ECON 510 - ALL

Chapter 1

Introduction

Overview

Economics and managerial decision making

Economics of a business

Review of economic terms

Learning objectives

define managerial economics

cite important types of resource allocation decisions

illustrate how economic changes affect a illustrate how economic changes affect a firm’s ability to earn an acceptable returnfirm’s ability to earn an acceptable return

apply to an individual firm the three basic apply to an individual firm the three basic questions faced by a countryquestions faced by a country


• Economics

The study of the behavior of humanbeings in producing, distributing andconsuming material goods andservices in a world of scarceresources


• Management

The science of organizing and allocating a firm’s scarce resources to achieve its desired objectives


• Managerial economics

The use of economic analysis to makebusiness decisions involving the best use(allocation) of an organization’s scarceresources

• Douglas - “Managerial economics is .. the application of economic principles and methodologies to the decision-making process within the firm or organization.”

• Pappas & Hirschey - “Managerial economics applies economic theory and methods to business and administrative decision-making.”

• Salvatore - “Managerial economics refers to the application of economic theory and the tools of analysis of decision science to examine how an organisation can achieve its objectives most effectively.”


Examples• How to use economic theory to set prices that maximize

profits.• How to use economic theory to choose the cost-minimizing

production technique for a given scale of output.• How to use economic theory to select the “optimal” location

for a new restaurant, grocery store, etc.• How to use economic theory to forecast near-term demand

for goods and services.

Examples of Management Decisions

1. What rates should Cingular charge for its wireless telephone service?2. How many iPods should Apple manufacture in the current quarter?3. Should Red Lobster locate a restaurant in Jonesboro?4. Should Time-Warner leave the cable TV business?5. Should Minneapolis build a new baseball facility for the Twins?6. Should Ohio Edison scrap its coal-fired power plants in favor of oil fired plants to

comply with regulatory controls on sulfur emissions? Or should it install expensive “scrubbing” equipment to its existing plants?

7. Should ASU charge differential tuition for business, nursing, and engineering courses?

8. Should the City of Jonesboro offer bus service?9. Should H & R Block outsource tax preparation to India?

Six Steps to Decision Making

1. Defining the Problem

2. Determining the Objective

3. Exploring the Alternatives

4. Predicting the consequences

5. Making a choice

6. Performing sensitivity analysis

Defining the problem

“I have a flat tire.”

Defining the objective

The spare won’t last long I gotta get this tire replaced!

Exploring the alternatives

• Go to Sears• Go to Wal-Mart

Predicting the consequences

• Wal-Mart is closer and cheaper.• Sears has a sale on Michelin radials.

Making a choice

I’m going to Wal-Mart

How can managerial economics assist decision-makings by firms?

• Adopt a general perspective, not a sample of one

• Simple models provide stepping stone to more complexity and realism

• Thinking logically has value itself and can expose sloppy thinking

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Why Managerial Economics?

• A powerful “analytical engine”.• A broader perspective on the firm.

• what is a firm?• what are the firm’s overall objectives?• what pressures drive the firm towards profit and

away from profit

• The basis for some of the more rigourous analysis of issues in Marketing and Strategic Management.


• Relationship to other business disciplines

Marketing: demand, price elasticity

Finance: capital budgeting, breakevenanalysis, opportunity cost, value added

Management science: linearprogramming, regression analysis,forecasting


• Relationship to other business disciplines

Strategy: types of competition,structure-conduct-performanceanalysis

Managerial accounting: relevantcost, breakeven analysis, incrementalcost analysis, opportunity cost


• Questions that managers must answer:

– What are the economic conditions in our particular market?• market structure?• supply and demand?• technology?



– What are the economic conditions in our particular market?• government regulations?• international dimensions?• future conditions?• macroeconomic factors?



– Should our firm be in this business?• if so, at what price?• and at what output level?



– How can we maintain a competitive advantage over other firms?• cost-leader?• product differentiation?• market niche?• outsourcing, alliances, mergers?• international perspective?



– What are the risks involved?• shifts in demand/supply conditions?• technological changes?• the effect of competition?• changing interest rates and inflation rates?



– What are the risks involved?• exchange rates (for companies in

international trade)?• political risk (for firms with foreign

operations)?

Risk is the chance that actual future outcomes will differ from those expected


• The economics of a business refers to the key factors that affect the firm’s ability to earn an acceptable rate of return on its owners’ investment

The most important of these factors are– competition– technology– customers

Economics of a business• Change: the four-stage model

– Stage I (the ‘good old days’)• market dominance• high profit margin• cost plus pricing

… changes in technology, competition, customers force firm into Stage II ..

Economics of a business• Change: the four-stage model

– Stage II (crisis)• cost management• downsizing• restructuring

… ‘re-engineering’ to deal with changes and move firm into Stage III ..


• Change: the four-stage model

– Stage III (reform)• revenue management• cost cutting has limited benefit

… focus on ‘top-line’ growth ..


• Change: the four-stage model

– Stage IV (recovery)• revenue plus

… revenue grows profitably

Economics of a business• Example: Avon

• well established company, in stage I until late 1970s• found itself in Stage II during 1980s• since mid 1990s, entered stage III• expanded into emerging markets and updated its image

Economics of a business• Example: Sears, Kmart

• Wal-Mart effect• Sears pushed down to number three in late 1980s … repositioned itself as a clothing store• Kmart filed for bankruptcy in 2002 … plan to acquire Sears


• Example: Kodak

• struggled to transition from chemical-based film to digital imaging• responded by developing strong cash flows in new product range


• Microeconomics is the study of individual consumers and producers in specific markets, especially:

• supply and demand• pricing of output• production process• cost structure• distribution of income


• Macroeconomics is the study of the aggregate economy, especially:

• national output (GDP)• unemployment• inflation• fiscal and monetary policies• trade and finance among nations


• Resources are inputs (factors) of production, notably:

• land• labor• capital• entrepreneurship (management skills)


• Scarcity is the condition in which resources are not available to satisfy all the needs and wants of a specified group of people

• Opportunity cost is the amount (or subjective value) that must be sacrificed in choosing one activity over the next best alternative


• Allocation decisions must be made because of scarcity. Three choices:

What should be produced?

How should it be produced?

For whom should be produced?


• Economic decisions of the Firm

What - begin or stop providing goods/services (production)How - hiring, staffing, capital budgeting (resourcing)For whom – target the customers most likely to purchase (marketing)


• Entrepreneurship is the willingness to take certain risks in the pursuit of goals

• Management is the ability to organize resources and administer tasks to achieve objectives

Global application

• Example: Western Union

• began over 100 years ago• huge changes in technology• to survive, the company branched out

Chapter 2

The Firm and Its Goals

The Firm and Its Goals

• The Firm• Economic Goal of the Firm• Goals Other Than Profit• Do Companies Maximize Profits?• Maximizing the Wealth of Stockholders• Economic Profits

Learning Objectives• Understand reasons for existence of firms and meaning

of transaction costs• Explain economic goals and optimal decision making• Describe meaning of “principal-agent” problem• Distinguish between “profit maximization” and

“shareholder wealth maximization”• Demonstrate usefulness of Market Value Added and

Economic Value Added

Theory of Firm• Why do firms exist?

– Coase (1937) • Transaction costs (Market vs. Firm): company compares costs of organizing an activity

internally with the cost of using the market system for its transaction.– Alchian & Demsetz (1972)

• Team production costs & monitoring• The firm emerges because extra output is provided by team production, but that the

success of this depends on being able to manage the team so that metering problems (it is costly to measure the marginal outputs of the co-operating inputs for reward purposes) and attendant shirking (the moral hazard problem) can be overcome, by estimating marginal productivity by observing or specifying input behavior.

– Schumpeter (1938)• Entrepreneur (lower opportunity costs)

Common Concern: Costs!!!Common Issue: Scarcity (Efficiency)

Coase (1937)

Transaction Costs

Why do Firms exist?

• Why is all economic activity not coordinated through markets?

• Firms exist because they offer cost advantages over market transactions- Transactions costs- Monitoring- Economies of scale or scope- Economies of team production

Why not Markets?

• Why is all economic activity not coordinated through organized firms (or just one giant firm)?

• Principal-agent and incentive problems• Problems of information and management

provide limits to firm size

The Firm’s Constraints• Available technology• Prices of inputs• Fixed capital in the short run• Degree of competition in the output market• Given these constraints the firm needs to choose

the method of production and output level that will maximize profit

• Maximizing profit implies minimizing the cost of production

The Firm

• A firm is a collection of resources that is transformed into products demanded by consumers.

• Profit is the difference between revenue received and costs incurred.

The Firm

• Transaction costs are incurred when entering into a contract.– Types of transaction costs

• Investigation• Negotiation• Enforcing contract and coordinating transactions

– Influences• Uncertainty• Frequency of recurrence• Asset specificity

The Firm

• Examples

Kodak – uses offshoring to source cameras IBM – manufacturing computers overseas Exult – third party services used in human

resources

The Firm

• Limits to Firm Size– tradeoff between external

transactions and the cost of internal operations

– Company chooses to allocate resources so total cost is minimum

– Outsourcing of peripheral, non-core activities

The Firm• Illustration: Coase and the Internet

• Ronald Coase wrote in 1937, pre-internet• but his ideas are still relevant today• tradeoff between internal costs and external

transactions• search costs

Economic Goal of the Firm

• Primary objective of the firm (to economists) is to maximize profits.– Profit maximization hypothesis– Other goals include market share, revenue

growth, and shareholder value• Optimal decision is the one that brings the

firm closest to its goal.

Economic Goal of the Firm

• Short-run vs. Long-run– Nothing to do directly with calendar time– Short-run: firm can vary amount of some

resources but not others (e.g. Labor, but not capital)

– Long-run: firm can vary amount of all resources– At times short-run profitability will be sacrificed

for long-run purposes

Goals Other Than Profit

• Economic Goals– Market share, Growth rate– Profit margin– Return on investment, Return on assets– Technological advancement– Customer satisfaction– Shareholder value

Goals Other Than Profit

• Non-economic Objectives– Good work environment– Quality products and services– Corporate citizenship, social responsibility

Do Companies Maximize Profit?

• Criticism: Companies do not maximize profits but instead their aim is to “satisfice.”– “Satisfice” is to achieve a set goal, even though

that goal may not require the firm to “do its best.”– Two components to “satisficing”:

• Position and power of stockholders• Position and power of professional management


• Position and power of stockholders– Medium-sized or large corporations are owned by

thousands of shareholders– Shareholders own only minute interests in the

firm– Shareholders diversify holdings in many firms– Shareholders are concerned with performance of

entire portfolio and not individual stocks.


• Position and power of stockholders– Most stockholders are not well informed on how

well a corporation can do and thus are not capable of determining the effectiveness of management.

– Not likely to take any action as long as they are earning a “satisfactory” return on their investment.


• Position and power of professional management– High-level managers who are responsible for

major decision making may own very little of the company’s stock.

– Managers tend to be more conservative because jobs will likely be safe if performance is steady, not spectacular.


• Position and power of professional management– Management incentives may be misaligned

• E.g. incentive for revenue growth, not profits• Managers may be more interested in maximizing own

income and perks– Divergence of objectives is known as “principal-

agent” problem or “agency problem”

Maximizing the Wealth of Stockholders

• Counter-arguments which support the profit maximization hypothesis.– Large number of shares is owned by institutions (mutual funds,

banks, etc.) utilizing analysts to judge the prospects of a company.

– Stock prices are a reflection of a company’s profitability. If managers do not seek to maximize profits, stock prices fall and firms are subject to takeover bids and proxy fights.

– The compensation of many executives is tied to stock price.


• Views the firm from the perspective of a stream of earnings over time, i.e., a cash flow.

• Must include the concept of the time value of money.– Dollars earned in the future are worth less than

dollars earned today.


• Future cash flows must be discounted to the present (time value of money).

• The discount rate (minimum required rate of return on investment) is affected by risk.

• Two major types of risk:– Business Risk– Financial Risk


• Business risk involves variation in returns due to the ups and downs of the economy, the industry, and the firm (economy is cyclical).

• All firms face business risk to varying degrees.


• Financial Risk concerns the variation in returns that is induced by leverage.

• Leverage is the proportion of a company financed by debt (debt to equity ratio).

• The higher the leverage, the greater the potential fluctuations in stockholder earnings.

• Financial risk is directly related to the degree of leverage.


• The present price of a firm’s stock should reflect the discounted value of the expected future cash flows to shareholders (dividends).

• P = present price of the stock• D = dividends received per year• K = discount rate• N = life of firm in years

nn

kD

kD

kD

kDP

)1()1()1()1( 33

221


• If the firm is assumed to have an infinitely long life, the price of a share of stock which earns a dividend D per year is determined by the equation:

P = D/k


• Given an infinitely lived firm whose dividends grow at a constant rate (g) each year, the equation for the stock price becomes:

P = D1/(k-g)where D1 is the dividend to be paid during the coming year.

• Multiplying P by the number of shares outstanding gives total value of firm’s common equity (market capitalization).


• A simple example Assumption: Dividend PMT1=$4

Dividend Growth Rate=5%

Stockholder’s minimum required rate of return=12% What is the value of the company’s stock? P=4/(0.12-0.05)=4/0.7=$57.14


• Company tries to manage its business in such a way that the dividends over time paid from its earnings and the risk incurred to bring about the stream of dividends always create the highest price for the company’s stock.

• When stock options are substantial part of executive compensation, management objectives tend to be more aligned with stockholder objectives.


• Another measure of the wealth of stockholders is called Market Value Added (MVA)®.

• MVA represents the difference between the market value of the company and the capital that the investors have paid into the company.


• Market value includes value of both equity and debt.

• Capital includes book value of equity and debt as well as certain adjustments.– E.g. Accumulated R&D and goodwill.

• While the market value of the company will always be positive, MVA may be positive or negative.


• Another measure of the wealth of stockholders is called Economic Value Added (EVA)®.– EVA=(Return on Total Capital – Cost of Capital) x

Total Capital

if EVA > 0 shareholder wealth rising if EVA < 0 shareholder wealth falling

Economic Profits

• Economic profits and accounting profits are typically different.– Accounting treatments allowed by GAAP (Generally

Accepted Accounting principles)

– Accountants report cost on historical basis.– Economists are more concerned with opportunity

costs or alternative costs.

Economic Profits• Historical costs vs. replacement costs• Implicit costs and normal profits

– Return required by scarce resources to remain committed to a particular firm

• An economist includes costs that would be excluded by an accountant.

• Economic costs include historical and explicit costs (accounting) as well as replacement and implicit costs (normal profits)

• Economic profits is total revenue minus all economic costs

80

Economic Profits

• Explicit Costs– Actual payments made to factors of

production and other suppliers• Implicit Costs

– All the opportunity costs of the resources supplied by the firm’s owners

• Eg: opportunity cost of owner’s time• Eg: opportunity cost of owner-invested funds

Three Types of Profit

• Accounting Profit – Total Revenue – Explicit Costs

• Economic Profit – Total Revenue – Explicit Costs – Implicit Costs

• Normal Profit– The difference between accounting profit and economic

profit– The opportunity cost of the resources – How much accounting profit is needed for econ profit to

be exactly = 0?• Economic Loss

– An economic profit less than zero

The Difference Between Accounting Profit and Economic Profit

The Difference Between Accounting Profit and Economic Profit

• Revenue – Acct Costs = Acct Profit• Revenue – Econ Costs = Econ Profit

• Revenue – Explicit Costs = Acct Profit• Revenue – (Explicit + Implicit costs) = Econ Profit

• Acct Profit – Implicit Costs = Econ Profit• If Acct Profit exactly = Implicit Costs => Econ Profit = 0, and the

firm is said to be earning a “normal profit”

Econ vs. Acct Profits

• True or False: Economic profits are always less than or equal to accounting profits.

TRUE• If some implicit costs exist economic cost >

accounting cost Economic profit < accounting profit

(ie: we are subtracting more costs from the same revenue)

Example

• After graduation from the UNVA with a degree in MBA, you face the following job choice:

• Option 1: IBM in RTPSalary = $50K/year

• Option 2: Suntan shop in Key LargoSalary = $15K/year

• If you choose option 2, you have to drain your $10,000 savings to start the business. Assume that you could have earned 10% on that money.

86

Example continued• Suppose you choose option 2… 1st year analysis:Revenue = $50,000 Costs of inventory = $8,000

Labor expenses = $15,000Rent = $12,000

• acounting economic- inventory - inventory- rent - rent- wages for worker - wages for worker

- opp cost of Labor ($50k)- opp cost of funds = $1,000

the normal rate of return on capital.

Example continued

• Accounting profit = 50 – 8 – 15 – 12 = 15

• Economic profit = 50 – 8 – 15 – 12 – 50 – 1

= -36• Earning less than a normal profit

– How much is a normal profit for this firm?

Global application

• Other countries, other cultures

• foreign currencies• legal differences• language• attitudes• role of government

Chapter 3

Supply and Demand

Supply and Demand

• Market Demand• Market Supply• Market Equilibrium• Comparative Statics Analysis

– Short-run Analysis– Long-run Analysis

• Supply, Demand, and Price

Learning Objectives• Define supply, demand, and equilibrium price.• List and provide specific examples of nonprice determinants of supply

and demand.• Distinguish between short-run rationing function and long-run guiding

function of price• Illustrate how concepts of supply and demand can be used to analyze

market conditions in which management decisions about price and allocations must be made.

• Use supply and demand diagrams to show how determinants of supply and demand interact to determine price in the short and long run

Market Demand

• Demand for a good or service is defined as quantities of a good or service that people are ready (willing and able) to buy at various prices within some given time period, other factors besides price held constant.

Market Demand

Market demand is the sum of all the individual demands.

Example demand for pizza:

Market Demand

The inverse relationship between price and the quantity demanded of a good or service is called the Law of Demand.

Market Demand

• Changes in price result in changes in the quantity demanded.– This is shown as movement along the demand

curve.• Changes in nonprice determinants result in

changes in demand.– This is shown as a shift in the demand curve.

Market Demand

• Nonprice determinants of demand– Tastes and preferences– Income– Prices of related products– Future expectations– Number of buyers

Market Supply

• The supply of a good or service is defined as quantities of a good or service that people are ready to sell at various prices within some given time period, other factors besides price held constant.

Market Supply

• Changes in price result in changes in the quantity supplied.– This is shown as movement along the supply

curve.• Changes in nonprice determinants result in

changes in supply.– This is shown as a shift in the supply curve.

Market Supply

• Nonprice determinants of supply– Costs and technology– Prices of other goods or services offered by the

seller– Future expectations– Number of sellers– Weather conditions

Market Equilibrium

• Equilibrium price: The price that equates the quantity demanded with the quantity supplied.

• Equilibrium quantity: The amount that people are willing to buy and sellers are willing to offer at the equilibrium price level.

Market Equilibrium

• Shortage: A market situation in which the quantity demanded exceeds the quantity supplied.– A shortage occurs at a price below the equilibrium

level.• Surplus: A market situation in which the

quantity supplied exceeds the quantity demanded.– A surplus occurs at a price above the equilibrium

level.

Market Equilibrium

Comparative Statics Analysis• A commonly used method in economic analysis: a form of sensitivity,

or what-if analysis

• Process of comparative statics analysis– State all the assumptions needed to construct the model.– Begin by assuming that the model is in equilibrium.– Introduce a change in the model. In so doing, a condition of

disequilibrium is created.– Find the new point at which equilibrium is restored.– Compare the new equilibrium point with the original one.

Comparative Statics: Example

Step 2• Begin the analysis in

equilibrium as shown by Q1 and P1.


Step 2• Begin the analysis in

equilibrium as shown by Q1 and P1.


Step 3• Assume that a new study

shows pizza to be the most nutritious of all fast foods.

• Consumers increase their demand for pizza as a result.


Step 4• The shift in demand

results in a new equilibrium price, P2 , and quantity, Q2.


Step 5• Comparing the new

equilibrium point with the original one we see that both equilibrium price and quantity have increased.

Comparative Statics Analysis

• The short run is the period of time in which:– Sellers already in the market respond to a change

in equilibrium price by adjusting variable inputs.– Buyers already in the market respond to changes

in equilibrium price by adjusting the quantity demanded for the good or service.


• The rationing function of price is the change in market price to eliminate the imbalance between quantities supplied and demanded.

Short-run Analysis

• An increase in demand causes equilibrium price and quantity to rise.

Short-run Analysis

• A decrease in demand causes equilibrium price and quantity to fall.

Short-run Analysis

• An increase in supply causes equilibrium price to fall and equilibrium quantity to rise.

Short-run Analysis

• A decrease in supply causes equilibrium price to rise and equilibrium quantity to fall.


• The long run is the period of time in which:– New sellers may enter a market– Existing sellers may exit from a market– Existing sellers may adjust fixed factors of

production– Buyers may react to a change in equilibrium price

by changing their tastes and preferences or buying preferences


• The guiding or allocating function of price is the movement of resources into or out of markets in response to a change in the equilibrium price.

Long-run Analysis• Initial change: decrease in

demand from D1 to D2

• Result: reduction in equilibrium price and quantity, now P2,Q2

• Follow-on adjustment:– movement of resources out

of the market– leftward shift in the supply

curve to S2– Equilibrium price and

quantity now P3,Q3

Long-run Analysis• Initial change: increase in

demand from D1 to D2

• Result: increase in equilibrium price and quantity, now P2,Q2

• Follow-on adjustment:– movement of resources into

the market– rightward shift in the supply

curve to S2– Equilibrium price and

quantity now P3,Q3

Supply, Demand, and Price:The Managerial Challenge

• In the extreme case, the forces of supply and demand are the sole determinants of the market price.– This type of market is “perfect competition”

• In other markets, individual firms can exert market power over their price because of their:– dominant size.– ability to differentiate their product through advertising, brand

name, features, or services

Supply, demand, and price:the managerial challenge

• Example: coffee

• ‘buy low, sell high’• 2000: overproduction led to price falls• 2004: prices moved up again• Starbucks effects

Supply, demand, and price:the managerial challenge

• Example: air travel

• ‘buy high, sell low’• industry deregulated in late 1970s• tight competition• post 9/11, a low-cost structure is needed

Global application

• Example: the market for cobalt

• rare metal• produced as a by-product• strategic item• prices rising

Chapter 4

Demand Elasticity

Demand Elasticity

• The Economic Concept of Elasticity• The Price Elasticity of Demand• The Cross-Elasticity of Demand• Income Elasticity• Other Elasticity Measures• Elasticity of Supply

Learning Objectives

• Define and measure elasticity• Apply concepts of price elasticity, cross-

elasticity, and income elasticity• Understand determinants of elasticity• Show how elasticity affects revenue

The Economic Concept of Elasticity

• The demand curve sloped downward to the right (the lowered the price, the greater the quantity demanded)

• Elasticity: the percentage change in one variable relative to a percentage change in another.

Bin changepercent Ain changepercent Elasticity oft Coefficien

The Price Elasticity of Demand

• A firm contemplating lowering its price to counteract new competition

• Price elasticity of demand: The percentage change in quantity demanded caused by a 1 percent change in price.

Price %Quantity %E

p

Measurement of Price Elasticity • Arc elasticity: Elasticity which is measured over a

discrete interval of a demand (or a supply) curve.

• Ep = Coefficient of arc price elasticity• Q1 = Original quantity demanded• Q2 = New quantity demanded• P1 = Original price• P2 = New price

2/)(2/)( 21

12

21

12

PPPP

QQQQEp

Example

• P1=11 P2=12 Q1=7 Q2=6

Then what is EP ?


• Point elasticity: Elasticity measured at a given point of a demand (or a supply) curve.

1

1

εP

PdQ xdP Q

=


The point elasticity of a linear demand function can be expressed as:

1

1

QP

PQ

p

Example

• Q=18-P at when P=$12 and Q=6then what is EP ?

• Q=100-P2 when P1=5, then Q=75 then what is EP ?


• Some demand curves have constant elasticity over the relevant range

• Such a curve would look like:Q = aP-b

where –b is the elasticity coefficient• This equation can be converted to linear by

expressing it in logarithms:log Q = log a – b(log P)


• Elasticity differs along a linear demand curve


• Categories of Elasticity– Relative elasticity of demand: EP > 1

– Relative inelasticity of demand: 0 < EP < 1

– Unitary elasticity of demand: EP = 1

– Perfect elasticity: EP = ∞

– Perfect inelasticity: EP = 0

Special Cases

P

D

D

Q0 0 Q

Infinitely (price) elastic Infinitely price inelastic


• Factors affecting demand elasticity– Ease of substitution– Proportion of total expenditures– Durability of product

• Possibility of postponing purchase• Possibility of repair• Used product market

– Length of time period


• Derived demand: The demand for products or factors that are not directly consumed, but go into the production of a final product.

• The demand for such a product or factor exists because there is demand for the final product.


• The derived demand curve will be more inelastic:– the more essential is the component in question.– the more inelastic is the demand curve for the

final product.– the smaller is the fraction of total cost going to

this component.– the more inelastic is the supply curve of

cooperating factors.

Example

• Consider demand for residential housing (the final product) and the derived demand for one class of labor employed in construction , electricians ( the demand for electricians does not exist for its own sake)– Assumptions: 1. can not build a house without electricians

2. the cost of electricians is a relatively small percentage of the entire cost of the house


• A long-run demand curve will generally be more elastic than a short-run curve.

• As the time period lengthens consumers find way to adjust to the price change, via substitution or shifting consumption


• There is a relationship between the price elasticity of demand and revenue received.– Because a demand curve is downward sloping, a

decrease in price will increase the quantity demanded

– If elasticity is greater than 1, the quantity effect is stronger than the price effect, and total revenue will increase


• As price decreases– Revenue rises when

demand is elastic.– Revenue falls when it is

inelastic.– Revenue reaches it peak

when elasticity of demand equals 1.


• Marginal Revenue: The change in total revenue resulting from changing quantity by one unit.

QuantityMR

Revenue Total


• For a straight-line demand curve the marginal revenue curve is twice as steep as the demand


• At the point where marginal revenue crosses the X-axis, the demand curve is unitary elastic and total revenue reaches a maximum.


• Some sample elasticities– Coffee: short run -0.2, long run -0.33– Kitchen and household appliances: -0.63– Meals at restaurants: -2.27– Airline travel in U.S.: -1.98– Beer: -0.84, Wine: -0.55

The Cross-Elasticity of Demand

• Cross-elasticity of demand: The percentage change in quantity consumed of one product as a result of a 1 percent change in the price of a related product.

B

AX P

QE

%%


• Arc Elasticity

2/)(2/)( 21

12

21

12

BB

BB

AA

AAx PP

PPQQ

QQE


• Point Elasticity

B

B

A

AX P

PQQE


• The sign of cross-elasticity for substitutes is positive.

• The sign of cross-elasticity for complements is negative.

• Two products are considered good substitutes or complements when the coefficient is larger than 0.5.

Income Elasticity

• Income Elasticity of Demand: The percentage change in quantity demanded caused by a 1 percent change in income.

• Y is shorthand for Income

YQEY

%%

Income Elasticity

• Arc Elasticity

2/)(2/)( 21

12

21

12

YYYY

QQQQEY

Other Elasticity Measures

• Categories of income elasticity– Superior goods: EY > 1

– Normal goods: 0 >EY >1– Inferior goods – demand

decreases as income increases: EY < 0

Elasticity of Supply

• Elasticity is encountered every time a change in some variable affects quantities.– Advertising expenditure– Interest rates– Population size


• Price Elasticity of Supply: The percentage change in quantity supplied as a result of a 1 percent change in price.

• If the supply curve slopes upward and to the right, the coefficient of supply elasticity is a positive number.

Price %SuppliedQuantity %E

S


• Arc elasticity

2/)(2/)( 21

12

21

12

PPPP

QQQQEs

Demand Elasticity

• When the supply curve is more elastic, the effect of a change in demand will be greater on quantity than on the price of the product.

• With a supply curve of low elasticity, a change in demand will have a greater effect on price than on quantity.

Global application

• Example: price elasticities in Asia

– imports almost always price inelastic– if exports price inelastic, export earnings will rise

as prices rise– if exports price elastic, export earnings will rise

with world incomes

Optimization using Calculus

We will review some rules of differential

calculus that are especially useful for

management decision making

Suppose that a business firm has estimated its profit () function (based on marketing and production studies) as follows:

6.31.02 2 QQ

Where is profit (in thousands of dollars) and Q is quantity (in thousands of units).

Thus the problem for management is to set its quantity(Q) at the level that maximizes profits ().

The profit function

The profit function shows the relationship between the manager’s decision variable (Q) and her objective (). That is why we call it the

objective function.

What is an objective function?

The Profit Function

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

Quantity

Prof

it

Marginal profit at a particular output is given by the slope of a line tangent to the profit function

The profit function again

Quantity Profit(000s) (000s)

0.0 -3.62.0 0.04.0 2.86.0 4.88.0 6.0

10.0 6.412.0 6.014.0 4.816.0 2.818.0 0.020.0 -3.6

Computing profit at various output levels using a

spreadsheet• Recall our profit function is given by:

= 2Q - .1Q2 – 3.6• Fill in the “quantity” column with 0, 2, 4, . . .• Assume that you typed zero in column cell a3 of

your spreadsheet• Place your cursor in the the cell b3 (it now contains

the bolded number –3.6)—just to the right of cell a3.

• Type the following in the formula bar:

=(2*a3)-(.1*a3^2)-3.6

and click on the check mark to the left of the formula bar.

• Now move your cursor to the southeast corner of cell b3 until you see a small cross (+).Now move your cursor down through cells b4, b5, b6 . . . to compute profit at various levels of output.

Rule 1: The derivative of a constant is zero.

Example:

Let Y = 7

Thus:

dy/dx = 0

y

X0

7

Rules of calculus

Rule 2: The derivative of a constant times a variable is simply the constant.

Example:

Let y = 13x

Thus:

dy/dx = 13

y

x

13

26

210Rule 2

1 naxndxdy

Rule 3: A power function has the form y = axn, where a and n are constants. The derivative of a power function is:

Example:

Let y = 4x3

Thus:

Rule 3

212/ xdxdy

Special cases of the power function

Note the following:

y =1/x2 is equivalently written as y = x-2

and

xy can be written y = x1/2

Hence by rule 3 (or the power rule), the respective derivatives are given by:

dy/dx = -2x-3

And

dy/dx = .5x-1/2 = x/5. Power functions

Rule 4: Suppose the product of two functions :

y = f(x)g(x). Then we have:

)()( fdxdgg

dxdf

dxdy

Example:

Let y = (4x)(3x2)

Thus:

dy/dx = (4)(3x2) + (6x)(4x) = 36x2

Rule 4

Rule 5 Rule 5: The derivative of the sum of functions is equal to the sum of the derivatives.

If y = f(x) + g(x), then:dy/dx = df/dx + dg/dx

Example: Let: y = .1x2 – 2x3

Thus: dy/dx = .2x – 6x2

Rule 6: Suppose y is a quotient: y = f(x)/g(x). Then we have:

2

))(/())(/(g

fdxdggdxdfdxdy

Example

Suppose we have: y = x/(8 + x)

Thus:

dy/dx = [1 • (8 + x) – 1 • (x)]/ (8 + x)2 = 8/(8 + x)2

Rule 6

The marginal profit (M) function

Let the profit function be given by: = 2Q - .1Q2 – 3.6To obtain the marginal profit function, we take the first derivative of profit with respect to output (Q): M = d/dQ = 2 - .2QTo solve for the output level that maximizes profits, set M =0. 2 - .2Q = 0 Thus: Q = 10

The second derivative

We know that the slope of the profit function is zero at its maximum point. So the first derivative of the

profit function with respect to Q will be zero at that

output. Problem is, how do we know we have a

maximum instead of a minimum?

-20

-15

-10

-5

0

5

10

15

0 2 4 6 8 10 12 14

Ouput (Thousands)

Prof

it (T

hous

ands

)

A more complicated profit function

1061.8.1 32 QQQ

Notice this function has a slope of zero at two levels of output

Taking the second derivative

1061.8.1 32 QQQ

Our profit function () is given by:

Now let’s derive the marginal profit (M )function

63.6.3/ 2 QQdQd

We can verify that M = 0 when Q = 2 and Q = 10.

Maximum or minimum?

Notice at the minimum point of the function, the slope is turning from zero to positive. Notice also at the maximum point, the slope is changing from

zero to negative

To insure a maximum, check to see that the second derivative is negative

To take the second derivative of the profit function:

dQdM

dQdQdd

dQd

)/(

2

2

Thus we have:

QdQ

QQddQd 6.6.3)3.6.3( 2

2

2

Hence, when Q = 2, we find that d2/dQ2 = 3.6 - .6(2) = 2.4

When Q = 10, we find that d2/dQ2 = 3.6 - .6(10) = -2.4

Marginal Revenue and Marginal Cost

Hence to find the profit maximizing output, set the first derivative of the revenue function equal to the first derivative of the cost functions

Marginal profit (M) is zero when marginal revenue (MR) is equal to marginal cost (MC), or alternatively, when MR – MC = 0.

Solving for the profit maximizing output

Let (Q) = R(Q) – C(Q),

where R is sales revenue and C is cost

Thus we have

0 MCMRdQdC

dQdR

dQd

Multivariable functions

Suppose we have a multivariate function such as the following: = f(P, A), where P is market price and A is the advertising budget. Our function has been estimated as follows:

PAAAPP 242220 22

We would like to know:•How sensitive are profits to a change in price, other things being equal (or ceteris paribus)? •How sensitive are profits to a change in the advertising budget, ceteris paribus?

Partial derivativesWe get the answer to

question 1 by taking the first partial derivative of with respect to P.

ApP

242

We can find the answer to question 2 by taking the first partial derivative of with respect to A:

PAA

224

Solving for the P and A that maximize

2 – 4P + 2A= 0

4 - 2A + 2P = 0

The solution is:P = 3 and A = 5

We know that profits will be maximized when the first partial derivatives are equal to zero. Hence, we set them equal to zero and obtain a linear equation system with 2

unknowns (P and A)

Constrained optimization

Examples:

•Maximize profits subject to the constraint that output is equal to or greater than some minimum level.

•Maximize output subject to the constraint that cost must be equal to or less than some maximum value.

So far we have looked at problems in which the decision maker maximizes

some variable () but faces no constraints. We call this “unconstrained optimization.” Often, however, we seek to maximize (or minimize) some variable

subject to one or more constraints.

Example 1

Suppose we are seeking to maximize the following profit function subject to the constraint that Q 7.

2440 QQ

What happens if we take the first derivative, set equal to zero, and solve for Q?

QdQd 840

Solving to maximize , we get Q = 5. But that violates our constraint.

A firm has a limited amount of output and must decide what quantities (Q1 and Q2) to sell in two different market segments. Suppose its profit () function is given by:

)40()5.20( 222

211 QQQQ

The firm’s output cannot exceed 25—that is, it seeks to maximize subject to Q 25. If we set the marginal profit functions equal to zero and solved for Q1 and Q2, we would get: Q1 = 20 and Q2 = 20, so that Q1 + Q2 =40. Again, this violates the constraint that total output cannot exceed 25.

Another example

Method of Lagrange Multipliers

This technique entails creating a new

variable (the Lagrange multiplier) for each constraint. We then determine optimal

values for each decision variable and

the Lagrange multiplier.

Lagrange technique: Example 1

Recall example 1 . Our constraint was given by Q = 7. We can restate this constraint as:

7 – Q = 0Our new variable will be denoted by z. Our Lagrange (L) function can be written:

)7(440)7( 2 QzQQQzL

Taking the partials of L with respect to Q and z

07

0840

QzL

zQQL

Solving for Q and z simultaneously, we obtain:

Q = 7 and z = -16

Now we just take the first partial derivative of L

with respect to Q and z, set them

equal to zero, and solve.

Interpretation of the Lagrange multiplier (z)

You may interpret the result that z = -16 as

follows: marginal profit (M) at the

constrained optimum output is –16 —that is, the last unit produced subtracted $16 for our

profit

Lagrange technique: Example 2

Recall example 2 . Our constraint was given by:Q1 + Q2 = 25

Our Lagrange (L) function can be written as :

)25()40()5.20( 212

222

11 QQzQQQQL

Taking the partials of L with respect to Q1, Q2 and z

025

0240

020

21

22

11

QQzL

zQQL

zQQL

The solutions are:Q1 = 10Q2 = 15z = 10

This time we take the first partial derivative of L with respect to Q1, Q2, and z, set

them equal to zero, and solve.

-6

-4

-2

0

2

4

6

8

0 2 4 6 8 10 12 14 16 18 20

Output (000s Units)

Prof

it (0

00s)

Chapter 5

Demand Estimation and Forecasting

Demand Estimationand Forecasting

• Regression Analysis• Problems in Use of Regression Analysis• Subjects of Forecasts• Prerequisites of a Good Forecast• Forecasting Techniques

Learning Objectives

• Specify components of a regression model that can be used to estimate a demand equation

• Interpret regression results• Explain meaning of R2

• Evaluate statistical significance of regression coefficients using t-test and statistical significance of R2 using F-test

Learning Objectives

• Recognize challenges of obtaining reliable cross-sectional and time series data on consumer behavior that can be used in regression models of demand

• Understand importance of forecasting in business

• Describe six different forecasting techniques

Learning Objectives

• Show how to carry out least squares projections and decompose them into trends, seasonal, cyclical, and irregular movements

• Explain basic smoothing methods of forecasting, such as moving average and exponential smoothing

198

The Scientific Method1. Identify the Question and Define Relevant Variables

2. Specify Assumptions

or

3. Formulate a hypothesis

4. Test the hypothesis

Reject the hypothesis Use the hypothesis until a better one shows up

Modify Approach

Estimation

• Estimation: an attempt to quantify the links between the level of demand for a product (dependant variable) and the variables (independent variables) which determine it

• E.g., the demand for a hotel room depending upon– Their price– Household incomes– The weather

Estimation of Demand• Objective: Learn how to estimate a demand function using regression analysis, and interpret the results

• A chief uncertainty for managers - what will happen to their product. – forecasting, prediction & estimation– need for data

Data Collection

• Data for studies pertaining to countries, regions, or industries are readily available and reliable.

• Data for analysis of specific product categories may be more difficult to obtain.– Buy from data providers (e.g. ACNielsen, IRI)– Perform a consumer survey– Focus groups– Technology: Point-of-sale, bar codes, RFID(radio frequency

identification)

Regression Analysis

• Regression Analysis: A procedure commonly used by economists to estimate consumer demand with available data.– Cross-Sectional Data: provide information on

variables for a given period of time.– Time Series Data: give information about variables

over a number of periods of time.

Regression Analysis

• Regression equation: linear, additive– Y = a + b1X1 + b2X2 + b3X3 + b4X4

• Y: dependent variable, amount to be determined• a: constant value, y-intercept• Xn: independent, explanatory variables, used to explain

the variation in the dependent variable• bn: regression coefficients (measure impact of

independent variables)

Regression Analysis

• Regression Results– Negative coefficient shows that as the independent variable (Xn)

changes, the quantity demanded changes in the opposite direction.

– Positive coefficient shows that as the independent variable (Xn) changes, the quantity demanded changes in the same direction.

– Magnitude of regression coefficients is measured by elasticity of each variable.

Simple Linear Regression

• Qt = a + b Pt + t

• time subscripts & error term• Find “best fitting” line

t = Qt - a - b Pt t

2= [Qt - a - b Pt] 2

• mint 2= [Qt - a - b Pt]2

• Solution: b = Cov(Q,P)/Var(P) and a = mean(Q) - b•mean(P)

_P

Q

_Q

OLS --ordinaryleastsquares

Statistical Estimation of the a Demand Function

• Steps to take:– Specify the variables -- formulate the demand model, select

a Functional Form• Linear Q = a + b•P + c•I• double log ln Q = a + b•ln P + c•ln Iln Q = a + b•ln P + c•ln I• quadratic Q = a + b•P + c•I+ d•P2

– Estimate the parameters --• determine which are statistically significant• try other variables & other functional forms

– Develop forecasts from the model

Estimation Process

Regression ModelRegression Modelyy = = 00 + + 11xx + +

Regression EquationRegression EquationEE((yy) = ) = 00 + + 11xx

Unknown ParametersUnknown Parameters00, , 11

Sample Data:Sample Data:x yx y

xx11 y y11. .. . . .. . xxnn yynn

bb00 and and bb11provide estimates ofprovide estimates of

00 and and 11

EstimatedEstimatedRegression EquationRegression Equation

Sample StatisticsSample Statistics

bb00, , bb11

0 1y b b x

Regression Analysis• Statistical evaluation of regression results

– t-test: test of statistical significance of each estimated regression coefficient

– b: estimated coefficient– SEb: standard error of the estimated coefficient– Rule of 2: if absolute value of t is greater than 2, estimated

coefficient is significant at the 5% level– If coefficient passes t-test, the variable has a true impact on

demand

bSE

b t

T-tests

• Different samples would yield different coefficients

• Test the hypothesis that coefficient equals zero– Ho: b = 0– Ha: b 0

• RULE: If absolute value of the estimated t > Critical-t, then REJECT Ho. – It’s significant.

• estimated t = (b - 0) / b• critical t

– Large Samples, critical t2• N > 30

– Small Samples, critical t is on Student’s t-table• D.F. = # observations, minus number of

independent variables, minus one.• N < 30

Regression Analysis

• Statistical evaluation of regression results– Coefficient of determination (R2): percentage of variation in the

dependent variable (Y) accounted for by variation in all explanatory variables (Xn)

• Value ranges from 0.0 to 1.0• Closer to 1.0, the greater the explanatory power of the

regression equation– F-test: measures statistical significance of the entire regression

as a whole (not each coefficient)

Coefficients of Determination: R2

• R-square -- % of variation in dependent variable that is explained

^

• Ratio of [Qt -Qt] 2 to [Qt - Qt] 2

• As more variables are included, R-square rises

• Adjusted R-square, however, can decline _

P

Q

_Q

Qt

Regression Analysis

• Steps for analyzing regression results– Check signs and magnitudes– Compute elasticity coefficients– Determine statistical significance

Regression Problems

• Identification Problem: The estimation of demand may produce biased results due to simultaneous shifting of supply and demand curves.

• Advanced estimation techniques, such as two-stage least squares and indirect least squares, are used to correct this problem.

Plot Historical Data

• Look at the relationship of price and quantity over time

• Plot it– Is it a demand curve

or a supply curve?– Problem -- not held

other things equal

quantity

Price

92

9794

93

96

98

95

D? or S?

Identification Problem

• Q = a + b P can appear upward or downward sloping.

• Suppose supply varies and demand is FIXED.

• All points lie on the demand curve quantity

P

S1

S2

S3

Demand

Suppose SUPPLY is Fixed

• Let DEMAND shift and supply be FIXED.

• All points are on the SUPPLY curve.

• We say that the SUPPLY curveis identified.

quantity

P

D1

D2

D3

Supply

When both Supply and Demand Vary

• Often both supply and demand vary.

• Equilibrium points are in shaded region.

• A regression of Q = a + b P will be neither a demand nor a supply curve. quantity

P

D1

D2

S1

S2

Regression results

• Example: estimating demand for pizza

– demand for pizza affected by 1. price of pizza 2. price of complement (soda) – managers can expect price decreases to lead to

lower revenue– tuition and location are not significant

Regression Problems• Multicollinearity: two or more independent variables are highly

correlated, thus it is difficult to separate the effect each has on the dependent variable.

• Passing the F-test as a whole, but failing the t-test for each coefficient is a sign that multicollinearity exists.

• A standard remedy is to drop one of the closely related independent variables from the regression.

• E.g., If current income changes in the same way as inflation over time, then we will not be able to separate their impact on current consumption

Problems• Autocorrelation: also known as serial correlation, occurs when the

dependent variable relates to the independent variable according to a certain pattern.

• Possible causes:– Effects on dependent variable exist that are not accounted for by the

independent variables.– The relationship may be non-linear

• The Durbin-Watson statistic is used to identify the presence of autocorrelation.

• To correct autocorrelation consider: – Transforming the data into a different order of magnitude– Introducing leading or lagging data

Forecasting

• Forecasting: an attempt to predict the level of sales at some future date

• Plan for future scenarios• Can be quantitative or qualitative

Subjects of Forecasts

• Gross Domestic Product (GDP)• Components of GDP

– E.g. consumption expenditure, producer durable equipment expenditure, residential construction

• Industry Forecasts– Sales of products across an industry

• Sales of a specific product

Prerequisites of a Good Forecast

• A good forecast should:– be consistent with other parts of the business– be based on knowledge of the relevant past– consider the economic and political environment

as well as changes– be timely

Forecasting Techniques

• Factors in choosing the right forecasting technique:– Item to be forecast– Interaction of the situation with the

characteristics of available forecasting methods– Amount of historical data available– Time allowed to prepare forecast


• Expert opinion• Opinion polls and market research• Surveys of spending plans• Economic indicators• Projections• Econometric models

Forecasting Techniques• Qualitative forecasting is based on judgments of

individuals or groups.• Quantitative forecasting utilizes significant amounts of

prior data as a basis for prediction.• Naïve forecasting projects past data without explaining

future trends.• Causal (or explanatory) forecasting attempts to explain

the functional relationships between the dependent variable and the independent variables.


• Expert opinion techniques– Jury of executive opinion: Forecasts generated by a group of

corporate executives assembled together. The major drawback is that persons with strong personalities may exercise disproportionate influence.

– The Delphi Method: A form of expert opinion forecasting that uses a series of questions and answers to obtain a consensus forecast, where experts do not meet.

Forecasting Techniques• Opinion polls: Sample populations are surveyed to

determine consumption trends.– may identify changes in trends– choice of sample is important– questions must be simple and clear

• Market research is closely related to opinion polling. – Market research will indicate “not only why the consumer is or

is not buying, but also who the consumer is, how he or she is using the product, and what characteristics the consumer thinks are most important in the purchasing decision.”

Forecasting Techniques• Surveys of spending plans: seek information about

“macro-type” data relating to the economy.1. Consumer intentions– Survey of Consumers, Survey Research Center, University of

Michigan– Consumer Confidence Survey, The Conference Board2. Inventories and sales expectations– A monthly survey published by the National Association of

Purchasing Agents with a large sample of purchasing executives


• Economic Indicators: A barometric method of forecasting designed to alert business to changes in economic conditions.– Leading, coincident, and lagging indicators– One indicator may not be very reliable, but a

composite of leading indicators may be used for prediction.

Forecasting Techniques• Leading Indicators predict changes in future economic activity

– Average hours, manufacturing– Initial claims for unemployment insurance– Manufacturers’ new orders for consumer goods and materials– Vendor performance, slower deliveries diffusion index– Manufacturers’ new orders, nondefense capital goods– Building permits, new private housing units– Stock prices, 500 common stocks– Money supply, M2– Interest rate spread, 10-year Treasury bonds minus federal funds– Index of consumer expectations

Forecasting Techniques• Coincident Indicators identify trends in current economic activities

– Employees on nonagricultural payrolls– Personal income less transfer payments– Industrial production– Manufacturing and trade sales

• Lagging Indicators confirm swings in past economic activities– Average duration of unemployment, weeks– Ratio, manufacturing and trade inventories to sales– Change in labor cost per unit of output, manufacturing (%)– Average prime rate charged by banks– Commercial and industrial loans outstanding– Ratio, consumer installment credit outstanding to personal income– Change in consumer price index for services


• General rule of thumb: if, after a period of increases, the leading indicator index sustains three consecutive declines, a recession (or a slowing) will follow.

• Economic indicators have predicted each recession since 1948.


• Economic Indicators Drawbacks– Leading indicator index has forecast a recession

when none ensued. – A change in the index does not indicate the

precise size of the decline or increase.– The data are subject to revision in the ensuing

months.


• Trend projections: A form of naïve forecasting that projects trends from past data without taking into consideration reasons for the change.– Compound growth rate– Visual time series projections– Least squares time series projection


• Compound growth rate: Forecasting by projecting the average growth rate of the past into the future. – Calculate the constant growth rate using available data, then

project this constant growth rate into the future. – Provides a relatively simple and timely forecast– Appropriate when the variable to be predicted increases at a

constant percentage


• General compound growth rate formula:

E = B(1+i)n

• E = final value• n = years in the series• B = beginning value• i = constant growth rate


• Visual Time Series Projections: plotting observations on a graph and viewing the shape of the data and any trends.


• Time series analysis: A naïve method of forecasting from past data by using least squares statistical methods.

• Data collected of a number of periods usually exhibit certain characteristics:– Trends– Cyclical fluctuations– Seasonal fluctuations– Irregular movements

– Trends: direction of movement of the data over relatively long period of time, either upward or downward

– Cyclical fluctuations: deviation from the trend due to general economic conditions

– Seasonal fluctuations: a pattern that repeats annually, e.g., Christmas

– Irregular movements: random occurrence of an event



• Time Series Analysis Advantages – easy to calculate– does not require much judgment or analytical skill– describes the best possible fit for past data– usually reasonably reliable in the short run


Yt = f(Tt, Ct, St, Rt)

• Yt = Actual value of the data at time t

• Tt = Trend component at t

• Ct = Cyclical component at t

• St = Seasonal component at t

• Rt = Random component at t

• Additive form: Yt = Tt + Ct + St + Rt

• Multiplicative form: Yt = (Tt)(Ct)(St)(Rt)


• Must decompose the time series into its four components– Remove seasonality– Compute trend– Isolate cycle– Cannot do anything with random component


• Seasonality: need to identify and remove seasonal factors, using moving averages to isolate those factors.

• Remove seasonality by dividing data by seasonal factor


• Trend Line: use least squares method• Possible best-fit line styles:

– Straight Line: Y = a + b(t)– Exponential Line: Y = abt

– Quadratic Line: Y = a + b(t) + c(t)2

• Choose style with a balance of high R2 and high t-statistics


• Cycle and Random Elements– Random factors cannot be predicted and should

be ignored– Isolate cycle by smoothing with a moving average


• Smoothing Techniques– Moving Average– Exponential Smoothing

• Work best when:– No strong trend in series (random variation)– Infrequent changes in direction of series– Fluctuations are random rather than seasonal or

cyclical

Forecasting Techniques• Moving Average: average of actual past results used to forecast one

period ahead

Et+1 = (Xt + Xt-1 + … + Xt-N+1)/N

• Et+1 : forecast for next period• Xt, Xt-1 : actual values at their respective times• N: number of observations included in average• E.g., twelve months moving average forecast for sales of a product for

March 2009 is the average of sales for the previous twelve month (March 2008 - Feb 2009)


• Exponential Smoothing: allows for decreasing importance of information in the more distant past, through geometric progression

Et+1 = w·Xt + (1-w) · Et

• w: weight assigned to an actual observation at period t


• Econometric Models: causal or explanatory models of forecasting– Regression analysis– Multiple equation systems

• Endogenous variables: comparable to dependent variables of single-equation model, but may influence other endogenous variables

• Exogenous variables: from outside the system, truly independent variables

Forecasting techniques• Example: econometric model

– Suits (1958) forecast demand for new automobiles ∆R = a0 + a1 ∆Y + a2 ∆P/M + a3 ∆S + a4 ∆X R = retail salesY = real disposable incomeP = real retail price of carsM = average credit termsS = existing stockX= dummy variable

Global application

• Example: forecasting exchange rates– The forward exchange rate is a predictor of a the

spot exchange rate, if today’s spot rate is $1.998 for 1 British Pound and the 90-forward rate is $1.989, then what?

– GDP– interest rates– inflation rates– balance of payments

Estimating Demand

Outline

•Where do demand functions come from?

•Sources of information for demand estimation

•Cross-sectional versus time series data

•Estimating a demand specification using the ordinary least squares (OLS) method.

•Goodness of fit statistics.

The goal of forecasting

To transform available data into equations that provide the best possible forecasts of economic variables—e.g., sales revenues and costs of production—that are crucial for management.

Demand for air travel Houston to Orlando

Q = 25 + 3Y + PO – 2P

Recall that our demand function was estimated as follows:

[4.1]

Where Q is the number of seats sold; Y is a regional income index; P0 is the fare charged by a rival airline, and P is the airline’s own fare.

Now we will explain how we estimated this

demand equation

Questions managers should ask about a forecasting equations

1. What is the “best” equation that can be obtained (estimated) from the available data?

2. What does the equation not explain?

3. What can be said about the likelihood and magnitude of forecast errors?

4. What are the profit consequences of forecast errors?

How do get the data to estimate demand forecasting equations?

•Customer surveys and interviews.

•Controlled market studies.

•Uncontrolled market data.

Campbell’s soup estimates demand functions from

data obtained from a survey of more than 100,000

consumers

Survey pitfalls

• Sample bias• Response bias • Response accuracy• Cost

Time -series data: historical data--i.e., the data sample consists of a series of daily, monthly, quarterly, or annual data for variables such as prices, income , employment , output , car sales, stock market indices, exchange rates, and so on.

Cross-sectional data: All observations in the sample are taken from the same point in time and represent different individual entities (such as households, houses, etc.)

Types of data

Year Month Day Won per Dollar1997 3 10 8771997 3 11 880.51997 3 12 879.51997 3 13 880.51997 3 14 881.51997 3 17 8821997 3 18 8851997 3 19 8871997 3 20 886.51997 3 21 8871997 3 24 8901997 3 25 891

Time series data: Daily observations, Korean Won per dollar

Student ID Sex Age Height Weight

777672431 M 21 6’1” 178 lbs.

231098765 M 28 5’11” 205 lbs.

111000111 F 19 5’8” 121 lbs.

898069845 F 22 5’4” 98 lbs.

000341234 M 20 6’2” 183 lbs

Example of cross sectional data

Estimating demand equations using regression analysis

Regression analysis is a statistical technique that allows us to quantify

the relationship between a dependent variable and one or

more independent or “explanatory” variables.

Y

X0

X and Y are notperfectly correlated.However, there is on average a positiverelationshipbetween Y and X

X1 X2

Regression theory

1

Y1

E(Y|X1)

Y

X0 X1

E(Y |Xi) = 0 + 1Xi

1 = Y1 - E(Y|X1)

We assume that expected conditional values

of Y associated with alternative values of X

fall on a line.

Our model is specified as follows:

Q = f (P) where Q is ticket sales and P is the fare

Specifying a single variable model

Q is the dependent variable—that is, we think that variations in Q can be

explained by variations in P, the “explanatory” variable.

ii PQ 10

0 and 1 are called parameters or population parameters.

We estimate these parameters using the data we have available

iii PQ 10

Estimating the single variable model

[1]

[2]

Since the datapoints are unlikely to fall

exactly on a line, (1)must be modified

to include a disturbanceterm (εi)

Estimated Simple Linear Regression Estimated Simple Linear Regression EquationEquation

The The estimated simple linear regression equationestimated simple linear regression equation

0 1y b b x

• is the estimated value of is the estimated value of yy for a given for a given xx value. value.y• bb11 is the slope of the line. is the slope of the line.• bb00 is the is the yy intercept of the line. intercept of the line.

• The graph is called the estimated regression line.The graph is called the estimated regression line.

Estimation Process

Regression ModelRegression Modelyy = = 00 + + 11xx + +

Regression EquationRegression EquationEE((yy) = ) = 00 + + 11xx

Unknown ParametersUnknown Parameters00, , 11

Sample Data:Sample Data:x yx yxx11 y y11. .. . . .. . xxnn yynn

bb00 and and bb11provide estimates ofprovide estimates of

00 and and 11

EstimatedEstimatedRegression EquationRegression Equation

Sample StatisticsSample Statistics

bb00, , bb11

0 1y b b x

Least Squares Method

• Least Squares Criterion

min (y yi i )2

where:where:yyii = = observedobserved value of the dependent variable value of the dependent variable for the for the iith observationth observation

^ yyii = = estimatedestimated value of the dependent variable value of the dependent variable for the for the iith observationth observation

• Slope for the Estimated Regression Equation

1 2( )( )

( )i i

i

x x y yb

x x

Least Squares Method

yy-Intercept for the Estimated Regression Equation-Intercept for the Estimated Regression Equation

Least Squares MethodLeast Squares Method

0 1b y b x

where:where:xxii = value of independent variable for = value of independent variable for iithth observationobservation

nn = total number of observations = total number of observations

__yy = mean value for dependent variable = mean value for dependent variable

__xx = mean value for independent variable = mean value for independent variable

yyii = value of dependent variable for = value of dependent variable for iithth observationobservation

Line of best fit

The line of best fit is the one that minimizes the squared sum of the vertical distances of the

sample points from the line

1. Specification

2. Estimation

3. Evaluation

4. Forecasting

The 4 steps of demand estimation using regression

Year and Average Number AverageQuarter Coach Seats Fare

97-1 64.8 25097-2 33.6 26597-3 37.8 26597-4 83.3 24098-1 111.7 23098-2 137.5 22598-3 109.6 22598-4 96.8 22099-1 59.5 23099-2 83.2 23599-3 90.5 24599-4 105.5 24000-1 75.7 25000-2 91.6 24000-3 112.7 24000-4 102.2 235

Mean 87.3 239.7Std. Dev. 27.9 13.1

Table 4-2Ticket Prices and Ticket Sales along an Air Route

Simple linear regression begins by plotting Q-P values on a scatter diagram to determine if there exists an approximate linear relationship:

Scatter plot diagram

Passengers

16014012010080604020

Fare

290

280

270

260

250

240

230

220

210

Scatter plot diagram with possible line of best fit

Average One-way Fare

7

6

5

4

3

2

$ 2 0

2 0

2 0

2 0

2 0

2 0

Demand curve: Q = 330- P

500 100 150Number of Seats Sold per Flight

Note that we use X to denote the explanatoryvariable and Y is the dependent variable.

So in our example Sales (Q) is the “Y” variable and Fares (P) is the “X” variable.

Q = Y

P = X

Computing the OLS estimators

We estimated the equation using the statistical software package SPSS. It generated the following output:

Coefficientsa

478.690 88.036 5.437 .000-1.633 .367 -.766 -4.453 .001

(Constant)FARE

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: PASSa.

Reading the SPSS Output

From this table we see that our estimate of 0 is 478.7 and our estimate of 1 is

–1.63.

Thus our forecasting equation is given by:

ii PQ 63.17.478ˆ

Step 3: Evaluation

Now we will evaluate the forecasting equation using standard goodness of fit statistics, including:

1. The standard errors of the estimates.

2. The t-statistics of the estimates of the coefficients.

3. The standard error of the regression (s)

4. The coefficient of determination (R2)

•We assume that the regression coefficients are normally distributed variables.

•The standard error (or standard deviation) of the estimates is a measure of the dispersion of the estimates around their mean value.

•As a general principle, the smaller the standard error, the better the estimates (in terms of yielding accurate forecasts of the dependent variable).

Standard errors of the estimates

The following rule-of-thumb is useful: The standard error of the regression coefficient should be less than half of the size of the corresponding regression coefficient.

2ˆˆ 11 ss

2

22ˆ1

i

i

xkne

s

Note that:

XXx ii

1sLet denote the standard error of our estimate of 1

Thus we have:

Where:

and

iii QQe ˆ

and

k is the number of estimated coefficients

Computing the standard error of 1

Coefficientsa

478.690 88.036 5.437 .000-1.633 .367 -.766 -4.453 .001

(Constant)FARE

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: PASSa.

By reference to the SPSS output, we see that the standard error of our estimate

of 1 is 0.367, whereas the (absolute value)our estimate of 1 is 1.63 Hence our estimate is about 4 ½

times the size of its standard error.

The SPSS output tells us that the t statistic for the the fare coefficient (P)

is –4.453 The t test is a wayof comparing the errorsuggested by the null

hypothesis to the standard error of the estimate.

To test for the significance of our estimate of 1, we set the following null hypothesis, H0, and the alternative hypothesis, H1

H0: 1 0

H1: 1 < 0

The t distribution is used to test for statistical significance of the estimate:

45.4049.0

063.1ˆ

1ˆ

11

s

t

The t test

The coefficient of determination, R2, is defined as the proportion of the total variation in the dependent variable (Y) "explained" by the regression of Y on the independent variable (X). The total variation in Y or the total sum of squares (TSS) is defined as:

n

i

i

n

i

i yYYTSS1

22

1

The explained variation in the dependent variable(Y) is called the regression sum of squares (RSS) and is given by:

n

i

i

n

i

i yYYRSS1

22

1

ˆˆ

Note: YYy ii

Coefficient of determination (R2)

What remains is the unexplained variation in the dependent variable or the error sum of squares (ESS)

n

i

i

n

i

i eYYESS1

22

1

ˆ

We can say the following:

•TSS = RSS + ESS, or

•Total variation = Explained variation + Unexplained variation

R2 is defined as:

n

i

i

n

i

i

n

i

i

n

i

i

y

e

y

y

RSSESS

TSSRSSR

1

2

1

2

1

2

1

2

2 1ˆ

1

We see from the SPSS model summary table that R2 for this model is .586

ANOVAb

6863.624 1 6863.624 19.826 .001a

4846.816 14 346.20111710.440 15

RegressionResidualTotal

Model1

Sum ofSquares df

MeanSquare F Sig.

Predictors: (Constant), FAREa.

Dependent Variable: PASSb.

Model Summary

.766a .586 .557 18.6065Model1

R R SquareAdjusted R

Square

Std. Errorof the

Estimate


Note that: 0 R2 1

If R2 = 0, all the sample points lie on a horizontal line or in a circle

If R2 = 1, the sample points all lie on the regression line

In our case, R2 0.586, meaning that 58.6 percent of the variation in the dependent variable (consumption) is explained by the regression.

Notes on R2

This is not a particularly good fit based on R2 since 41.4 percent of the variation in the dependent variable is unexplained.

The standard error of the regression (s) is given by:

kn

es

n

i

i

1

2

Standard error of the regression

The model summary tells us that s = 18.6Regression is based on the assumption that the error term is normally distributed, so that 68.7% of the actual values of the dependent variable (seats sold) should be within one standard error ($18.6 in our example) of their fitted value.

Also, 95.45% of the observed values of seats sold should be within 2 standard errors of their fitted values (37.2).

Model Summary

.766a .586 .557 18.6065Model1

R R SquareAdjusted R

Square

Std. Errorof the

Estimate


Step 4: Forecasting

ii PQ 63.17.478ˆ

Recall the equation obtained from the regression results is :

Our first step is to perform an “in-sample”

forecast.

At the most basic level, forecasting consists of inserting forecasted values of the explanatory variable P (fare) into the forecasting equation to obtain forecasted values of the dependent

variable Q (passenger seats sold).

Year and Predicted Actual Quarter Sales (Q*) Sales (Q) Q* - Q (Q* - Q)sq

97-1 64.8 70.44 5.64 31.8197-2 33.6 45.94 12.34 152.2897-3 37.8 45.94 8.14 66.2697-4 83.3 86.77 3.47 12.0498-1 111.7 103.1 -8.6 73.9698-2 137.5 111.26 -26.24 688.5498-3 109.6 111.26 1.66 2.7698-4 96.8 119.43 22.63 512.1299-1 59.5 103.1 43.6 1900.9699-2 83.2 94.94 11.74 137.8399-3 90.5 78.61 -11.89 141.3799-4 105.5 86.77 -18.73 350.8100-1 75.7 70.44 -5.26 27.6700-2 91.6 86.77 -4.83 23.3300-3 112.7 86.77 -25.93 672.3600-4 102.2 94.94 -7.26 52.71

Sum of Squared Errors 4846.80

In-Sample Forecast of Airline Sales

In-Sample Forecast of Airline Sales

Year/Quarter

00.300.199.399.198.398.197.397.1

Pass

enge

rs160

140

120

100

80

60

40

20

Actual

Fitted

Our ability to generate accurate forecasts of the dependent variable depends on two factors:

•Do we have good forecasts of the explanatory variable?

•Does our model exhibit structural stability, i.e., will the causal relationship between Q and P expressed in our forecasting equation hold up over time? After all, the estimated coefficients are average values for a specific time interval (1987-2001). While the past may be a serviceable guide to the future in the case of purely physical phenomena, the same principle does not necessarily hold in the realm of social phenomena (to which economy belongs).

Can we make a good forecast?

Single Variable Regression Using Excel

We will estimate an equation and use it to predict home prices in two cities. Our data set is on

the next slide

City Income Home Price

Akron, OH 74.1 114.9

Atlanta, GA 82.4 126.9

Birmingham, AL 71.2 130.9

Bismark, ND 62.8 92.8

Cleveland, OH 79.2 135.8

Columbia, SC 66.8 116.7

Denver, CO 82.6 161.9

Detroit, MI 85.3 145

Fort Lauderdale, FL 75.8 145.3

Hartford, CT 89.1 162.1

Lancaster, PA 75.2 125.9

Madison, WI 78.8 145.2

Naples, FL 100 173.6

Nashville, TN 77.3 125.9

Philadelphia, PA 87 151.5

Savannah, GA 67.8 108.1

Toledo, OH 71.2 101.1

Washington, DC 97.4 191.9

•Income (Y) is average family income in 2003

•Home Price (HP) is the average price of a new or existing home in 2003.

Model Specification

YbbHP 10

Scatter Diagram: Income and Home Prices

80

100

120

140

160

180

200

50 60 70 80 90 100 110

Income

Hom

e Pr

ices

Regression Statistics

Multiple R 0.906983447

R Square 0.822618973

Adjusted R Square 0.811532659

Standard Error 11.22878416

Observations 18

CoefficientsStandard

Error t Stat

Intercept -48.11037724 21.58459326 -2.228922114

Income 2.332504769 0.270780116 8.614017895

ANOVA

df SS

Regression 19355.71550

2

Residual 162017.36949

8

Total 17 11373.085

Excel Output

YHP 33.211.48

City Income Predicted HP

Meridian, MS 59,600 $ 138,819.89

Palo Alto, CA 121,000 $ 281,881.89

Equation and prediction

Chapter 6

The Theory and Estimation of Production

The Theory and Estimation of Production

The Production FunctionShort-Run Analysis of Total, Average, and

Marginal ProductLong-Run Production FunctionEstimation of Production FunctionsImportance of Production Functions in

Managerial Decision Making

Learning Objectives

Define production function and explain difference between short-run and long-run production function

Explain “law of diminishing returns” and how it relates to the Three Stages of Production

Define the Three Stages of Production and explain why a rational firm always tries to operate in Stage II

Learning Objectives

Provide examples of types of inputs that might go into a production function for a manufacturing or service company

Describe various forms of a production function that are used in statistical estimation of these functions

Briefly describe the Cobb-Douglas function and cite a few statistical studies that used this particular functional form in their analysis

The Production Function

Production function: defines the relationship between inputs and the maximum amount that can be produced within a given period of time with a given level of technology.

Mathematically, the production function can be expressed as

Q=f(X1, X2, ..., Xk) Q: level of output X1, X2, ..., Xk: inputs used in the production process


Key assumptionsSome given “state of the art” in the production

technology.Whatever input or input combinations are included in

a particular function, the output resulting from their utilization is at the maximum level.


For simplicity we will often consider a production function of two inputs:

Q=f(X, Y)Q: outputX: LaborY: Capital


The short-run production function shows the maximum quantity of good or service that can be produced by a set of inputs, assuming the amount of at least one of the inputs used remains unchanged.

The long-run production function shows the maximum quantity of good or service that can be produced by a set of inputs, assuming the firm is free to vary the amount of all the inputs being used.

Short-Run and Long-Run Production

In the short run some inputs are fixed and some variable e.g. the firm may be able to vary the

amount of labor, but cannot change the amount of capital

in the short run we can talk about factor productivity

In the long run all inputs become variable e.g. the long run is the period in which a firm

can adjust all inputs to changed conditions

in the long run we can talk about returns to scale (compare latter with economies of scale, which is a cost related concept)

Short-Run and Long-Run Production

Short-Run Changes in Production(Factor Productivity)

Units of KEmployed Output Quantity (Q)

8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8Units of L Employed

How much does the quantity of Q change, when the quantity of L is increased?

Short-Run Analysis of Total,Average, and Marginal Product

Alternative terms in reference to inputs Inputs Factors Factors of production Resources

Alternative terms in reference to outputs Output Quantity (Q) Total product (TP) Product


Marginal product (MP): change in output (or Total Product) resulting from a unit change in a variable input.

Average Product (AP): Total Product per unit of input used.

XQMPX

XQAPX


If MP > AP then AP is rising.

If MP < AP then AP is falling.

MP=AP when AP is maximized.

30

90

130161184196 Total Product

Q from hiring fourth worker

Q from hiring third worker

Q from hiring second worker

Q from hiring first worker

increasing marginal returns

diminishing marginal returns

Units of Output

Number of Workers62 3 4 51



Law of Diminishing Returns: As additional units of a variable input are combined with a fixed input, at some point the additional output (i.e., marginal product) starts to diminish. Nothing says when diminishing returns will start to take effect,

only that it will happen at some point. All inputs added to the production process are exactly the same

in individual productivity


The Three Stages of Production in the Short RunStage I: From zero units of the variable input to where

AP is maximized (where MP=AP)Stage II: From the maximum AP to where MP=0Stage III: From where MP=0 on

AP,MP

X

Stage I Stage II Stage III

APX

MPXFixed input grossly underutilized; specialization and teamwork cause AP to increase when additional X is used

Specialization and teamwork continue to result in greater output when additional X is used; fixed input being properly utilized

Fixed input capacity is reached; additional X causes output to fall



In the short run, rational firms should only be operating in Stage II.

Why not Stage III? Firm uses more variable inputs to produce less output

Why not Stage I? Underutilizing fixed capacity Can increase output per unit by increasing the amount of the

variable input


What level of input usage within Stage II is best for the firm?

The answer depends upon how many units of output the firm can sell, the price of the product, and the monetary costs of employing the variable input.


Total Revenue Product (TRP): market value of the firm’s output, computed by multiplying the total product by the market price. TRP = Q · P

Marginal Revenue Product (MRP): change in the firm’s TRP resulting from a unit change in the number of inputs used. MRP = = MP · P

XTRP


Total Labor Cost (TLC): total cost of using the variable input, labor, computed by multiplying the wage rate by the number of variable inputs employed. TLC = w · X

Marginal Labor Cost (MLC): change in total labor cost resulting from a unit change in the number of variable inputs used. Because the wage rate is assumed to be constant regardless of the number of inputs used, MLC is the same as the wage rate (w).


Summary of relationship between demand for output and demand for input A profit-maximizing firm operating in perfectly competitive

output and input markets will be using the optimal amount of an input at the point at which the monetary value of the input’s marginal product is equal to the additional cost of using that input.

MRP = MLC

Example 1

Table 7.6 Combining Marginal Revenue Product (MRP) with Marginal Labor Cost (MLC)Total Marginal Total Marginal

Labor Total Average Marginal Revenue Revenue Labor LaborUnit Product Product Product Product Product Cost Cost(X) (Q or TP) (AP) (MP) (TRP) (MRP) (TLC) (MLC) TRP-TLC MRP-MLC0 0 0 0 0 0 01 10000 10000 10000 20000 20000 10000 10000 10000 100002 25000 12500 15000 50000 30000 20000 10000 30000 200003 45000 15000 20000 90000 40000 30000 10000 60000 300004 60000 15000 15000 120000 30000 40000 10000 80000 200005 70000 14000 10000 140000 20000 50000 10000 90000 100006 75000 12500 5000 150000 10000 60000 10000 90000 07 78000 11143 3000 156000 6000 70000 10000 86000 -40008 80000 10000 2000 160000 4000 80000 10000 80000 -6000

Note: P = Product Price = $2W = Cost per unit of labor = $10000TRP = TP x P, MRP = MP x PTLC = X x WMLC = TLC / X


Multiple variable inputs Consider the relationship between the ratio of the marginal product of

one input and its cost to the ratio of the marginal product of the other input(s) and their cost.

e.g., country 1 w=$2 MP(L)=2 country 2 w=$4 MP(L)=6 Then, where to produce a product? Other factors may outweigh this relationship

Political/Economic risk factors`

k

k

wMP

wMP

wMP

2

2

1

1

The Long-Run Production Function

In the long run, a firm has enough time to change the amount of all its inputs.Effectively, all inputs are variable.

The long run production process is described by the concept of returns to scale.


If all inputs into the production process are doubled, three things can happen:output can more than double

increasing returns to scale (IRTS)output can exactly double

constant returns to scale (CRTS)output can less than double

decreasing returns to scale (DRTS)


One way to measure returns to scale is to use a coefficient of output elasticity:

If EQ > 1 then IRTS If EQ = 1 then CRTS If EQ < 1 then DRTS

inputsallinchangePercentageQinchangePercentage

QE


Returns to scale can also be described using the following equation

hQ = f(kX, kY)

If h > k then IRTSIf h = k then CRTSIf h < k then DRTS

Long-Run Changes in Production(Returns to Scale)


8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8Units of L Employed

How much does the quantity of Q change, when the quantity of both L and K is increased?


Graphically, the returns to scale concept can be illustrated using the following graphs.

Q

X,Y

IRTSQ

X,Y

CRTSQ

X,Y

DRTS

Estimation of Production Functions

Forms of Production Functions Short run: existence of a fixed factor to which is added a variable

factorOne variable, one fixed factorQ = f(L)K

Increasing marginal returns followed by decreasing marginal returnsCubic functionQ = a + bL + cL2 – dL3

Diminishing marginal returns, but no Stage IQuadratic functionQ = a + bL - cL2


Forms of Production FunctionsPower function

Q = aLb

If b > 1, MP increasingIf b = 1, MP constantIf b < 1, MP decreasingCan be transformed into a linear equation when

expressed in logarithmic terms logQ = loga + bLogL


Forms of Production Functions Cobb-Douglas Production Function: Q = aLbKc

Both capital and labor inputs must exist for Q to be a positive number

Can be increasing, decreasing, or constant returns to scale b + c > 1, IRTS b + c = 1, CRTS b + c < 1, DRTS

Permits us to investigate MP for any factor while holding all others constant

Elasticities of factors are equal to their exponents


Forms of Production Functions Cobb-Douglas Production Function

Can be estimated by linear regression analysisCan accommodate any number of independent variablesDoes not require that technology be held constantShortcomings:

Cannot show MP going through all three stages in one specification Cannot show a firm or industry passing through increasing, constant,

and decreasing returns to scale Specification of data to be used in empirical estimates`


Statistical Estimation of Production Functions Inputs should be measured as “flow” rather than

“stock” variables, which is not always possible.Usually, the most important input is labor.Most difficult input variable is capital.Must choose between time series and cross-sectional

analysis.


Aggregate Production Functions Many studies using Cobb-Douglas did not deal with individual

firms, rather with aggregations of industries or an economy. Gathering data for aggregate functions can be difficult.

For an economy: GDP could be usedFor an industry: data from Census of Manufactures or

production index from Federal Reserve BoardFor labor: data from Bureau of Labor Statistics

Importance of Production Functions in Managerial Decision Making

Production levels do not depend on how much a company wants to produce, but on how much its customers want to buy.

Capacity Planning: planning the amount of fixed inputs that will be used along with the variable inputs. Good capacity planning requires: Accurate forecasts of demand Effective communication between the production and marketing

functions

How to determine optimal combination of inputs in the long-run

• To illustrate this case, use “production isoquants”

An isoquant is a curve showing all possible combinations of inputs physically capable of producing a given fixed level of output

Example 2 Production Table


8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8Units of K Employedof L

IsoquantUnits of KEmployed

An Isoquant Curve

Graph of Isoquant

0

1

2

3

4

5

6

7

1 2 3 4 6 X

Y

Q=52

Substituting Inputs

There exists some degree of substitutability between inputs.

Different degrees of substitution:

Sugar

a) Perfect substitution b) Perfect complementarity

All other ingredients

Natural flavoring

Q

Q

Capital

Labor L1 L2 L3 L4

K1 K

2 K

3

K4

Cornsyrup

c) Imperfect substitution

Substituting Inputs (continued)

In case the two inputs are imperfectly substitutable, the optimal combination of inputs depends on the degree of substitutability and on the relative prices of the inputs

Substituting Inputs (continued)• The degree of imperfection in substitutability

is measured with marginal rate of technical substitution (MRTS):

MRTS = Y/X (in this MRTS some of L is removed from the production and substituted by K to maintain the same level of output)

Law of Diminishing Marginal Rate of Technical Substitution:

Table Input Combinationsfor Isoquant Q = 52Combination Y X

A 6 2B 4 3C 3 4D 2 6E 2 8

Y X MRTS -2 1 2 -1 1 1 -1 2 1/2 0 2

Law of Diminishing Marginal Rate of Technical Substitution (continued)

0

1

2

3

4

5

6

7

2 3 4 6 8 X

Y

X = 2Y = -1

X = 1Y = -1X = 1

Y =- 2

A

BC

D E

MRTS = Y/X = MPX /MPY

Units of YEmployed Output Quantity (Q)

8 37 60 83 96 107 117 127 1287 42 64 78 90 101 110 119 1206 37 52 64 73 82 90 97 1045 31 47 58 67 75 82 89 954 24 39 52 60 67 73 79 853 17 29 41 52 58 64 69 732 8 18 29 39 47 52 56 521 4 8 14 20 27 24 21 17

1 2 3 4 5 6 7 8Units of K Employedof X

MPX / MPY in Relation to MRTS (X for Y)Combination Q Y MPx X Mpy MRTS (L for K) MPL / MPK

A 52 6 2B 52 4 13 3 6.5 2 2C 52 3 11 4 11 1 1D 52 2 6.5 6 13 1/2 1/2

MRTS = Y/X = MPX /MPY

Importance of production functions in managerial decision making

• Example: cell phones

• Asian consumers want new phone every 6 months

• demand for 3G products• Nokia, Samsung, SonyEricsson must be speedy

and flexible (lean manufacturing)


• Example: Zara

• Spanish fashion retailer• factories located close to stores• quick response time to suggestion in 2-4 weeks

(competitors’ responding time in 4 to 12 months)


• Application: call centers

• service activity• production function is Q = f(X,Y) where Q = number of calls X = variable inputs Y = fixed input


• Application: China’s workers

• is China running out of workers?• industrial boom• e.g., bicycle factory in Guangdong Province

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