Managerial Economics

The Nature and Scopeof ManagerialEconomics

Thomas F. Rutherford

1.1

The Course Lectures Hirschey

Lecture 1The Nature and Scope of ManagerialEconomicsGetting Started with Economics

Managerial Economics

September 23, 2010

Thomas F. RutherfordCenter for Energy Policy and Economics

Department of Management, Technology and EconomicsETH Zürich



1.2


Introduction to Ideas

Let us begin with the ideas of economics in general and then link inmanagerial economics.

Economics is: “the study of how people allocate scarceresources.”

Managerial economics focuses on how managers allocate theirscarce resources:

• People• Skills• Office equipment• Warehouses• Machinery• Raw materials



1.2









1.2









1.2









1.3


Subjects in Business Administration

Courses dealing with the functions of a business:• Production (operations)• Human resources management• Marketing• Finance

There are separate courses for each of these areas, but managerial

economics is not in this list.



1.3


Subjects in Business Administration

Courses dealing with the functions of a business:• Production (operations)• Human resources management• Marketing• Finance





1.4


Courses Dealing with the Business Environment

• Ethics• Legal issues• International business• Information technology





1.4


Courses Dealing with the Business Environment

• Ethics• Legal issues• International business• Information technology





1.5


Courses Dealing with Methodology

• Quantitative methods• Decision theory and management science• Game theory

• Managerial economics



1.5


Courses Dealing with Methodology

• Quantitative methods• Decision theory and management science• Game theory• Managerial economics



1.6


What is the idea of managerial economics

Use economic princples to solve the problems which managersencounter when running their businesses:

• Tend to be more technical• Involves more mathematics and statistics than other courses.



1.7


Scope of Managerial Economics

What are the elements of managerial economics and how are theseinter-related?

• The core focus of ME is pricing (price theory),• But this view can be misleading (too narrow) – ME is generally

concerned with all aspects of firm operation which affect profit.



1.8


A Taxonomy

Joint dependence of demand and supply:

PRICING

/ \

DEMAND SUPPLY

Basic ideas here:• Supply is “cost theory”• Demand is “theory of the consumer”.



1.9


Other Ideas

• Supply is also affected by behavior of producers – includingcompeting firms (production theory and strategy).

• Theory of the firm (nature, objectives) interrelates game theoryand business strategy.

• Theory of markets concerns the nature of competition (how areprices and profits determined in different types of competitivesituations?



1.10


Diagrammatic Perspective

Government

|

----------------------------------------

| |

Theory --------- Pricing ------------ Competitionn

Firm / \ Theory

/ \

Demand Supply

| |

Consumer Production

Theory Theory



1.11


Lecture Sequence

1 Introductory concepts: scope and context, demand, supply andmarket equilibrium.

2 Demand theory and estimation (marketing)3 Cost and market structure (strategic decisions)4 Decision making with risk (investment under uncertainty)



1.12


Outline of Lectures

9/23 The nature and scope of managerial economics (MH 1,2)

9/30 Demand, supply and market equilibrium (MH 3, HW #1)

10/7 Budget constraints, preferences and consumer choice (MH 4)

10/14 Demand functions – price and income elasticities (HW # 2)

10/21 Demand estimation and forecasting (MH 5 and 6)

10/28 Case study: marketing (Professor Hoffman)

11/4 Firm level cost minimization (MH: 7 and 8)

11/11 Competitive markets (MH: 10 and 11; HW # 3)

11/18 Case study: firm-level decisions (Professor Hoffman)

11/25 Imperfectly competitive markets (MH 12 and 13)

12/2 Game theory and pricing (MH 14 and 15, HW # 4)

12/9 Risk and uncertainty (MH 16)

12/16 Case study: investment under uncertainty (ProfessorHoffman)

12/23 Final Examination Review



1.13


Assessment

• Homeworks (40%)• Final examination in early January, 2011 (60%).



1.14


Other information

Instructors: Thomas F. Rutherford and Volker HoffmanEmail: [email protected]

Office: ZUE (E7)Phone: 044 632 6359Office Hours: Wednesday mornings and by appointment.

Course Web Page:http://ethz.ch/cepe/education/managerialeconomics



1.15


Textbook

• The course will be based on the textbook Managerial Economics

(12th edition) by Mark Hirschey (MH).• Additional readings will be periodically assigned from Managerial

Economics: A Problem-Solving Approach (2nd edition) by Froeband McCann, South-Western.

• Copies of these texts are available for short-term loan from mysecretary Rina Fichtl, ZUE E8 ([email protected]).



1.16


Objectives of the Course

• Learn foundations of economics• Appreciate the role of economic ideas in managerial decisions.• Learn some formal models and methods of analysis in

economics and management science.



1.17


Key Ideas from Herschey Chapter 1

• How Is Managerial Economics Useful?• Theory of the Firm• Profit Measurement• Why Do Profits Vary among Firms?• Role of Business in Society



1.18


Key Concepts

managerial economics

theory of the firm

expected value maximization

value of the firm

present value

optimize

satisfice

business profit

normal rate of return

economic profit

profit margin

return on stockholders’ equity

frictional profit theory

monopoly profit theory

innovation profit theory

compensatory profit theory



1.19


How is Managerial Economics Useful?

• Evaluating Choice Alternatives• Identify ways to efficiently achieve goals.• Specify pricing and production strategies.• Spell out production and marketing rules to maximize profits.

• Making the Best Decision• Managerial economics helps meet management objectives

efficiently.• Managerial economics shows the logic of consumer, and

government decisions



1.20




1.21


Theory of the Firm

• Expected Value Maximization• Owner-managers maximize short-run profits.• Primary goal is long-term expected value maximization.

• Constraints and the Theory of the Firm• Resource constraints.• Social constraints.

• Limitations of the Theory of the Firm• Alternative theory adds perspective.• Competition forces efficiency.• Hostile takeovers threaten inefficient managers.



1.22


Measuring Profit

• Business Versus Economic Profit• Business (accounting) profit reflects explicit costs and revenues.• Economic profit.

• Profit above a risk-adjusted normal return.• Considers cash and noncash items.

• Variability of Business Profits• Business profits vary widely



1.23


Why Do Profits Vary Among Firms?

• Disequilibrium Profit Theories• Unexpected revenue growth.• Unexpected cost savings.

• Compensatory Profit Theories• Profits accrue to firms that are better, faster, or cheaper than the

competition.



1.24


Role of Business in Society

• Why Firms Exist• Businesses help satisfy consumer wants.• Businesses contributes to social welfare

• Social Responsibility of Business• Serve customers.• Provide employment opportunities.• Play by the rules (laws and regulations)



1.25


Froeb and McCann: Managerial Economics and Problem Solving

• Problem�solving�requires�two�steps:�First,�figure�out�why�mistakes�are�being�made;�and�then�figure�out�how�to�make�them�stop.�

• The�rationalͲactor�paradigm�assumes�that�people�act�rationally,�optimally,�and�selfͲinterestedly.�To�change�behavior,�you�have�to�change�incentives.

• Good�incentives�are�created�by�rewarding�good�performance.

• A�wellͲdesigned�organization�is�one�in�which�employee�incentives�are�aligned�with�organizational�goals.�By�this�we�mean�that�employees�have�enough�information�to�make�good�decisions,�and�the�incentive�to�do�so.�

• You�can�analyze�any�problem�by�asking�three�questions:�(1)�Who�is�making�the�bad�decision?;�(2)�Does�the�decision�maker�have�enough�information�to�make�a�good�decision?;�and�(3)�the�incentive�to�do�so?�

• Answers�to�these�questions�will�suggest�solutions�centered�on�(1)�letting�someone�else�make�the�decision,�someone�with�better�information�or�incentives;�(2)�giving�the�decision�maker�more�information;�or�(3)�changing�the�decision�maker’s�incentives.

Problem:�OverͲbidding�OVI�gas�tract

• A�young�geologist�was�preparing�a�bid�recommendation�for�an�oil�tract�in�the�Gulf�of�Mexico.�

• With�knowledge�of�the�productivity�of�neighboring�tracts�also�owned�by�company,�the�geologist�recommended�a�bid�of�$5�million.

• Senior�management,�though,�bid�$20�million�Ͳ far�over�the�next�highestͲbid�of�$750,000.

• What,�if�anything,�is�wrong?• The�goal�of�this�text�is�to�provide�tools�to�help�diagnose�and�solve�

problems�like�this.

2

Problem�solving

• Two�distinct�steps:• Figure�out�what’s�wrong,�i.e.,�why�the�bad�decision�was�made

• Figure�out�how�to�fix�it• Both�steps�require�a�model�of�behavior

• Why�are�people�making�mistakes?• What�can�we�do�to�make�them�change?

• Economists�use�the�rational�actor�paradigm�to�model�behavior.�The�rational�actor�paradigm�states:• People�act�rationally,�optimally,�selfͲinterestedly

• i.e.,�they�respond�to�incentives�– to�change�behavior�you�must�change�incentives.

3

How�to�figure�out�what�is�wrong

• Under�the�rational�actor�paradigm,�mistakes�are�made�for�one�of�two�reasons:�• lack�of�information�or

• bad�incentives.��

• To�diagnose�a�problem,�ask�3�questions:1.�Who�is�making�bad�decision?

2.�Do�they�have�enough�info�to�make�a�good�decision?

3.�Do�they�have�the�incentive�to�do�so?

4

How�to�fix�it

• The�answers�will�suggest�one�or�more�solutions:�1.�Let�someone�else�make�the�decision,�someone�with�better�

information�or�incentives.

2.�Change�the�information�flow.

3.�Change�incentives• Change�performance�evaluation�metric

• Change�reward�scheme

• Use�benefitͲcost�analysis�to�choose�the�best�(most�profitable?)�solution

5

Presenter

Discussion: Is the best solution the most profitable?

Keep�the�ultimate�goal�in�mind

For�a�business�or�organization�to�operate�profitably�and�efficiently�the�incentives�of�individuals�need�to�be�aligned�with�the�goals�of�the�company.�

• How�do�we�make�sure�employees�have�the�informationnecessary�to�make�good�decisions?

• And�the�incentive to�do�so?

6

Analyze�the�overͲbidding�mistake

• Another�clue:• After�winning�the�bid,�the�geologist�increased�the�estimated�

reserves�of�the�company.• But,�after�a�dry�well�was�drilled,�the�reserve�estimates�were�

decreased.• Senior�Management�stepped�in�and�ordered�an�increase�in�the�

reserve�estimate.• Last�clue:

• Senior�management�resigned�several�months�later.

7

Presenter

For the OVI story: (1) Senior Management made bad decision (2) They had enough information to make a good decision (3) The incentives provided did not support the good decision The answer is to change the incentives of senior management.

ANSWER:�Manager�bonuses�for�increasing�reserves

• The�bonus�system�created�incentives�to�overͲbid.�• Senior�managers�were�rewarded�for�acquiring�reserves�

regardless�of�their�profitability

• Bonuses�also�created�incentive�to�manipulate�the�reserve�estimate.

• Now�that�we�know�what�is�wrong,�how�do�we�fix�it?• Let�someone�else�decide?

• Change�information�flow?

• Change�incentives?• Performance�evaluation�metric

• Reward�scheme

8

Ethics

• Does�the�rationalͲactor�paradigm�encourage�selfͲinterested,�selfish�behavior?

• NO!• Opportunistic�behavior�is�a�fact�of�life.

• You�need�to�understand�it�in�order�to�control�it.

• The�rationalͲactor�paradigm�is�a�tool�for�analyzing�behavior,�not�a�prescription�for�how�to�live�your�life.

9

Economic Analysis ofCompetitive Markets


2.4

Lecture Overview Microeconomics Review Examples of applied price theory Building a Microeconomic ModelDemand and Supply Equilibrium Taxation Elasticities Consumer and Producer Surplus

Market equilibrium

• A market is in equilibrium when total quantity demanded bybuyers equals total quantity supplied by sellers.

• An equilibrium is supported by market prices.• At equilibrium prices, the market is made up of voluntary

participants.• Market prices reflect marginal willingness to accept (by firms)

and marginal willingness to pay (by consumers).



2.5


Willingness to Pay (=Marginal Value)

p

D(p)

Market Demand

q = D(p)



2.6


Willingness to Accept (=Marginal Cost)

p

S(p)

Market Supply

q = S(p)



2.7


Equilibrium

p

S(p),D(p)

Market Supply

q = S(p)

Market Demand

q = D(p)

p∗

q∗



2.8


Disequilibrium Price Above Equilibrium: Excess Supply

p

S(p),D(p)

Market Supply

q = S(p)

Market Demand

q = D(p)

p′

D(p′) S(p′)

p∗D(p′) < S(p′):Excess supply



2.9


Disequilibrium Price Below Equilibrium: Excess Demand

p

S(p),D(p)

Market Supply

q = S(p)

Market Demand

q = D(p)p′

D(p′) S(p′)

p∗D(p′) > S(p′):

Excess demand



2.10


Equilibrium in a Linear Model

An example of calculating a market equilibrium when the marketdemand and supply curves are linear:

D(p) = a− bp

S(p) = c + dp

Hence:a− bp∗ − c + dp∗

and the equilibrium price is:

p∗ =a− cb + d

and the equilibrum quantity is:

q∗ = D(p∗) = S(p∗) =ad + bcb + d



2.10




D(p) = a− bp

S(p) = c + dp



p∗ =a− cb + d


q∗ = D(p∗) = S(p∗) =ad + bcb + d



2.10




D(p) = a− bp

S(p) = c + dp



p∗ =a− cb + d


q∗ = D(p∗) = S(p∗) =ad + bcb + d



2.10




D(p) = a− bp

S(p) = c + dp



p∗ =a− cb + d


q∗ = D(p∗) = S(p∗) =ad + bcb + d



2.11


Inverse Demand and Supply

Willingness to pay can be characterized by representing price as aninverse function of quantity:

q = D(p) = a− bp ⇔ p =a− q

b= D−1(q)

and willingness to accept is likewise defined:

q = S(p) = c + dp ⇔ p =−c + q

d= S−1(q)



2.12


D−1(q)

S−1(q)

S(p),D(p)

Market Inverse SupplyS−1(q) = (−c + q)/d

D−1(q) = (a− q)/b

MarketInverseDemand

p∗

q∗



2.13


Dual Formulation

In equilibrium, we have firms supply to the point where market priceequals willingness to supply:

p = S−1(q) =−c + q

d

and households consume goods to the point where market priceequals willingness to pay:

p = D−1(q) =a− q

b= S−1(q) =

−c + qd

Hence, in equilibriumS−1(q) = D−1(q)

andq∗ =

ad + bcb + d

sop∗ = D−1(q∗) = S−1(q∗) =

a− cb + d



2.13


Dual Formulation


p = S−1(q) =−c + q

d


p = D−1(q) =a− q

b= S−1(q) =

−c + qd


andq∗ =

ad + bcb + d

sop∗ = D−1(q∗) = S−1(q∗) =

a− cb + d



2.13


Dual Formulation


p = S−1(q) =−c + q

d


p = D−1(q) =a− q

b= S−1(q) =

−c + qd


andq∗ =

ad + bcb + d

sop∗ = D−1(q∗) = S−1(q∗) =

a− cb + d



2.13


Dual Formulation


p = S−1(q) =−c + q

d


p = D−1(q) =a− q

b= S−1(q) =

−c + qd


andq∗ =

ad + bcb + d

so

p∗ = D−1(q∗) = S−1(q∗) =a− cb + d



2.13


Dual Formulation


p = S−1(q) =−c + q

d


p = D−1(q) =a− q

b= S−1(q) =

−c + qd


andq∗ =

ad + bcb + d

sop∗ = D−1(q∗) = S−1(q∗) =

a− cb + d



2.14


Special Case 1: Fixed Supply Quantity

Hal Varian, Intermediate Microeconomics – Norton



2.15


Fixed Supply Equilibrium

Supply is fixed (q∗ = c), hence price is determined by the inversedemand curve:

p∗ = D−1(q∗) =a− cb + d

Notice that this equilibrium outcome describes a situtation in whichfirms are unable to respond to changes in market price, as is quitecommon in short-run situations – particularly for energy markets inwhich changes to infrastructure require many years.



2.16


Special Case 2: Fixed Supply Price




2.17


Quantity Taxes

• A quantity tax levied at a rate of t is a tax of t CHF paid on eachunit traded.

• If the tax is levied at on sellers then it is an excise tax.• If the tax is levied on buyers then it is a sales tax.• When a tax is denominated in currency units, it is a specific tax.

When it is denominted as a percentage of the sales value, it isreferred to as an ad-valorem tax.



2.18


Quantity Taxes

Typical questions which arise concerning quantity taxes:• What is the effect of a quantity tax on a market’s equilibrium?• How are prices affected?• How is the quantity traded affected?• Who pays the tax?• How are gains-to-trade altered?



2.19


Market Equilibrium with Quantity Taxes

A tax rate t makes the price paid by buyers, pb, higher by t than theprice received by sellers, ps:

pb = ps − t

Even with a tax, market clear.I.e. quantity demanded by buyers at price pb must equal quantitysupplied by sellers at price ps:

D(pb) = S(ps)

The market equilibrium then involves two equations in two unknowns.Notice that these two conditions apply regardless of whether the taxis levied on sellers or on buyers. Hence, a sales tax rate $t has thesame effect as an excise tax rate $t.



2.19




pb = ps − t


D(pb) = S(ps)




2.19




pb = ps − t


D(pb) = S(ps)

The market equilibrium then involves two equations in two unknowns.

Notice that these two conditions apply regardless of whether the taxis levied on sellers or on buyers. Hence, a sales tax rate $t has thesame effect as an excise tax rate $t.



2.19




pb = ps − t


D(pb) = S(ps)




2.20


Geometry of Taxation




2.21


Geometry of Taxation




2.22


Equivalent Impacts of Sales and Excise Taxes




2.23


Tax Incidence

• Who pays the tax of $t per unit traded?• The division of the $t between buyers and sellers is the incidence

of the tax.



2.24


Algebra of Tax IncidenceEquilibrium conditions:

pb = ps + t

a− bpb = c + dps

Substitute for pb in the second equation:

a− b(ps + t) = c + dps

⇒ ps =a− c − bt

b + d.

Substitute into the demand or supply function to obtain:

qt =ad + bc − bdt

b + d

andpb = ps + t =

a− c + dtb + d

Note that as t → 0, pb → p∗, the equilibrium price without taxes, andqt → ad+bc

b+d



2.24



pb = ps + t

a− bpb = c + dps




b + d.


qt =ad + bc − bdt

b + d

andpb = ps + t =

a− c + dtb + d


b+d



2.24



pb = ps + t

a− bpb = c + dps




b + d.


qt =ad + bc − bdt

b + d

andpb = ps + t =

a− c + dtb + d


b+d



2.24



pb = ps + t

a− bpb = c + dps




b + d.


qt =ad + bc − bdt

b + d

andpb = ps + t =

a− c + dtb + d


b+d



2.25


Comparative Statics

ps =a− c − bt

b + d

qt =ad + bc − bdt

b + d

pb = ps + t =a− c + dt

b + d

As t increases:• ps falls,• pb rises,• qt falls.



2.25


Comparative Statics

ps =a− c − bt

b + d

qt =ad + bc − bdt

b + d

pb = ps + t =a− c + dt

b + dAs t increases:• ps falls,• pb rises,• qt falls.



2.26


Algebraic Incidence

ps =a− c − bt

b + d

qt =ad + bc − bdt

b + d

pb =a− c + dt

b + d

The tax paid per unit by the buyer is

pb − p∗ =a− c + dt

b + d− a− c

b + d=

dtb + d

The tax paid per unit by the seller is:

p∗ − ps =a− cb + d

− a− c − btb + d

=bt

b + d



2.26


Algebraic Incidence

ps =a− c − bt

b + d

qt =ad + bc − bdt

b + d

pb =a− c + dt

b + d

The tax paid per unit by the buyer is

pb − p∗ =a− c + dt

b + d− a− c

b + d=

dtb + d

The tax paid per unit by the seller is:

p∗ − ps =a− cb + d

− a− c − btb + d

=bt

b + d



2.27


Price Responses are Inversely Proportional to Elasticities

Demand response:

εD ≈∆qq∗

pb−p∗p∗

⇒ pb − p∗ ≈ ∆q × p∗

εD × q∗

Supply response:

εS =≈∆qq∗

ps−p∗p∗

⇒ ps − p∗ ≈ ∆q × p∗

εS × q∗



2.28


Tax Incidence and Relative Responsiveness

Define tax incidence as:I =

pb − p∗

p∗ − ps

where:

pb − p∗ ≈ ∆q × p∗

εD × q∗

ps − p∗ ≈ ∆q × p∗

εS × q∗

Hence

I =pb − p∗

p∗ − ps≈ − εS

εD



2.28


Tax Incidence and Relative Responsiveness

Define tax incidence as:I =

pb − p∗

p∗ − ps

where:

pb − p∗ ≈ ∆q × p∗

εD × q∗

ps − p∗ ≈ ∆q × p∗

εS × q∗

Hence

I =pb − p∗

p∗ − ps≈ − εS

εD



2.29


Tax Incidence with Perfect Elastic or Perfectly Inelastic Supply




2.30


Geometry of Tax Incidence




2.31


Tax Incidence and Responsiveness of Supply and Demand

• The fraction of a $t quantity tax paid by buyers rises as supplybecomes more own-price elastic or as demand becomes lessown-price elastic.

• When εD = 0 and εS > 0, buyers pay the entire tax, even thoughit is levied on the sellers.

• When εS = 0 and εD > 0, sellers pay the entire tax, even thoughit is levied on the buyers.



2.32


Deadweight Loss

A quantity tax imposed on a competitive market reduces the quantitytraded and so reduces gains-to-trade (i.e. the sum of Consumers’and Producers’ Surpluses).

The lost total surplus is the tax’s deadweight loss, or excess burden.



2.33


Tax Revenue

ps =a− c − bt

b + d

qt =ad + bc − bdt

b + d

pb =a− c + dt

b + d

The total tax is then

T = tqt = tad + bc − bdt

b + d

Note that this is a concave quadratic form. When bd > 0 there existsa tax rate, t∗ which maximizes T . For t > t∗, tax revenue decreaseswith the tax rate.



2.33


Tax Revenue

ps =a− c − bt

b + d

qt =ad + bc − bdt

b + d

pb =a− c + dt

b + d



b + d




2.33


Tax Revenue

ps =a− c − bt

b + d

qt =ad + bc − bdt

b + d

pb =a− c + dt

b + d



b + d




2.34


Tax Revenue




2.35


Consumer and Producer Surplus




2.36


A Tax Affects Both Consumer and Producer Surplus




2.37


Deadweight Loss Measures Value of Trades which Disappear




2.38


Deadweight Loss is Zero When Quantities are Fixed




2.39


Deadweight Loss and Own-Price Elasticities

• Deadweight loss due to a quantity tax rises as either marketdemand or market supply becomes more own-price elastic.

• If either εD = 0 or εS = 0 then the deadweight loss is zero.• Analysis of an economic policy proposal involves assessment of

both equity and efficiency. In the Marshallian model, equityimpacts are evaluated on the basis of either (i) surplus(consumer and producer) or (ii) tax incidence. Efficiency in theMarshallian model is assessed on the basis of the deadweightloss.

Market Forces: Demandand Supply


2a.1

Concepts

Lecture 2aMarket Forces: Demand and SupplyMarshallian Economics

Managerial EconomicsSeptember 30, 2011





2a.2

Concepts

Overview

I. Market Demand Curve- The Demand Function- Determinants of Demand- Consumer Surplus

II. Market Supply Curve- The Supply Function- Supply Shifters- Producer Surplus

III. Market EquilibriumIV. Price RestrictionsV. Comparative Statics



2a.3

Concepts

Market Demand Curve

• Shows the amount of a good that will be purchased at alternativeprices, holding other factors constant.

• Law of Demand• The demand curve is downward sloping.



2a.4

Concepts

Determinants of Demand

• Income• Normal good• Inferior good

• Prices of Related Goods• Prices of substitutes• Prices of complements

• Advertising and consumer tastes• Population• Consumer expectations



2a.5

Concepts

The Demand Function

• A general equation representing the demand curve

Qdx = f (Px ,Py ,M,H)

- Qdx = quantity demand of good X .

- Px = price of good X .- Py = price of a related good Y .

o Substitute good.o Complement good.

- M = income.o Normal good.o Inferior good.

- H = any other variable affecting demand.



2a.6

Concepts

Inverse Demand Function: “Willingness to pay”

- Price as a function of quantity demanded.- Example:

- Demand FunctionQd

x (p) = 10 � 2p

- Inverse Demand Function:

Px(q) = 5 � q/2



2a.7

Concepts

Change in Quantity Demanded



2a.8

Concepts

Change in Demand



2a.9

Concepts

Consumer Surplus

• The value consumers get from a good but do not have to pay for.• Consumer surplus will prove particularly useful in marketing and

other disciplines emphasizing strategies like value pricing andprice discrimination.



2a.10

Concepts

I got a great deal!

• That company offers a lot of bang for thebuck!

• Amazon provides good value.• Total value greatly exceeds total amount

paid.• Consumer surplus is large.



2a.11

Concepts

I got a lousy deal!

• That car dealer drives a hard bargain!• I almost decided not to buy it!• They tried to squeeze the very last cent from

me!• Total amount paid is close to total value.• Consumer surplus is low.



2a.12

Concepts

Consumer Surplus: Discrete Case



2a.13

Concepts

Consumer Surplus: Continuous Case



2a.14

Concepts

Market Supply Curve

• The supply curve shows the amount of a good that will beproduced at alternative prices.

• Law of Supply- The supply curve is upward sloping.



2a.15

Concepts

Supply Shifters

• Input prices• Technology or government regulations• Number of firms

- Entry- Exit

• Substitutes in production• Taxes

- Excise tax- Ad valorem tax

• Producer expectations



2a.16

Concepts

The Supply Function

• An equation representing the supply curve:

Qsx = f (Px ,Pr ,W ,H)

- Qsx = quantity supplied of good X .

- Px = price of good X .- Pr = price of a production substitute.- W = price of inputs (e.g., wages).- H = other factors affecting supply.



2a.17

Concepts

Inverse Supply Function: “Willingness to accept”

• Price as a function of quantity supplied.• Example:

- Supply FunctionQs

x = 10 + 2Px

- Inverse Supply Function:

Px(q) = 5 + 0.5q



2a.18

Concepts

Change in Supply Quantity



2a.19

Concepts

Change in Market Supply



2a.20

Concepts

Producer Surplus

• The amount producers receive in excess of the amountnecessary to induce them to produce the good.



2a.21

Concepts

Market Equilibrium

• The Price (P) that Balances supply anddemand

- QSx = Qd

x

- No shortage or surplus• Steady-state



2a.22

Concepts

If price is too low ...



2a.23

Concepts

If price is too high ...



2a.24

Concepts

Price Restrictions

• Price Ceilings- The maximum legal price that can be charged.- Examples:

o Gasoline prices in the 1970s.o Housing in New York City.o Proposed restrictions on ATM fees.

• Price Floors- The minimum legal price that can be charged.- Examples:

o Minimum wage.o Agricultural price supports.



2a.25

Concepts

Impact of a Price Ceiling



2a.26

Concepts

Full Economic Price

• The dollar amount paid to a firm under a price ceiling, plus thenon-pecuniary price.

PF = Pc + µ

- PF = full economic price- Pc = price ceiling- µ = nonpecuniary price



2a.27

Concepts

An Example from the 1970s

• Ceiling price of gasoline: $1.• 3 hours in line to buy 15 gallons of gasoline:

- Opportunity cost: $5/hr.- Total value of time spent in line: 3 ? $5 = $15.- Non-pecuniary price per gallon: $15/15 = $1.

• Full economic price of a gallon of gasoline: $1+$1=2.



2a.28

Concepts

Impact of a Price Floor



2a.29

Concepts

Comparative Static Analysis

• How do the equilibrium price and quantity change when adeterminant of supply and/or demand change?



2a.30

Concepts

Applications: Demand and Supply Analysis

• Event: The WSJ reports that the prices of PC components areexpected to fall by 5-8 percent over the next six months.

• Scenario 1: You manage a small firm that manufactures PCs.• Scenario 2: You manage a small software company.



2a.31

Concepts

Use Comparative Static Analysis to see the Big Picture!

• Comparative static analysis shows how the equilibrium price andquantity will change when a determinant of supply or demandchanges.



2a.32

Concepts

Scenario 1: Implications for a Small PC Maker

• Step 1: Look for the “Big Picture.”• Step 2: Organize an action plan (worry about details).



2a.33

Concepts

Big Picture: Impact of decline in component prices on PC market



2a.34

Concepts

Big Picture Analysis: PC Market

• Equilibrium price of PCs will fall, and equilibrium quantity ofcomputers sold will increase.

• Use this to organize an action plan:- contracts/suppliers?- inventories?- human resources?- marketing?- do I need quantitative estimates?



2a.35

Concepts

Scenario 2: Software Maker

• More complicated chain of reasoning to arrive at the “Big Picture.”• Step 1: Use analysis like that in Scenario 1 to deduce that lower

component prices will lead to- a lower equilibrium price for computers.- a greater number of computers sold.

• Step 2: How will these changes affect the “Big Picture” in thesoftware market?



2a.36

Concepts

Big Picture: Impact of lower PC prices on the software market



2a.37

Concepts

Big Picture Analysis: Software Market

• Software prices are likely to rise, and more software will be sold.• Use this to organize an action plan.



2a.38

Concepts

Conclusion

• Use supply and demand analysis to- clarify the “big picture” (the general impact of a current event on

equilibrium prices and quantities).- organize an action plan (needed changes in production,

inventories, raw materials, human resources, marketing plans,etc.).

A Market Model with

Excel


Lecture 2b.1

Lecture Lecture 2bA Market Model with ExcelManagerial Economics

September 30, 2011



A Market Model with

Excel


Lecture 2b.2

Calibrated Demand: Elasticity

The elasticity of demand (⌘D

> 0) is formally define as:

✏D

=% change quantity

% change price=

�Q

Q

D

�P

D

P

D

This elasticity is a local approximation of the responsiveness ofquantity to price. The elasticity characaterizes the slope of thedemand function at a given price level.

A Market Model with

Excel


Lecture 2b.3

Linear Demand

We can use ✏D

together with a reference price and reference quantity

calibrate a demand function. The linear demand model based onthese data can be written as:

Q

D

= Q

D

✓1 � |✏

D

|✓

P

D

P

D

� 1◆◆

in whichQ

d

is the reference demand quantityP

D

is the reference demand price

A Market Model with

Excel


Lecture 2b.4

Isoelastic Demand

The same input data can used to calibrate an isoelastic demandfunction:

Q

D

= Q

D

✓P

D

P

D

◆�|✏D

|

In the neighborhood of P

D

, these functions are identical, yet as pricesdepart from the reference point, the two functions may departsignificantly.

A Market Model with

Excel


Lecture 2b.5

Digression: The Revenue Function

The revenue function at a given price is defined as:

R(p) = pQ(p)

Irregardless of the value of ✏, revenue is a concave parabolic functionof price in the linear model. Let Q

⇤ denote the quantity for whichR(Q) is maximal. When ✏ < 1, maximal revenue occurs for Q

⇤ < Q.When ✏ > 1, Q

⇤ > Q.

A Market Model with

Excel


Lecture 2b.6

Revenue – Isoelastic Model

When ✏D

= 1, revenue is constant in the isoelastic model. Otherwise,when ✏ < 1, Q

⇤ # 0, and when ✏ > 1, Q

⇤ " 1.

A Market Model with

Excel


Lecture 2b.7

Revenue Calculation Worksheet

A Market Model with

Excel


Lecture 2b.8

Inelastic Demand

A Market Model with

Excel


Lecture 2b.9

Elastic Demand

A Market Model with

Excel


Lecture 2b.10

A Coal Market Market

1 Find data on base year production, consumption and prices ofcoal in a collection of countries which collectively representglobal coal supply and demand.

2 Calibrate a model to these data.3 Perform counterfactural analysis by applying excise taxes in a

subset of regions, corresponding to the Annex-B member states.4 Assume that coal supply is price elasticity (in the range of 1 to 2).5 Assume that coal demand is price in-elastic (in the range of 0.5).6 Evaluate the global leakage rate:

` =% increase in coal use in non-Annex B states

% decrease in coal use in Annex B states

7 Does the leakage rate exceed 100% as is claimed by somecritical of climate policy?

8 Remember that The most interesting answer to any question in

economics is: It depends.

A Market Model with

Excel


Lecture 2b.10



2 Calibrate a model to these data.

3 Perform counterfactural analysis by applying excise taxes in asubset of regions, corresponding to the Annex-B member states.

4 Assume that coal supply is price elasticity (in the range of 1 to 2).5 Assume that coal demand is price in-elastic (in the range of 0.5).6 Evaluate the global leakage rate:






A Market Model with

Excel


Lecture 2b.10




subset of regions, corresponding to the Annex-B member states.

4 Assume that coal supply is price elasticity (in the range of 1 to 2).5 Assume that coal demand is price in-elastic (in the range of 0.5).6 Evaluate the global leakage rate:






A Market Model with

Excel


Lecture 2b.10




subset of regions, corresponding to the Annex-B member states.4 Assume that coal supply is price elasticity (in the range of 1 to 2).

5 Assume that coal demand is price in-elastic (in the range of 0.5).6 Evaluate the global leakage rate:






A Market Model with

Excel


Lecture 2b.10




subset of regions, corresponding to the Annex-B member states.4 Assume that coal supply is price elasticity (in the range of 1 to 2).5 Assume that coal demand is price in-elastic (in the range of 0.5).

6 Evaluate the global leakage rate:






A Market Model with

Excel


Lecture 2b.10










A Market Model with

Excel


Lecture 2b.10










A Market Model with

Excel


Lecture 2b.10









economics is:

It depends.

A Market Model with

Excel


Lecture 2b.10










A Market Model with

Excel


Lecture 2b.11

Energy Data and Models

1 With the worldwide web, there are many data sources.

2 The data required for academic research is fundmentallydifferent than the data required by market participants, many ofwhom are trying to shave small price differences.

3 Data is not very valuable without a model.4 Economics offers several alternative approaches for modeling:

•Econometrics works with large quantities of data and often veryfew parametric assumptions.

•Calibrated microeconomic models begin with an explicit theory andrelatively few data are required.

• Econometrics can be concerned with measuring elasticities whilecalibrated policy analysis seeks to assess the policy implications ofa given set of benchmark data values and elasticity assumptions.

5 Highschool students and naive undergraduates are typicallypreoccupied with data. PhD students and professionresearchers are typically preoccpied with models.

A Market Model with

Excel


Lecture 2b.11


1 With the worldwide web, there are many data sources.2 The data required for academic research is fundmentally

different than the data required by market participants, many ofwhom are trying to shave small price differences.






A Market Model with

Excel


Lecture 2b.11




3 Data is not very valuable without a model.

4 Economics offers several alternative approaches for modeling:•

Econometrics works with large quantities of data and often veryfew parametric assumptions.




A Market Model with

Excel


Lecture 2b.11









A Market Model with

Excel


Lecture 2b.11









A Market Model with

Excel


Lecture 2b.11









A Market Model with

Excel


Lecture 2b.11









A Market Model with

Excel


Lecture 2b.12

Supply Elasticity

The elasticity of supply (⌘S

> 0) is formally defined as:

⌘S

=% change quantity

% change price=

�Q

Q

s

�P

s

P

s

The elasticity is a dimensionless representation of the slope of thesupply curve.

For calibrated policy analysis models, the elasticity of supply is amodel input. In many econometric exercies, the elasticity of supply isa model output.

A Market Model with

Excel


Lecture 2b.12

Supply Elasticity



⌘S

=% change quantity

% change price=

�Q

Q

s

�P

s

P

s



A Market Model with

Excel


Lecture 2b.12

Supply Elasticity



⌘S

=% change quantity

% change price=

�Q

Q

s

�P

s

P

s



A Market Model with

Excel


Lecture 2b.13

Calibrated Linear Supply Functions

In calibrated equilibrium models we can use a reference price,reference quantity and an elasticity of supply to define a linear supplyfunction. That is, we can write:

Q

s

= Q

s

✓1 + ⌘

S

✓P

s

P

s

� 1◆◆

where:Q

s

is the reference supply quantityP

s

is the reference supply price⌘

S

is the price elasticity of supply

Note that when P

s

= P

s

, Q

s

= Q

s

.

A Market Model with

Excel


Lecture 2b.14

Calibrated Iso-Elastic Supply Functions

A simple alternative to the linear model is the iso-elastic model :

Q

s

= Q

s

✓P

s

P

s

◆⌘S

A Market Model with

Excel


Lecture 2b.15

A Simple Model of the Global Coal Market

The basic structure of the model is summarized by the equation:X

r

S

r

(p) =X

r

D

r

(p, tr

)

in whichp is the world market price of coal

S

r

(p) is coal supply in region r .t

r

is the specific tax on coal in region r .D

r

(p, tr

) is coal demand in region r .The demand and supply functions employed in the model are linear,hence:

S

r

(p) = a

r

+ b

r

p

andD

r

(p, tr

) = ↵r

� �r

(p + t

r

)

A Market Model with

Excel


Lecture 2b.16

Implementation in Excel

• We will illustrate how this simple coal model can be implemented

in Excel.

• The model consists of an Excel worksheet with regional data.One cell in the sheet measures the equilibrium price.

• Model benchmark inputs include base year supply, demand andtax rates.

• Model econometric inputs include elasticities of supply anddemand in each of the regions.

• Model policy inputs include specific tax rates.• Model equilibrium is defined by a single variable: the

international coal price.• A model equilibrium determines supply and demand for each of

the regions.• A model equilibrium also determines the leakage rate.

A Market Model with

Excel


Lecture 2b.16



in Excel.• The model consists of an Excel worksheet with regional data.

One cell in the sheet measures the equilibrium price.

• Model benchmark inputs include base year supply, demand andtax rates.





A Market Model with

Excel


Lecture 2b.16




One cell in the sheet measures the equilibrium price.• Model benchmark inputs include base year supply, demand and

tax rates.





A Market Model with

Excel


Lecture 2b.16





tax rates.• Model econometric inputs include elasticities of supply and

demand in each of the regions.




A Market Model with

Excel


Lecture 2b.16






demand in each of the regions.• Model policy inputs include specific tax rates.

• Model equilibrium is defined by a single variable: theinternational coal price.

• A model equilibrium determines supply and demand for each ofthe regions.

• A model equilibrium also determines the leakage rate.

A Market Model with

Excel


Lecture 2b.16






demand in each of the regions.• Model policy inputs include specific tax rates.• Model equilibrium is defined by a single variable: the

international coal price.

• A model equilibrium determines supply and demand for each ofthe regions.

• A model equilibrium also determines the leakage rate.

A Market Model with

Excel


Lecture 2b.16






demand in each of the regions.• Model policy inputs include specific tax rates.• Model equilibrium is defined by a single variable: the



A Market Model with

Excel


Lecture 2b.17

The Model Worksheet

• The model worksheet is displayed below.

• The market price variable is specified in B6 which has theassigned range name “P”. This cell is used to define equilibriumdemand and supply values in columns G and H.

• The equilibrium values depend on the assigned policyparameters, consumption tax rates which appear in column F.The sum of squares market balance is defined as:

� =X

r

(Sr

� D

r

)2

This is displayed in cell B7.• If, for example, a tax rate is changed, then the model is out of

equilibrium and resulting imbalance is displayed in B7.

A Market Model with

Excel


Lecture 2b.17

The Model Worksheet

• The model worksheet is displayed below.• The market price variable is specified in B6 which has the

assigned range name “P”. This cell is used to define equilibriumdemand and supply values in columns G and H.


� =X

r

(Sr

� D

r

)2



A Market Model with

Excel


Lecture 2b.17

The Model Worksheet



• The equilibrium values depend on the assigned policyparameters, consumption tax rates which appear in column F.

The sum of squares market balance is defined as:

� =X

r

(Sr

� D

r

)2



A Market Model with

Excel


Lecture 2b.17

The Model Worksheet




� =X

r

(Sr

� D

r

)2

This is displayed in cell B7.

• If, for example, a tax rate is changed, then the model is out ofequilibrium and resulting imbalance is displayed in B7.

A Market Model with

Excel


Lecture 2b.17

The Model Worksheet




� =X

r

(Sr

� D

r

)2



A Market Model with

Excel


Lecture 2b.18

The Model

A Market Model with

Excel


Lecture 2b.19

The Excel Solver

• The model is solved using the Excel solver add-in (Tools >Solver ...). In order to use the model, you may need to add areference to the Solver VBA add-in functions.

• To use the solver we choose our target cell, the square marketimbalance, to be the "Target Cell" and choose the "Min" option(see Figure 2 below).

• Our only design variable is P, so the only cell we are going tochange is B6 (range name P).

• Having specified these items, we click on the Solve button. Themodel is solved instantaneously, and we are then presented witha dialogue box asking whether to accept the solution (Figure 3).

A Market Model with

Excel


Lecture 2b.19

The Excel Solver





A Market Model with

Excel


Lecture 2b.19

The Excel Solver





A Market Model with

Excel


Lecture 2b.19

The Excel Solver





A Market Model with

Excel


Lecture 2b.20

The Solver Dialogue

A Market Model with

Excel


Lecture 2b.21

Accepting a Solution

A Market Model with

Excel


Lecture 2b.22

What is the insight?

Supply and demand elasticities for coal are low, and leakage ratesrarely exceed 10% for any sort of climate policies currently underdiscussion.

A Market Model with

Excel


Lecture 2b.23

Quantitative DemandAnalysis


3a.1

Lecture 3aQuantitative Demand AnalysisElasticities and Estimation

Managerial EconomicsOctober 7, 2011





3a.2

Overview

I. The Elasticity Concept- Own Price Elasticity- Elasticity and Total Revenue- Cross-Price Elasticity- Income Elasticity

II. Demand Functions- Linear- Log-Linear

III. Regression Analysis



3a.3

The Elasticity Concept

• How responsive is variable G to a change in variable S

EG,S =%�G%�S

• If EG,S > 0, then S and G are directly related.• If EG,S < 0, then S and G are inversely related.• If EG,S = 0, then S and G are unrelated.



3a.4

Formal Definition of Elasticity

• An alternative way to measure the elasticity of a functionG = f (S) is

EG,S =dGdS

SG

• If EG,S > 0, then S and G are directly related.• If EG,S < 0, then S and G are inversely related.• If EG,S = 0, then S and G are unrelated.



3a.5

Own Price Elasticity of Demand

EQx ,Px =%�Qd

x

%�Px

• Should be negative according to the “law of demand.”• Elastic:

|EQx ,Px | > 1

• Inelastic:|EQx ,Px | < 1

• Unitary:|EQx ,Px | = 1



3a.6

Perfectly Elastic & Inelastic Demand



3a.7

Own-Price Elasticity and Total Revenue

• Elastic• Increase (a decrease) in price leads to a decrease (an increase) in

total revenue.• Inelastic

• Increase (a decrease) in price leads to an increase (a decrease) intotal revenue.

• Unitary• Total revenue is maximized at the point where demand is unitary

elastic.



3a.8

Elasticity, Total Revenue and Linear Demand



3a.9




3a.10




3a.11




3a.12




3a.13




3a.14




3a.15




3a.16

Demand, Marginal Revenue (MR) and Elasticity

• For a linear inverse demand function,

MR(Q) = a + 2bQ,

where b < 0.• When

• MR > 0, demand is elastic;• MR = 0, demand is unit elastic;• MR < 0, demand is inelastic.



3a.17

Factors Affecting the Own-Price Elasticity

• Available Substitutes• The more substitutes available for the good, the more elastic the

demand.• Time

• Demand tends to be more inelastic in the short term than in thelong term.

• Time allows consumers to seek out available substitutes.• Expenditure Share

• Goods that comprise a small share of consumer’s budgets tend tobe more inelastic than goods for which consumers spend a largeportion of their incomes.



3a.18

Cross-Price Elasticity of Demand

EQx ,Py =%�Qd

x

%�Py

If EQx ,Py > 0 then X and Y are substitutes.If EQx ,Py < 0 then X and Y are complements.



3a.19

Predicting Revenue Changes from Two Products

Suppose that a firm sells two related goods, X and Y . If the price ofX is change, then total revenue will change by:

�R =�RX (1 + EQx ,Px ) + RY EQy ,Px

�⇥%�PX



3a.20

Income Elasticity

EQx ,M =%�Qd

x

%�M

If EQx ,M > 0, then X is a normal good.If EQx ,M < 0, then X is a inferior good.



3a.21

Uses of Elasticities

• Pricing.• Managing cash flows.• Impact of changes in competitors’ prices.• Impact of economic booms and recessions.• Impact of advertising campaigns.• And lots more!



3a.22

Example 1: Pricing and Cash Flows

• According to an FTC Report by Michael Ward, AT&T’s own priceelasticity of demand for long distance services is -8.64.

• AT&T needs to boost revenues in order to meet it’s marketinggoals.

• To accomplish this goal, should AT&T raise or lower it’s price?



3a.23

Answer: Lower price!

• Since demand is elastic, a reduction in price will increasequantity demanded by a greater percentage than the pricedecline, resulting in more revenues for AT&T.



3a.24

Example 2: Quantifying the Change

• If AT&T lowered price by 3 percent, what would happen to thevolume of long distance telephone calls routed through AT&T?



3a.25

Answer: Calls Increase!

Calls would increase by 26 percent!

EQx ,Px = �8.64 =%�Qd

x

%�Px

�8.64 =%�Qd

x

�3%) %�Qd

x = 26%



3a.26

Example 3: Impact of a Change in a Competitor’s Price

• According to an FTC Report by Michael Ward, AT&T’s crossprice elasticity of demand for long distance services is 9.06.

• If competitors reduced their prices by 4 percent, what wouldhappen to the demand for AT&T services?



3a.27

Answer: AT&T’s Demand Falls!

AT&T’s demand would fall by 36 percent!

EQx ,Py = 9.06 =%�Qd

x

%�Py

9.06 =%�Qd

x

�4%) %�Qd

x = �36%



3a.28

Interpreting Demand Functions

• Mathematical representations of demand curves.• Example:

Qdx = 10 � 2Px + 3Py � 2M

• Law of demand holds (coefficient of Px is negative).• X and Y are substitutes (coefficient of Py is positive).• X is an inferior good (coefficient of M is negative).



3a.29

Linear Demand Functions and Elasticities

• General Linear Demand Function and Elasticities:

Qdx = ↵0 + ↵xPx + ↵y Py + ↵MM + ↵HH

• Own-Price Elasticity:

EQx ,Px = ↵xPx

Qx

• Cross-Price Elasticity:

EQx ,Py = ↵yPy

Qx

• Income Elasticity:

EQx ,M = ↵MMQx



3a.30

Example of Linear Demand

• Qd = 10 � 2P• Own-Price Elasticity: (�2)P/Q.• If P = 1, Q = 8 (since 10 � 2 = 8).• Own price elasticity at P = 1, Q = 8:

(�2)(1)/8 = �0.25

.



3a.31

Log-Linear Demand

• General Log-Linear Demand Function:

ln Qdx = �0 + �xPx + �y Py + �MM + �HH

• Own-Price Elasticity: �x

• Cross-Price Elasticity: �y

• Income Elasticity: �M



3a.32

Example of Log-Linear Demand

• ln(Qd ) = 10 � 2ln(P).

• Own Price Elasticity: �2.



3a.33

Graphical Representation of Linear and Log-Linear Demand



3a.34

Regression Analysis

• One use is for estimating demand functions.• Important terminology and concepts:

• Least Squares Regression model: Y = a + bX + e.• Least Squares Regression line: Y = a + bX• Confidence Intervals.• t-statistic.• R-square or Coefficient of Determination.• F-statistic.



3a.35

An Example

• We can use a spreadsheet to estimate the following log-lineardemand function.

Qx = �0 + �x ln Px + e



3a.36

Summary Output



3a.37

Interpreting the Regression Output

• The estimated log-linear demand function is:• ln(Qx) = 7.58 � 0.84ln(Px).• Own price elasticity: �0.84 (inelastic).

• How good is our estimate?• t-statistics of 5.29 and -2.80 indicate that the estimated coefficients

are statistically different from zero.• R-square of 0.17 indicates the ln(Px) variable explains only 17

percent of the variation in ln(Qx).• F-statistic significant at the 1 percent level.



3a.38

Conclusion

• Elasticities are tools you can use to quantify the impact ofchanges in prices, income, and advertising on sales andrevenues.

• Given market or survey data, regression analysis can be used toestimate:

• Demand functions.• Elasticities.• A host of other things, including cost functions.

• Managers can quantify the impact of changes in prices, income,advertising, etc.

Demand Estimation in

Economics


3b.1

Lecture 3bDemand Estimation in EconomicsIntro to Econometrics

Managerial Economics

October 7, 2011




Economics


3b.2

Econometric modeling

• Economists use two main type of statistical models to forecastand provide policy analysis.

1 Single-equation models study a variable of interest with a single(linear or non-linear) function of a number of explanatory variables.

2 In multiple or simultaneous equation models, the variable ofinterest is a function of several explanatory variables which arerelated to each other with a set of equations.

• Specific estimation techniques may be needed depending on thedata type:

1 A times series is a time-ordered (daily, weekly, . . . ) sequence ofdata (price, income, . . . ) which often requires special statisticaltreatment.

2 a cross section refers to data collected by observing manysubjects (individuals, firms or countries) at the same point in time.Its analysis usually consists of comparing the differences amongthe subjects.

• Here we provide some background on demand estimation andregression analysis in the context of a single-equation approach.


Economics


3b.3

Simple Linear Demand Estimation

• "Nobody employs expensive, time-consuming and complicateddemand estimation techniques when inexpensive and simplemethods work just fine.", Hirschey (2009, p.162).

•Example 1: Grasshopper (GZ), one of Zurich’s soccer teamsplaying in the Swiss Soccer Super League, offered CHF 5 off theCHF 20 regular price of reserved seats. Sales increased from6’000 to 7’000 seats per game. What is the demand for GZ’sgame tickets? Assuming a linear relationship:

Q = a + bP )(

6000 = a + b(20)7000 = a + b(15)

Solving for a and b gives the deterministic demand relationship:

Q = 10000 + -200P| {z }demand

or, equivanlently, P = 50 +�0.005Q| {z }inverse demand

(1)


Economics


3b.4

Price Elasticity of Demand

• From Example 1, we notice that the slope of the demand functionbeing negative, GZ’s games are a normal good!

• We can also compute the price elasticity resulting directly fromthe price change (arc elasticty):

Q1�Q0Q0

P1�P0P0

=7000�6000

600015�20

20= �2

3

• Note that in the context of a linear function, the arc elasticty isequal to the point elasticity:

@Q

@P

P0

Q0= �200

206000

= �23

• Economists usually plot the inverse demand, i.e., the pricevariable is on the y -axis. The inverse demand function is usefulin several contexts.

Inverse Linear Demand Function

• Economists usually plot demand functions with the price variableis on the y -axis and the quantities in the x-axis:


Economics


3b.6

Revenue-maximizing output level

• If the cost of producing an additional soccer game for GZ is fixed,we can use the inverse ticket demand function (1) to find therevenue-maximizing price level :

T = P ⇥ Q = (50 � 0.005Q| {z }inverse demand

) = 50Q � 0.005Q

2

• Let’s maximize T with respect to Q:

FOC: (@T/@Q) = 0 ) 50 � 0.01Q = 0 ) Q

⇤ = 5000SOC: (@2

T/@Q

2) < 0 ) �0.01 < 0 ) Q

⇤ is max!Price at Q

⇤: P

⇤ = 50 � 0.005(5000) = 25

• Verify on slide 3 that at P = 20, T = 120000. Reducing the priceto P = 15 (-25%) increased the ticket sales in a lower proportion(+16.6%) to 6000. Therefore, T dropped to 105000. Setting theprice to 25 could have generated 125000 in ticket revenues.

• Would " P have been judicious for GZ? Well, less costumers(-1000) means less high margin products (sodas, beers,burgers,. . . ) sold!

Identification Problem

Estimating demand relations can be complicated because of the interplaybetween demand and supply.

The dashed AB line is not a demand. Advanced statistical techniques arerequired to identify demand in that case.

C. Ordas Criado (CEPE-ETH) Managerial Economics (Fall 2010) Demand Estimation 9 / 40

Statistical Relation

A deterministic relation is an association between variables that is knownwith certainty.

Economic relationships are not deterministic in nature because they cannotbe predicted with absolute accuracy.

Real world economic data are rather of statistical type :


Linear Models

A statistical model in the context of demand estimation for good x coud beof the form :

Qx = a0 + a1Px + a2m + a3Py + ! (3)

Qx = b0Pb1x mb2Pb3

y e! (4)

where ! and " are random terms that follow some statistical distribution.

Equation (3) is clearly linear. Some nonlinear functions, such as (4), arelinear in the parameters. To see why, note that:

logQx! "# $

Qx

= log b0! "# $

"0

+b1 logPx! "# $

Px

+b2 logm! "# $

m

+b3 logPy! "# $

Py

+" !

Qx = #0 + b1Px + b2m + b3Py + " (5)

The parameters of model (4) could be estimated with the linear model (5).

The most popular technique to estimate the coe!cients of functional formswhich are linear in the parameters is linear regression.


Linear Regression

Linear regression consists in finding the best-fitting line that minimizes thesum of squared deviations between the regression line and the set oforiginal data points. This technique is also know as the Ordinary LeastSquares (OLS) method.


Ordinary Least Squares (OLS)

Consider the following multiple regression model:

yi = #0 + #1xi1 + . . .+ #pxip + ! (6)

with n observations (i = 1, 2, . . . , n), p explanatory variables and K = p + 1coe!cents (the #ps plus the intercept #0, where k = 0, 1, 2, . . . ,K ).

The OLS method finds the # parameters (called #) such that :

min"0,"1,...,"p

n%

i=1

(!i )2 =

n%

i=1

(yi " #0 " #1xi1 " . . ." #1xip)2 (7)

Problem (7) has a closed form and unique solution when the explanatoryvariables are linearly independent, i.e., no exact linear relationships existbetween two or more explanatory variables.

Most statistical softwares possess pre-implemented routines/functions toperform regression analysis (Excel, Matlab, R, SPSS, S-Plus, Stata, . . . )


Fundamental OLS Assumptions

Four fundamental assumptions are necessary to get unbiased estimates ofthe parameters and to carry statistical inference with a regression model:

1 the model is correctly specified, i.e., the relationship is linear in theregression parameters #.

2 each term !i comes from a normal distribution with mean 0 andconstant variance $2 and it is independent of each other;

3 the explanatory variables x1, x2, . . . , xp are nonrandom, measuredwithout errors and independent of each other and of the intercept;

4 the error !i is uncorrelated with the observations xip for all p.

These assumptions can be formally verified (out-of-scope of this lecture). Ifthey are plausible, you can interpret the regression results.


Correlation Coe!cient

The goodness of fit of the regression estimates must be evaluated beforeinterpreting the regression coe!cients.

The most straightforward measure is simply the correlation coe!cientbetween the y data and their fitted counterpart, called y :

R = cor(y , y) =

&

(yi " y)(yi " ¯y)2'&

(yi " y)2&

(yi " ¯yi )2(8)

where y is the mean of the yi s and ¯y is the mean of the fitted values (yi s).

Note that R # [0, 1]. The closer R is to 1, the better the fit.


Explained and Unexplained Variation of Regression Fits

Other important goodness of fit measures (the R2 and the F -statistic) relyon a decomposition of the variation of the dependent variable y into ‘total’,‘explained’ and ‘unexplained’ variation:

SST =n!

i=1

(yi ! y)2 sum of squared deviations in y " total variation (9)

SSR =n!

i=1

(yi ! y)2 sum of squares of regression " explained variation (10)

SSE =n!

i=1

(yi ! yi )2 sum of squared errors " unexplained variation (11)

where yi is the regression estimate of yi and y is the mean of the yi s.

It is not di!cult to show that SST = SSR + SSE .


Explained and Unexplained Variation of Regression Fits

Goodness-of-Fit of the Regression Line: R2

The R2 captures the proportion of total variation of the dependent variabley ‘explained’ by the full set of independent variables and it is defined as

R2 =SSR

SST= 1"

SSE

SST. (12)

The R2 in (12) is equal to the square of R in (8) only when regression (6)includes an intercept. The closer the R2 is to 1, the larger the share ofvariation explained by the model.

Note that adding explanatory variables to the regression never penalizes theR2.

The R2 can be compared across models as long as the y variable shares thesame units of measurement.


Goodness-of-Fit of the Regression Line: Adjusted R2

A downward-adjusted version of the R2, called adjusted R2, exists toaccount for the degrees of freedom, i.e. the number of observationsbeyond the minimum needed to calculate the regression statistic. Theadjusted R2 is

R2adj = 1"

SSE/(n " K )

SST/(n " 1)= 1"

n " 1

n " K(1" R2). (13)

Note that R2adj is not the share of total variance explained by the regression

model (it can be negative even in the presence of an intercept).

Preference should be given to the R2adj when comparing regression models

with di"erent number of predictors.

The closer R2adj to 1, the better the model.


Global Significance of the Regressors - the F -test

The F -statistic tells if the explanatory variables as a group explain astatistically significant share of the variation in the dependent variable :

F =SSR/K " 1

SSE/(n " K )=

MSR

MSE=

R2/(K " 1)

(1" R2/(n " K ))(14)

MSR = (SSR/K " 1) is also called Mean Squares of Regression andMSE = (SSE/n " K ) is the Mean Squared Errors. The term df 1 = K " 1corresponds to the numerator’s degrees of freedom while df 2 = n " K is thedenominator’s degrees of freedom.

Note that F $ 0. If R2 = 0, then F = 0 and y is statistically unrelated to xvariables.

Data series always display some (weak) statistical relationships.

How large should F be to ensure that at least some of the explanatoryvariables explain a statistically significant portion of the variation in y ?


Building the F -distribution

F is a random variable whose statistical distribution can be determinedunder some assumptions.

Recall from equation (14) that F depends on the fitted values of theregression model (through the SS terms) and on two di"erent numbers ofdegrees of freedom.

Under the assumptions that :

1 the regression errors are normally distributed (see slide 14),2 #1 = #2 = . . . = #p = 0 in regression (6),

we can get statistical distributions of F , called F -distributions, whichdepend on the two numbers of degrees of freedom.

Assumption (2) above is the null hypothesis under which the F -distributionis derived. It assumes that none of the explanatory variables x has asignificant relationship with y .

The F -distributions are in general highly skewed to the right and theybecome more symmetric as the sample size increases.


F-distributions

The F -distribution depends on the degrees of freedom.

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

1.2

F

Density

F(df1=1,df2=10,alpha=5%)F(df1=2,df2=10,alpha=5%)F(df1=5,df2=10,alpha=5%)F(df1=5,df2=100,alpha=5%)

F-statistics above the colored thresholds suggest significant contribution ofthe explanatory variables at the 5% significance level.

Thresholds for the F -test

The size of the F -statistic from equation (14) is then compared to theF -values derived under the null hypothesis. If the F -statistic lies far in theright tail of the F -distribution, the null hypothesis is unlikely to be true forthe investigated dataset.

Statisticians usually consider that a F -statistic which has only 5% chances(or lower) to be observed under the null hypothesis is su!cient evidence toreject the null hypothesis. This rejection level is called significance leveland it is noted %.

Statistical tables of F -distributions exist for di"erent % levels. Theyprovide critical F -values, noted F !

(df 1,df 2,#), for a large range of degrees of

freedom. They report P(F > F !

(df 1,df 2)) = %. To reject the null hypothesis

at the significance level of % = 5%, the following criteria must hold:

F > F !

(df 1,df 2,0.05) ! P(F ) < 0.05 (15)

The term P(F ) in (15), called the p-value of the F computed with (14),corresponds to the probability that a F -statistic at least as extreme as F isobserved under the null hypothesis. Both criteria in (15) are identical.


Precision of the Regression Coe!cients

Rejecting the null hypothesis of the F -test ensures that the regression’spredictors as a whole contribute to explain a statistically significant portionof the variation in the dependent variable y . We can then proceed toanalyze the relationship between each explanatory variable and y .

Before interpreting their sign and magnitude, the precision andreliability of each individual coe!cient can be assessed with the help of:

1 its standard error or standard deviation, denoted se"k;

2 its t-statistic t = #k/se"k.

The detailed calculation of se"kis not shown here (it is part of any standard

regression output).

When the size of a coe!cient (or some deviation from it) is large ascompared to its standard deviation, the relationship between xk and y isexpected to be strong.


Testing the Regression Coe!cients: 2-tail or 1-tail t-tests

Two-tailed t-test: If we want to assess whether an individual coe!cient #k

is significantly di!erent from some arbitrary (possibly null) #!, we can

derive the theoretical distribution of t = "k""!

se!k

under the null hypothesis

that #k = #!. We then construct an interval around #! (called confidenceinterval) which contains with probability 1" % the true value #!. If thet-statistic that we obtain from the regression with the observed data doesnot fall within the confidence interval, we reject the null in favor of thealternative #k %= #!.

One-tailed t-test: Other alternative hypotheses can be of interest, inparticular, #k > #! or #k < #! at a significance level of %. Such tests simply

require the absolute value of t = "k""!

se!k

to be larger than some theoretical

threshold.

The appropriate distribution for the one-tailed or two-tailed t-tests when theOLS assumptions (slide 14) hold is the Student’s t-distribution. Therelated critical value is denoted t!(n"K ,#) and depends on the degrees offreedom n " K .


Student distribution

The t-statistic can be shown to be distributed as a Student’s t-distributioncentered on #! (here below #! = 0):

−10 −5 0 5 10

0.0

0.1

0.2

0.3

0.4

t

Density

stud(df=1,alpha=5%)stud(df=3,alpha=5%)norm(0,1,alpha=5%)

The colored dots are two-tails critical values for % = 5%. Note that thet-distribution tends toward the Normal shape as n " K increases.

Significance Level of the Regression Coe!cients

Again, we can rely indi"erently on either critical values of the t-statistic,noted t!(n"K ,#), or on a p-value of the t-statistic.

For testing the null hypothesis that #k = #! at the significance level %against the alternative #k %= #! (2-tail t-test):

compute t = "k""!

se!k

;

if |t| > t!(n"K ,#/2) or if P(|t|) < %, reject the null hypothesis in favor ofthe alternative of significant di"erence at the significance level of %.

For testing the null hypothesis that #k = #! against the alternative #k > #!

or #k < #! at the significance level % (1-tail t-test),

use the former t ratioif |t| > t!(n"K ,#) or P(|t|) < %, reject the null hypothesis in favor of thechosen unilateral alternative at the significance level %.

Tables of the t-distribution may report 1-tail p-values 1"P(t & t!n"K ,#) = %or 2-tail p-values 1" P(|t!n"K ,#/2| & t) = %. Be aware of what you use.


Confidence Interval around the Regression Coe!cients

Confidence intervals at the 1" % level can also be constructed around #k .If you use a table of the t-distribution (2-tail t-test):

#k ± t!(n"K ,#/2)se"k(16)

If that interval does not include some arbitrary (and possibly null) value #!,the regression coe!cient is significantly di"erent from #! at the %significance level.

Once you have carried out the appropriate individual t-tests on the #ks, youcan proceed to interpret the coe!cients.


Interpreting Regression Coe!cients

Regression coe!cients are parameters of a functional relationship, so theyare straightforward to interpret!

For the linear demand function :

Q = #0 + #1P + #2m + . . . ' #1 =&Q

&P; #2 =

&Q

&m. . . (17)

' #1 is the change in Q corresponding to a unit change in P when all otherexplanatory variables are kept constant.

If the variables are in logarithms, e.g. Q = logQ, P = logP , m = logm inequation (17), remember that the coe!cients are elasticities:

#1 =& logQ

& logP=

1Q&Q

1P&P

=$QQ$PP

; . . . (18)

Note that when you have more than one explanatory variable in a regression,the regression coe!cients are partial regression coe"cients, i.e.,#1 %= cor(Q, P) in equation (17).


A Regression Example with Excel 2007/2010

To replicate this example, use the file regression.xlsx from the coursewebsite. These data are from Hirschey (2009, P.190).

We estimate the following single equation demand model:

UNIT SOLD = #0 + #1PRICE + #2ADVERT + #3PERS SELL+ ! (19)

For performing regression analysis with Excel 2007/2010, you need first toenable Excel’s Data Analysis Toolbox:

1 go the the File tab or click the O!ce button and then click on Options2 click on Add-Ins, select ‘Analysis Toolpak’ in the ‘Inactive Application

Add-ins’ and click on the ‘Go. . . ’ button3 The ‘Add-Ins’ window will pop up. Select ‘Analysis Toolpak’ and click

OK

You can check that the Data Analysis Toolbox has been properly enabled byselecting the Data tab in Excel and checking that the ‘Data Analysis’ optionis available under the ‘Analysis’ buttons.

Then open regression.xlsx in Excel and use the data in the ‘data’ sheet.


A Regression Example with Excel - Steps (1) and (2)

Select ‘Analysis Toolpak’ and click on the ‘Go. . . ’ button

A Regression Example with Excel - Step (3)

Select ‘Analysis Toolpak’ and press OK

A Regression Example with Excel - Regression Tool

Click on the ‘Data Analysis’ button and select Regression

A Regression Example with Excel - Regression Window

To replicate the results, use the same options in the Regression Window.

A Regression Example with Excel - Regression Output

A Regression Example with Excel - Regression Output

The Excel regression output generated above is divided in 4 main parts:

1 Regression statistics (R , R2, R2adj , sereg , obs.)

2 SST, SSR, SSE and F-test, called (Analysis of Variance or ANOVA)

3 Regression Coe!cients

4 Residuals

The link between the Excel output and the formulas from the former slides isemphasized below. The sheet ‘regression (2)’ in regression.xlsx providesfurther formulas’ checks (in yellow) that can be of interest.


A Regression Example with Excel - ‘Regression Statistics’

Let’s focus on the ‘Regression statistics’:

The multiple correlation coe!cient R = cor(y , y) = 0.98 is very high.This is not too surprising when time-series are employed.The R square indicates that the regression explains 97% of the totalvariance. It can be computed either by squaring the above R (becausethe regression includes a constant: 0.982 = 0.97) or with informationfrom the ANOVA table (try to apply equation (12)).In regressions based on cross-sectional data, R2 > 0.5 is already a goodfitting performance.The R2

adj = 0.958 is pretty close to the R2 which indicates that thepenalization linked to the degrees of freedom is not large.The regression standard error (123.92) corresponds to the denominatorin equation (14). You can check this by typing in an Excel cell:

=sqrt(sumsq(resid range)/(12-4))

(replacing resid range with the appropriate range, check the sheet‘regression (2)’).


A Regression Example with Excel - ANOVA Table

The ANOVA table :

The figures reported in column Sum of Squares (SS) correspond toSSR, SSE and SST from equations (9) to (11).The MSR (Mean Square Regression) and MSE (Mean Square Errors)figures correspond to numerator and denominator from equation (14)and F = MSR/MSE = 85.40.To judge if F is large enough (is the contribution of all the predictorsto the explained variation is significant?), we can check whether or notP(F ) < 0.05: ' ‘Significance F’ being very small, we reject the nullhypothesis at the 5% significance level and conclude that the predictors(price, advertising and personal sells) contribute to explain thevariation of quantity sold.Note that you can get the P(F ) with the following Excel function:

=fdist(F,df1,df2)

(replace the F, df1, df2 with appropriate information from theANOVA table, check the sheet ‘regression (2)’)


A Regression Example with Excel - Coe!cients’ Results

Regarding the regression coe!cients:

The p-value of the t-stat is lower than 5% for most coe!cients, whichmeans that they are significantly di"erent from 0 at that significancelevel. You can check that the ‘T-stat’ column is the ‘Coe!cients’column divided by the ‘Standard error’ column.The Excel functions that provides the 1-tail or 2-tail p-values of theStudent’s t-distribution is ‘tdist()’ and the one for getting t!(n"K ,#) ist.inv().We can test if the price coe!cient (-0.296) is significantly lower than0 by simply comparing |t| = 2.908 with the unilateral cuto"t!12"4,0.05 = 1.86 as indicated in slide 27. As |t| > t! we reject the nullin favor of a significantly negative coe!cient.We also notice that for CHF 100 spent in Advertising we get anaverage of 3.6 units sold/month.

Finally note that Excel provides 95% confidence intervals around thecoe!cients. They correspond to those described in equation (16).


References

Hirschey M., Managerial Economics, 12th Edition, Ch.5.

Chatterjee S., Hadi A., Price B., Regression Analysis by Example, 3rdEdition, Ch.3.

W. Greene, Econometric Analysis, 6th Edition, Ch.3.


Theory of Individual Behavior

Managerial Economics October 14, 2011

Overview I. Consumer Behavior

– Indifference Curve Analysis. – Consumer Preference Ordering.

II. Constraints – The Budget Constraint. – Changes in Income. – Changes in Prices.

III. Consumer Equilibrium IV. Indifference Curve Analysis & Demand Curves

– Individual Demand. – Market Demand.

Consumer Behavior • Consumer Opportunities

– The possible goods and services consumer can afford to consume.

• Consumer Preferences – The goods and services consumers actually

consume. • Given the choice between 2 bundles of goods

a consumer either: – Prefers bundle A to bundle B: A B. – Prefers bundle B to bundle A: A B. – Is indifferent between the two: A B.

Indifference Curve Analysis

Indifference Curve – A curve that defines the

combinations of 2 or more goods that give a consumer the same level of satisfaction.

Marginal Rate of Substitution – The rate at which a consumer is

willing to substitute one good for another and maintain the same satisfaction level.

I. II.

III.

Good Y

Good X

Consumer Preference Ordering Properties

• Completeness • More is Better • Diminishing Marginal Rate of Substitution • Transitivity

Complete Preferences • Completeness Property

– Consumer is capable of expressing preferences (or indifference) between all possible bundles. (“I don’t know” is NOT an option!) • If the only bundles available to

a consumer are A, B, and C, then the consumer

– is indifferent between A and C (they are on the same indifference curve).

– will prefer B to A. – will prefer B to C.

I. II.

III.

Good Y

Good X

A

C

B

More Is Better! • More Is Better Property

– Bundles that have at least as much of every good and more of some good are preferred to other bundles. • Bundle B is preferred to A since B

contains at least as much of good Y and strictly more of good X.

• Bundle B is also preferred to C since B contains at least as much of good X and strictly more of good Y.

• More generally, all bundles on ICIII are preferred to bundles on ICII or ICI. And all bundles on ICII are preferred to ICI.

I. II.

III.

Good Y

Good X

A

C

B

1

33.33

100

3

Diminishing MRS • MRS

– The amount of good Y the consumer is willing to give up to maintain the same satisfaction level decreases as more of good X is acquired.

– The rate at which a consumer is willing to substitute one good for another and maintain the same satisfaction level.

• To go from consumption bundle A to B the consumer must give up 50 units of Y to get one additional unit of X.

• To go from consumption bundle B to C the consumer must give up 16.67 units of Y to get one additional unit of X.

• To go from consumption bundle C to D the consumer must give up only 8.33 units of Y to get one additional unit of X.

I. II.

III.

Good Y

Good X 1 3 4 2

100

50

33.33 25

A

B

C D

Consistent Bundle Orderings • Transitivity Property

– For the three bundles A, B, and C, the transitivity property implies that if C B and B A, then C A.

– Transitive preferences along with the more-is-better property imply that • indifference curves will not

intersect. • the consumer will not get

caught in a perpetual cycle of indecision.

I. II.

III.

Good Y

Good X 2 1

100

5

50

7

75

A

B

C

The Budget Constraint • Opportunity Set

– The set of consumption bundles that are affordable. • PxX + PyY M.

• Budget Line – The bundles of goods that exhaust a

consumers income. • PxX + PyY = M.

• Market Rate of Substitution – The slope of the budget line

• -Px / Py.

Y

X

The Opportunity Set

Budget Line

Y = M/PY – (PX/PY)X M/PY

M/PX

Changes in the Budget Line • Changes in Income

– Increases lead to a parallel, outward shift in the budget line (M1 > M0).

– Decreases lead to a parallel, downward shift (M2 < M0).

• Changes in Price – A decreases in the price of

good X rotates the budget line counter-clockwise (PX0

> PX1

). – An increases rotates the

budget line clockwise (not shown).

X

Y

X

Y New Budget Line for a price decrease.

M0/PY

M0/PX

M2/PY

M2/PX

M1/PY

M1/PX

M0/PY

M0/PX0 M0/PX1

Consumer Equilibrium

• The equilibrium consumption bundle is the affordable bundle that yields the highest level of satisfaction. – Consumer equilibrium

occurs at a point where MRS = PX / PY.

– Equivalently, the slope of the indifference curve equals the budget line.

I. II.

III.

X

Y

Consumer Equilibrium

M/PY

M/PX

Price Changes and Consumer Equilibrium

• Substitute Goods – An increase (decrease) in the price of good X leads to an

increase (decrease) in the consumption of good Y. • Examples:

– Coke and Pepsi. – Verizon Wireless or AT&T.

• Complementary Goods – An increase (decrease) in the price of good X leads to a

decrease (increase) in the consumption of good Y. • Examples:

– DVD and DVD players. – Computer CPUs and monitors.

Complementary Goods

When the price of good X falls and the consumption of Y rises, then X and Y are complementary goods. (PX1

> PX2)

Pretzels (Y)

Beer (X)

II

I 0

Y2

Y1

X1 X2

A

B

M/PX1 M/PX2

M/PY1

Income Changes and Consumer Equilibrium

• Normal Goods – Good X is a normal good if an increase

(decrease) in income leads to an increase (decrease) in its consumption.

• Inferior Goods – Good X is an inferior good if an increase

(decrease) in income leads to a decrease (increase) in its consumption.

Normal Goods

An increase in income increases the consumption of normal goods.

(M0 < M1).

Y

II

I

0

A

B

X

M0/Y

M0/X

M1/Y

M1/X X0

Y0

X1

Y1

Decomposing the Income and Substitution Effects

Initially, bundle A is consumed. A decrease in the price of good X expands the consumer’s opportunity set.

The substitution effect (SE) causes the consumer to move from bundle A to B.

A higher “real income” allows the consumer to achieve a higher indifference curve.

The movement from bundle B to C represents the income effect (IE). The new equilibrium is achieved at point C.

Y

II

I

0

A

X

C

B

SE

IE

A Classic Marketing Application

Other goods (Y)

II

I

0

A

C

B F

D E

Pizza (X)

0.5 1 2

A buy-one, get-one free pizza deal.

Individual Demand Curve

• An individual’s demand curve is derived from each new equilibrium point found on the indifference curve as the price of good X is varied.

X

Y

$

X

D

II

I

P0

P1

X0 X1

Market Demand • The market demand curve is the horizontal

summation of individual demand curves. • It indicates the total quantity all consumers would

purchase at each price point.

Q

$ $

Q

50

40

D2 D1

Individual Demand Curves

Market Demand Curve

1 2 1 2 3 DM

Conclusion

• Indifference curve properties reveal information about consumers’ preferences between bundles of goods. – Completeness. – More is better. – Diminishing marginal rate of substitution. – Transitivity.

• Indifference curves along with price changes determine individuals’ demand curves.

• Market demand is the horizontal summation of individuals’ demands.

Optimization Tools


4.1

Lecture 4Optimization ToolsLagrangian Methods

Managerial EconomicsOctober 14, 2011



Optimization Tools


4.2

Good Mathematical References for Economics

• Mathematics for Economists by Carl P. Simon and LawrenceBloom, Norton, 1994. (an essential reference)

• Optimization in Economic Theory by Avinash K. Dixit, Oxford,1975. (a sentimental favorite)

• Mathematical methods for economic theory: a tutorial by MartinJ. Osborne, econoimcs.utoronto.ca/osborne (openaccess, very nicely organized)

• Microeconomic Analysis by Hal Varian, Chapters 26 and 27(terse but useful)

Optimization Tools


4.3

The First Derivative

Let f : R ! R. The derivative of f at x⇤ be denoted Df (x⇤). Becausef (x) is a scalar function, we have:

Df (x⇤) =df (x)

dx

The first derivative can be used to approximate the value of f atpoints close to x⇤. For small departures distances t , we have

f (x⇤ + t) ⇡ f (x⇤) + Df (x⇤)t .

Alternatively, we might write:

f (x) ⇡ L(x |x⇤) ⌘ f (x⇤) + Df (x⇤)(x � x⇤)

where L(x |x⇤) denotes the linear approximation to f anchored at x⇤.

Optimization Tools


4.4

An Example of Linear Approximation

To illustrate how a linear approximation works, suppose thatf (x) = sin(x). We have Df (x) = cos(x). A local approximation to f (x)is then

L(x |x) = sin(x) + cos(x)(x � x)

Optimization Tools


4.5

Alternative Linear Approximations to sin(x)

Optimization Tools


4.6

Second Order Approximations

A second order Taylor series approximation can be employed whenthe function to be approximated has continuous second derivatives.We can define a quadratic approximation to f (x) as:

Q(x ; x⇤) = f (x⇤) + Df (x⇤)(x � x⇤) +12(x � x⇤)D2f (x⇤)(x � x⇤)

The following figure illustrates the relationship between the underlingsine function and three different quadratic approximations.

Optimization Tools


4.7

Alternative Quadratic Approximations

Optimization Tools


4.8

The Gradient Vector

When f (x) is a scalar function with vector arguments, e.g. m = 1 orf : Rn ! R, the gradient of f at x⇤ is a vector whose coordinates arethe partial derivatives of f at x⇤:

D(f (x⇤)) =

✓@f (x⇤)

@x1, . . . ,

@f (x⇤)

@xn

◆

The gradient vector is also denoted rf (x⇤).

Optimization Tools


4.9

Definition

A quadratic form on Rn is a real-valued function of the form:

Q(x1, . . . , xn) =X

ij

aij xixj

in which each term is monomial of degree two.We can write this type of function compactly with vector-matrixnotation, i.e.

Q(x) = xT Ax

in which A is a symmetric matrix.

Optimization Tools


4.10

Quadratic Forms – Two Dimensions

When n = 2, we have:

Q(x) = a11x21 + a12x1x2 + a22x2

2

provided that

A =

✓a11

12 a12

12 a12 a22

◆

The Jacobian matrix of a given function provides a typical symmetricmatrices which appears in quadratic forms.

Note that if A is a non-symmetric square matrix, the associatedquadratic form has the same value as the related symmetric matrix:

A0 =12

A +12

AT

Optimization Tools


4.11

Definiteness and Quadratic Forms

Recall our quadratic approximation to a function f :

f (x) ⇡ f (x⇤) + Df (x⇤)(x � x⇤) +12(x � x⇤)0D2f (x⇤)(x � x⇤))

Suppose that we have selected an x⇤ such that Df (x⇤) = 0. Then thevalue of f (x) is given by:

f (x) ⇡ f (x⇤) + (x � x⇤)0A(x � x⇤))

where A = 12 D2f (x⇤).

• If A is positive definite then x⇤ is a local minimizer of f ().• If A is negative definite then x⇤ is a local maximizer of f ().

Optimization Tools


4.12

Concavity

A function of one variable is concave if

f (tx + (1 � t)y) � tf (x) + (1 � t)f (y)

For example, the sin(x) function is concave between x = 0.2 andy = 1.6, as illustrated in the following figure.

Optimization Tools


4.13

Local Concavity of the Sine Function

Optimization Tools


4.14

Convexity

1 If f is a convex function, then f 00(x) � 0 for all x2 If f is a convex function, then

f (x) � f (y) + f 0(y)|x � y |

3 If f is a convex function, and f 0(x⇤) = 0, then x⇤ minimizes thefunction f .

Optimization Tools


4.15

Unconstrained minimization

If f is differentiable at a local minimum x 2 U (open), then

rf (x) = 0.

This is a necessary condition – not a sufficient condition. (All localminima satisfy this condition, but there exist points which are not localminima which also satisfy this condition, e.g. local maxima or saddlepoints.)

Optimization Tools


4.16

Descent directions

• f : U ! R differentiable• x 2 U (open)

If rf (x)v < 0 then 9⌧ 2 R such that

f (x + ⌧v) < f (x) 8⌧ 2 (0, ⌧)

The vector v (above) is a descent direction at x .

Recall that if rf (x) 6= 0 then rf (x) is the direction of steepest ascentat x .

This follows from the Cauchy-Swartz inequality

|xy | = ||x ||||y || cos(✓) ||x ||||y ||

Optimization Tools


4.17

Equality Constrained Optimization

min f (x)

subject to:g(x) = 0 (P)

x 2 Rn

where• f and g are differentiable on Rn.• g : Rn ! Rm m n

Optimization Tools


4.18

Lagrange’s Theorem

TheoremLagrange If x is a local minimum of (P), and the Jacobian matrixrg(x) has rank m, then there exist numbers �1, . . . , �m such that

rf (x) +mX

i=1

�irgi(x) = 0

The numbers �1, . . . , �m are called Lagrange multipliersThe function L(x ,�) = f (x) +

Pmi=1 �i gi(x) is the Lagrangian for (P).

Optimization Tools


4.19

Practical usefulness of Lagrange’s method

Solution of a constrained optimization problem with n variables and mconstraints can be equivalent to solving a nonlinear system of n + mequations.

For economists, this result enormously simplifies the formulation andsolution of market equilibrium models, because we are able toincorporate multiple agents, each of which optimizes a separateobjective function subject to constraints.

Optimization Tools


4.20

Geometry of Constrained Optimization

Optimization Tools


4.21

Need for the regularity condition

The assumption that rank rg(x) = m is a regularity condition.

Lagrange’s theorem is not valid unless the regularity condition holds.

EXAMPLE:

min x1

(P) subject tox2

1 + (x2 � 1)2 = 1

x21 + (x2 + 1)2 = 1

Note: (P) has only one feasible point x = (0, 0).

rf (x) = (1, 0)

rg1(x) = (0,�2)

rg2(x) = (0, 2)

The Lagrange multipliers cannot exist here.

Optimization Tools


4.22

Irregular Example: No Multipliers Exist

5aConsumer Choice ExamplesOptimization and Human Behavior

Handout for Managerial Economics October 21, 2011

Thomas F. Rutherford, Center for Energy Policy and Economics, ETH Zürich

5a.1

A Choice Experiment

Thomas lives in Ann Arbor where he currently spends 30% of his income on rent. He has anemployment offer in Zürich which pays 50% more than he currently earns, but he is hesitantto take the job because rental rates in Zürich are three times higher than in Ann Arbor. Assum-ing that Thomas has CES preferences with elasticity of substitution σ ; on purely economicgrounds, should he move?

As is the case for all interesting questions in economics, the only good answer to this problem is “Itdepends.”. 5a.2

IntuitionThomas’s offer in Zürich does not pay him enough to live exactly the lifestyle that he enjoys in Ann

Arbor, as he would need a 60% raise to cover rent and consumption. The elasticity of substitution is key.If it is high, he more willing substitutes consumption of goods and services for housing and thereby lowershis cost of living in Zürich. On the other hand, if the elasticity is low, he is “stuck in his ways”, and themove is a bad idea. 5a.3

Calibration to a Benchmark EquilibriumWe are given information about Thomas’s choices in Ann Arbor. This information is essentially an

observation of a benchmark equilibrium, consisting of the prevailing prices and quantities of goods demand.The benchmark equilibrium data together with assumptions about elasticities are used to evaluate Thomas’schoices after a discrete change in the economic environment. The steps involved in solving this littletextbook model are identical to those typically employed in applied general equilibrium analysis. 5a.4

Graphical Representation

5a.5

1

PreferencesThe utility function:

U(C,H) = (αCρ +(1−α)Hρ )1/ρ

Exponent ρ is defined by the elasticity of substitution, σ , as

ρ = 1−1/σ .

The model of consumer choice is:

maxU(C,H) s.t. C+ pHH = M

5a.6

DemandDerivation of demand functions which solve the utility maximization problem involves solving two

equations in two unknowns:

∂U/∂H∂U/∂C

=(1−α)Hρ−1

αCρ−1 = pH ;

henceHC

=

(1−α

α pH

)σ

Substituting into the budget constraint, we have:

H =M

pH +(

α pH1−α

)σ =(1−α)σ Mp−σ

H

ασ +(1−α)σ p1−σ

H

and

C =M

1+ pH

(1−α

α pH

)σ =ασ M


H

5a.7

CalibrationIt is conventional in applied general equilibrium analysis to employ exogenous elasticities and calibrated

value values. If we follow this approach, σ is then exogenous and α is calibrated.Choosing units so that the benchmark price of housing (pH ) is unity, we have:

θ = pH H/M

Substitute into the demand function:

1+(

α

1−α

)σ

=MH

=1θ

;

and then solve for the preference parameter α:

α =(1−θ)1/σ

θ 1/σ +(1−θ)1/σ.

5a.8

Money Metric UtilitySubstitute for α in U(C,H), and denoting the base year expenditure on other goods as C = (1−θ)M,

we have

U(C,H) = κ

((1−θ)1/σCρ +θ

1/σ Hρ)1/ρ

where the κ is a constant which may take on any positive value without altering the preference ordering.It is convenient to assign this value to the benchmark expenditure, so that utility can be measured in money-metric units at benchmark prices.

Noting that θ 1/σ = θ 1−ρ , we then can write the utility function as:

U(C,H) = M((1−θ)

(CC

)ρ

+θ

(HH

)ρ)1/ρ

5a.9

2

Indirect UtilityFormally, we have:

V (pH ,M) =U(C(pH ,M),M(pH ,M)) =M(


H

)1/(1−σ)

In money-metric terms, we can use benchmark income to normalize the utility function:

V (pH ,M) =M

(1−θ +θ p1−σ

H )1/(1−σ)

5a.10

Demand Functions – Calibrated Share Form

H = HV (pH ,M)

M

(pU

pH

)σ

C = CV (pH ,M)

M

( pU

1

)σ

where

pU =(

1−θ +θ p1−σ

H

)1/(1−σ)

5a.11

Should Thomas Move?Thomas’s welfare level in Zürich can be easily computed in money-metric terms as:

V (pH = 3,M = 1.5) =1.5(

0.7+0.3×31−σ)1/(1−σ)

This expression cannot (to my knowledge) be solved in closed form, however it is easily to solve usingExcel. The critical value for σ is that which equates welfare in Zürich with welfare level in Ann Arbor, i.e.V = 1. The numerical value is found to be σ∗ = 0.441. The general dependence of welfare on the θ and σ

can be illustrated in a contour diagram. 5a.12

Dependence of Welfare on Benchmark Shares and Elasticity

5a.13

Multivariable OptimizationThe concept of multivariate optimization is important in managerial economics because many demand

and supply relations involve more than two variables. In demand analysis, it is typical to consider thequantity sold as a function of the price of the product itself, the price of other goods, advertising, income,and other factors. In cost analysis, cost is determined by output, input prices, the nature of technology, andso on.. 5a.14

3

Optimal AdvertisingTo explore the concepts of multivariate optimization and the optimal level of advertising, consider

a hypothetical multivariate product demand function for CSI, Inc., where the demand for product Q isdetermined by the price charged, P, and the level of advertising, A:

Q = 5,000−10P+40A+PA−0.8A2−0.5P2

Determine the joint optimal price (P∗) and level of advertising (A∗) which maximize CSI output. 5a.15

First Order ConditionsBegin by calculating partial derivates of demand with respect to price and level of advertising:

∂Q∂P

=−10+A−P

∂Q∂A

= 40+P−1.6A

First order conditions for maximization of demand are:

∂Q∂P

= 0

∂Q∂A

= 05a.16

Optimization = Solving Simultaneous EquationsHence, the optimal level of price and advertising solve:

−10+A−P = 0

40+P−1.6A = 0

Hence, P∗ = 40, A∗ = $5,000 and the maximal output is Q∗ = 5,800.Note that in subsequent chapters we will learn that the policies which maximize output may differ from

those which maximal profit, depending on how production cost relates to output. 5a.17

Nonlinear PricingConsider a consumer choice model in which the two goods consist of telecommunication services (x)

and all other goods (y). Let the price of other goods is fixed at unity. Telecommunication services aresomewhat special in that due to economies of scale, these are offered with potentially substantial quantitydiscounts. Once a subscription fee of f CHF is made, services are offerred at a substantially reduced price.In the absence of the connection fee, px = 1. Telecommunication services made to customers who havepaid the connection fee are offered at a price of px.

The consumer is assumed to have the following utility function:

maxU(x,y) = xα y1−α

5a.18

A. Ignoring the subscription plan, solve for the quantity of telecommunication services demanded bythe consumer.

The standard consumer model is one of budget-constrained utility maximization. Hence, we solvemaxU(x,y) s.t. pxx+ y = M. The first order condition is:

∂U(x,y)/∂x∂U(x,y)/∂y

=px

1

Hence,

x∗ =αMpx

and

y∗ =(1−α)M

15a.19

4

B. Assuming that the consumer chooses to buy a subscription. Show that she will buy the followingquantities:

x∗ = αM− f

px

y∗ = (1−α)(M− f )

If the consumer buys a subscription, the purchase quantity solves:

maxU(x,y)

s.t.pxx+ y = M− f .

The first order conditions are identical to the previous case, except that M is replaced by M− f andpx is replaced by px.

5a.20

C. Holding p fixed, what is the critical value of f such that the consumer is indifferent about buying asubscription.

The critical value of f is that for which:

U(x∗,y∗) =U(x∗, y∗)

Substituting for U() we have:(α

Mpx

)α ((1−α)

M1

)1−α

=

(α

M− fpx

)α ((1−α)

M− f1

)1−α

Thus,Mpα

x=

M− fpα

x

and

f ∗ = M(

1−(

px

px

)α)5a.21

D. Sketch the budget constraint and the optimal choice for a consumer who chooses not to accept thesubscription.

If a consumer buys a subscription, the maximum amount she can purchase of other goods is M− f .The slope of the budget line is −p. If the optimal point on the subscription-based budget constraintis associated with a lower indifference curve, then the consumer will not purchase a subscription:

5a.22

E. Holding f fixed, graphically find the maximum discount price level which would induce this con-sumer to purchase additional units of telecommunication services (p∗x).

Here we rotate the subscription based budget constraint around the y axis intercept to the point that itis just tangent to the original indifference curve:

5

5a.23

F. Solve for p∗h( f ) analytically.

As shown above, the price which makes the consumer indifferent between taking a subscription ornot is:

p∗( f ) =(

1− fM

)1/α

5a.24

G. Suppose that the marginal cost of supply for telecom services is 1. What combination of f and pxmaximizes firm profit?

maxf

Π( f ) = f +(p( f )−1)x = f +(p−1)αM− fp( f )

The first order for profit maximization is:

dΠ

d f= 1−α− α

p− α f

p2dp( f )

d f= 0

Applying the basic rules of calculus, we have:

dp( f )d f

=1α

(1− f

M

)1/α−1(−1M

)=

−p1−α

αM

Hence:

dp( f )d f

= 1−α +α

p+

α fp2 = 0⇒ p = 1

5a.25Basic idea: nonlinear pricing does not provide a means of increasing firm profits in the case of Cobb-

Douglas demand. The optimal subscription rate is zero ( f ∗ = 0), and it is optimal to price at marginal cost(p∗ = 1). 5a.26

Cobb-Douglas CalibrationSuzy consumes ice cream (x1) and soda (x2) for lunch every day, and she currently has one ice cream

and two sodas per week when they both cost 1 CHF. What Cobb-Douglas utility function is consistent withSuzy’s choices over ice cream and soda. Write down demand functions which could extrapolate her optimalchoices to any expenditure (m) and prices (p1 and p2). 5a.27

A Cobb-Douglas Calibration Exercise: AnswerBased current choices, we observe that Suzy’s budget shares for ice cream and sodas are 1/3 and 2/3,

respectively. The Cobb-Douglas utility function which describes her preferences is:

U(x1,x2) = x1/31 x2/3

2

and demand functions arex1 =

Y3p1

andx2 =

2Y3p2

5a.28

6

Calibration Exercise #2Suppose that irregardless of relative prices, Suzy always has one soda before and one soda after eating

an ice cream. What utility function is consistent with these choices? Write down demand functions whichcould extrapolate her optimal choices to any expenditure (m) and prices (p1 and p2). 5a.29

Exercise # 2: SolutionPerfect complement preferences have the form:

U(x1,x2) = min(x1

a1,

x2

a2)

in which the ratio a1a2

determines the ratio in which goods 1 and 2 are consumed. In the present example,we have:

U(x1,x2) = min(x1,x2

2)

and demand functions given by:

x1 =Y

p1 +2p2

andx2 = 2

Yp1 +2p2

5a.30

Calibration Exercise #3When Suzy gets to the lunch counter, she always asks about the price of ice cream and the price of soda.

If two sodas cost less than one ice cream, she has spends all of her money on soda. Otherwise she buys icecream. What utility function is consistent with these choices? Write down demand functions which couldextrapolate her optimal choices to any expenditure (m) and prices (p1 and p2). 5a.31

Calibration Exercise #3 SolutionGeneral perfect substitues preferences have the form:

U(x1,x2) = a1x1 +a2x2

in which the ratio a1a2

represents the marginal rate of substitution of good 1 for good 2. The demand functionsfor these preferences are given by:

x1 =

{0 when p1

p2> a1

a2Mp1

otherwise

x2 =

{0 when p1

p2< a1

a2Mp2

otherwise5a.32

7

Chapter 5: Production

Managerial Economics Lecture Notes

Friday, October 21, 2011

Overview I. Production Analysis

– Total Product, Marginal Product, Average Product. – Isoquants. – Isocosts. – Cost Minimization

II. Cost Analysis – Total Cost, Variable Cost, Fixed Costs. – Cubic Cost Function. – Cost Relations.

III. Multi-Product Cost Functions

Production Analysis • Production Function

– Q = F(K,L) • Q is quantity of output produced. • K is capital input. • L is labor input. • F is a functional form relating the inputs to output.

– The maximum amount of output that can be produced with K units of capital and L units of labor.

• Short-Run vs. Long-Run Decisions • Fixed vs. Variable Inputs

Production Function Algebraic Forms

• Linear production function: inputs are perfect substitutes.

• Leontief production function: inputs are used in fixed proportions.

• Cobb-Douglas production function: inputs have a degree of substitutability.

baLKLKFQ ,

bLaKLKFQ ,

cLbKLKFQ ,min,

Productivity Measures: Total Product

• Total Product (TP): maximum output produced with given amounts of inputs.

• Example: Cobb-Douglas Production Function: Q = F(K,L) = K.5 L.5

– K is fixed at 16 units. – Short run Cobb-Douglass production function:

Q = (16).5 L.5 = 4 L.5

– Total Product when 100 units of labor are used?

Q = 4 (100).5 = 4(10) = 40 units

Productivity Measures: Average Product of an Input

• Average Product of an Input: measure of output produced per unit of input. – Average Product of Labor: APL = Q/L.

• Measures the output of an “average” worker. • Example: Q = F(K,L) = K.5 L.5

– If the inputs are K = 16 and L = 16, then the average product of labor is APL = [(16) 0.5(16)0.5]/16 = 1.

– Average Product of Capital: APK = Q/K. • Measures the output of an “average” unit of capital. • Example: Q = F(K,L) = K.5 L.5

– If the inputs are K = 16 and L = 16, then the average product of capital is APK = [(16)0.5(16)0.5]/16 = 1.

Productivity Measures: Marginal Product of an Input

• Marginal Product on an Input: change in total output attributable to the last unit of an input. – Marginal Product of Labor: MPL = Q/L

• Measures the output produced by the last worker. • Slope of the short-run production function (with respect to

labor). – Marginal Product of Capital: MPK = Q/K

• Measures the output produced by the last unit of capital. • When capital is allowed to vary in the short run, MPK is the

slope of the production function (with respect to capital).

Q

L

Q=F(K,L)

Increasing Marginal Returns

Diminishing Marginal Returns

Negative Marginal Returns

MP

AP

Increasing, Diminishing and Negative Marginal Returns

Guiding the Production Process

• Producing on the production function – Aligning incentives to induce maximum worker effort.

• Employing the right level of inputs – When labor or capital vary in the short run, to

maximize profit a manager will hire: • labor until the value of marginal product of labor equals the

wage: VMPL = w, where VMPL = P x MPL. • capital until the value of marginal product of capital equals

the rental rate: VMPK = r, where VMPK = P x MPK .

Isoquant

• Illustrates the long-run combinations of inputs (K, L) that yield the producer the same level of output.

• The shape of an isoquant reflects the ease with which a producer can substitute among inputs while maintaining the same level of output.

Marginal Rate of Technical Substitution (MRTS)

• The rate at which two inputs are substituted while maintaining the same output level.

K

LKL MP

MPMRTS

Linear Isoquants

• Capital and labor are perfect substitutes – Q = aK + bL – MRTSKL = b/a – Linear isoquants imply that

inputs are substituted at a constant rate, independent of the input levels employed.

Q3 Q2 Q1

Increasing Output

L

K

Leontief Isoquants

• Capital and labor are perfect complements.

• Capital and labor are used in fixed-proportions.

• Q = min {bK, cL} • Since capital and labor are

consumed in fixed proportions there is no input substitution along isoquants (hence, no MRTSKL).

Q3 Q2

Q1

K

Increasing Output

L

Cobb-Douglas Isoquants

• Inputs are not perfectly substitutable.

• Diminishing marginal rate of technical substitution. – As less of one input is used in

the production process, increasingly more of the other input must be employed to produce the same output level.

• Q = KaLb

• MRTSKL = MPL/MPK

Q1 Q2

Q3

K

L

Increasing Output

Isocost • The combinations of inputs

that produce a given level of output at the same cost:

wL + rK = C

• Rearranging, K= (1/r)C - (w/r)L

• For given input prices, isocosts farther from the origin are associated with higher costs.

• Changes in input prices change the slope of the isocost line.

K

L C1

L

K New Isocost Line for a decrease in the wage (price of labor: w0 > w1).

C1/r

C1/w C0

C0/w

C0/r

C/w0 C/w1

C/r

New Isocost Line associated with higher costs (C0 < C1).

Cost Minimization

• Marginal product per dollar spent should be equal for all inputs:

• But, this is just

rw

MPMP

rMP

wMP

K

LKL

rwMRTSKL

Cost Minimization

Q

L

K

Point of Cost Minimization

Slope of Isocost =

Slope of Isoquant

Optimal Input Substitution

• A firm initially produces Q0 by employing the combination of inputs represented by point A at a cost of C0.

• Suppose w0 falls to w1. – The isocost curve rotates

counterclockwise; which represents the same cost level prior to the wage change.

– To produce the same level of output, Q0, the firm will produce on a lower isocost line (C1) at a point B.

– The slope of the new isocost line represents the lower wage relative to the rental rate of capital.

Q0

0

A

L

K

C0/w1 C0/w0 C1/w1 L0 L1

K0

K1 B

Cost Analysis

• Types of Costs – Short-Run

• Fixed costs (FC) • Sunk costs • Short-run variable

costs (VC) • Short-run total costs

(TC) – Long-Run

• All costs are variable • No fixed costs

Total and Variable Costs

C(Q): Minimum total cost of producing alternative levels of output:

C(Q) = VC(Q) + FC

VC(Q): Costs that vary with output. FC: Costs that do not vary with output.

$

Q

C(Q) = VC + FC

VC(Q)

FC

0

Fixed and Sunk Costs

FC: Costs that do not change as output changes. Sunk Cost: A cost that is forever lost after it has been paid. Decision makers should ignore sunk costs to maximize profit or minimize losses

$

Q

FC

C(Q) = VC + FC

VC(Q)

Some Definitions

Average Total Cost ATC = AVC + AFC ATC = C(Q)/Q

Average Variable Cost AVC = VC(Q)/Q

Average Fixed Cost AFC = FC/Q

Marginal Cost MC = DC/DQ

$

Q

ATC AVC

AFC

MC

MR

Fixed Cost

$

Q

ATC

AVC

MC

ATC

AVC

Q0

AFC Fixed Cost

Q0(ATC-AVC) = Q0 AFC = Q0(FC/ Q0) = FC

Variable Cost

$

Q

ATC

AVC

MC

AVC Variable Cost

Q0

Q0AVC = Q0[VC(Q0)/ Q0] = VC(Q0)

Minimum of AVC

$

Q

ATC

AVC

MC

ATC

Total Cost

Q0

Q0ATC = Q0[C(Q0)/ Q0] = C(Q0)

Total Cost

Minimum of ATC

Cubic Cost Function

• C(Q) = f + a Q + b Q2 + cQ3

• Marginal Cost? MC(Q) = dC/dQ = a + 2bQ + 3cQ2

An Example – Total Cost: C(Q) = 10 + Q + Q2 – Variable cost function:

VC(Q) = Q + Q2

– Variable cost of producing 2 units: VC(2) = 2 + (2)2 = 6

– Fixed costs: FC = 10

– Marginal cost function: MC(Q) = 1 + 2Q

– Marginal cost of producing 2 units: MC(2) = 1 + 2(2) = 5

Long-Run Average Costs

LRAC

$

Q

Economies of Scale

Diseconomies of Scale

Q*

Multi-Product Cost Function

• C(Q1, Q2): Cost of jointly producing two outputs.

• General function form:

22

212121, cQbQQaQfQQC

Economies of Scope

• C(Q1, 0) + C(0, Q2) > C(Q1, Q2). – It is cheaper to produce the two outputs jointly

instead of separately.

• Example: – It is cheaper for Time-Warner to produce Internet

connections and Instant Messaging services jointly than separately.

Cost Complementarity

• The marginal cost of producing good 1 declines as more of good two is produced:

MC1Q1,Q2) /Q2 < 0.

• Example:

– Cow hides and steaks.

Quadratic Multi-Product Cost Function

• C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2 • MC1(Q1, Q2) = aQ2 + 2Q1 • MC2(Q1, Q2) = aQ1 + 2Q2 • Cost complementarity: a < 0 • Economies of scope: f > aQ1Q2

C(Q1 ,0) + C(0, Q2 ) = f + (Q1 )2 + f + (Q2)2 C(Q1, Q2) = f + aQ1Q2 + (Q1 )2 + (Q2 )2 f > aQ1Q2: Joint production is cheaper

A Numerical Example:

• C(Q1, Q2) = 90 - 2Q1Q2 + (Q1 )2 + (Q2 )2 • Cost Complementarity? Yes, since a = -2 < 0 MC1(Q1, Q2) = -2Q2 + 2Q1 • Economies of Scope? Yes, since 90 > -2Q1Q2

Conclusion

• To maximize profits (minimize costs) managers must use inputs such that the value of marginal of each input reflects price the firm must pay to employ the input.

• The optimal mix of inputs is achieved when the MRTSKL = (w/r).

• Cost functions are the foundation for helping to determine profit-maximizing behavior in future chapters.

Date post:	30-Oct-2014
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Managerial Economics

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