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

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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

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Persons

Prof. Dr. Uwe Walz

Professor of Industrial Organisation

Goethe University Frankfurt am Main

Email: uwalz@econ.uni-frankfurt.de

Teaching Assistant: Jan Krzyzanowski

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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

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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.

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• 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

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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

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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

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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?

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• 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.

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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

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-

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

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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

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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).

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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

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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

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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.

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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.

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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?

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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?

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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

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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

.......

......

.......

......

.......

......

.......

......

.......

......

.......

......

.......

......

.......

......

.......

......

.......

......

.......

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.......

......

.......

......

.......

......

.......

......

.......

......

.......

......

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.

.......

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.......

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.......

......

.......

......

.......

......

......

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....................................................................................................

.......

.

.......

.

.......

.

.......

.

.......

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.......

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...

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...............................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................∂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

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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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

.......

.

.......

.

.......

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.

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

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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............. ............. ............. ............. ............. ............. ............. .................................................................................................

............. ............. ............. ............. .........................................................................................................................................

•

•

•A

B

C

PS

CS

xS(p)

xD(p)

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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

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............. ............. ............. ............. ............. ............. ............. ...

............. ............. ............. ............. ............. .............

xS(pP )

xD(pP + t)

Consumer pays the tax

x′

p′P

p∗D

p′D

-

6

pD

x

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..............................................................................................................................................................................................................................................................................................................

6t

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............. ............. ............. ............. ............. .............

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

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.................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................

•

•

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

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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

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6

pD

x

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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

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=⇒ 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

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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

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................

................................................................

................................................................

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................................................................

.......

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............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ...........

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•

•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

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•

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x(p)

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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

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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•

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BC c′(x)

x(p)

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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

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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:

•

•

•

•

•

•

•

..........................................................................................................................................................................................................................................................................................................................................................................

.............................

.............................

.............................

.............................

.............................

...............

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.............................

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....................

................................................................................................................................................................

(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

.............................

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............................

.............................

............................

..

..............................................................................................................................................................................................................................................................................................................................................................................................................pA(pB)

pB(pA)

............. ............. ............. ............. ............. ........................................................................................

..................................

...........................................................................................................................................

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...........................................................................................................................................

...........................................................................................................................................

...........................

................................................................................................................

...........................

................................................................................................................

...........................

................................................................................................................

...........................

................................................................................................................

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

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6

CostsUtility

y.................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

C1(y)

B(y∗)

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.• •

Group 1

0 12 16-

6

CostsUtility

y..............................................................................................................................................................................................................................................................................................................................................

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C2(y)B(y∗)

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.

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.

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.• •

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

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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-�

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...................................................................................................................................................................................................................................................................................................................................................................................................................

.......

.

.......

.

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.

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.

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A

Comp. A

Illustration:

x∗/2 x∗

-

6MC

x........................................

........................................

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..................

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.......

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.....

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.

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.

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.

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.

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

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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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...................................................................................................................................................................................................................................................................................................................................................................................................................

.......

.

.......

.

.......

.

.......

.

.......

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.......

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.......

......... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........

At

Comp. A

xB

-

6MC

x.............................

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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

...........................................................................................................................................................................................................................................................................................................................................................................................................................................

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•

•

•

•

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