marked supply function reflects the marginal costs of the company. The area below
the supply function therefore reflects the variable costs of the company.
Definition: Producer surplus = income − (variable) costs
The income is given by p · y while the variable costs are represented by the area beneath
the supply function ABy0. Accordingly, the producer surplus amounts in the diagram to
p · y − ABy0 = ABp. In the industry equilibrium with free market entry, the producer
surplus just corresponds with the sum of the fixed costs of all the companies in the market.
As just discussed, the number of companies determines the free market entry: companies
continue to enter the market until the equilibrium market price just corresponds with the
minimum of the average total costs.
155
An advanced question: In a case where there are constant economies of scale everywhere,
(i) why is the producer surplus zero and (ii) why is the number of companies undetermined?
Summary
1. The total supply function of an industry is generally gained by aggregating the
individual supply functions of the participating companies.
2. On the other hand a distinction is made between the short-term and the long-term
supply function. While the number of companies is fixed in the short term, the
number of companies is determined in the long term by possible market entries.
Companies continue to enter until losses can be expected upon entry.
3. In the industry equilibrium perfect competition and zero profits still occur if the
number of supplying companies is large enough. Market entry can, however, be
inhibited, for example by network effects. In these industries, higher-than-normal
profits are made, which can stimulate the innovative activity of the companies.
4. The concept of the producer surplus can describe the profit situation of the com-
panies active in the market.
156
V Market equilibrium with perfect competition
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 9
After the brief introduction to the market mechanism in Chapter II and the presentation
of the derivation of the demand and supply functions in Chapters III and IV, we now come
to a more detailed observation of the entire market process. Until now individual business
decisions formed the core of the analysis. In the process we restricted ourselves to only
one side of the market (demand or supply). In contrast, we shall concentrate below on the
interaction of supply and demand on markets, which leads to market equilibrium. We
are thus concerned with the analysis of the market equilibrium and its changes when the
framework conditions of the market situation change.
To repeat once again the characteristics of the observed perfect market: many suppliers,
many demanders; the agents take the market price as given; a completely homogenous
good (which is in no way differentiated) is supplied (no personal, material, time-related
or spatial preferences); there is perfect information on both sides of the market. These
assumptions will be suspended successively in the following chapters.
V.1 Market equilibrium and efficiency
We shall now observe the market equilibrium. The following diagram shows the supply
function xS = xS(p) and the demand function xD = xD(p):
x∗
p∗
-
6
p
x
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ............. ............. .................................................................................................
•
xS(p)
xD(p)
157
Market equilibrium exists when the supplied and demanded quantities are equal:
xD(p) = xS(p)
The unambiguousness of the equilibrium (only precisely one equilibrium exists) is guar-
anteed by the monotony of the demand curve xD and the supply curve xS as well a
xS(0) < xD(0) und xS(∞) > xD(∞).
Solution of a paradox: Walrasian Auctioneer
We have explained previously that with perfect competition, each market participant ac-
cepts the price as given. But how does the market price then come about? The solution
to the problem can be found with the help of the theoretical construct of the Walrasian
Auctioneer. He continues to call out prices until equilibrium is reached. No trade is
conducted while there is disequilibrium, i.e. no trade at non-market-clearing prices. If, for
example, demand exceeds supply at a price that has been called out, a new (higher) price
is called out. This process is repeated until the market is cleared (demand=supply).
Nobel Prize winner Paul Samuelsons comments on the concept of the “Walrasian Auction-
eer: A Great Myth. Emphasize both words.”
Pareto efficiency
In economic terms efficiency mean Pareto efficiency. The objective is the „correct“ use of
scarce means, i.e. no waste.
Definition: An allocation (i.e. a certain use of scarce resources) is (Pareto) efficient,
if there is no other allocation that better improves the situation of an economic entity
without disadvantaging another.
The efficiency objective is intended to achieve the economic principle: maximum output
at a given input or minimum input at a given output. Pareto efficiency is also socially
acceptable: after all, nobody’s situation should be made less favourable. If there are many
Pareto efficient states, the question of distribution arises, which must be decided politically.
158
Example of Pareto efficiency:
There are ten loaves of bread and two persons, A and B. In this case there are eleven
Pareto-efficient situations (allocations), if we leave the loaves as a whole. These are: A
receives no loaf, B ten loaves; A receives one loaf, B nine loaves, etc. There is a Pareto-
inefficient situation when A receives four loaves and B three loaves (while three go stale).
Naturally the situation is generally much more complicated, but the example demonstrates
the basic idea of Pareto efficiency.
Some economic concepts in detail
The central problem in economics is the efficient use of scarce resources. A certain use
of scarce resources is known as resource allocation or, in brief allocation. If the use of
the resources is (Pareto) efficient, as defined above, we call this an efficient allocation
(of resources). Along with the question of allocation, the discipline of economics is also
concerned with the distribution of resources (in politics this question is often at the
forefront). Speaking figuratively, allocation questions are about the maximum size of the
cake and distribution questions are about the size of the individual slices of cake. If a
given distribution of goods is changed (for example due to government policy), we call this
redistribution. From an economic perspective, this should be done efficiently (or with
the smallest loss of efficiency). This means, in the above example: The initial state is A
(three loaves) and B (seven loaves). A redistribution should ensure that both receive five
loaves, and not that A gets four and B gets four.
For individuals we use the utility function as a measure of wellbeing. The existence of
many different individuals creates the problem of the interpersonal utility comparison.
For this reason, a social welfare function is often used. This undertakes a weighting of
individual utilities, e.g. a higher weighting of the utility of poor households, pensioners,
women, nationals, etc. Many measures (such as tax policy) lead to changes in the resource
endowment of individual households. If these are assessed for utility we speak of the
welfare effects of each political measure (more on that later!).
159
Market equilibrium and Pareto efficiency
1. Main theory of the welfare economy: Market equilibrium with perfect competition
is Pareto-efficient.
With market equilibrium Pareto efficiency is reached when the sum of consumer surplus
and producer surplus is at its maximum:
x′ x∗
p∗
p′
-
6
p
x
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ............. ............. .................................................................................................
............. ............. ............. ............. .........................................................................................................................................
•
•
•A
B
C
PS
CS
xS(p)
xD(p)
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
.
..
..
..
.............
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
.
..
........
Economic intuition:
At x < x∗ one producer is always willing to produce and supply one unit at a lower price
than a consumer is willing to pay. That is why additional production and subsequent
exchange increase welfare. At x > x∗, on the other hand, the costs of additional produc-
tion are greater than the consumers marginal willingness to pay. Therefore, production
exceeding x∗ does not increase welfare.
V.2 Interventions in the market equilibrium
“In this world nothing can be said to be certain, except death and taxes.”
(Mark Twain)
There are a number of various interventions in the market process, such as regulation, laws
and prohibitions, rules and much more. The most important and the most sensible to
analyse here must surely be the effect of taxes. We shall look a little closer at them below.
160
Recent example:
In the current political debate in Germany it is often claimed that an increase in the
general sales tax would be a good instrument to finance state spending and to reduce
the budget deficit. If we follow the discussion more closely, the effects of taxation on the
sales quantities of companies and the question of load distribution form the center of the
discussion. We will now address precisely these questions.
V.2.1 Effects of taxation
We start with an analysis of the effects caused by taxation, and will then examine the
corresponding welfare effects. For the analysis we will apply the method of compara-
tive statics. The question is now equilibrium changes when one or more (exogenous)
parameters are changed.
In general we can distinguish between a quantity tax and a value tax. The quantity
tax (e.g. mineral oil tax) is charged per quantity unit sold. The tax rate t thus drives a
wedge between the demand price pD and producer price pP . Then the following applies
pP = pD + t, respectively pP = pD − t.
The value tax (e.g. value added tax), on the other hand, is collected as a percentage of
the value of the good sold. The following applies pD = pP (1 + τ) with the value tax rate
τ .
Effects of a quantity tax on the market equilibrium
xD(pD) = xS(pP ) (52)
with pP = pD − t (53)
(52) in (53): xD(pD) = xS(pD − t) (54)
161
It is irrelevant who pays the tax. For this, see the next two diagrams:
x′
p′P
p∗P
p′D
-
6
pP
x
...................................................................................................................................................................................................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
?t
............. ............. ............. ............. ............. ...........................................................................................................................
............. ............. ............. ............. ............. ............. ............. ...
............. ............. ............. ............. ............. .............
xS(pP )
xD(pP + t)
Consumer pays the tax
x′
p′P
p∗D
p′D
-
6
pD
x
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
6t
............. ............. ............. ............. ............. ...........................................................................................................................
............. ............. ............. ............. ............. ............. ............. ...
............. ............. ............. ............. ............. .............
xS(pD − t)
xD(pD)
Producer pays the tax
We can also describe the matter analytically. To do so we revert to a linear demand and
a linear supply function:
Demand function: xD = a− bpD (55)
Supply function: xS = c+ dpP (56)
Price relationship: pD = pP + t (57)
Market equilibrium: xS = xD (58)
162
By placing (55), (56) and (57) in (58) we get
xD︷ ︸︸ ︷a− bpD =
xS︷ ︸︸ ︷c+ d(pD − t)
p∗D =a− c+ dt
b+ d
In order to show what effect a tax increase has on the equilibrium demand price, we form
the 1st derivation of the demand price at the tax rate t:
∂p∗D∂t
=d
b+ d
The effect of the tax on the equilibrium price therefore depends on both the slope of the
demand function and the slope of the supply function. The flatter the (inverse) demand
curve, i.e. the greater b, is, the smaller is the price effect. The flatter the (inverse) supply
curve, i.e. the greater d is, the greater the positive effect on pD.
We can also determine the effect of a tax increase on the producer price. The equilibrium
produce price is:
p∗P = p∗D − t =a− c− bt
d+ b
The negative effect of the slope of the demand curve (b) thus remains. The flatter the
(inverse) supply curve, i.e. the greater d is, the smaller the (negative) effect of a tax
increase on the equilibrium producer price.
The shifting of taxes
From an economic perspective the question must be asked as to who carries the burden
of taxation. Ultimately it is irrelevant who actually pays the taxes in a purely technical
sense, as the burden of taxation can be passed on to the other side of the market by means
of price increases. The basic idea can be illustrated by a simple example: Assume the
original market price is 100. Now a tax of the amount of t = 10 is charged. There are two
possibilities that lead to the same result: firstly it could be the case that the consumers
pay the tax and the consumer price remains constant (at 100). The consumers then pay de
facto 110 per unit, while the producers continue to receive only 100. If, on the other hand,
the producers pay the tax and if they then raise the consumer price to 110, the result is
exactly the same.
163
Burden of taxation
The burden of taxation can be described by the extent of each price change (∆pD for
consumers, ∆pP for producers). A more exact measure, however, is the change in consumer,
or respectively producer surplus. In general the burden of taxation is carried to a certain
extent by the consumer and to a certain extent by the producer. The exact distribution
depends on the elasticity of demand or supply. The decisive question is to what extent are
the suppliers in a position to shift taxes. To clarify this we shall examine some extreme
cases. The depiction takes the case in which the suppliers pay the taxes. The economic
consequences will not be restricted by the assumptions listed above.
a) Perfectly elastic supply function:
p∗D
p∗D + t
-
6
pD
x
...................................................................................................................................................................................................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................
•
•
xS
xD
6t
=⇒ The tax is shifted fully to the consumers, who carry the complete burden of taxation.
The perfectly elastic supply function implies that the producers now supply at the equilib-
rium price p∗P = p∗S. If there is a tax increase, the suppliers will continue to demand p∗P . If
the net producer price lies below the original equilibrium price, there will be absolutely no
supply. That is why the new equilibrium price is calculated as the original price plus the
tax imposed upon it. The producer price remains unchanged, while the consumer price is
raised.
164
b) Perfectly inelastic supply function: (d = 0)
pP = p∗D − t
p∗D
-
6
pD
x
...................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
......
............. ............. ............. ............. ............. .............
............. ............. ............. ............. ............. .............•
xS
xD
6t
=⇒ The tax cannot be shifted and the supplier bears the burden of taxation.
The equilibrium consumer price p∗D now does not change at all, as the producers are willing
to supply xS at every price. The producer price pP therefore sinks by the amount of tax.
The empirically relevant case lies somewhere between a) and b):
Conclusion: The more elastic the supply function, the more tax burden can be shifted
from the supplier to the consumer.
c) Perfectly elastic demand function:
p∗D
-
6
pD
x
...................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................
xS(pD)xS(pD − t)
xD(pD)
=⇒ No shifting of tax, producers bear the full tax burden.
Changes to the effective consumer price pD would lead to too drastic (infinitely many)
demand quantity changes. The supply is restricted, as the effective (net) producer price is
reduced precisely by the tax rate t.
165
d) Perfectly inelastic demand: (b = 0)
p∗D
p∗D + t
-
6
pD
x
...................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
.......
...... xS(pD)xD
............. ............. ............. ............. ............. ............. ............. ...
............. ............. ............. ............. ............. ............. ............. ...
=⇒ Complete shifting of the burden of taxation to the consumers.
The demanded quantity does not react at all to price changes. Therefore the new consumer
price increases by the imposed tax rate.
The normal case lies between c) and d):
Conclusion: The more inelastic the demand function, the more tax burden can be shifted
from the supplier to the consumer.
To summarise, the distribution of tax burden depends on the elasticity of the demand
or the supply. The purely technical “payment“ of the tax is irrelevant from an economic
viewpoint. Further applications of the model can be found in the following areas: brokers’
fees, subventions (e.g. rent subsidies), tax write-off possibilities.
V.2.2 Welfare effects of taxation
The main argument of this section is that the taxation of consumer goods encroaches
into the efficient allocation that would occur with perfect competition and without state
intervention, and that there is a reduction in welfare as a result. Accordingly, not only
do redistribution effects occur, but also allocation effects, which reduce efficiency. The
economic reason for this can be found in the fact that a tax distorts the free decisions of
the private economic entity. The following diagram illustrates this connection:
166
-
6
xt x∗
pP = pD − t
p∗
pD
pD
x
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ............. ............. .................................................................................................
............. ............. ............. ............. .......
............. ............. ............. ............. .......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
.......
......
•
•
•
•
•
•
•
F
B
G
C
A
I
H
xS
xD
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
.
..
........
The original equilibrium without taxes occurs at (p∗, x∗) Total welfare is then calculated
as the sum of consumer surplus (CS) and producer surplus (PS): CS + PS = ABC.
The equilibrium after the tax imposition (t) is given at (pD, xt). The consumer surplus
then amounts to CS = CHG and the producer surplus to PS = AIF . The tax income
is described by the area FGHI. This produces a loss in welfare (additional burden) of
FGB, as the tax income can be interpreted as a positive welfare contribution.
The tax distorts the decisions of the economic entities. As well as a basic removal of
purchasing power from those taxed, an additional burden must also be borne by consumers
and producers, as less is consumed or supplied. If even if one side of the market technically
or monetarily pays nothing, it will still generally bear some of the burden. If, for example,
the consumers pay the taxes fully, but demand less because of the tax, producers sell less
and thus also carry some of the burden.
It is not important how much the individual economic entity pays to the state, but rather
how much it must pay in total (as consumer) or receives in total per unit (as supplier).
If, for example, the supplier pays the tax to the state, but the consumer price rises as a
reaction to that, the consumer must also carry part of the burden. All in all there is a
reduction in quantity and thus suppliers and consumers suffer losses.
Economic policy implication: Taxes should be imposed in such a manner that the loss
of efficiency is kept as low as possible.
167
Summary
1. With perfect competition a market equilibrium emerges, in which all suppliers and
demanders appear as so-called price takers. The theoretical construct of the Wal-
rasian Auctioneer can solve the paradox of how a market price comes about at all
in such a market.
2. According to the first principle of welfare theory, the competitive market
equilibrium is Pareto-efficient. That means that nobody can be advantaged
without worsening the situation of another. To put it another way: all resources are
used optimally and waste is excluded. In this condition the sum of consumer and
producer surplus is at a maximum.
3. Interventions by the state have an influence on the equilibrium quantity and
prices in a market. It should be taken into account who must bear the burden of
state intervention, in order to be able to conduct a proper political discussion.
4. The most important state intervention into the market is the imposition of taxes.
Consumer taxes are raised as quantity and value taxes. With the help of com-
parative statics, each of the quantity and price effects can be analysed. The slope
of the relevant demand and supply functions decides the amount of the price and
quantity effects of a tax.
5. From an economic viewpoint it is irrelevant whether or not the imposed tax is paid
by the consumers or the producers. Instead it must be clarified, whether the tax can
be shifted by the taxpayer to the other side of the market, in order to be able to
illustrate the distribution of a taxation burden.
6. The burden of taxation can be discerned roughly from the development of the con-
sumer and producer prices. A more exact measure, however, is the relevant effect
on the consumer or producer surplus. The economic analysis has shown that
especially elastic demand and supply functions allow the relevant side of the market
to shift the burden of taxation to the other side.
7. As well as redistribution effects, distorting taxes also lead to a reduction in
overall welfare. Defined as the sum of consumer surplus, producer surplus and
168
taxation, this occurs when the reduction of both surpluses is not compensated by
the tax revenue. Possible negative effects of taxation on the general economy must
be taken into account in the political decision making process.
169
VI Market equilibrium with imperfect competition
In this part of the course we complete our renunciation of the competitive market, i.e. we
abandon the assumption of perfect competition.
Definition: With imperfect competition there are only a few market participants on one
or both sides of the market.
Overview of the most important market forms:
1. Polypoly: many small suppliers, many demanders
2. Monopoly: only one supplier, many demanders, restricted market entry on the
supplier side
3. Oligopoly: few suppliers, many demanders, restricted market entry on the supplier
side
4. Monopolistic competition: few suppliers, many demanders, free market entry
A corresponding classification can also be made for the demand side. The market forms
are then known, analogous to the supplier side, as monopsony, oligopsony and monopsonic
competition.
There are naturally many mixed forms, such as partial monopoly (many small suppliers
and one large supplier) or bilateral monopoly (one supplier and one demander, e.g. tariff
parties). However, we will focus on market concentration on the supplier side. Accord-
ingly, we assume few suppliers and many small demanders. The most important difference
between a monopoly, or monopolistic competition, and an oligopoly is the strategic inter-
action in an oligopoly.
With strategic interaction the decision of the competitor have a noticeable effect on
one’s own profit. The oligopolist therefore takes the actions of the others into account
when making his decisions.
170
VI.1 Traditional monopoly theory
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapters 10 and 11
The most important characteristic of a monopolist is that he does not consider the market
price to be given, and the price is not a set figure in his optimisation calculations, as is the
case with perfect competition. The monopolist includes the falling demand curve in his
own calculations, as the price is a function of the quantity he is supplying. That means
that the monopolist is faced with a trade-off between higher quantities and a higher price.
The profit function of the monopolist follows as:
P = p(x)x− C(x)
with p(x) being the price function dependant on the supply quantity. Following from this
is the corresponding first order condition:
∂P
∂x= p
′(x) · x︸ ︷︷ ︸
b
+ p(x)− C′(x)︸ ︷︷ ︸
a
!= 0 (59)
The monopolist chooses de facto a profit-optimising point on the demand curve. It is
irrelevant whether optimisation occurs via the price p or the sales quantity x.
The optimality condition (59) shows the trade-off between quantity and price:
Term a: As long as p(x) − C′(x) > 0 a marginal expansion of quantity increases profit,
as the price achieved is greater than the marginal costs. The monopolist can therefore
achieve a unit cost profit margin by expanding the supply quantity. We call this positive
marginal profit.
Term b: An expansion of quantity is only possible by reducing the price. p′(x) · x < 0
shows which loss of profit the monopolist must accept if the price falls by a marginal unit
(p′(x)). This price reduction must be multiplied by the total supply quantity (x), as the
price reduction applies to all sold goods units.
Equation (59) produces: p′(x) · x+ p = C
′(x)
171
=⇒Marginalrevenue(MR) = Marginalcosts(MC)︸ ︷︷ ︸general optimality condition
If we solve this condition for the price it becomes clear that the monopolist is setter a
higher price than in a competitive situation:
p = C′(x)−p′
(x) · x︸ ︷︷ ︸>0
> C′(x)
The profit-maximising price in a monopoly therefore lies above the marginal costs.
Equation (59) can be reformulated in the so-called Amoroso-Robinson relation, which
establishes the connection of the monopoly price with the price elasticity of demand ϵ and
the marginal costs:
ϵ = −∂x
∂p
p
x
=⇒ p
(1− 1
ϵ
)= C
′(x)
With perfectly elastic demand (ϵ→∞) the optimality conditions becomes
p = C′(x). This case is then analogous to perfect competition. In general, the more elastic
the demand function, the lower the monopolist’s room for manoeuvre. Intuitively, the
monopolist then loses many customers when there is a price increase.
Furthermore, the monopolist produces only in the elastic area of the demand function
(ϵ > 1). At (ϵ < 1), C′(x) = p(1 − 1
ϵ) becomes negative and is therefore excluded by
definition. The economic intuition behind this is as follows: a demand elasticity smaller
than one implies that is worthwhile raising the price. Why? A price increase implies that
the quantity is decreasing and thus also the costs. However, if ϵ < 1, this also means that
a price change will lead to a disproportionately low decrease in quantity. This has the
consequence that sales increase when the price increases. At the same time, as we have
established, costs decrease. At ϵ < 1 it is therefore never optimal to remain at this point,
it is always worthwhile raising the price (until we land on the elastic part of the demand
function; see our discussion in III.5.2).
172
Graphical representation:
x∗
pC
p∗
-
6
p
x
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................
................................................................
................................................................
................................................................
................................................................
.......
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ...........
............. ............. ............. ............. ............. ...........................................................................................................................
•
•A
c′(x)
xD(p)
MR
The profit-maximising point (profit-maximising supply quantity x∗) lies at the intercept
of the marginal cost and marginal revenue curve (MR). If we go from this intercept to the
demand curve, we get the profit-maximising price (p∗). The profit-maximising point A
(p∗, x∗) is often called the Cournot Point.
The monopolist’s profit
With perfect competition the market continues to be entered until zero-profits arise: if a
positive profit exists, this is an incentive for new companies to enter the market. Because
there is by definition no market entry in a monopoly, positive profits can occur.
173
First the graphic illustration:
x∗
p∗
-
6
p
x
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................
................................................................
................................................................
................................................................
................................................................
.......
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ...........................................................................................................................
............. ............. ............. ............. ............. .............•
•
• c′(x)
x(p)
MR
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
G
P: variable profits minus
possible fixed costs
We can also analyse the matter for the case of a linear demand function and constant
marginal costs:
x = 1− p =⇒ p = 1− x
PM = (1− x)x− cx
∂PM
∂x= (1− 2x)− c(x) = 0
1− 2x = c
xM =1− c
2
→ p = 1− x =1 + c
2
PM =1 + c
2· 1− c
2− c · 1− c
2=
[1− c
2
]2> 0
Note: The marginal revenue curve (MR) has double the slope of the demand function.
174
Why does a monopoly exist?
When there are positive profits, why do other companies not enter the market? Reasons
for the existence of a monopoly could be due to (a) state regulation (prohibition of com-
petition), (b) the existence of a natural monopoly or (c) strategic market entry deterrents
on the part of the monopolist.
a) State market entry barriers:
Examples can be found above all in the energy sector, in the transport system and in
telecommunications. However, state regulation is in decline internationally. In the Euro-
pean Union the member states increasingly must convert the corresponding guidelines into
national laws.
b) Natural monopoly:
With natural monopoly there are technological reasons for the monopolisation of the mar-
ket. Usually those industries with increasing economies of scale are affected. The average
cost minimum is then only reached with very large sales quantities, relative to the size of
the market. However, this argument often applies only to a degree. In the telecommuni-
cations sector, for example, it only applies to the physical telephone network, but not to
the service provision of telephoning in the fixed network. The following diagram illustrates
the typical average cost path that leads to a natural monopoly.
-
6
p
x
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
....................................................................................................................................................................................................................................................................................................
............................................................................................
x(p)
LACM
LACC
LACC : long-term average costs in the case of competition
LACC-minimum: small individual sales quantities x, relative to market size
175
LACM : long-term average costs in the case of a monopoly
LACM -minimum: large sales quantity x, relative to market size
c) Strategic market entry deterrents:
The monopolist prevents possible market entry by means of strategic investments. This
might include, for example, the development of high capacities, the strengthening of the
brand name or strategic patents (dormant patents, the use of which is threatened). All of
the measures signal credibly to potential market entrants that a market entry would not
be profitable.
(In-)Efficiency of the monopoly
The competitive solution is (statically) efficient. Because the monopoly price is greater
than the equilibrium price with perfect competition (pM > pC = C′) the monopoly solution
is inefficient. Conversely, this means that with a change in quantity (specifically: increase
in quantity), the sum of producer and consumer surplus can be increased. In this course
we use a watered-down efficiency concept: the so-called compensation criterion. If the
demanders were to compensate the monopolist for his losses (reduction of the producer
quantity), a surplus would remain when transferring to the competitive solution, which
could be distributed on both sides of the market. We call this remainder the efficiency
gain. Illustration:
xM
pM
-
6
p
x
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
................................................................
................................................................
................................................................
................................................................
................................................................
.......
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............. ............. ............. ............. ............. ...........................................................................................................................
............. ............. ............. ............. ............. .............•
•
•
A
BC c′(x)
x(p)
MR
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
.
..
..
..
..............
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
PS
CS
176
The welfare-related difference to the competitive solution is shown by the area ABC. The
economic cause can be found in the higher price in the monopoly, which leads to a lower
sales quantity. This results in a shortage in supply to consumers.
Important distinction: Static efficiency vs. dynamic efficiency
In the dynamic context, i.e. with the existence of research and development (R&D), the
following applies: future profits form an incentive for innovators to engage in R&D today.
On the other hand, with perfect competition (zero profit), there is no incentive to invest
in R&D. Therefore a monopoly can possibly be dynamically efficient, even if the there is
no static efficiency.
Price discrimination in the monopoly
Until now we have observed the case that the monopolist sets only one price for all con-
sumers. In reality, however, we can often observe price discrimination. Special prices are
set for certain groups of buyers, or bulk discounts are given, for example. In general, three
types of price discrimination can be distinguished:
(a) Perfect price discrimination (first-degree price discrimination)
(b) Price discrimination according to demander group (third-degree price discrimination)
(c) Price discrimination as a self-selection mechanism (second-degree price discrimina-
tion)
a) Perfect price discrimination
Perfect price discrimination is an idealised concept (theoretical extreme situation). The
monopolist completely exhausts the consumer surplus. The individual price that is ul-
timately demanded from the single consumer corresponds to his marginal willingness to
pay. The prerequisite for the exhaustion of the consumer surplus is the assumption that no
goods arbitrage is possible. It is therefore excluded that a consumer can buy at the (lower)
price of another consumer. First-degree price discrimination would only be possible with
different consumers (with a different marginal willingness to pay) if each consumer had his
177
individual marginal willingness to pay written on his forehead, i.e. he could not pretend
anything else.
The situation is somewhat different when all individuals are identical. Assuming that each
of the n consumers has the same downward-sloping demand curve:
-
6
p
x
...................................................................................................................................................................................................................................................................................................................................................................................................................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
............. ............. ............. ............. ............. .............•
c′(x)
x(p)
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
.
..
..
..
..
..
..
..
..
.
..
..
.
..
..
..
..............
CS
Then the tariff that leads to perfect price discrimination would look as follows:
T (x) =
A+ pcxc for x = xc
0 else(60)
with xC being the equilibrium quantity in competition and A = CS the consumer surplus.
By these means the consumer is relieved of his entire surplus by the producer. However, the
efficient allocation is implemented, because the competitive solution is realised. Therefore
there is no difference with regard to the efficient allocation result (quantity and price), but
there is with regard to the distribution. With the competitive solution, both the consumer
surplus and the producer surplus are positive, i.e. both groups receive a portion of the
total welfare cake. With first-degree price discrimination, on the other hand, the producers
get the entire surplus, while the consumers are left empty-handed.
b) Price discrimination according to demander groups
Assume there are two groups of demanders, which can be differentiated by objective char-
acteristics. Examples include students/non-students, women/men, etc.
The two demand functions are given as:
x1 = x1(p1) (61)
x2 = x2(p2) (62)
178
The profit function of the monopolist is then:
P = p1x1(p1) + p2x2(p2)− C(x1(p1) + x2(p2)) (63)
The profit is calculated as the sum of each profit in the submarkets. This produces the
following first order conditions:
x1 + p1∂x1
∂p1− C
′ ∂x1
∂p1
!= 0 (64)
x2 + p2∂x2
∂p2− C
′ ∂x2
∂p2
!= 0 (65)
or, after reformulation:
p1
(1− 1
ϵ1
)= C
′(66)
p2
(1− 1
ϵ2
)= C
′(67)
whereby ϵi = −(pi/xi)(∂xi/∂pi) describes the relevant demand elasticity with i = 1, 2.
Because the right-hand side of both optimality conditions are identical, we get:
p1p2
=(1− 1
ϵ2)
(1− 1ϵ1)
(68)
The group with the more elastic demand is charged a lower price, for example ϵ2 > ϵ1 =⇒
p1 > p2. Economically, this is understandable: the more elastic the demand, the stronger it
will react to price increases and the less attractive are price increases from the company’s
perspective. This is how we can explain the existence of student discounts, as student
demand is generally more price elastic than that of professors. Further example: time-
differentiated offers (peak season/off-season), daily lunchtime offers, happy hour, etc
c) Price discrimination as a self-selection mechanism
With imperfect information, different output quantities can be sold at different prices.
The reasoning behind it is the objective of the profit-maximising (self-)selection of different
groups of demanders through bulk discounts. This form of price discrimination is addressed
in detail in the main degree course.
179
Summary
1. In contrast to polypoly (perfect competition) imperfect competition is charac-
terised by the fact that only a few participants are active on at least one side of the
market. On the supplier side we speak of a monopoly or monopolist competition
and oligopoly, while on the demand side we use the terms monopsony, or monop-
sonic competition and oligopsony. Strategic interaction among suppliers is the main
difference between a monopoly and an oligopoly.
2. The traditional monopolist appears as the only supplier of a good and therefore takes
account of the connection between price and supply quantity (falling demand curve)
in his profit maximisation calculations. The optimality condition of the monopolist is
marginal revenues equal marginal costs (Cournot Point). Compared to perfect
competition, there are higher equilibrium prices and lower quantities.
3. The Amoroso-Robinson relation shows the connection between the price elas-
ticity of demand, the marginal costs and the monopoly price: the lower the price
elasticity, the greater the difference between marginal costs and monopoly price.
4. The fact that a monopolist can achieve positive profits poses the question as to
why other companies do not enter the market. This, however, can be prevented by
state regulation, increasing economies of scale or strategic market entry deterrents.
5. The shortage of supply to consumers (scarcer supply) leads to the fact that a
monopolist market is statically inefficient. That means that the sum of producer
and consumer surplus (i.e. total welfare) is higher with perfect competition. From
a dynamic perspective, monopolist markets certainly can, however, be efficient, as
positive profits represent an incentive to invest in research and development.
6. In monopolist markets different types of price discrimination often occur. With
perfect price discrimination the monopolist exhausts the entire consumer surplus.
This, however, is only possible if the monopolist has perfect information about the
preferences of the consumers.
7. Price discrimination according to demander group is connected to less in-
formation. If the demanders can be differentiated according to obvious criteria, the
180
monopolist can demand corresponding prices. This is optimal for the monopolist, as
the price elasticity of demand differs according to the demander group. Furthermore,
self-selection mechanisms (e.g. bulk discounts) can also be implemented, which
can overcome the inherent information problem of price discrimination.
VI.2 Oligopoly and game theory
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapters 12 and 13
In the real world neither the competitive case nor the monopolist case occurs. Rather,
we frequently encounter an intermediate situation in which there are few companies and
many demanders (oligopoly). Examples of oligopolies include the civil aviation industry
(Airbus and Boeing), the automobile industry, food retailers, the field of internet software
and many more.
In an oligopoly, every company has an influence on the market price. This means that
strategic decisions are made by the companies, as the price and quantity decisions
of competitors can have a strong influence on one’s own profit. Below we shall restrict
ourselves for reasons of simplicity to the duopoly, in which only two companies are active.
Strategic decisions are generally analysed with the help of game theory. Here we shall use
the methods of non-cooperative game theory.
VI.2.1 Introduction to game theory
Game theory has a wide range of applications outside of microeconomics: macroeconomics
(e.g. central bank games), foreign trade theory (e.g. customs policy), tax competition,
corporate organisation, political analysis, etc.
181
Introductory example of strategic interaction:
a) Prisoner’s dilemma
A/B B:D B:C
A:D -1/-1 -30/0
A:C 0/-30 -15/-15
The table shows the payoff matrix for the situation described in the lecture. The number
of years to be spent in prison is entered as a negative number, so a higher number of points
is better for the prisoner than a lower one.
In equilibrium the non-cooperative solution (C,C) emerges, which is worse than (D,D).
If Prisoner A decides to deny (D) rather than confess (C), he is worse off both when B
confesses or denies. The same applies to Prisoner B. The strategy (C) is the so-called
dominant strategy for both players.
In many cases, however, there is no dominant strategy. A small example of a research and
development game can illustrate this (HRB= high research budget; LRB= low research
budget)
Company 1/2 2:HRB 2: LRB
1: HRB 40/200 100/60
1: LRB 30/0 80/40
Here Player 2 has no dominant strategy (if 1 chooses HRB is it optimal for Player 2
to also select HRB, if 1 chooses LRB is it optimal for Player 2 to also select LRB).
An alternative solution concept is then a non-cooperative Nash equilibrium (John Nash:
Nobel Prize winner for Economics in 1994; A Beautiful Mind!): a Nash equilibrium exists
when none of the players has an incentive to change his own strategy unilaterally. In this
game, this is the case with the strategy combination (HRB; HRB). Before we turn to the
Nash equilibrium in more detail, let us consider (i) the complexity of even easier strategic
situations (cf. sub-point b)) and (ii) sequential strategic decisions (cf. sub-point c)).
182
b) Strategic voting
The three members of the Dead Poets Club (Boris, Horace, Maurice) must decide on the
acceptance of a new member. There are three alternatives: Alice, Bob or nobody. The
following table shows the preferences of the current members:
Boris Horace Maurice
1. Alice 1. Nobody 1. Bob
2. Nobody 2. Bob 2. Alice
3. Bob 3. Alice 3. Nobody
There are two rounds of voting: first the choice will be made between Alice and Bob. Then
the second round pits the winner of the first round against the option „Nobody".
What will the result look like? To what extent is the vote strategic? Assuming you are
Maurice, would you design the voting round differently?
c) Pricing
For some agents/companies, their own level of profits is influenced directly by the decisions
of the other agents. As a formula: Pi = Pi(pi, pj). The following diagram shows a decision
tree, which shows how high the payoff is for two producers, when high or low prices (pH
bzw. pL) are set:
•
•
•
•
•
•
•
..........................................................................................................................................................................................................................................................................................................................................................................
.............................
.............................
.............................
.............................
.............................
...............
..................................................................................................................................................................................................................................
.............................
.............................
.............................
.............................
....................
................................................................................................................................................................
(100, 100)
(0, 200)
(10, 100)
(50, 50)
1
2
2
pH
pH
pH
pL
pL
pL
183
What is the right decision: pL or pH ?
Approach: The first producer (1) anticipates the decision of the second producer (2),
depending on his own decision, and then chooses the optimum price. As Producer 1s
payoff depends on the price of Producer 2, strategic interaction takes place. Game theory
is concerned with how we behave in such situations.
We shall now address in detail the central Nash equilibrium concept.
Nash equilibrium for the two-player case
First, some terms and variables must be defined:
• Players (i): agents who are involved in the matter (companies, countries, etc.)
• Strategy (a): decision variable of the players (price, quantity)
• Strategy set (s): quantity of all possible strategies (e.g. p ∈ [0, 100])
• Payoff (G): result for the players (e.g. profits)
Definition: A strategy combination (a∗1, a∗2) is a Nash equilibrium when no player has an
incentive to change his strategy unilaterally.
Formal definition of the conditions of a Nash equilibrium:
P1(a∗1, a
∗2) ≥ P1(a1, a
∗2) ∀ a1 = a∗1
∧ P2(a∗2, a
∗1) ≥ P2(a2, a
∗1) ∀ a2 = a∗2
Example 1:
A / B pH pL
pH 2/3 5/2
pL 0/1 6/0
184
The matrix shows which payoff structure emerges for the different combinations of higher
and lower prices, whereby the first (second) number refers to Player A (B). There is a
dominant strategy only for Player B (pH). A Nash equilibrium occurs at (pH , pH) because
it is only with this constellation that there is no incentive to change strategy unilaterally.
Example 2:
A / B pH pM pL
pH 2/3 3/1 2/2
pM 1,5/2 1/3 4/0
pL 1/1 0/0 0/2
Here, too, the only Nash equilibrium is given at (pH , pH).
Example 3: Battle of the sexes
M / F Tennis Theatre
Tennis 2/3 1/1
Theatre 1/1 3/2
In this example there are two possible Nash equilibriums: tennis-tennis or theatre-theatre.
In games with many decision levels we speak of subgame perfect solutions, i.e. there must
be a Nash equilibrium at every stage of the game. With sequential decisions a backward
solution method is always chosen in general. This means beginning at the last stage of the
game and then working ahead (backwards) logically to the first stage of the game. Below
we shall apply the basic concepts of game theory just shown to oligopoly theory.
VI.2.2 Oligopoly theory
A distinction will first be made between quantity strategy (quantity as a decision vari-
able) and price strategy (price as a decision variable). Furthermore it is important to
185
classify markets according to homogenous and differentiated goods, as each of the strategic
interactions greatly differ from the other.
Homogenous quantity duopoly with simultaneous decisions
The quantity competition is often called the Cournot competition. We assume that there
are two identical companies in the market, A and B (symmetrical duopoly). Both suppliers
produce the same homogenous good. Furthermore, we assume constant marginal costs of
10. The demand function of the households is given at p = 100− (xA + xB). The decision
variable of the oligopolist is the supply quantity.
From the assumptions made, this is the profit-maximisation problem of the producers:
Pi = (p− 10) · xi = (100− (xi + xj)− 10)xi i, j = A,B i = j (69)
∂Pi
∂xi
= 100− 2xi − xj − 10!= 0 (70)
Company A: 100− 2xA − xB − 10!= 0 (71)
Reaction function: xA =90− xB
2(71’)
Company B: 100− 2xB − xA − 10!= 0 (72)
Reaction function: xB =90− xA
2(72’)
The best answer in each case to the actions of the competitor is described by the reaction
function. For alternative quantities of the competitor, the optimal own quantities are
calculated: xA = xA(xB) and xB = xB(xA).
186
x∗A
x∗B
-
6
xB
xA
..............................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................
xA(xB)
xB(xA)
............. ............. ............. ............. ..............................................................................
Graphical Illustration:
•
Nash-equilibrium
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
P 1A
P 2A
...........................
.....................................................................................
..................
.........
...........................
.....................................................................................
..................
.........
...........................
.....................................................................................
..................
.........
P 1B
P 2B
The maximum profit with alternative supply quantities of the competitor is reached on
the best-answer curve (reaction curve). The further the iso-profit functions PA and PB
are removed from the origin, the lower the level of profit. This is because a higher supply
quantity of the competitor reduces one’s own profit. A Nash equilibrium is given at the
intercept of the two best-answer curves. In this point, neither of the companies has an
incentive to change strategy unilaterally. At x∗B the optimum supply quantity of A is x∗
A
and vice versa.
If we place the reaction function (72’) in the reaction function (71)′ we get:
xA =90
2− 90− xA
4(73)
=90 + xA
4(74)
This we get the equilibrium supply quantity of Company A xA = 30. Placing this in the
reaction function of Company B then produces the supply quantity xB = 30. The identical
supply quantities x∗A = x∗
B are due to the symmetrical assumptions made above.
Price duopoly with differentiated goods
We again observe a market with two companies, A and B (symmetrical duopoly). Now,
however, the suppliers produce differentiated goods, which are substitutes. Accordingly,
187
the demand function for the goods of Company A (B) is xA = 100 − pA + pB (xB =
100 − pB + pA) and marginal costs are constant at 10. This time the strategic decision
variable is the price.
The profit functions with differentiated goods are:
Pi = (pi − 10)xi (75)
= (pi − 10)(100− pi + pj) i, j = A,B i = j
with first order conditions:
∂Pi
∂pi= 100− pi + pj − (pi − 10) (76)
= 100− 2pi + pj + 10!= 0
From the FOCs the reaction functions follow:
pi =100 + pj + 10
2(77)
and with that pA =100 + pB + 10
2(78)
pB =100 + pA + 10
2(79)
The reaction function (79) placed in (78) yields the equilibrium price:
pA =100 + (100+pA+10)
2+ 10
2
=330
3= 110 (80)
Due to the symmetry properties the result pA = pB can be derived directly.
188
p∗A
p∗B
-
6
pB
pA
.............................
............................
............................
............................
.............................
............................
............................
............................
.............................
............................
............................
............................
.............................
............................
..
..............................................................................................................................................................................................................................................................................................................................................................................................................pA(pB)
pB(pA)
............. ............. ............. ............. ............. ........................................................................................
..................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................................................................................................................................
...........................
................................................................................................................
...........................
................................................................................................................
...........................
................................................................................................................
...........................
................................................................................................................
Graphical illustration:
Again the Nash equilibrium is given at the intercept of the two reaction curves. Neither
of the two duopolists has an incentive to move away from the price combination (p∗A, p∗B).
From A’s perspective the price p∗A is optimal given B’s optimal price (p∗B). From B’s
perspective the price p∗B is optimal given A’s optimal price (p∗A).
Price duopoly with homogenous goods
Again there is a simultaneous determination of the prices of both companies. The so-
called Bertrand duopoly delivers important knowledge that can often be very helpful as an
approximation of the actual market situation. With homogenous goods, the company with
the higher price does not receive any demand, even if the prices differ only marginally.
Proposition: In the Nash equilibrium both suppliers set the same price, which corresponds
to the marginal costs c: pA = pB = c.
Intuition:
With a market price that lies above the marginal costs, it is always worthwhile to underbid,
so that the entire demand can be exploited. Ultimately, the consumer does not care about
who sells him the homogeneous good. Only at pA = pB = c does no company have any
incentive to change strategy.
189
More detailed evidence:
At pA > pB > c there are absolutely no sales and zero profit for Company A: xA = 0 and
GA = 0. In undercutting the price of Company B by a very small amount ϵ (pA = pB−ϵ >
c) Company A can achieve a positive profit. For this reason, the original price pA cannot
have been optimal.
If both companies set pA = pB = p > c, Company A will make the following profit:
PA = (p− c) · x(p)/2 > 0
By slightly reducing the price under p (pA = p− ϵ), Company A makes a profit of
PA = (p− ϵ− c) · x(p− ϵ)
For small ϵ this expression is larger than the one above and greater profits could be made
than at the starting point. The profit contribution falls only marginal, but the sales
quantity increases discretely and strongly.
In order to show that both companies set the price equal to the marginal costs, we assume
that
pA > p∗B = c
Then Supplier B always has the possibility to raise the price marginally and to make a
profit. Only at p∗A = p∗B = c does neither company have an incentive to change strategy
unilaterally. Naturally, in this Nash equilibrium the companies also do not make a profit.
Implication:
Even with only a few companies (in our example only two), price competition leads to com-
petitive solutions with homogenous goods. This result is the so-called Bertrand solution
(“Bertrand-paradox"): “Two is enough for competition."
Application examples:
The classic example of duopolistic price competition with homogenous goods are fuel sta-
tions located opposite each other on an arterial road. More up-to-date examples can be
found in the field of e-commerce: if no product differentiation can be achieved, this leads
to intensive price competition. As the competitor is only one click away, there are often
190
no profits. A further application example is Call-by-Call in the telephone network: tele-
phoning is a completely homogenous good. With Call-by-Call suppliers the customer has
absolutely no switching costs, which leads to stronger price competition.
Quantities or prices as strategic variables?
Sales quantities are usually interpreted as capacities. There is no general answer to the
question of whether companies compete in prices or in capacities. This depends ultimately
on the industry in question. There are, however, some basic patterns: in very intensely
competitive industries a price model tends to be used more. In less intensely competitive
industries, on the other hand, a quantity model seems to make more sense. Our results
(lower profit, higher competition) also suggest that companies in intensely competitive
industries have an incentive to use other instruments, such as advertising, to increase their
sales quantities and profits.
Homogenous quantity duopoly with sequential decisions
In this section we address a so-called sequential game. In general this is the modelling of
interactions between oligopolists, not only at one point in time. The simplest multi-stage
game contains precisely two game stages and leads to the Stackelberg solution.
The starting point for this is a homogenous duopoly in which there is quantity competition
and the inverse demand function
p = a− (x1 + x2) (81)
is given. Due to institutional reasons that will not be specified here, one of the two
duopolists is in a position to make his quantity decision before the competitor. He is
therefore also known as the Stackelberg leader. Then, in the second stage of the game, the
Stackelberg follower must determine his quantities x2, dependent on the quantity decision
x1 by the leader. As there is perfect information, the leader can also anticipate the be-
haviour of the follower and thus adapt his quantity decision to it from the start.
The two-stage game is solved by means of backward induction and fulfils the criterion of
subgame perfectness.
First the quantity decision of the Stackelberg follower is derived. He maximises his profit
191
as follows:
maxx2
Π2(x2, x1) = (a− (x1 + x2)− c) x2 (82)
We calculate
∂Π2
∂x2
= a− 2x2 − x1 − c!= 0 (83)
and get the solution for x2, dependent on x1
x∗2 =
a− x1 − c
2(84)
We get the optimal quantity of the leader by placing (84), i.e. the optimal quantity of the
follower, into the profit function (Π1 = (a − (x1 + x2) − c)x1) and maximizes the profit
with respect to x1:
maxx1
Π1 =
(a− (x1 +
a− x1 − c
2)− c
)x1 (85)
∂Π1
∂x1
!= 0 (86)
This results in:
x∗1 = a−c
2(87)
and finally x∗2 = a−c
4(88)
The following is produced for the profits of leader and follower
P1 =(a− c)2
8(89)
P2 =(a− c)2
16(90)
The leader can sell a larger quantity than the follower and attains a higher profit. We
speak here of a First Mover Advantage. Indeed the leader has the incentive to produce a
larger quantity than in the Cournot duopoly, while the follower produced a lower quantity.
Summary
1. An oligopoly leads to strategic supply behaviour, as the decisions of the compa-
nies have an influence on the objective of the competition. The strategic interaction
on these markets can be analysed with the help of game theory.
192
2. If dominant strategies exist on the sides of all game participants, the determina-
tion of the equilibrium is no problem. However, if there is no dominant strategy, we
must revert to the Nash equilibrium concept. An equilibrium is then defined by
the fact that none of the participants has an incentive to change strategy unilaterally.
To solve multi-stage games it is necessary to proceed backwards. We start with the
last game stage and then work back logically to the first.
3. With oligopolistic competition the market result depends greatly on whether the
suppliers compete in quantities or prices. Furthermore a distinction must be made
between homogeneous and differentiated goods.
4. In the quantity duopoly with simultaneous decisions and homogeneous
goods a Nash equilibrium can be calculated by means of deriving the reaction func-
tions. If both suppliers are identical (same marginal costs, etc.) the total demand
quantity is divided exactly in two halves. In the symmetrical price duopoly with
differentiated goods a symmetrical Nash equilibrium also emerges, in which both
suppliers set the same price.
5. In a symmetrical price duopoly with homogeneous goods we get the astonishing
market result that the producers in equilibrium set the price equal to the marginal
costs. This produces therefore the same efficient allocation as in perfect competition,
although there are only two suppliers with a corresponding supposed market power
(Bertrand paradox).
6. In general price models tend to describe and explain intensely competitive mar-
kets. Quantity models, on the other hand, suit markets in which there is little
competition.
7. The Stackelberg model is an example of sequential games. There are two stages,
whereby one of the players (Stackelberg leader) can determine his strategy first. The
Stackelberg leader anticipates the behaviour of the Stackelberg follower and can, as
a result, increase his quantities and his profits (compared to the quantity duopoly
with simultaneous decisions).
193
VII Asymmetric information
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 17
Until now we have assumed that there is either perfect or imperfect information, but that
all market participants have an identical information status. This is often not realistic,
especially regarding the quality of the goods, the factor of work or loans (borrowers). In
these and very many other cases, the information status of the contractual parties is often
very asymmetrical.
Where there is asymmetric information, one of the market participants has better in-
formation than the opposite party. This distribution of information is an essential problem
for the efficient functioning of markets. The classic example of asymmetric information
distribution is the used car market: The first essay by George Akerlof (1970) - Market for
„ Lemons“ - was on this subject.
There are two different forms of asymmetric information. With adverse selection the
asymmetric information exist prior to the conclusion of the contract. With moral hazard
asymmetric information emerges only after the signing of the contract.
VII.1 Asymmetric information and market failure
The problem of asymmetric information will be illustrated below based on Akerlof’s used
car example. The basic idea can be represented by a very simple model. This case involves
specifically the problem of adverse selection.
Model assumptions:
The following assumptions apply: the suppliers and demanders in the used car market have
different information. The suppliers have better information about the used cars than the
demanders. We assume 100 suppliers and 100 demanders, and each of them wants to buy
or sell a car.
194
Generally accessible information: 50 cars of good quality and 50 cars of poor quality are
in offer. The suppliers of a good (bad) car are willing to sell for 15,000 monetary units
(8,000 monetary units). The demanders, for their part, are willing to pay 16,000 monetary
units (9,000 monetary units) for a good (bad) car.
How does the market equilibrium look like?
1. With perfect information on both sides all of the cars would be sold. The
equilibrium prices are set according to negotiating power at pG ∈ [15.000, 16.000] or
pS ∈ [8.000, 9.000]. It is therefore a functioning market.
2. With asymmetric information the demanders cannot assign a particular quality to
the individual cars, while the suppliers know the quality of their cars. Assuming that the
demanders are willing to pay just the expectation value: pE = 0, 5 · 16.000 + 0, 5 · 9.000 =
12.500. At this price, however, only the owner of poor cars are willing to sell. This
willingness therefore gives the demander a signal for the poor quality of the car offered.
As the demanders are not prepared to pay pE for a bad car, a lower equilibrium price is
produced for the bad cars: pS ∈ [8.000, 9.000]. Good cars will not be sold at all, as no
good car is available for pS ∈ [8.000, 9.000]. The following applies to p > 15.000 50/50
chance of getting a good car. The demanders, however, are not willing to pay 15,000 for
this chance. Therefore, asymmetric information leads to market failure.
Reasons for the market failure
The suppliers of bad cars exercise an indirect (negative) effect on the suppliers of good
cars. The demanders are therefore only willing to pay very little for a used car. This
is because the demanders are not clear about the origin of the cars. If there were only
suppliers of good cars, there would be no market failure.
195
VII.2 Adverse selection and signals
We shall first provide three important examples of adverse selection.
a) Health insurance:
The insured party has better information about his health than the insurer can ever dis-
cover. A problem then arises when insuring good risks: it is difficult or almost impossible
to prove one’s own good health. The insurer always has the problem that an illness may
exist. The evidence can be problematic in individual cases. Uniform tariffs might then lead
to a situation where health insured parties (good risks) do not participate voluntarily in
the insurance. State measures (compulsory insurance) are frequently justified economically
and legally by highlighting potential adverse selection.
b) Labour market:
There is asymmetric information with regard to the „quality“ of the employee, such as
motivation, resilience, team capabilities, etc. The selection of „good“ employees is very
difficult and costly. If the risk of hiring a „bad“ employee is too high, unemployment might
rise.
c) Financial market:
Debt/loan: The risk of a loan or a financial project is better known to a borrower
than to the lender. Adverse selection often occurs in new companies, in particular start-up
enterprises. The probability of default is difficult to estimate, or only at high cost. Lenders
(e.g. banks) are interested in low risk, while the borrower is interested in high risk. The
reason for this is the special payout structure of loans. While the lender nears the total
risk of default, he benefits only to a limited degree from any possible profits.
196
......................................................................................................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................
-K
0
(1 + r)K
-
6Y B
Project return
Illustration:
The diagram shows the payout structure of a loan dependent on the project return (start
up). With their revenue of Y B the banks do not participate in the success that exceeds
loan repayment and interest payments. They are therefore not included in revenue on the
dotted line, yet carry the risk (continuous line with slope 1). With asymmetric information,
banks may therefore not be willing to give loans, which can lead to credit rationing. In
order to encourage market efficiency on credit markets with adverse selection, so-called
convertible bonds are frequently used.
Solutions for adverse selection
a) State regulation by means of compulsory insurance (health insurance)
b) Separate contracts: Lenders can try to separate good borrowers from bad, for ex-
ample using the instrument of credit securities. The idea behind this is self-selection
by the different groups by creating different contracts. Each group then has an in-
centive to choose the contract that has been designed for them and not one of the
others.
c) Building up a reputation makes particular sense with quality information prob-
lems such as with restaurant meals, holidays, new issues of shares, etc.
197
d) Signals: Sellers (who are better informed) can send signals to potential buyers that
contain information about the product quality. This model is attributable to Michael
Spence (1979). Training and education can serve as signals about the quality of a
new employee.
Now: Signals to solve adverse selection
To analyse signals in adverse selection we shall now observe a simple labour market model.
Model framework:
There are two equally-sized groups of employees in the labour market: Group 1 employees
have low productivity and produce goods with an annual net value of 40,000 monetary
units; Group 2 employees have high productivity and produce 60,000 monetary units per
year.
If a company were to pay just according to productivity, the wages would be w1 = 40, 000
monetary units and w2 = 60, 000 monetary units. We also assume that Group 1 employees
have a reservation wage of 40,000 monetary units and Group 2 employees have one of
60,000 monetary units. The types of employee cannot be identified, and the average wage
w0 = 50, 000 is therefore paid. However, only Group 1 employees are willing to work for
this wage rate. This therefore leads to adverse selection with equilibrium wage w1 and
simultaneous non-employment of Group 2 employees.
Type signalisation by means of training:
The decisive assumption of the model is that the investment in training for Group 2
employees is cheaper (learning comes more easily) than for Group 1 employees. The costs
of training are a function of the number of training years (y):
• C1(y) = 40, 000y for workers with low productivity
• C2(y) = 25, 000y for workers with high productivity
198
Training has absolutely no production-increasing utility (naturally only in the model!).
We shall now look at the following selection equilibrium: the company employs all workers
with y ≥ y∗ at w2 and all with y < y∗ at w1.
Which number of training years y∗ is a selection equilibrium?
Let us first look at the cost-utility considerations of the employee: if we assume a working
duration of 20 years (no discounting for reasons of simplicity), the utility from y > y∗
for both groups is By>y∗ = 400, 000 (20 working years multiplied by the wage difference
20, 000). Group 1 will then not make any investment for y > 10 and Group 2 for y > 16.
A selection equilibrium y∗ establishes itself between 10 and 16 training years. The diagram
illustrates the connection using the example y∗ = 12:
0 10 12
400.000
-
6
CostsUtility
y.................................................................................................................................................................................................................................................................................................................................................................
.............................................................................................
C1(y)
B(y∗)
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.• •
Group 1
0 12 16-
6
CostsUtility
y..............................................................................................................................................................................................................................................................................................................................................
.............................................................................................
C2(y)B(y∗)
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.• •
Group 2
Group 1 will not invest in training (y∗1 = 0), while Group 2 invests exactly 12 years in
training (y∗2 = 12). All employees with y ≥ y∗ can then be identified as Group 2 employees
and receive the wage rate w2. All employees with y < y∗ are Group 1 employees and
receive w1.
Conclusion:
Training serves as a signal of work productivity. The only purpose of training in the model
is the partial solving of the asymmetric information problem.
199
Further examples:
1. Guarantees and rights of return: Only suppliers of good-quality products can afford
to provide these additional services.
2. More shareholding of the management in a company: This signals the high quality
of the investment projects.
Concluding question: Why do lawyers in small towns spend much less on suits that in big
cities?
VII.3 Moral hazard and incentives
Moral hazard
With moral hazard the informational asymmetry arises after conclusion of the contract:
the actions of the opposite party (e.g. manager, worker or borrower) cannot be observed
or checked. We then speak of a hidden action problem or principal-agent problem:
The conclusion of a contract changes the behaviour of the contractual parties. The worse-
informed of the two is known as the principal and the better-informed is the agent.
Examples:
1. Full health insurance changes demand behaviour (variable price equals zero).
2. Rented apartments are treated differently than owned apartments.
3. Financial crisis: Disincentives to equity shareholders of banks with high debt-to-
equity ratio (e.g. at UBS 30:1), especially with the state protection of borrowed
capital (deposit insurance); incentive to excessive risk-taking; incentive passed on to
bank manager.
4. National debt crisis: Bailout approval for states reduces saving incentives.
5. Moral hazard can also be found in many of life’s circumstances: hiring handymen,
subletting apartments, lending the car, behaviour in the workplace, getting married,
etc.
200
Incentives
In order to solve this problem, incentives must be put in place to ensure that the contractual
partner (whose actions cannot be fully checked) behaves in the interests of all. This can
be done with explicit or implicit contracts.
Example of an implicit (employment) contract: An employer pays an employee a wage that
increases with company affiliation. The behaviour of the employee is checked sporadically.
If he behaves correctly, the employment relationship continues. Otherwise the employee is
reprimanded and, after several such occurrence, fired. He then loses the higher wage that
came about as a result of long company affiliation.
Theory of optimal contracts
The theory of optimal contracts pursues the goal of designing contracts (credit agreements,
insurance contracts, investment contracts, employment contracts, marriage contracts) in
such a manner that they are incentive compatible. The basic idea is as follows: If
the agent remains completely unaffected by the outcome of a process, he will not make
any effort. However, if his income (utility) is dependent on success, he will try to achieve
a positive result. An optimal incentive would therefore require the full participation of
the agent in the success. Often, however, the principal is risk-neutral and the agent risk-
adverse. Efficient risk sharing then leads to the fact that the agent is not at all involved
in the success. Thus, there is a trade off: Optimal risk sharing ←→ incentives.
Example: Manager-owner relationship
We want to illustrate the theory of optimal contracts using the example of the manager-
owner relationship. The manager (agent) is a specialist in company management and is
expected to maximise the value of the company. However, he has his own interests, such
as building up prestige (such as constructing the Empire State Building, expensive office
furnishings, etc.). The owner (principal) cannot fully observe the efforts of the manager,
as profit is also influenced by other things (e.g. general economic situation). In a results-
independent contract the manager will not engage himself, or not enough, in pursuing the
interests of the owner. With asymmetric information, therefore, a performance-related con-
201
tract is necessary (bonus contract, share options, etc.). With an exclusively performance-
related payment, the manager is fully exposed to the risk that, despite his best efforts,
a poor industrial economic situation greatly reduces profits. If he is risk-adverse and the
owner (who has invested in many shares) risk-neutral, this is not an optimal design of
payment. Therefore a mix of fixed salary and profit-related remuneration would appear to
be the best solution, and indeed can often be found in practice.
In the financial crisis: intensive discussion about whether such incentive contracts might
have led to excessive risk-taking by bank managers. Current state of discussion: yes,
certainly, but this was also in the interests of the bank shareholders. That means that
the correct answer is not to regulate remuneration, but to introduce much stricter equity
capital rules for banks.
Summary
1. Asymmetric information distribution can lead to an inefficient market alloca-
tion. Adverse selection means that asymmetric information exists prior to the
conclusion of a contract, and moral hazard is when this emerges only after (due
to) the conclusion of the contract.
2. Akerlof’s lemon market example illustrates the negative effects of adverse se-
lection. In the basic model the market for high-quality cars collapses due to the
simultaneous supply of lower-quality cars, as demanders have no information about
the product quality of each car.
3. The problem of adverse selection can be decreases by state regulation, separate con-
tracts, reputation-building or signal setting. Training and/or education, for exam-
ple, can send signals to potential employers, which contain additional information
about the quality of the labour being offered.
4. Inefficiencies that arise from incorrect behaviour in cases of moral hazard can be
mitigated by setting relevant incentives. The theory of optimal contracts
deals with the efficient arrangement of risk distribution and incentive structure.
202
VIII Theory of externalities
Literature for preparation and follow-up:
Pindyck/Rubinfeld, Chapter 18
The initial basic analysis of a market economy was carried out in a model world with perfect
competition (abandoned in Chapter VI), perfect information (abandoned in Chapter VII)
and without considering externalities. However, the existence of externalities is one of the
most important causes of market failure.
Until now we proceeded as if everyone bore the consequences of his own actions. If a
consumer, for example, consumed some more of a good, he bears completely the costs of
the additional consumption. If a producer expands supply, he must himself cover all the
costs of production.
Definition: Externalities exist when an agent does not bear the full consequences of his
actions, i.e. he pays (receives) no price for parts of his activities.
Examples of externalities:
1. Beekeeper ←→ Fruit grower
2. Environmental pollution
3. Smoking ←→ Passive smoking
4. Innovation capability due to a spill-over of knowledge
5. High risk of a bank leads to (systemic) risk and thus to risks for other banks/banking
systems
6. Competition between central banks (depreciation race)
Terminology
First a distinction can be made between a consumer externality (consumer bears the
consequence of the actions of an agent, for example smoking) and a producer externality
203
(producer bears the consequence of the actions of another agent, for example knowledge
spill-over). Furthermore there are positive externalities (beekeeper example or knowledge
spill-over) and negative externalities (for example smoking or environmental pollution).
Finally, pecuniary externalities (passed on via the price system) must be differentiated
from non-pecuniary or technological externalities (no effect on market prices). Below we
only refer to the non-pecuniary externalities. To identify an externality it is very helpful
to ask whether the effect concerned has an impact on price.
We first begin with the analysis of externalities and the inefficiency of the market system.
Then we shall address the possibilities to correct externalities.
VIII.1 Externalities and the inefficiency of the market mechanism
Until now we have been concerned primarily with an efficient market system, in which the
price mechanism acts as an invisible hand, leading to an optimal allocation of resources.
We will now demonstrate that the price system no longer works when externalities arise.
To this purpose we will use a simple model of a negative externality in the area of envi-
ronmental pollution.
We observe two companies that are both situated at a river. A chemicals company (C)
is located at the upper course of the river, and a fishery (F) is downstream. The revenue
of the fishery is affected negatively by the production activities of the chemicals company.
Assume that labour is the only production factor in both companies.
The production functions of each are:
Chemical company C = C(LC); ∂C/∂Lc > 0 (91)
Fishery F = F (LF , C); ∂F/∂LF > 0, ∂F/∂C < 0 (92)
The profit functions can then be formulated as follows:
Chemical company PC = pc · C(LC)− w · LC (93)
Fishery PF = pF · F (LF , C(LC))− w · LF (94)
204
1. Joint profit maximisation:
We first look at a fictional planning solution, which can be interpreted as an integrated
company:
PG = PC + PF
= pc · C(Lc)− w · LC + pF · F (LF , C(LC))− w · LF (95)
The first order conditions are:
∂PG
∂LC
= pC ·∂C
∂LC
− w + pF ·∂F
∂C· ∂C∂LC
!= 0 (96)
∂PG
∂LF
= pF ·∂F
∂LF
− w!= 0 (97)
This solution describes the total economic optimum and thus the optimal allocation of
resources.
2. Single economic optimum
If the two companies act separately, the first order conditions are:
∂PC
∂LC
= pC ·∂C
∂LC
− w!= 0 (98)
∂PF
∂LF
= pF ·∂F
∂LF
− w!= 0 (99)
3. Comparison of both solutions
We now compare the optimality conditions with joint optimisation, (96) and (97), with
the optimality conditions when optimisation is done separately, (98) and (99). As the
chemical production is assumed to have an externality on the fishery, the sign of ∂F∂C
is
negative. At the same time, the standard assumption of positive marginal productivity of
the factor of labour applies to the chemical production, i.e. ∂C∂LC
> 0. A comparison of the
205
optimality conditions (96) and (98) now shows the difference between the social and the
single economic optimum:
pF ·∂F
∂C
∂C
∂LC
< 0 (100)
Consequently, too much labour LC is used in the single economic optimum and therefore
too much is produced. The extent of the externality can also be derived from equation
(100). The following diagram illustrates again the connection described, based on the
marginal profit of the chemicals company.
-
6
Marginal profit
LC
.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........
........
........
........
........
6
?
pF∂F∂C
∂C∂LC
...................................................................................................................................................
(96) (98)
Economic intuition:
Negative externality means that the chemicals company (C) does not “internalise“ the
negative effect on the profits of the fishery (F ). From a total economic perspective, its
realised (perceived) costs are too low. The result is too much chemical production measured
against the socially optimal level of output.
The output of the fishery (F ) is also not optimal, despite the optically identical form of
the first order conditions, as ∂F (LF ,C)∂LF
is also dependent on the production activity of the
chemicals firm.
206
Similar considerations apply in the case of a positive externality. The result then, however,
is too little production.
Conclusion:
The existence of externalities leads to market failure, as the market solution is no longer
efficient. The economic reason for this is that there is no price for the externality (price of
clean water). The actual problem is that no water ownership rights have been defined.
VIII.2 Strategies to internalise externalities
A distinction can be made between the following fundamental strategy types:
1. State strategies (prohibitions or rules, taxes)
2. Market-based strategies
VIII.2.1 Prohibitions or laws and taxes
Let us consider a market with negative externalities. As in the example above, the marginal
costs of the supplier causing the externality are too low. Formulated more exactly, this
means that the private marginal costs are lower than the social ones: MCpriv < MCsoc. We
already know that this circumstance leads to a production level that is too high: Xp > X∗.
The following diagram once again illustrates the negative externality.
x∗ xp
-
6
p
x
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
-�
ext. effect
....................................................................................................................................................................................
D
D’
MCsoc MCpriv
207
Interventions by means of state prohibitions or laws:
If, for example, there are two companies causing the externality, each is bound by state
regulation to produce a maximum of only x∗/2. The disadvantage of this approach is that,
generally, production specified independently of the private company is not cost-minimising
and can thus lead to inefficiency.
x∗/2
-
6MC
x...................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
A
Comp. A
Illustration:
x∗/2 x∗
-
6MC
x........................................
........................................
........................................
........................................
........................................
........................................
........................................
........................................
..................
.......
.
.......
.
.......
.
.....
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
B C
Comp. B
The overall costs of production in this example represent the area A+B. However, if the
optimal social production quantity were to be produced by company B alone, the resulting
overall costs would be lower: B + C < A+B.
Interventions by means of Pigou taxes:
The basic idea of a Pigou tax is that a statutory price is set for a good that does not
have a market price. This passes the burden onto whoever caused the externality by his
economic activities.
208
-
6
p
x
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................................................................................................................................................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
x∗ xP
6
? Tax
...........................................................................................................................................
D
D’
MCsoc = MCpriv(t)
MCpriv
Graphical illustration:
The (Pigou) tax rate increases the private marginal costs of the company and thus leads
to an optimum social solution: Xp(t) = X∗. The advantage of a Pigou tax can be seen in
the cost-minimising allocation of production activity.
p− t
xA
-
6MC
x...................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........
At
Comp. A
xB
-
6MC
x.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.............................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
Bt
Comp. B
A disadvantage with imperfect information, however, is the insufficient accuracy of the
tax. In the case of externalities that are very strong, such as highly toxic substances,
prohibition might be better than a Pigou tax. An optimal policy usually consists of a
combination of taxes and prohibitions. This can be implemented by means of certificates
(especially environmental certification).
209
VIII.2.2 Market-based solution
We first looked at active interventions by the state in the market and price systems, which
might counteract externalities. An alternative solution to the externality problem goes
back one step further, to the cause of the problem: a lack of ownership rights.
The so-called Coase solution (Coase, R.H.: emeritus professor Univ. Chicago, 1991,
Nobel Prize for Economics) shows that the market itself can lead to an efficient allocation
when ownership rights are defined. This can be illustrated by the above example. If the
fishery had ownership rights to (clean) water, the chemicals firm would have to compensate
the fishery for the permission to pollute. If the chemical company owned the rights, the
fishery would have to pay the chemicals firm a settlement for the desired non-pollution.
Coase theorem
The allocation of ownership rights leads to a solution of the externality problem and to a
(Pareto) efficient allocation. If the allocation of ownership rights enables the existence of
a market (e.g. for pollution), this leads to an efficient allocation. The exact distribution
of the ownership rights is irrelevant from an allocation perspective (the same amount of
pollution occurs in both cases). Accordingly, the distribution of ownership rights merely
has distribution effects.
Illustration of the Coase theorem:
First we shall simplify our initial example: In the situation without ownership rights the
chemicals company has the marginal revenue function MR = MR(C) with MR′(C) <
0, i.e. it produces with falling marginal sales. In the single economic equilibrium the
produced quantity Cpriv is chosen, so that:
MR = pc∂C
∂LC
= MC,
with MC = const. > 0 being the private marginal costs.
In the ideal case (integrated company) the first order condition for the chemicals firm is,
in contrast:
MR = pc∂C
∂LC
= MC +MS,
210
with MS = const. > 0 being the marginal social costs (marginal damage to the fishery).
This results in C∗ as the optimum social production quantity. To illustrate this graphically:
MC
MS
MS +MC
C∗ Cpriv
-
6
C
...........................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
.......
.
•
•
•
•
A
E
D
F
0
pC∂C∂LC
pC∂C∂LC−MS
..............................................................................................................................................................................
Distribution of ownership rights
1. Ownership rights with the fishery - liability of the chemicals company
In order to be able to pollute (i.e. produce), the chemicals company must pay liability to
the fishery. Otherwise the chemicals company can be sued. For every unit produced, the
chemicals firm must pay compensation to the amount of the marginal damage (0A). Up to
a quantity of C∗ it is optimal to expand production. From the perspective of the chemicals
company, at C∗ marginal costs + compensation per unit (MS) =MR, which represents the
optimality condition with joint optimisation. The fishery receives payments to the amount
of the area 0ADC∗. The chemicals companys profit represents the area between MR and
MC +MS (up to C∗).
2. Ownership rights with the chemicals company
The fishery pays compensation to the chemicals company for the reduction of the chem-
icals production. The fishery is willing to pay marginal compensation to the amount of
the marginal damage (MD). The chemicals firm, for its part, is then willing to reduce
211
production from Cpriv to C∗: the following applies below (above) C∗:
MS +MC < (>)MR,
i.e. the savings from the marginal reduction in productions are smaller (greater) than the
reduction in sales. Altogether the fishery pays 0ADC∗ to the chemicals company.
Conclusion: In both cases a Pareto-efficient allocation occurs. The equilibrium thus
established is, from an allocation viewpoint, identical =⇒ normal distribution difference.
Critique
If the transaction costs of the negotiations are too high, no negotiated solution will be
found. Therefore, the Coase theorem is only relevant for markets in which there are very
few market participants. Further problems arise from the non-consideration of income
effects on the economic entities and the possible existence of asymmetric information.
Coase theorem in the narrowest sense
In the absence of transaction costs and income effects and with symmetrical information,
the awarding of ownership rights leads to a Pareto-efficient allocation. This allocation
result is independent of the distribution of the ownership rights.
Summary
1. Among the most important causes of market failure are externalities. These arise
when a market participant is not burdened fully with the consequences of his eco-
nomic activities by means of the price mechanism.
2. It can be shown by means of a comparison between the social and the single economic
optimization problems that, from a total economic perspective, externalities lead to
an inefficient allocation of resources. Where there are negative externalities,
there is a tendency to produce or consume too much, while positive externalities
lead to too little production or consumption.
212
3. Attempts can be made by means of state interventions to achieve the optimum
social allocation. This can be done either directly by means of state regulation
of production quantities (consumption quantities) or by means of so-called Pigou
taxes. The latter places the burden of the social costs of the externality on whoever
has caused them. The tax option generally leads to higher market efficiency, while
prohibitions or laws are more to the point.
4. Alternatively, the total economic efficient allocation can be achieved by specifying
ownership rights. This can create a market for externalities, thus involving the
advantage of the price mechanism.
5. In this context, the Coase theorem, in its narrow sense, says that the granting
of ownership rights leads to a Pareto-efficient allocation, as long as there are no
transaction costs, income effects or asymmetric information. In particular, the actual
distribution of the income rights is irrelevant for the allocation result.
213