kuliah teori ekonomi... · 1 rancangan pembelajaran semester ( rps) program studi /jurusan :...

TEORI EKONOMI MIKRO

DOSEN:

DR. ARDITO BHINADI, SE., M.SI

JURUSAN ILMU EKONOMI, FAKULTAS EKONOMI, UPN “VETERAN” YOGYAKARTA

2013

1

RANCANGAN PEMBELAJARAN SEMESTER ( RPS)

Program Studi /Jurusan : EKONOMI PEMBANGUNAN/ILMU EKONOMI Matakuliah / Kode : TEORI EKONOMI MIKRO / SKS / Semester : 3 (tiga x 50 menit)/ II (dua) Mata Kuliah Prasyarat : Ekonomi Mikro Pengantar Dosen : Dr. H. Ardito Bhinadi, M.Si

I.Deskripsi Mata Kuliah:

Matakuliah ini membahas sejumlah teori ekonomi mikro dari teori konsumen, teori produsen, berbagai bentuk pasar dan eksternalitas.

II.Kompetensi Umum :

Pada akhir perkuliahan mahasiswa diharapkan mampu memahami dan menjelaskan model-model ekonomi, pilihan dan permintaan, produksi dan penawaran, pasar kompetitif, kekuatan pasar, penetapan harga di pasar input, dan kegagalan pasar.

III. Analisis Instruksional Terlampir IV. Strategi Pembelajaran :

Pembelajaran menggunakan metoda ceramah dan diskusi dengan harapan muncul sensitifitas mahasiswa terhadap masalah mikro ekonomi. Materi perkuliahan didasarkan pada beberapa buku dan studi kasus yang harus difahami oleh mahasiswa. Dosen menyampaikan materi dalam bentuk dalam power point. Media yang digunakan adalah papan tulis, LCD, dan Laptop.

V. Rencana Pembelajaran Mingguan

Pertemuan Ke

Kompetensi Pokok/Sub-pokok Bahasan

Metoda Pembelajaran

Media Pembelajaran

Metoda Evaluasi

Referensi

1

(Satu)

Mahasiswa mampu memahami berbagai model ekonomi.

Model-Model Ekonomi

Ceramah dan diskusi

Papan tulis, LCD, Laptop,

Pertanyaan kuis/umpan balik

Ch1

2

(Dua)

Mahasiswa mampu memahami preferensi dan utilitas konsumen.

Preferensi dan Utilitas

Mahasiswa Presentasi, Ceramah dan diskusi



Ch 3

3

(Tiga)

Mahasiswa mampu efek substitusi dan pendapatan.

Efek Substitusi dan Pendapatan




Ch 5

4

(Empat)

Mahasiswa mampu memahami hubungan permintaan antar barang.

Hubungan Permintaan Antar Barang




Ch 6

2

5

(Lima)

Mahasiswa mampu memahami fungsi-fungsi produksi

Fungsi-Fungsi Produksi




Ch 9

6

(Enam)

Mahasiswa mampu memahami fungsi-fungsi biaya.

Fungsi-Fungsi Biaya.




Ch 10

7

(Tujuh)

Mahasiwa mampu menghitung maksimisasi laba.

Maksimisasi Laba Mahasiswa Presentasi, Ceramah dan Diskusi

Papan tulis, LCD, Laptop

Pertanyaan umpan balik

Ch 11

Ujian Tengah Semester

Pertemuan Ke

Kompetensi Pokok/Sub-pokok Bahasan

Metoda Pembelajaran

Media Pembelajaran

Metoda Evaluasi

Referensi

8

(Delapan)

Mahasiwa mampu memahami model persaingan keseimbangan parsial.

Model Persaingan Keseimbangan Parsial

Diskusi dan Kuis



Ch 12

9

(Sembilan)

Mahasiwa mampu memahami keseimbangan umum dan kesejahteraan.

Keseimbangan Umum dan Kesejahteraan

Diskusi dan Kuis



Ch 13

10

(Sepuluh)

Mahasiwa mampu memahami monopoli.

Monopoli Diskusi dan Kuis



Ch 14

11

(Sebelas)

Mahasiwa mampu memahami persaingan tidak sempurna.

Persaingan Tidak Sempurna

Diskusi dan Kuis



Ch 15

12

(Dua Belas)

Mahasiwa mampu memahami pasar tenaga kerja

Pasar Tenaga Kerja Diskusi dan Kuis



Ch 16

13

(Tiga Belas)

Mahasiwa mampu memahami informasi asimetris.

Asimetris Informasi Diskusi dan Kuis



Ch 18

3

14

(Empat Belas)

Mahasiwa mampu memahami eksternalitas dan barang publik.

Eksternalitas dan Barang Publik

Diskusi dan Kuis



Ch 19

Ujian Akhir Semester

1. Sumber Referensi

Nicholson, Walter and Christopher Snyder, 2008. Microeconomic Theory, Basic Principles and Extensions, Tenth Edition, Thomson South-Western, United Stated of America.

2. Komponen Penilaian

1.Ujian Tengah Semester = 30% 2.Ujian Akhir Semester = 30% 3.Partisipasi Kelas = 20% 4.Tugas-Tugas = 20%

1

1

Microeconomic Theory Basic Principles and Extensions, 9e

Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.

By

WALTER NICHOLSON

Slides prepared by

Linda Ghent Eastern Illinois University

2

Chapter 1

ECONOMIC MODELS

3

Theoretical Models

• Economists use models to describe

economic activities

• While most economic models are

abstractions from reality, they provide

aid in understanding economic behavior

4

Verification of Economic Models

• There are two general methods used to

verify economic models:

– direct approach

• establishes the validity of the model’s

assumptions

– indirect approach

• shows that the model correctly predicts real-

world events

2

5

Verification of Economic Models

• We can use the profit-maximization model

to examine these approaches

– is the basic assumption valid? do firms really

seek to maximize profits?

– can the model predict the behavior of real-world

firms?

6

Features of Economic Models

• Ceteris Paribus assumption

• Optimization assumption

• Distinction between positive and

normative analysis

7

Ceteris Paribus Assumption

• Ceteris Paribus means “other things the

same”

• Economic models attempt to explain

simple relationships

– focus on the effects of only a few forces at a

time

– other variables are assumed to be unchanged

during the period of study

8

Optimization Assumptions

• Many economic models begin with the

assumption that economic actors are

rationally pursuing some goal

– consumers seek to maximize their utility

– firms seek to maximize profits (or minimize

costs)

– government regulators seek to maximize

public welfare

3

9

Optimization Assumptions

• Optimization assumptions generate

precise, solvable models

• Optimization models appear to be

perform fairly well in explaining reality

10

Positive-Normative Distinction

• Positive economic theories seek to

explain the economic phenomena that

is observed

• Normative economic theories focus on

what “should” be done

11

The Economic Theory of Value

• Early Economic Thought

– “value” was considered to be synonymous

with “importance”

– since prices were determined by humans,

it was possible for the price of an item to

differ from its value

– prices > value were judged to be “unjust”

12


• The Founding of Modern Economics

– the publication of Adam Smith’s The Wealth of Nations is considered the beginning of modern

economics

– distinguishing between “value” and “price”

continued (illustrated by the diamond-water

paradox)

• the value of an item meant its “value in use”

• the price of an item meant its “value in exchange”

4

13


• Labor Theory of Exchange Value

– the exchange values of goods are determined by

what it costs to produce them

• these costs of production were primarily affected by

labor costs

• therefore, the exchange values of goods were

determined by the quantities of labor used to produce

them

– producing diamonds requires more labor than

producing water

14


• The Marginalist Revolution

– the exchange value of an item is not determined

by the total usefulness of the item, but rather

the usefulness of the last unit consumed

• because water is plentiful, consuming an additional

unit has a relatively low value to individuals

15


• Marshallian Supply-Demand Synthesis

– Alfred Marshall showed that supply and demand

simultaneously operate to determine price

– prices reflect both the marginal evaluation that

consumers place on goods and the marginal

costs of producing the goods

• water has a low marginal value and a low marginal

cost of production Low price

• diamonds have a high marginal value and a high

marginal cost of production High price 16

Supply-Demand Equilibrium

Quantity per period

Price

P*

Q*

D

The demand curve has a

negative slope because

the marginal value falls as

quantity increases

S

The supply curve has a positive

slope because marginal cost

rises as quantity increases

Equilibrium

QD = Qs

5

17


qD = 1000 - 100p

qS = -125 + 125p

Equilibrium qD = qS

1000 - 100p = -125 + 125p

225p = 1125

p* = 5

q* = 500 18


• A more general model is

qD = a + bp

qS = c + dp

Equilibrium qD = qS

a + bp = c + dp

bd

cap

*

19


A shift in demand will lead to a new equilibrium:

Q’D = 1450 - 100P

Q’D = 1450 - 100P = QS = -125 + 125P

225P = 1575

P* = 7

Q* = 750

20


S

Quantity per period

D

Price

5

500

7

750

D’

An increase in demand...

…leads to a rise in the

equilibrium price and

quantity.

6

21

• General Equilibrium Models

– the Marshallian model is a partial

equilibrium model

• focuses only on one market at a time

– to answer more general questions, we

need a model of the entire economy

• need to include the interrelationships between

markets and economic agents


22

• The production possibilities frontier can

be used as a basic building block for

general equilibrium models

• A production possibilities frontier shows

the combinations of two outputs that

can be produced with an economy’s

resources


23

Quantity of clothing

(weekly)

Quantity of food

(weekly)

10 9.5

4

2

Opportunity cost of

clothing = 1/2 pound of food

Opportunity cost of

clothing = 2 pounds of food

3 4 12 13

A Production Possibility Frontier

24

• The production possibility frontier

reminds us that resources are scarce

• Scarcity means that we must make

choices

– each choice has opportunity costs

– the opportunity costs depend on how much

of each good is produced


7

25


• Suppose that the production possibility

frontier can be represented by

2252 22 yx

• To find the slope, we can solve for Y

22225 xy

• If we differentiate

y

x

y

xxx

dx

dy 2

2

4)4()2225(

2

1 2/12

26


• when x=5, y=13.2, the slope= -2(5)/13.2= -0.76

• when x=10, y=5, the slope= -2(10)/5= -4

• the slope rises as y rises

y

x

y

xxx

dx

dy 2

2

4)4()2225(

2

1 2/12

27

• Welfare Economics

– tools used in general equilibrium analysis have

been used for normative analysis concerning

the desirability of various economic outcomes

• economists Francis Edgeworth and Vilfredo Pareto

helped to provide a precise definition of economic

efficiency and demonstrated the conditions under

which markets can attain that goal


28

Modern Tools

• Clarification of the basic behavioral

assumptions about individual and firm

behavior

• Creation of new tools to study markets

• Incorporation of uncertainty and imperfect

information into economic models

• Increasing use of computers to analyze

data

8

29

Important Points to Note:

• Economics is the study of how scarce

resources are allocated among

alternative uses

– economists use simple models to

understand the process

30


• The most commonly used economic

model is the supply-demand model

– shows how prices serve to balance

production costs and the willingness of

buyers to pay for these costs

31


• The supply-demand model is only a

partial-equilibrium model

– a general equilibrium model is needed to

look at many markets together

32


• Testing the validity of a model is a

difficult task

– are the model’s assumptions

reasonable?

– does the model explain real-world

events?

1

1

Chapter 3

PREFERENCES AND UTILITY

Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved. 2

Axioms of Rational Choice

• Completeness

– if A and B are any two situations, an

individual can always specify exactly one of

these possibilities:

• A is preferred to B

• B is preferred to A

• A and B are equally attractive

3


• Transitivity

– if A is preferred to B, and B is preferred to

C, then A is preferred to C

– assumes that the individual’s choices are

internally consistent

4


• Continuity

– if A is preferred to B, then situations suitably

“close to” A must also be preferred to B

– used to analyze individuals’ responses to

relatively small changes in income and

prices

2

5

Utility • Given these assumptions, it is possible to

show that people are able to rank in order

all possible situations from least desirable

to most

• Economists call this ranking utility

– if A is preferred to B, then the utility assigned

to A exceeds the utility assigned to B

U(A) > U(B)

6

Utility • Utility rankings are ordinal in nature

– they record the relative desirability of

commodity bundles

• Because utility measures are not unique,

it makes no sense to consider how much

more utility is gained from A than from B

• It is also impossible to compare utilities

between people

7

Utility • Utility is affected by the consumption of

physical commodities, psychological

attitudes, peer group pressures, personal

experiences, and the general cultural

environment

• Economists generally devote attention to

quantifiable options while holding

constant the other things that affect utility

– ceteris paribus assumption 8

Utility

• Assume that an individual must choose

among consumption goods x1, x2,…, xn

• The individual’s rankings can be shown

by a utility function of the form:

utility = U(x1, x2,…, xn; other things)

– this function is unique up to an order-

preserving transformation

3

9

Economic Goods • In the utility function, the x’s are assumed

to be “goods” – more is preferred to less

Quantity of x

Quantity of y

x*

y*

Preferred to x*, y*

?

? Worse

than

x*, y* 10

Indifference Curves • An indifference curve shows a set of

consumption bundles among which the

individual is indifferent

Quantity of x

Quantity of y

x1

y1

y2

x2

U1

Combinations (x1, y1) and (x2, y2)

provide the same level of utility

11

Marginal Rate of Substitution • The negative of the slope of the

indifference curve at any point is called

the marginal rate of substitution (MRS)

Quantity of x

Quantity of y

x1

y1

y2

x2

U1

1

UUdx

dyMRS

12

Marginal Rate of Substitution • MRS changes as x and y change

– reflects the individual’s willingness to trade y

for x

Quantity of x

Quantity of y

x1

y1

y2

x2

U1

At (x1, y1), the indifference curve is steeper.

The person would be willing to give up more

y to gain additional units of x

At (x2, y2), the indifference curve

is flatter. The person would be

willing to give up less y to gain

additional units of x

4

13

Indifference Curve Map • Each point must have an indifference

curve through it

Quantity of x

Quantity of y

U1 < U2 < U3

U1

U2

U3

Increasing utility

14

Transitivity • Can any two of an individual’s indifference

curves intersect?

Quantity of x

Quantity of y

U1

U2

A

B C

The individual is indifferent between A and C.

The individual is indifferent between B and C.

Transitivity suggests that the individual

should be indifferent between A and B

But B is preferred to A

because B contains more

x and y than A

15

Convexity • A set of points is convex if any two points

can be joined by a straight line that is

contained completely within the set

Quantity of x

Quantity of y

U1

The assumption of a diminishing MRS is

equivalent to the assumption that all

combinations of x and y which are

preferred to x* and y* form a convex set

x*

y*

16

Convexity • If the indifference curve is convex, then

the combination (x1 + x2)/2, (y1 + y2)/2 will

be preferred to either (x1,y1) or (x2,y2)

Quantity of x

Quantity of y

U1

x2

y1

y2

x1

This implies that “well-balanced” bundles are preferred

to bundles that are heavily weighted toward one

commodity

(x1 + x2)/2

(y1 + y2)/2

5

17

Utility and the MRS • Suppose an individual’s preferences for

hamburgers (y) and soft drinks (x) can

be represented by

yx 10 utility

• Solving for y, we get

y = 100/x

• Solving for MRS = -dy/dx:

MRS = -dy/dx = 100/x2

18

Utility and the MRS

MRS = -dy/dx = 100/x2

• Note that as x rises, MRS falls

– when x = 5, MRS = 4

– when x = 20, MRS = 0.25

19

Marginal Utility • Suppose that an individual has a utility

function of the form

utility = U(x,y)

• The total differential of U is

dy

y

Udx

x

UdU

• Along any indifference curve, utility is

constant (dU = 0)

20

Deriving the MRS • Therefore, we get:

y

Ux

U

dx

dyMRS

constantU

• MRS is the ratio of the marginal utility of

x to the marginal utility of y

6

21

Diminishing Marginal Utility and the MRS

• Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS – diminishing MRS requires that the utility

function be quasi-concave

• this is independent of how utility is measured

– diminishing marginal utility depends on how utility is measured

• Thus, these two concepts are different

22

Convexity of Indifference Curves

• Suppose that the utility function is

yx utility

• We can simplify the algebra by taking the

logarithm of this function

U*(x,y) = ln[U(x,y)] = 0.5 ln x + 0.5 ln y

23


x

y

y

x

y

Ux

U

MRS

5.0

5.0

*

*

• Thus,

24


• If the utility function is

U(x,y) = x + xy + y

• There is no advantage to transforming this utility function, so

x

y

y

Ux

U

MRS

1

1

7

25



22 utility yx

• For this example, it is easier to use the

transformation

U*(x,y) = [U(x,y)]2 = x2 + y2

26


y

x

y

x

y

Ux

U

MRS

2

2

*

*

• Thus,

27

Examples of Utility Functions

• Cobb-Douglas Utility

utility = U(x,y) = xy

where and are positive constants

– The relative sizes of and indicate the

relative importance of the goods

28


• Perfect Substitutes

utility = U(x,y) = x + y

Quantity of x

Quantity of y

U1 U2

U3

The indifference curves will be linear.

The MRS will be constant along the

indifference curve.

8

29


• Perfect Complements

utility = U(x,y) = min (x, y)

Quantity of x

Quantity of y The indifference curves will be

L-shaped. Only by choosing more

of the two goods together can utility

be increased.

U1

U2

U3

30

Examples of Utility Functions • CES Utility (Constant elasticity of

substitution)

utility = U(x,y) = x/ + y/

when 0 and

utility = U(x,y) = ln x + ln y

when = 0 – Perfect substitutes = 1

– Cobb-Douglas = 0

– Perfect complements = -

31

Examples of Utility Functions • CES Utility (Constant elasticity of

substitution)

– The elasticity of substitution () is equal to

1/(1 - )

• Perfect substitutes =

• Fixed proportions = 0

32

Homothetic Preferences

• If the MRS depends only on the ratio of

the amounts of the two goods, not on

the quantities of the goods, the utility

function is homothetic

– Perfect substitutes MRS is the same at

every point

– Perfect complements MRS = if y/x >

/, undefined if y/x = /, and MRS = 0 if

y/x < /

9

33

Homothetic Preferences

• For the general Cobb-Douglas function,

the MRS can be found as

x

y

yx

yx

y

Ux

U

MRS

1

1

34

Nonhomothetic Preferences

• Some utility functions do not exhibit

homothetic preferences

utility = U(x,y) = x + ln y

y

yy

Ux

U

MRS

1

1

35

The Many-Good Case

• Suppose utility is a function of n goods

given by

utility = U(x1, x2,…, xn)

• The total differential of U is

n

n

dxx

Udx

x

Udx

x

UdU

...2

2

1

1

36

The Many-Good Case

• We can find the MRS between any two

goods by setting dU = 0

j

i

i

j

ji

x

U

x

U

dx

dxxxMRS

) for (

j

j

i

i

dxx

Udx

x

UdU

0

• Rearranging, we get

10

37

Multigood Indifference Surfaces

• We will define an indifference surface

as being the set of points in n

dimensions that satisfy the equation

U(x1,x2,…xn) = k

where k is any preassigned constant

38

Multigood Indifference Surfaces

• If the utility function is quasi-concave,

the set of points for which U k will be

convex

– all of the points on a line joining any two

points on the U = k indifference surface will

also have U k

39

Important Points to Note: • If individuals obey certain behavioral

postulates, they will be able to rank all

commodity bundles

– the ranking can be represented by a utility

function

– in making choices, individuals will act as if

they were maximizing this function

• Utility functions for two goods can be

illustrated by an indifference curve map 40


• The negative of the slope of the

indifference curve measures the marginal

rate of substitution (MRS)

– the rate at which an individual would trade

an amount of one good (y) for one more unit

of another good (x)

• MRS decreases as x is substituted for y

– individuals prefer some balance in their

consumption choices

11

41


• A few simple functional forms can capture

important differences in individuals’

preferences for two (or more) goods

– Cobb-Douglas function

– linear function (perfect substitutes)

– fixed proportions function (perfect

complements)

– CES function

• includes the other three as special cases

42


• It is a simple matter to generalize from

two-good examples to many goods

– studying peoples’ choices among many

goods can yield many insights

– the mathematics of many goods is not

especially intuitive, so we will rely on two-

good cases to build intuition

1

1

Chapter 5

INCOME AND SUBSTITUTION

EFFECTS


Demand Functions • The optimal levels of x1,x2,…,xn can be

expressed as functions of all prices and

income

• These can be expressed as n demand

functions of the form:

x1* = d1(p1,p2,…,pn,I)

x2* = d2(p1,p2,…,pn,I) • • •

xn* = dn(p1,p2,…,pn,I)

3

Demand Functions • If there are only two goods (x and y), we

can simplify the notation

x* = x(px,py,I)

y* = y(px,py,I)

• Prices and income are exogenous

– the individual has no control over these

parameters

4

Homogeneity • If we were to double all prices and

income, the optimal quantities demanded

will not change

– the budget constraint is unchanged

xi* = di(p1,p2,…,pn,I) = di(tp1,tp2,…,tpn,tI)

• Individual demand functions are

homogeneous of degree zero in all prices

and income

2

5

Homogeneity • With a Cobb-Douglas utility function

utility = U(x,y) = x0.3y0.7

the demand functions are

• Note that a doubling of both prices and

income would leave x* and y*

unaffected

xpx

I3.0*

ypy

I7.0*

6

Homogeneity • With a CES utility function

utility = U(x,y) = x0.5 + y0.5

the demand functions are

• Note that a doubling of both prices and

income would leave x* and y*

unaffected

xyx pppx

I

/1

1*

yxy pppy

I

/1

1*

7

Changes in Income

• An increase in income will cause the

budget constraint out in a parallel

fashion

• Since px/py does not change, the MRS

will stay constant as the worker moves

to higher levels of satisfaction

8

Increase in Income • If both x and y increase as income rises,

x and y are normal goods

Quantity of x

Quantity of y

C

U3

B

U2

A

U1

As income rises, the individual chooses

to consume more x and y

3

9

Increase in Income • If x decreases as income rises, x is an

inferior good

Quantity of x

Quantity of y

C

U3

As income rises, the individual chooses

to consume less x and more y

Note that the indifference

curves do not have to be

“oddly” shaped. The

assumption of a diminishing

MRS is obeyed.

B

U2

A U1

10

Normal and Inferior Goods

• A good xi for which xi/I 0 over some

range of income is a normal good in that

range

• A good xi for which xi/I < 0 over some

range of income is an inferior good in

that range

11

Changes in a Good’s Price

• A change in the price of a good alters

the slope of the budget constraint

– it also changes the MRS at the consumer’s

utility-maximizing choices

• When the price changes, two effects

come into play

– substitution effect

– income effect

12


• Even if the individual remained on the same

indifference curve when the price changes,

his optimal choice will change because the

MRS must equal the new price ratio

– the substitution effect

• The price change alters the individual’s

“real” income and therefore he must move

to a new indifference curve

– the income effect

4

13


Quantity of x

Quantity of y

U1

A

Suppose the consumer is maximizing

utility at point A.

U2

B

If the price of good x falls, the consumer

will maximize utility at point B.

Total increase in x 14


U1

Quantity of x

Quantity of y

A

To isolate the substitution effect, we hold

“real” income constant but allow the

relative price of good x to change

Substitution effect

C

The substitution effect is the movement

from point A to point C

The individual substitutes

good x for good y

because it is now

relatively cheaper

15


U1

U2

Quantity of x

Quantity of y

A

The income effect occurs because the

individual’s “real” income changes when

the price of good x changes

C

Income effect

B

The income effect is the movement

from point C to point B

If x is a normal good,

the individual will buy

more because “real”

income increased

16


U2

U1

Quantity of x

Quantity of y

B

A

An increase in the price of good x means that

the budget constraint gets steeper

C The substitution effect is the

movement from point A to point C

Substitution effect

Income effect

The income effect is the

movement from point C

to point B

5

17

Price Changes for Normal Goods

• If a good is normal, substitution and

income effects reinforce one another

– when price falls, both effects lead to a rise in

quantity demanded

– when price rises, both effects lead to a drop

in quantity demanded

18

Price Changes for Inferior Goods

• If a good is inferior, substitution and

income effects move in opposite directions

• The combined effect is indeterminate

– when price rises, the substitution effect leads

to a drop in quantity demanded, but the

income effect is opposite

– when price falls, the substitution effect leads

to a rise in quantity demanded, but the

income effect is opposite

19

Giffen’s Paradox

• If the income effect of a price change is

strong enough, there could be a positive

relationship between price and quantity

demanded

– an increase in price leads to a drop in real

income

– since the good is inferior, a drop in income

causes quantity demanded to rise

20

A Summary • Utility maximization implies that (for normal

goods) a fall in price leads to an increase in

quantity demanded

– the substitution effect causes more to be purchased as the individual moves along an indifference curve

– the income effect causes more to be purchased

because the resulting rise in purchasing power

allows the individual to move to a higher

indifference curve

6

21

A Summary

• Utility maximization implies that (for normal

goods) a rise in price leads to a decline in

quantity demanded

– the substitution effect causes less to be

purchased as the individual moves along an

indifference curve

– the income effect causes less to be purchased

because the resulting drop in purchasing

power moves the individual to a lower

indifference curve 22

A Summary

• Utility maximization implies that (for inferior

goods) no definite prediction can be made

for changes in price

– the substitution effect and income effect move

in opposite directions

– if the income effect outweighs the substitution

effect, we have a case of Giffen’s paradox

23

The Individual’s Demand Curve

• An individual’s demand for x depends

on preferences, all prices, and income:

x* = x(px,py,I)

• It may be convenient to graph the

individual’s demand for x assuming that

income and the price of y (py) are held

constant

24

x

…quantity of x

demanded rises.


Quantity of y

Quantity of x Quantity of x

px

x’’

px’’

U2

x2

I = px’’ + py

x’

px’

U1

x1

I = px’ + py

x’’’

px’’’

x3

U3

I = px’’’ + py

As the price

of x falls...

7

25


• An individual demand curve shows the

relationship between the price of a good

and the quantity of that good purchased by

an individual assuming that all other

determinants of demand are held constant

26

Shifts in the Demand Curve

• Three factors are held constant when a

demand curve is derived

– income

– prices of other goods (py)

– the individual’s preferences

• If any of these factors change, the

demand curve will shift to a new position

27

Shifts in the Demand Curve

• A movement along a given demand

curve is caused by a change in the price

of the good

– a change in quantity demanded

• A shift in the demand curve is caused by

changes in income, prices of other

goods, or preferences

– a change in demand

28

Demand Functions and Curves

• If the individual’s income is $100, these

functions become

xpx

I3.0*

ypy

I7.0*

• We discovered earlier that

xpx

30*

ypy

70*

8

29

Demand Functions and Curves

• Any change in income will shift these

demand curves

30

Compensated Demand Curves

• The actual level of utility varies along

the demand curve

• As the price of x falls, the individual

moves to higher indifference curves

– it is assumed that nominal income is held

constant as the demand curve is derived

– this means that “real” income rises as the

price of x falls

31


• An alternative approach holds real income

(or utility) constant while examining

reactions to changes in px

– the effects of the price change are

“compensated” so as to constrain the

individual to remain on the same indifference

curve

– reactions to price changes include only

substitution effects

32

Compensated Demand Curves • A compensated (Hicksian) demand curve

shows the relationship between the price

of a good and the quantity purchased

assuming that other prices and utility are

held constant

• The compensated demand curve is a two-

dimensional representation of the

compensated demand function

x* = xc(px,py,U)

9

33

xc

…quantity demanded

rises.


Quantity of y

Quantity of x Quantity of x

px

U2

x’’

px’’

x’’

y

x

p

pslope

''

x’

px’

y

x

p

pslope

'

x’ x’’’

px’’’ y

x

p

pslope

'''

x’’’

Holding utility constant, as price falls...

34

Compensated & Uncompensated Demand

Quantity of x

px

x

xc

x’’

px’’

At px’’, the curves intersect because

the individual’s income is just sufficient

to attain utility level U2

35


Quantity of x

px

x

xc

px’’

x* x’

px’

At prices above px2, income

compensation is positive because the

individual needs some help to remain

on U2

36


Quantity of x

px

x

xc

px’’

x*** x’’’

px’’’

At prices below px2, income

compensation is negative to prevent an

increase in utility from a lower price

10

37


• For a normal good, the compensated

demand curve is less responsive to price

changes than is the uncompensated

demand curve

– the uncompensated demand curve reflects

both income and substitution effects

– the compensated demand curve reflects only

substitution effects

38

Compensated Demand Functions

• Suppose that utility is given by

utility = U(x,y) = x0.5y0.5

• The Marshallian demand functions are

x = I/2px y = I/2py

• The indirect utility function is

5.05.02

),,( utility yx

yxpp

ppVI

I

39


• To obtain the compensated demand

functions, we can solve the indirect

utility function for I and then substitute

into the Marshallian demand functions

5.0

5.0

x

y

p

Vpx 5.0

5.0

y

x

p

Vpy

40


• Demand now depends on utility (V)

rather than income

• Increases in px reduce the amount of x

demanded

– only a substitution effect

5.0

5.0

x

y

p

Vpx 5.0

5.0

y

x

p

Vpy

11

41

A Mathematical Examination of a Change in Price

• Our goal is to examine how purchases of

good x change when px changes

x/px

• Differentiation of the first-order conditions

from utility maximization can be performed

to solve for this derivative

• However, this approach is cumbersome

and provides little economic insight 42


• Instead, we will use an indirect approach

• Remember the expenditure function

minimum expenditure = E(px,py,U)

• Then, by definition

xc (px,py,U) = x [px,py,E(px,py,U)]

– quantity demanded is equal for both demand

functions when income is exactly what is

needed to attain the required utility level

43


• We can differentiate the compensated

demand function and get

xc (px,py,U) = x[px,py,E(px,py,U)]

xxx

c

p

E

E

x

p

x

p

x

xx

c

x p

E

E

x

p

x

p

x

44


• The first term is the slope of the

compensated demand curve

– the mathematical representation of the

substitution effect

xx

c

x p

E

E

x

p

x

p

x

12

45


• The second term measures the way in

which changes in px affect the demand

for x through changes in purchasing

power

– the mathematical representation of the

income effect

xx

c

x p

E

E

x

p

x

p

x

46

The Slutsky Equation

• The substitution effect can be written as

constant

effect onsubstituti

Uxx

c

p

x

p

x

• The income effect can be written as

xx p

Ex

p

E

E

x

I effect income

47


• Note that E/px = x

– a $1 increase in px raises necessary

expenditures by x dollars

– $1 extra must be paid for each unit of x

purchased

48

The Slutsky Equation • The utility-maximization hypothesis

shows that the substitution and income

effects arising from a price change can be

represented by

I

xx

p

x

p

x

p

x

Uxx

x

constant

effect income effect onsubstituti

13

49


• The first term is the substitution effect

– always negative as long as MRS is

diminishing

– the slope of the compensated demand curve

must be negative

I

xx

p

x

p

x

Uxx constant

50


• The second term is the income effect

– if x is a normal good, then x/I > 0

• the entire income effect is negative

– if x is an inferior good, then x/I < 0

• the entire income effect is positive

I

xx

p

x

p

x

Uxx constant

51

A Slutsky Decomposition

• We can demonstrate the decomposition

of a price effect using the Cobb-Douglas

example studied earlier

• The Marshallian demand function for

good x was

x

yxp

ppxI

I5.0

),,(

52


• The Hicksian (compensated) demand

function for good x was

5.0

5.0

),,(x

y

yx

c

p

VpVppx

• The overall effect of a price change on

the demand for x is

2

5.0

xx pp

x I

14

53


• This total effect is the sum of the two

effects that Slutsky identified

• The substitution effect is found by

differentiating the compensated demand

function

5.1

5.05.0 effect onsubstituti

x

y

x

c

p

Vp

p

x

54


• We can substitute in for the indirect utility

function (V)

25.1

5.05.05.025.0)5.0(5.0

effect onsubstitutixx

yyx

pp

ppp II

55


• Calculation of the income effect is easier

2

25.05.05.0 effect income

xxx ppp

xx

II

I

• Interestingly, the substitution and income

effects are exactly the same size

56

Marshallian Demand Elasticities

• Most of the commonly used demand

elasticities are derived from the

Marshallian demand function x(px,py,I)

• Price elasticity of demand (ex,px)

x

p

p

x

pp

xxe x

xxx

px x

/

/,

15

57

Marshallian Demand Elasticities

• Income elasticity of demand (ex,I)

x

xxxex

I

IIII

/

/,

• Cross-price elasticity of demand (ex,py)

x

p

p

x

pp

xxe

y

yyy

px y

/

/,

58

Price Elasticity of Demand

• The own price elasticity of demand is

always negative

– the only exception is Giffen’s paradox

• The size of the elasticity is important

– if ex,px < -1, demand is elastic

– if ex,px > -1, demand is inelastic

– if ex,px = -1, demand is unit elastic

59

Price Elasticity and Total Spending

• Total spending on x is equal to

total spending =pxx

• Using elasticity, we can determine how

total spending changes when the price of

x changes

]1[)(

,

xpx

x

x

x

x exxp

xp

p

xp

60

Price Elasticity and Total Spending

• The sign of this derivative depends on

whether ex,px is greater or less than -1

– if ex,px > -1, demand is inelastic and price and

total spending move in the same direction

– if ex,px < -1, demand is elastic and price and

total spending move in opposite directions

]1[)(

,

xpx

x

x

x

x exxp

xp

p

xp

16

61

Compensated Price Elasticities

• It is also useful to define elasticities

based on the compensated demand

function

62


• If the compensated demand function is

xc = xc(px,py,U)

we can calculate

– compensated own price elasticity of

demand (exc,px)

– compensated cross-price elasticity of

demand (exc,py)

63

Compensated Price Elasticities • The compensated own price elasticity of

demand (exc,px) is

c

x

x

c

xx

ccc

pxx

p

p

x

pp

xxe

x

/

/,

• The compensated cross-price elasticity

of demand (exc,py) is

c

y

y

c

yy

ccc

pxx

p

p

x

pp

xxe

y

/

/,

64


• The relationship between Marshallian

and compensated price elasticities can

be shown using the Slutsky equation

I

xx

x

p

p

x

x

pe

p

x

x

p x

x

c

c

xpx

x

x

x,

I,,, xx

c

pxpx eseexx

• If sx = pxx/I, then

17

65


• The Slutsky equation shows that the

compensated and uncompensated price

elasticities will be similar if

– the share of income devoted to x is small

– the income elasticity of x is small

66

Homogeneity

• Demand functions are homogeneous of

degree zero in all prices and income

• Euler’s theorem for homogenous

functions shows that

II

x

p

xp

p

xp

y

y

x

x 0

67

Homogeneity

• Dividing by x, we get

I,,,0 xpxpx eeeyx

• Any proportional change in all prices

and income will leave the quantity of x

demanded unchanged

68

Engel Aggregation

• Engel’s law suggests that the income

elasticity of demand for food items is

less than one

– this implies that the income elasticity of

demand for all nonfood items must be

greater than one

18

69

Engel Aggregation

• We can see this by differentiating the

budget constraint with respect to

income (treating prices as constant)

II

yp

xp yx1

III

I

II

I

I,,1 yyxxyx eses

y

yyp

x

xxp

70

Cournot Aggregation

• The size of the cross-price effect of a

change in the price of x on the quantity

of y consumed is restricted because of

the budget constraint

• We can demonstrate this by

differentiating the budget constraint with

respect to px

71

Cournot Aggregation

x

y

x

x

x p

ypx

p

xp

p

0

I

y

yp

p

yp

px

x

xp

p

xp x

x

yxx

x

x

III0

xx pyyxpxx esses ,,0

xpyypxx sesesxx

,,

72

Demand Elasticities

• The Cobb-Douglas utility function is

U(x,y) = xy (+=1)

• The demand functions for x and y are

xpx

I

ypy

I

19

73

Demand Elasticities

• Calculating the elasticities, we get

1 2,

x

x

x

x

x

px

p

p

px

p

p

xe

x I

I

00 ,

x

p

x

p

p

xe

yy

y

px y

1 ,

x

x

x

p

px

xe

I

II

II

74

Demand Elasticities

• We can also show

– homogeneity

0101,,, Ixpxpx eeeyx

– Engel aggregation

111,, II yyxx eses

– Cournot aggregation

xpyypxx sesesxx

0)1(,,

75

Demand Elasticities

• We can also use the Slutsky equation to

derive the compensated price elasticity

1)1(1,,, Ixxpx

c

px eseexx

• The compensated price elasticity

depends on how important other goods

(y) are in the utility function

76

Demand Elasticities

• The CES utility function (with = 2,

= 5) is

U(x,y) = x0.5 + y0.5

• The demand functions for x and y are

)1( 1

yxx pppx

I

)1( 1

yxy pppy

I

20

77

Demand Elasticities

• We will use the “share elasticity” to

derive the own price elasticity

xxx px

x

x

x

xps e

s

p

p

se ,, 1

• In this case,

11

1

yx

xx

pp

xps

I

78

Demand Elasticities

• Thus, the share elasticity is given by

1

1

1121

1

,1)1()1(

yx

yx

yx

x

yx

y

x

x

x

xps

pp

pp

pp

p

pp

p

s

p

p

se

xx

• Therefore, if we let px = py

5.1111

11,,

xxx pspx ee

79

Demand Elasticities

• The CES utility function (with = 0.5,

= -1) is

U(x,y) = -x -1 - y -1

• The share of good x is

5.05.01

1

xy

xx

pp

xps

I

80

Demand Elasticities

• Thus, the share elasticity is given by

5.05.0

5.05.0

15.05.025.05.0

5.15.0

,

1

5.0

)1()1(

5.0

xy

xy

xy

x

xy

xy

x

x

x

xps

pp

pp

pp

p

pp

pp

s

p

p

se

xx

• Again, if we let px = py

75.012

5.01,,

xxx pspx ee

21

81

Consumer Surplus

• An important problem in welfare

economics is to devise a monetary

measure of the gains and losses that

individuals experience when prices

change

82

Consumer Welfare • One way to evaluate the welfare cost of a

price increase (from px0 to px

1) would be

to compare the expenditures required to

achieve U0 under these two situations

expenditure at px0 = E0 = E(px

0,py,U0)

expenditure at px1 = E1 = E(px

1,py,U0)

83

Consumer Welfare

• In order to compensate for the price rise,

this person would require a

compensating variation (CV) of

CV = E(px1,py,U0) - E(px

0,py,U0)

84

Consumer Welfare

Quantity of x

Quantity of y

U1

A

Suppose the consumer is maximizing

utility at point A.

U2

B

If the price of good x rises, the consumer

will maximize utility at point B.

The consumer’s utility

falls from U1 to U2

22

85

Consumer Welfare

Quantity of x

Quantity of y

U1

A

U2

B

CV is the amount that the

individual would need to be

compensated

The consumer could be compensated so

that he can afford to remain on U1

C

86

Consumer Welfare

• The derivative of the expenditure function

with respect to px is the compensated

demand function

),,(),,(

0

0Uppx

p

UppEyx

c

x

yx

87

Consumer Welfare

• The amount of CV required can be found

by integrating across a sequence of

small increments to price from px0 to px

1

1

0

1

0

),,( 0

x

x

x

x

p

p

p

p

xyx

c dpUppxdECV

– this integral is the area to the left of the

compensated demand curve between px0

and px1

88

welfare loss

Consumer Welfare

Quantity of x

px

xc(px…U0)

px1

x1

px0

x0

When the price rises from px0 to px

1,

the consumer suffers a loss in welfare

23

89

Consumer Welfare

• Because a price change generally

involves both income and substitution

effects, it is unclear which compensated

demand curve should be used

• Do we use the compensated demand

curve for the original target utility (U0) or

the new level of utility after the price

change (U1)?

90

The Consumer Surplus Concept

• Another way to look at this issue is to

ask how much the person would be

willing to pay for the right to consume all

of this good that he wanted at the

market price of px0

91

The Consumer Surplus Concept

• The area below the compensated

demand curve and above the market

price is called consumer surplus

– the extra benefit the person receives by

being able to make market transactions at

the prevailing market price

92

Consumer Welfare

Quantity of x

px

xc(...U0)

px1

x1

When the price rises from px0 to px

1, the actual

market reaction will be to move from A to C

xc(...U1)

x(px…)

A

C

px0

x0

The consumer’s utility falls from U0 to U1

24

93

Consumer Welfare

Quantity of x

px

xc(...U0)

px1

x1

Is the consumer’s loss in welfare

best described by area px1BApx

0

[using xc(...U0)] or by area px1CDpx

0 [using xc(...U1)]?

xc(...U1)

A

B C

D

px0

x0

Is U0 or U1 the

appropriate utility

target?

94

Consumer Welfare

Quantity of x

px

xc(...U0)

px1

x1

We can use the Marshallian demand

curve as a compromise

xc(...U1)

x(px…)

A

B C

D

px0

x0

The area px1CApx

0

falls between the

sizes of the welfare

losses defined by

xc(...U0) and

xc(...U1)

95

Consumer Surplus

• We will define consumer surplus as the

area below the Marshallian demand

curve and above price

– shows what an individual would pay for the

right to make voluntary transactions at this

price

– changes in consumer surplus measure the

welfare effects of price changes

96

Welfare Loss from a Price Increase

• Suppose that the compensated demand

function for x is given by

5.0

5.0

),,(x

y

yx

c

p

VpVppx

• The welfare cost of a price increase

from px = 1 to px = 4 is given by

4

1

5.05.0

4

1

5.05.0 2

x

X

p

pxyxy pVppVpCV

25

97


• If we assume that V = 2 and py = 2,

CV = 222(4)0.5 – 222(1)0.5 = 8

• If we assume that the utility level (V) falls to 1 after the price increase (and used this level to calculate welfare loss),

CV = 122(4)0.5 – 122(1)0.5 = 4

98

Welfare Loss from Price Increase

• Suppose that we use the Marshallian

demand function instead

15.0),,( -

xyx pppx II

• The welfare loss from a price increase

from px = 1 to px = 4 is given by

4

1

1

4

1

ln5.05.0

x

x

p

pxx

-

x pdppLoss II

99


• If income (I) is equal to 8,

loss = 4 ln(4) - 4 ln(1) = 4 ln(4) = 4(1.39) = 5.55

– this computed loss from the Marshallian demand function is a compromise between the two amounts computed using the compensated demand functions

100

Revealed Preference and the Substitution Effect

• The theory of revealed preference was

proposed by Paul Samuelson in the late

1940s

• The theory defines a principle of

rationality based on observed behavior

and then uses it to approximate an

individual’s utility function

26

101


• Consider two bundles of goods: A and B

• If the individual can afford to purchase

either bundle but chooses A, we say that

A had been revealed preferred to B

• Under any other price-income

arrangement, B can never be revealed

preferred to A

102


Quantity of x

Quantity of y

A

I1

Suppose that, when the budget constraint is

given by I1, A is chosen

B

I3

A must still be preferred to B when income

is I3 (because both A and B are available)

I2

If B is chosen, the budget

constraint must be similar to

that given by I2 where A is not

available

103

Negativity of the Substitution Effect

• Suppose that an individual is indifferent

between two bundles: C and D

• Let pxC,py

C be the prices at which

bundle C is chosen

• Let pxD,py

D be the prices at which

bundle D is chosen

104


• Since the individual is indifferent between

C and D

– When C is chosen, D must cost at least as

much as C

pxCxC + py

CyC ≤ pxCxD + py

CyD

– When D is chosen, C must cost at least as

much as D

pxDxD + py

DyD ≤ pxDxC + py

DyC

27

105


• Rearranging, we get

pxC(xC - xD) + py

C(yC -yD) ≤ 0

pxD(xD - xC) + py

D(yD -yC) ≤ 0

• Adding these together, we get

(pxC – px

D)(xC - xD) + (pyC – py

D)(yC - yD) ≤ 0

106


• Suppose that only the price of x changes

(pyC = py

D)

(pxC – px

D)(xC - xD) ≤ 0

• This implies that price and quantity move

in opposite direction when utility is held

constant

– the substitution effect is negative

107

Mathematical Generalization

• If, at prices pi0 bundle xi

0 is chosen

instead of bundle xi1 (and bundle xi

1 is

affordable), then

n

i

n

i

iiii xpxp1 1

1000

• Bundle 0 has been “revealed preferred”

to bundle 1

108

Mathematical Generalization

• Consequently, at prices that prevail

when bundle 1 is chosen (pi1), then

n

i

n

i

iiii xpxp1 1

1101

• Bundle 0 must be more expensive than

bundle 1

28

109

Strong Axiom of Revealed Preference

• If commodity bundle 0 is revealed

preferred to bundle 1, and if bundle 1 is

revealed preferred to bundle 2, and if

bundle 2 is revealed preferred to bundle

3,…,and if bundle K-1 is revealed

preferred to bundle K, then bundle K

cannot be revealed preferred to bundle 0

110


• Proportional changes in all prices and

income do not shift the individual’s

budget constraint and therefore do not

alter the quantities of goods chosen

– demand functions are homogeneous of

degree zero in all prices and income

111


• When purchasing power changes

(income changes but prices remain the

same), budget constraints shift

– for normal goods, an increase in income

means that more is purchased

– for inferior goods, an increase in income

means that less is purchased

112


• A fall in the price of a good causes

substitution and income effects

– for a normal good, both effects cause more

of the good to be purchased

– for inferior goods, substitution and income

effects work in opposite directions

• no unambiguous prediction is possible

29

113


• A rise in the price of a good also

causes income and substitution effects

– for normal goods, less will be demanded

– for inferior goods, the net result is

ambiguous

114


• The Marshallian demand curve

summarizes the total quantity of a good

demanded at each possible price

– changes in price prompt movements

along the curve

– changes in income, prices of other goods,

or preferences may cause the demand

curve to shift

115


• Compensated demand curves illustrate

movements along a given indifference

curve for alternative prices

– they are constructed by holding utility

constant and exhibit only the substitution

effects from a price change

– their slope is unambiguously negative (or

zero)

116


• Demand elasticities are often used in

empirical work to summarize how

individuals react to changes in prices

and income

– the most important is the price elasticity of

demand

• measures the proportionate change in quantity

in response to a 1 percent change in price

30

117


• There are many relationships among

demand elasticities

– own-price elasticities determine how a

price change affects total spending on a

good

– substitution and income effects can be

summarized by the Slutsky equation

– various aggregation results hold among

elasticities 118


• Welfare effects of price changes can

be measured by changing areas below

either compensated or ordinary

demand curves

– such changes affect the size of the

consumer surplus that individuals receive

by being able to make market transactions

119


• The negativity of the substitution effect

is one of the most basic findings of

demand theory

– this result can be shown using revealed

preference theory and does not

necessarily require assuming the

existence of a utility function

1

1

Chapter 6

DEMAND RELATIONSHIPS

AMONG GOODS


The Two-Good Case

• The types of relationships that can

occur when there are only two goods

are limited

• But this case can be illustrated with two-

dimensional graphs

3

Gross Complements

Quantity of x

Quantity of y

x1 x0

y1

y0

U1

U0

When the price of y falls, the substitution

effect may be so small that the consumer

purchases more x and more y

In this case, we call x and y gross

complements

x/py < 0

4

Gross Substitutes

Quantity of x

Quantity of y

In this case, we call x and y gross

substitutes

x1 x0

y1

y0

U0

When the price of y falls, the substitution

effect may be so large that the consumer

purchases less x and more y

U1

x/py > 0

2

5

A Mathematical Treatment

• The change in x caused by changes in py

can be shown by a Slutsky-type equation

I

xy

p

x

p

x

Uyy

constant

substitution

effect (+)

income effect

(-) if x is normal

combined effect

(ambiguous) 6

Substitutes and Complements

• For the case of many goods, we can

generalize the Slutsky analysis

I

ij

Uj

i

j

i xx

p

x

p

x

constant

for any i or j

– this implies that the change in the price of

any good induces income and substitution

effects that may change the quantity of

every good demanded

7

Substitutes and Complements

• Two goods are substitutes if one good

may replace the other in use

– examples: tea & coffee, butter & margarine

• Two goods are complements if they are

used together

– examples: coffee & cream, fish & chips

8

Gross Substitutes and Complements

• The concepts of gross substitutes and

complements include both substitution

and income effects

– two goods are gross substitutes if

xi /pj > 0

– two goods are gross complements if

xi /pj < 0

3

9

Asymmetry of the Gross Definitions

• One undesirable characteristic of the gross

definitions of substitutes and complements

is that they are not symmetric

• It is possible for x1 to be a substitute for x2

and at the same time for x2 to be a

complement of x1

10


• Suppose that the utility function for two

goods is given by

U(x,y) = ln x + y

• Setting up the Lagrangian

L = ln x + y + (I – pxx – pyy)

11


gives us the following first-order conditions:

L/x = 1/x - px = 0

L/y = 1 - py = 0

L/ = I - pxx - pyy = 0

• Manipulating the first two equations, we get

pxx = py

12


• Inserting this into the budget constraint, we

can find the Marshallian demand for y

pyy = I – py

– an increase in py causes a decline in spending

on y

• since px and I are unchanged, spending on x must

rise ( x and y are gross substitutes)

• but spending on y is independent of px ( x and y

are independent of one another)

4

13

Net Substitutes and Complements

• The concepts of net substitutes and complements focuses solely on substitution effects

– two goods are net substitutes if

0

constant

Uj

i

p

x

0

constant

Uj

i

p

x

– two goods are net complements if

14

Net Substitutes and Complements

• This definition looks only at the shape of

the indifference curve

• This definition is unambiguous because

the definitions are perfectly symmetric

constantconstant

Ui

j

Uj

i

p

x

p

x

15

Gross Complements

Quantity of x

Quantity of y

x1 x0

y1

y0

U1

U0

Even though x and y are gross

complements, they are net substitutes

Since MRS is diminishing,

the own-price substitution

effect must be negative so

the cross-price substitution

effect must be positive

16

Substitutability with Many Goods

• Once the utility-maximizing model is

extended to may goods, a wide variety

of demand patterns become possible

• According to Hicks’ second law of

demand, “most” goods must be

substitutes

5

17


• To prove this, we can start with the

compensated demand function

xc(p1,…pn,V)

• Applying Euler’s theorem yields

0...2

2

1

1

n

c

in

c

i

c

i

p

xp

p

xp

p

xp

18


• In elasticity terms, we get

0...21 c

in

c

i

c

i eee

• Since the negativity of the substitution

effect implies that eiic 0, it must be the

case that

0ij

c

ije

19

Composite Commodities

• In the most general case, an individual

who consumes n goods will have

demand functions that reflect n(n+1)/2

different substitution effects

• It is often convenient to group goods

into larger aggregates

– examples: food, clothing, “all other goods”

20

Composite Commodity Theorem

• Suppose that consumers choose among n

goods

• The demand for x1 will depend on the

prices of the other n-1 commodities

• If all of these prices move together, it may

make sense to lump them into a single

composite commodity (y)

6

21


• Let p20…pn

0 represent the initial prices of

these other commodities

– assume that they all vary together (so that the

relative prices of x2…xn do not change)

• Define the composite commodity y to be

total expenditures on x2…xn at the initial

prices

y = p20x2 + p3

0x3 +…+ pn0xn

22


• The individual’s budget constraint is

I = p1x1 + p20x2 +…+ pn

0xn = p1x1 + y

• If we assume that all of the prices p20…pn

0

change by the same factor (t > 0) then the

budget constraint becomes

I = p1x1 + tp20x2 +…+ tpn

0xn = p1x1 + ty

– changes in p1 or t induce substitution effects

23


• As long as p20…pn

0 move together, we can

confine our examination of demand to

choices between buying x1 and

“everything else”

• The theorem makes no prediction about

how choices of x2…xn behave

– only focuses on total spending on x2…xn

24

Composite Commodity

• A composite commodity is a group of

goods for which all prices move together

• These goods can be treated as a single

commodity

– the individual behaves as if he is choosing

between other goods and spending on this

entire composite group

7

25

Example: Composite Commodity

• Suppose that an individual receives

utility from three goods:

– food (x)

– housing services (y), measured in

hundreds of square feet

– household operations (z), measured by

electricity use

• Assume a CES utility function 26


• The Lagrangian technique can be used

to derive demand functions

zyxzyxU

111 ),,( utility

zxyxx pppppx

I

zyxyy pppppy

I

yzxzz pppppz

I

27


• If initially I = 100, px = 1, py = 4, and pz = 1, then

• x* = 25, y* = 12.5, z* = 25

– $25 is spent on food and $75 is spent on

housing-related needs

28


• If we assume that the prices of housing

services (py) and electricity (pz) move

together, we can use their initial prices to

define the “composite commodity”

housing (h)

h = 4y + 1z

• The initial quantity of housing is the total

spent on housing (75)

8

29


• If I = 100, px = 1, py = 4, and ph = 1, then

x* = 25 and spending on housing (h*) =

75

hxy pppx

3

I

• Now x can be shown as a function of I, px, and ph

30


• If py rises to 16 and pz rises to 4 (with px

remaining at 1), ph would also rise to 4

• The demand for x would fall to

7

100

431

100*

x

• Housing purchases would be given by

7

600

7

100100* hPh

31


• Since ph = 4, h* = 150/7

• If I = 100, px = 1, py = 16, and pz = 4, the

individual demand functions show that

x* = 100/7, y* = 100/28, z* = 100/14

• This means that the amount of h that is

consumed can also be computed as

h* = 4y* + 1z* = 150/7 32

Household Production Model • Assume that individuals do not receive

utility directly from the goods they

purchase in the market

• Utility is received when the individual

produces goods by combining market

goods with time inputs

– raw beef and uncooked potatoes yield no

utility until they are cooked together to

produce stew

9

33

Household Production Model • Assume that there are three goods that

a person might want to purchase in the

market: x, y, and z

– these goods provide no direct utility

– these goods can be combined by the

individual to produce either of two home-

produced goods: a1 or a2

• the technology of this household production

can be represented by a production function

34

Household Production Model • The individual’s goal is to choose x,y,

and z so as to maximize utility

utility = U(a1,a2)

subject to the production functions

a1 = f1(x,y,z)

a2 = f2(x,y,z)

and a financial budget constraint

pxx + pyy + pzz = I

35

Household Production Model • Two important insights from this general

model can be drawn

– because the production functions are

measurable, households can be treated as

“multi-product” firms

– because consuming more a1 requires more

use of x, y, and z, this activity has an

opportunity cost in terms of the amount of a2

that can be produced

36

The Linear Attributes Model • In this model, it is the attributes of

goods that provide utility to individuals

• Each good has a fixed set of attributes

• The model assumes that the production

equations for a1 and a2 have the form

a1 = ax1x + ay

1y + az1z

a2 = ax2x + ay

2y + az2z

10

37

The Linear Attributes Model

a1

a2

0

x

The ray 0x shows the combinations of a1 and a2

available from successively larger amounts of good x

y

The ray 0y shows the combinations of

a1 and a2 available from successively

larger amounts of good y

z

The ray 0z shows the

combinations of a1 and

a2 available from

successively larger

amounts of good z

38


• If the individual spends all of his or her

income on good x

x* = I/px

• That will yield

a1* = ax1x* = (ax

1I)/px

a2* = ax2x* = (ax

2I)/px

39


a1

a2

0

x

y

z

x* is the combination of a1 and a2 that would be

obtained if all income was spent on x

x*

y*

y* is the combination of a1 and a2 that

would be obtained if all income was

spent on y

Z*

z* is the combination of

a1 and a2 that would be

obtained if all income was

spent on z

40


a1

a2

0

x

y

z

x*

y*

z*

All possible combinations from mixing the

three market goods are represented by

the shaded triangular area x*y*z*

11

41

A utility-maximizing individual would never

consume positive quantities of all three

goods


a1

a2

0

x

y

z

Individuals with a preference toward

a1 will have indifference curves similar

to U0 and will consume only y and z

U0

Individuals with a preference

toward a0 will have

indifference curves similar

to U1 and will consume only

x and y

U1

42

The Linear Attributes Model • The model predicts that corner solutions

(where individuals consume zero amounts

of some commodities) will be relatively

common

– especially in cases where individuals attach

value to fewer attributes than there are

market goods to choose from

• Consumption patterns may change

abruptly if income, prices, or preferences

change

43


• When there are only two goods, the

income and substitution effects from the

change in the price of one good (py) on

the demand for another good (x) usually

work in opposite directions

– the sign of x/py is ambiguous

• the substitution effect is positive

• the income effect is negative

44


• In cases of more than two goods,

demand relationships can be specified

in two ways

– two goods are gross substitutes if xi /pj

> 0 and gross complements if xi /pj < 0

– because these price effects include

income effects, they may not be

symmetric

• it is possible that xi /pj xj /pi

12

45


• Focusing only on the substitution

effects from price changes does

provide a symmetric definition

– two goods are net substitutes if xi c/pj >

0 and net complements if xi c/pj < 0

– because xic /pj = xj

c /pi, there is no

ambiguity

– Hicks’ second law of demand shows that

net substitutes are more prevalent

46


• If a group of goods has prices that

always move in unison, expenditures

on these goods can be treated as a

“composite commodity” whose “price”

is given by the size of the proportional

change in the composite goods’ prices

47


• An alternative way to develop the

theory of choice among market goods

is to focus on the ways in which

market goods are used in household

production to yield utility-providing

attributes

– this may provide additional insights into

relationships among goods

1

1

Chapter 9

PRODUCTION FUNCTIONS


Production Function

• The firm’s production function for a

particular good (q) shows the maximum

amount of the good that can be produced

using alternative combinations of capital

(k) and labor (l)

q = f(k,l)

3

Marginal Physical Product • To study variation in a single input, we

define marginal physical product as the

additional output that can be produced by

employing one more unit of that input

while holding other inputs constant

kk fk

qMP

capital of product physical marginal

lll

fq

MP

labor of product physical marginal

4

Diminishing Marginal Productivity

• The marginal physical product of an input

depends on how much of that input is

used

• In general, we assume diminishing

marginal productivity

0112

2

ff

k

f

k

MPkk

k 0222

2

ff

fMPll

l

ll

2

5

Diminishing Marginal Productivity

• Because of diminishing marginal

productivity, 19th century economist

Thomas Malthus worried about the effect

of population growth on labor productivity

• But changes in the marginal productivity of

labor over time also depend on changes in

other inputs such as capital

– we need to consider flk which is often > 0

6

Average Physical Product

• Labor productivity is often measured by

average productivity

l

l

ll

),(

input labor

output kfqAP

• Note that APl also depends on the

amount of capital employed

7

A Two-Input Production Function

• Suppose the production function for

flyswatters can be represented by

q = f(k,l) = 600k 2l2 - k 3l3

• To construct MPl and APl, we must

assume a value for k

– let k = 10

• The production function becomes

q = 60,000l2 - 1000l3 8


• The marginal productivity function is

MPl = q/l = 120,000l - 3000l2

which diminishes as l increases

• This implies that q has a maximum value:

120,000l - 3000l2 = 0

40l = l2

l = 40

• Labor input beyond l = 40 reduces output

3

9


• To find average productivity, we hold

k=10 and solve

APl = q/l = 60,000l - 1000l2

• APl reaches its maximum where

APl/l = 60,000 - 2000l = 0

l = 30

10


• In fact, when l = 30, both APl and MPl are

equal to 900,000

• Thus, when APl is at its maximum, APl

and MPl are equal

11

Isoquant Maps

• To illustrate the possible substitution of

one input for another, we use an

isoquant map

• An isoquant shows those combinations

of k and l that can produce a given level

of output (q0)

f(k,l) = q0

12

Isoquant Map

l per period

k per period

• Each isoquant represents a different level

of output

– output rises as we move northeast

q = 30

q = 20

4

13

Marginal Rate of Technical Substitution (RTS)

l per period

k per period

q = 20

- slope = marginal rate of technical

substitution (RTS)

• The slope of an isoquant shows the rate

at which l can be substituted for k

lA

kA

kB

lB

A

B

RTS > 0 and is diminishing for

increasing inputs of labor

14

Marginal Rate of Technical

Substitution (RTS)

• The marginal rate of technical

substitution (RTS) shows the rate at

which labor can be substituted for

capital while holding output constant

along an isoquant

0

) for ( qqd

dkkRTS

ll

15

RTS and Marginal Productivities • Take the total differential of the production

function:

dkMPdMPdkk

fd

fdq k

ll

ll

• Along an isoquant dq = 0, so

dkMPdMP k ll

kqq MP

MP

d

dkkRTS l

ll

0

) for (

16

RTS and Marginal Productivities

• Because MPl and MPk will both be

nonnegative, RTS will be positive (or zero)

• However, it is generally not possible to

derive a diminishing RTS from the

assumption of diminishing marginal

productivity alone

5

17


• To show that isoquants are convex, we

would like to show that d(RTS)/dl < 0

• Since RTS = fl/fk

ll

l

d

ffd

d

dRTS k )/(

2)(

)]/()/([

k

kkkkk

f

ddkfffddkfff

d

dRTS ll

l

lllll

18

RTS and Marginal Productivities • Using the fact that dk/dl = -fl/fk along an

isoquant and Young’s theorem (fkl = flk)

3

22

)(

)2(

k

kkkkk

f

fffffff

d

dRTS lllll

l

• Because we have assumed fk > 0, the

denominator is positive

• Because fll and fkk are both assumed to be

negative, the ratio will be negative if fkl is

positive

19


• Intuitively, it seems reasonable that fkl = flk

should be positive

– if workers have more capital, they will be

more productive

• But some production functions have fkl < 0

over some input ranges

– when we assume diminishing RTS we are

assuming that MPl and MPk diminish quickly

enough to compensate for any possible

negative cross-productivity effects 20

A Diminishing RTS • Suppose the production function is

q = f(k,l) = 600k 2l 2 - k 3l 3

• For this production function

MPl = fl = 1200k 2l - 3k 3l 2

MPk = fk = 1200kl 2 - 3k 2l 3

– these marginal productivities will be

positive for values of k and l for which

kl < 400

6

21

A Diminishing RTS

• Because

fll = 1200k 2 - 6k 3l

fkk = 1200l 2 - 6kl 3

this production function exhibits

diminishing marginal productivities for

sufficiently large values of k and l

– fll and fkk < 0 if kl > 200

22

A Diminishing RTS

• Cross differentiation of either of the

marginal productivity functions yields

fkl = flk = 2400kl - 9k 2l 2

which is positive only for kl < 266

23

A Diminishing RTS

• Thus, for this production function, RTS is

diminishing throughout the range of k and l

where marginal productivities are positive

– for higher values of k and l, the diminishing

marginal productivities are sufficient to

overcome the influence of a negative value for

fkl to ensure convexity of the isoquants

24

Returns to Scale

• How does output respond to increases

in all inputs together?

– suppose that all inputs are doubled, would

output double?

• Returns to scale have been of interest

to economists since the days of Adam

Smith

7

25

Returns to Scale

• Smith identified two forces that come

into operation as inputs are doubled

– greater division of labor and specialization

of function

– loss in efficiency because management

may become more difficult given the larger

scale of the firm

26

Returns to Scale

• If the production function is given by q =

f(k,l) and all inputs are multiplied by the

same positive constant (t >1), then

Effect on Output Returns to Scale

f(tk,tl) = tf(k,l) Constant

f(tk,tl) < tf(k,l) Decreasing

f(tk,tl) > tf(k,l) Increasing

27

Returns to Scale

• It is possible for a production function to

exhibit constant returns to scale for some

levels of input usage and increasing or

decreasing returns for other levels

– economists refer to the degree of returns to

scale with the implicit notion that only a

fairly narrow range of variation in input

usage and the related level of output is

being considered

28

Constant Returns to Scale • Constant returns-to-scale production

functions are homogeneous of degree

one in inputs

f(tk,tl) = t1f(k,l) = tq

• This implies that the marginal

productivity functions are homogeneous

of degree zero

– if a function is homogeneous of degree k,

its derivatives are homogeneous of degree

k-1

8

29

Constant Returns to Scale

• The marginal productivity of any input

depends on the ratio of capital and labor

(not on the absolute levels of these

inputs)

• The RTS between k and l depends only

on the ratio of k to l, not the scale of

operation

30


• The production function will be

homothetic

• Geometrically, all of the isoquants are

radial expansions of one another

31


l per period

k per period

• Along a ray from the origin (constant k/l), the RTS will be the same on all isoquants

q = 3

q = 2

q = 1

The isoquants are equally

spaced as output expands

32

Returns to Scale • Returns to scale can be generalized to a

production function with n inputs

q = f(x1,x2,…,xn)

• If all inputs are multiplied by a positive

constant t, we have

f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq

– If k = 1, we have constant returns to scale

– If k < 1, we have decreasing returns to scale

– If k > 1, we have increasing returns to scale

9

33

Elasticity of Substitution • The elasticity of substitution () measures

the proportionate change in k/l relative to

the proportionate change in the RTS along

an isoquant

RTS

k

k

RTS

dRTS

kd

RTS

k

ln

)/ln(

/

)/(

%

)/(%

l

l

ll

• The value of will always be positive

because k/l and RTS move in the same

direction 34

Elasticity of Substitution

l per period

k per period

• Both RTS and k/l will change as we

move from point A to point B

A

B q = q0

RTSA

RTSB

(k/l)A

(k/l)B

is the ratio of these

proportional changes

measures the

curvature of the

isoquant

35


• If is high, the RTS will not change

much relative to k/l

– the isoquant will be relatively flat

• If is low, the RTS will change by a

substantial amount as k/l changes

– the isoquant will be sharply curved

• It is possible for to change along an

isoquant or as the scale of production

changes 36


• Generalizing the elasticity of substitution

to the many-input case raises several

complications

– if we define the elasticity of substitution

between two inputs to be the proportionate

change in the ratio of the two inputs to the

proportionate change in RTS, we need to

hold output and the levels of other inputs

constant

10

37

The Linear Production Function

• Suppose that the production function is

q = f(k,l) = ak + bl

• This production function exhibits constant

returns to scale

f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l)

• All isoquants are straight lines

– RTS is constant

– =

38

The Linear Production Function

l per period

k per period

q1 q2 q3

Capital and labor are perfect substitutes

RTS is constant as k/l changes

slope = -b/a =

39

Fixed Proportions


q = min (ak,bl) a,b > 0

• Capital and labor must always be used

in a fixed ratio

– the firm will always operate along a ray

where k/l is constant

• Because k/l is constant, = 0

40

Fixed Proportions

l per period

k per period

q1

q2

q3

No substitution between labor and capital

is possible

= 0

k/l is fixed at b/a

q3/b

q3/a

11

41

Cobb-Douglas Production Function


q = f(k,l) = Akalb A,a,b > 0

• This production function can exhibit any

returns to scale

f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l)

– if a + b = 1 constant returns to scale

– if a + b > 1 increasing returns to scale

– if a + b < 1 decreasing returns to scale

42

Cobb-Douglas Production Function

• The Cobb-Douglas production function is

linear in logarithms

ln q = ln A + a ln k + b ln l

– a is the elasticity of output with respect to k

– b is the elasticity of output with respect to l

43

CES Production Function • Suppose that the production function is

q = f(k,l) = [k + l] / 1, 0, > 0

– > 1 increasing returns to scale

– < 1 decreasing returns to scale

• For this production function

= 1/(1-)

– = 1 linear production function

– = - fixed proportions production function

– = 0 Cobb-Douglas production function 44

A Generalized Leontief Production Function


q = f(k,l) = k + l + 2(kl)0.5

• Marginal productivities are

fk = 1 + (k/l)-0.5

fl = 1 + (k/l)0.5

• Thus,

5.0

5.0

)/(1

)/(1

l

ll

k

k

f

fRTS

k

12

45

Technical Progress

• Methods of production change over time

• Following the development of superior

production techniques, the same level

of output can be produced with fewer

inputs

– the isoquant shifts in

46

Technical Progress


q = A(t)f(k,l)

where A(t) represents all influences that

go into determining q other than k and l

– changes in A over time represent technical

progress

• A is shown as a function of time (t)

• dA/dt > 0

47

Technical Progress

• Differentiating the production function

with respect to time we get

dt

kdfAkf

dt

dA

dt

dq ),(),(

ll

dt

df

dt

dk

k

f

kf

q

A

q

dt

dA

dt

dq l

ll),(

48

Technical Progress

• Dividing by q gives us

dt

d

kf

f

dt

dk

kf

kf

A

dtdA

q

dtdq l

l

l

l

),(

/

),(

///

l

l

l

l

ll

dtd

kf

f

k

dtdk

kf

k

k

f

A

dtdA

q

dtdq /

),(

/

),(

//

13

49

Technical Progress

• For any variable x, [(dx/dt)/x] is the

proportional growth rate in x

– denote this by Gx

• Then, we can write the equation in terms

of growth rates

l

l

l

llG

kf

fG

kf

k

k

fGG kAq

),(),(

50

Technical Progress

• Since

llGeGeGG qkkqAq ,,

kqeq

k

k

q

kf

k

k

f,

),(

l

l

l

ll

l

l,

),(qe

q

q

kf

f

51

Technical Progress in the Cobb-Douglas Function


q = A(t)f(k,l) = A(t)k l 1-

• If we assume that technical progress

occurs at a constant exponential () then

A(t) = Ae-t

q = Ae-tk l 1-

52


• Taking logarithms and differentiating

with respect to t gives the growth

equation

qGq

tq

t

q

q

q

t

q

/lnln

14

53


l

l

l

GGtt

kt

ktAG

k

q

)1(ln

)1(ln

)ln)1(ln(ln

54


• If all but one of the inputs are held

constant, a relationship between the

single variable input and output can be

derived

– the marginal physical productivity is the

change in output resulting from a one-unit

increase in the use of the input

• assumed to decline as use of the input

increases

55


• The entire production function can be

illustrated by an isoquant map

– the slope of an isoquant is the marginal

rate of technical substitution (RTS)

• it shows how one input can be substituted for

another while holding output constant

• it is the ratio of the marginal physical

productivities of the two inputs

56


• Isoquants are usually assumed to be

convex

– they obey the assumption of a diminishing

RTS

• this assumption cannot be derived exclusively

from the assumption of diminishing marginal

productivity

• one must be concerned with the effect of

changes in one input on the marginal

productivity of other inputs

15

57


• The returns to scale exhibited by a

production function record how output

responds to proportionate increases in

all inputs

– if output increases proportionately with input

use, there are constant returns to scale

58


• The elasticity of substitution ()

provides a measure of how easy it is to

substitute one input for another in

production

– a high implies nearly straight isoquants

– a low implies that isoquants are nearly

L-shaped

59


• Technical progress shifts the entire

production function and isoquant map

– technical improvements may arise from the

use of more productive inputs or better

methods of economic organization

1

1

Chapter 10

COST FUNCTIONS

Copyright ©2005 by South-western, a division of Thomson learning. All rights reserved. 2

Definitions of Costs

• It is important to differentiate between

accounting cost and economic cost

– the accountant’s view of cost stresses out-

of-pocket expenses, historical costs,

depreciation, and other bookkeeping

entries

– economists focus more on opportunity cost

3

Definitions of Costs

• Labor Costs

– to accountants, expenditures on labor are

current expenses and hence costs of

production

– to economists, labor is an explicit cost

• labor services are contracted at some hourly

wage (w) and it is assumed that this is also

what the labor could earn in alternative

employment

4

Definitions of Costs • Capital Costs

– accountants use the historical price of the

capital and apply some depreciation rule to

determine current costs

– economists refer to the capital’s original price

as a “sunk cost” and instead regard the

implicit cost of the capital to be what

someone else would be willing to pay for its

use

• we will use v to denote the rental rate for capital

2

5

Definitions of Costs • Costs of Entrepreneurial Services

– accountants believe that the owner of a firm

is entitled to all profits

• revenues or losses left over after paying all input

costs

– economists consider the opportunity costs of

time and funds that owners devote to the

operation of their firms

• part of accounting profits would be considered as

entrepreneurial costs by economists

6

Economic Cost

• The economic cost of any input is the

payment required to keep that input in

its present employment

– the remuneration the input would receive in

its best alternative employment

7

Two Simplifying Assumptions

• There are only two inputs

– homogeneous labor (l), measured in labor-

hours

– homogeneous capital (k), measured in

machine-hours

• entrepreneurial costs are included in capital costs

• Inputs are hired in perfectly competitive

markets

– firms are price takers in input markets 8

Economic Profits

• Total costs for the firm are given by

total costs = C = wl + vk

• Total revenue for the firm is given by

total revenue = pq = pf(k,l)

• Economic profits () are equal to

= total revenue - total cost

= pq - wl - vk

= pf(k,l) - wl - vk

3

9

Economic Profits

• Economic profits are a function of the

amount of capital and labor employed

– we could examine how a firm would choose

k and l to maximize profit

• “derived demand” theory of labor and capital

inputs

– for now, we will assume that the firm has

already chosen its output level (q0) and

wants to minimize its costs

10

Cost-Minimizing Input Choices

• To minimize the cost of producing a

given level of output, a firm should

choose a point on the isoquant at which

the RTS is equal to the ratio w/v

– it should equate the rate at which k can be

traded for l in the productive process to the

rate at which they can be traded in the

marketplace

11


• Mathematically, we seek to minimize

total costs given q = f(k,l) = q0

• Setting up the Lagrangian:

L = wl + vk + [q0 - f(k,l)]

• First order conditions are

L/l = w - (f/l) = 0

L/k = v - (f/k) = 0

L/ = q0 - f(k,l) = 0 12


• Dividing the first two conditions we get

) for ( /

/kRTS

kf

f

v

wl

l

• The cost-minimizing firm should equate

the RTS for the two inputs to the ratio of

their prices

4

13


• Cross-multiplying, we get

w

f

v

fk l

• For costs to be minimized, the marginal

productivity per dollar spent should be

the same for all inputs

14


• Note that this equation’s inverse is also

of interest

kf

v

f

w

l

• The Lagrangian multiplier shows how

much in extra costs would be incurred

by increasing the output constraint

slightly

15

q0

Given output q0, we wish to find the least costly

point on the isoquant

C1

C2

C3

Costs are represented by

parallel lines with a slope of -

w/v


l per period

k per period

C1 < C2 < C3

16

C1

C2

C3

q0

The minimum cost of producing q0 is C2


l per period

k per period

k*

l*

The optimal choice

is l*, k*

This occurs at the

tangency between the

isoquant and the total cost

curve

5

17

Contingent Demand for Inputs

• In Chapter 4, we considered an

individual’s expenditure-minimization

problem

– we used this technique to develop the

compensated demand for a good

• Can we develop a firm’s demand for an

input in the same way?

18


• In the present case, cost minimization

leads to a demand for capital and labor

that is contingent on the level of output

being produced

• The demand for an input is a derived

demand

– it is based on the level of the firm’s output

19

The Firm’s Expansion Path

• The firm can determine the cost-

minimizing combinations of k and l for

every level of output

• If input costs remain constant for all

amounts of k and l the firm may

demand, we can trace the locus of cost-

minimizing choices

– called the firm’s expansion path

20


l per period

k per period

q00

The expansion path is the locus of cost-

minimizing tangencies

q0

q1

E

The curve shows

how inputs increase

as output increases

6

21


• The expansion path does not have to be

a straight line

– the use of some inputs may increase faster

than others as output expands

• depends on the shape of the isoquants

• The expansion path does not have to be

upward sloping

– if the use of an input falls as output expands,

that input is an inferior input 22

Cost Minimization


Cobb-Douglas:

q = k l

• The Lagrangian expression for cost

minimization of producing q0 is

L = vk + wl + (q0 - k l )

23

Cost Minimization

• The first-order conditions for a minimum

are

L/k = v - k -1l = 0

L/l = w - k l -1 = 0

L/ = q0 - k l = 0

24

Cost Minimization

• Dividing the first equation by the second

gives us

RTS

k

k

k

v

w

ll

l1

1

• This production function is homothetic

– the RTS depends only on the ratio of the two

inputs

– the expansion path is a straight line

7

25

Cost Minimization


CES:

q = (k + l )/

• The Lagrangian expression for cost

minimization of producing q0 is

L = vk + wl + [q0 - (k + l )/]

26

Cost Minimization

• The first-order conditions for a minimum

are

L/k = v - (/)(k + l)(-)/()k-1 = 0

L/l = w - (/)(k + l)(-)/()l-1 = 0

L/ = q0 - (k + l )/ = 0

27

Cost Minimization

• Dividing the first equation by the second

gives us

/1111

ll

kk

kv

w

• This production function is also

homothetic

28

Total Cost Function

• The total cost function shows that for

any set of input costs and for any output

level, the minimum cost incurred by the

firm is

C = C(v,w,q)

• As output (q) increases, total costs

increase

8

29

Average Cost Function

• The average cost function (AC) is found

by computing total costs per unit of

output

q

qwvCqwvAC

),,(),,( cost average

30

Marginal Cost Function

• The marginal cost function (MC) is

found by computing the change in total

costs for a change in output produced

q

qwvCqwvMC

),,(),,( cost marginal

31

Graphical Analysis of Total Costs

• Suppose that k1 units of capital and l1

units of labor input are required to

produce one unit of output

C(q=1) = vk1 + wl1

• To produce m units of output (assuming

constant returns to scale)

C(q=m) = vmk1 + wml1 = m(vk1 + wl1)

C(q=m) = m C(q=1) 32


Output

Total

costs

C

With constant returns to scale, total costs

are proportional to output

AC = MC

Both AC and

MC will be

constant

9

33


• Suppose instead that total costs start

out as concave and then becomes

convex as output increases

– one possible explanation for this is that

there is a third factor of production that is

fixed as capital and labor usage expands

– total costs begin rising rapidly after

diminishing returns set in

34


Output

Total

costs

C

Total costs rise

dramatically as

output increases

after diminishing

returns set in

35


Output

Average

and

marginal

costs MC

MC is the slope of the C curve

AC

If AC > MC,

AC must be

falling

If AC < MC,

AC must be

rising min AC

36

Shifts in Cost Curves

• The cost curves are drawn under the

assumption that input prices and the

level of technology are held constant

– any change in these factors will cause the

cost curves to shift

10

37

Some Illustrative Cost Functions

• Suppose we have a fixed proportions

technology such that

q = f(k,l) = min(ak,bl)

• Production will occur at the vertex of the

L-shaped isoquants (q = ak = bl)

C(w,v,q) = vk + wl = v(q/a) + w(q/b)

b

w

a

vaqvwC ),,(

38


• Suppose we have a Cobb-Douglas

technology such that

q = f(k,l) = k l

• Cost minimization requires that

l

k

v

w

l

v

wk

39


• If we substitute into the production

function and solve for l, we will get

//

/

/1 vwql

• A similar method will yield

//

/

/1 vwqk

40


• Now we can derive total costs as

///1),,( wBvqwvkqwvC l

where

//)(B

which is a constant that involves only

the parameters and

11

41


• Suppose we have a CES technology

such that

q = f(k,l) = (k + l )/

• To derive the total cost, we would use

the same method and eventually get

/)1(1/1//1 )(),,( wvqwvkqwvC l

1/111/1 )(),,( wvqqwvC42

Properties of Cost Functions

• Homogeneity

– cost functions are all homogeneous of

degree one in the input prices

• cost minimization requires that the ratio of input

prices be set equal to RTS, a doubling of all

input prices will not change the levels of inputs

purchased

• pure, uniform inflation will not change a firm’s

input decisions but will shift the cost curves up

43


• Nondecreasing in q, v, and w

– cost functions are derived from a cost-

minimization process

• any decline in costs from an increase in one of

the function’s arguments would lead to a

contradiction

44


• Concave in input prices

– costs will be lower when a firm faces input

prices that fluctuate around a given level

than when they remain constant at that

level

• the firm can adapt its input mix to take

advantage of such fluctuations

12

45

C(v,w,q1)

Since the firm’s input mix

will likely change, actual

costs will be less than

Cpseudo such as C(v,w,q1)

Cpseudo

If the firm continues to

buy the same input mix

as w changes, its cost

function would be Cpseudo

Concavity of Cost Function

w

Costs

At w1, the firm’s costs are C(v,w1,q1)

C(v,w1,q1)

w1 46


• Some of these properties carry over to

average and marginal costs

– homogeneity

– effects of v, w, and q are ambiguous

47

Input Substitution

• A change in the price of an input will

cause the firm to alter its input mix

• We wish to see how k/l changes in

response to a change in w/v, while

holding q constant

v

w

k

l

48

Input Substitution

• Putting this in proportional terms as

)/ln(

)/ln(

/

/

)/(

)/(

vw

k

k

vw

vw

ks

l

l

l

gives an alternative definition of the

elasticity of substitution

– in the two-input case, s must be nonnegative

– large values of s indicate that firms change

their input mix significantly if input prices

change

13

49

Partial Elasticity of Substitution

• The partial elasticity of substitution

between two inputs (xi and xj) with

prices wi and wj is given by

)/ln(

)/ln(

/

/

)/(

)/(

ij

ji

ji

ij

ij

ji

ijww

xx

xx

ww

ww

xxs

• Sij is a more flexible concept than

because it allows the firm to alter the

usage of inputs other than xi and xj

when input prices change 50

Size of Shifts in Costs Curves

• The increase in costs will be largely

influenced by the relative significance of

the input in the production process

• If firms can easily substitute another

input for the one that has risen in price,

there may be little increase in costs

51

Technical Progress

• Improvements in technology also lower

cost curves

• Suppose that total costs (with constant

returns to scale) are

C0 = C0(q,v,w) = qC0(v,w,1)

52

Technical Progress

• Because the same inputs that produced

one unit of output in period zero will

produce A(t) units in period t

Ct(v,w,A(t)) = A(t)Ct(v,w,1)= C0(v,w,1)

• Total costs are given by

Ct(v,w,q) = qCt(v,w,1) = qC0(v,w,1)/A(t)

= C0(v,w,q)/A(t)

14

53

Shifting the Cobb-Douglas Cost Function

• The Cobb-Douglas cost function is ///1),,( wBvqwvkqwvC l

where //)(B

• If we assume = = 0.5, the total cost

curve is greatly simplified:

5.05.02),,( wqvwvkqwvC l54


• If v = 3 and w = 12, the relationship is

qqqC 12362),12,3(

– C = 480 to produce q =40

– AC = C/q = 12

– MC = C/q = 12

55


• If v = 3 and w = 27, the relationship is

qqqC 18812),27,3(

– C = 720 to produce q =40

– AC = C/q = 18

– MC = C/q = 18

56


• Contingent demand functions for all of

the firms inputs can be derived from the

cost function

– Shephard’s lemma

• the contingent demand function for any input is

given by the partial derivative of the total-cost

function with respect to that input’s price

15

57


• Suppose we have a fixed proportions

technology

• The cost function is

b

w

a

vaqvwC ),,(

58


• For this cost function, contingent

demand functions are quite simple:

a

q

v

qwvCqwvk c

),,(),,(

b

q

w

qwvCqwvc

),,(),,(l

59


• Suppose we have a Cobb-Douglas

technology

• The cost function is

///1),,( wBvqwvkqwvC l

60


• For this cost function, the derivation is

messier:

/

/1

///1

),,(

v

wBq

wBvqv

Cqwvk c

16

61


/

/1

///1

),,(

v

wBq

wBvqw

Cqwvcl

• The contingent demands for inputs

depend on both inputs’ prices

62

Short-Run, Long-Run Distinction

• In the short run, economic actors have

only limited flexibility in their actions

• Assume that the capital input is held

constant at k1 and the firm is free to

vary only its labor input

• The production function becomes

q = f(k1,l)

63

Short-Run Total Costs

• Short-run total cost for the firm is

SC = vk1 + wl

• There are two types of short-run costs:

– short-run fixed costs are costs associated

with fixed inputs (vk1)

– short-run variable costs are costs

associated with variable inputs (wl)

64


• Short-run costs are not minimal costs

for producing the various output levels

– the firm does not have the flexibility of input

choice

– to vary its output in the short run, the firm

must use nonoptimal input combinations

– the RTS will not be equal to the ratio of

input prices

17

65


l per period

k per period

q0

q1

q2

k1

l1 l2 l3

Because capital is fixed at k1,

the firm cannot equate RTS

with the ratio of input prices

66

Short-Run Marginal and Average Costs

• The short-run average total cost (SAC)

function is

SAC = total costs/total output = SC/q

• The short-run marginal cost (SMC) function

is

SMC = change in SC/change in output = SC/q

67

Relationship between Short-Run and Long-Run Costs

Output

Total

costs

SC (k0)

SC (k1)

SC (k2)

The long-run

C curve can

be derived by

varying the

level of k

q0 q1 q2

C

68


Output

Costs

The geometric

relationship

between short-

run and long-run

AC and MC can

also be shown

q0 q1

AC

MC SAC (k0) SMC (k0)

SAC (k1) SMC (k1)

18

69


• At the minimum point of the AC curve:

– the MC curve crosses the AC curve

• MC = AC at this point

– the SAC curve is tangent to the AC curve

• SAC (for this level of k) is minimized at the same

level of output as AC

• SMC intersects SAC also at this point

AC = MC = SAC = SMC

70


• A firm that wishes to minimize the

economic costs of producing a

particular level of output should

choose that input combination for

which the rate of technical substitution

(RTS) is equal to the ratio of the

inputs’ rental prices

71


• Repeated application of this

minimization procedure yields the

firm’s expansion path

– the expansion path shows how input

usage expands with the level of output

• it also shows the relationship between output

level and total cost

• this relationship is summarized by the total

cost function, C(v,w,q)

72


• The firm’s average cost (AC = C/q)

and marginal cost (MC = C/q) can

be derived directly from the total-cost

function

– if the total cost curve has a general cubic

shape, the AC and MC curves will be u-

shaped

19

73


• All cost curves are drawn on the

assumption that the input prices are

held constant

– when an input price changes, cost curves

shift to new positions

• the size of the shifts will be determined by the

overall importance of the input and the

substitution abilities of the firm

– technical progress will also shift cost

curves 74


• Input demand functions can be derived

from the firm’s total-cost function

through partial differentiation

– these input demands will depend on the

quantity of output the firm chooses to

produce

• are called “contingent” demand functions

75


• In the short run, the firm may not be

able to vary some inputs

– it can then alter its level of production

only by changing the employment of its

variable inputs

– it may have to use nonoptimal, higher-

cost input combinations than it would

choose if it were possible to vary all

inputs

1

1

Chapter 11

PROFIT MAXIMIZATION


The Nature of Firms

• A firm is an association of individuals

who have organized themselves for the

purpose of turning inputs into outputs

• Different individuals will provide different

types of inputs

– the nature of the contractual relationship

between the providers of inputs to a firm

may be quite complicated

3

Contractual Relationships

• Some contracts between providers of

inputs may be explicit

– may specify hours, work details, or

compensation

• Other arrangements will be more

implicit in nature

– decision-making authority or sharing of

tasks

4

Modeling Firms’ Behavior

• Most economists treat the firm as a

single decision-making unit

– the decisions are made by a single

dictatorial manager who rationally pursues

some goal

• usually profit-maximization

2

5

Profit Maximization

• A profit-maximizing firm chooses both

its inputs and its outputs with the sole

goal of achieving maximum economic

profits

– seeks to maximize the difference between

total revenue and total economic costs

6

Profit Maximization

• If firms are strictly profit maximizers,

they will make decisions in a “marginal”

way

– examine the marginal profit obtainable

from producing one more unit of hiring one

additional laborer

7

Output Choice

• Total revenue for a firm is given by

R(q) = p(q)q

• In the production of q, certain economic

costs are incurred [C(q)]

• Economic profits () are the difference

between total revenue and total costs

(q) = R(q) – C(q) = p(q)q –C(q)

8

Output Choice

• The necessary condition for choosing the

level of q that maximizes profits can be

found by setting the derivative of the

function with respect to q equal to zero

0)('

dq

dC

dq

dRq

dq

d

dq

dC

dq

dR

3

9

Output Choice

• To maximize economic profits, the firm

should choose the output for which

marginal revenue is equal to marginal

cost

MCdq

dC

dq

dRMR

10

Second-Order Conditions

• MR = MC is only a necessary condition

for profit maximization

• For sufficiency, it is also required that

0)('

**

2

2

qqqqdq

qd

dq

d

• “marginal” profit must be decreasing at

the optimal level of q

11

Profit Maximization

output

revenues & costs

R

C

q*

Profits are maximized when the slope of

the revenue function is equal to the slope of

the cost function

The second-order

condition prevents us

from mistaking q0 as

a maximum

q0 12

Marginal Revenue

• If a firm can sell all it wishes without

having any effect on market price,

marginal revenue will be equal to price

• If a firm faces a downward-sloping

demand curve, more output can only be

sold if the firm reduces the good’s price

dq

dpqp

dq

qqpd

dq

dRqMR

])([)( revenue marginal

4

13

Marginal Revenue

• If a firm faces a downward-sloping

demand curve, marginal revenue will be

a function of output

• If price falls as a firm increases output,

marginal revenue will be less than price

14

Marginal Revenue

• Suppose that the demand curve for a sub

sandwich is q = 100 – 10p

• Solving for price, we get

p = -q/10 + 10

• This means that total revenue is

R = pq = -q2/10 + 10q

• Marginal revenue will be given by

MR = dR/dq = -q/5 + 10

15

Profit Maximization

• To determine the profit-maximizing

output, we must know the firm’s costs

• If subs can be produced at a constant

average and marginal cost of $4, then

MR = MC

-q/5 + 10 = 4

q = 30

16

Marginal Revenue and Elasticity

• The concept of marginal revenue is

directly related to the elasticity of the

demand curve facing the firm

• The price elasticity of demand is equal

to the percentage change in quantity

that results from a one percent change

in price

q

p

dp

dq

pdp

qdqe pq

/

/,

5

17


• This means that

pqep

dq

dp

p

qp

dq

dpqpMR

,

111

– if the demand curve slopes downward,

eq,p < 0 and MR < p

– if the demand is elastic, eq,p < -1 and

marginal revenue will be positive

• if the demand is infinitely elastic, eq,p = - and

marginal revenue will equal price 18


eq,p < -1 MR > 0

eq,p = -1 MR = 0

eq,p > -1 MR < 0

19

The Inverse Elasticity Rule

• Because MR = MC when the firm

maximizes profit, we can see that

pqepMC

,

11

pqep

MCp

,

1

• The gap between price and marginal

cost will fall as the demand curve facing

the firm becomes more elastic 20


pqep

MCp

,

1

• If eq,p > -1, MC < 0

• This means that firms will choose to

operate only at points on the demand

curve where demand is elastic

6

21

Average Revenue Curve

• If we assume that the firm must sell all

its output at one price, we can think of

the demand curve facing the firm as its

average revenue curve

– shows the revenue per unit yielded by

alternative output choices

22

Marginal Revenue Curve

• The marginal revenue curve shows the

extra revenue provided by the last unit

sold

• In the case of a downward-sloping

demand curve, the marginal revenue

curve will lie below the demand curve

23


output

price

D (average revenue)

MR

q1

p1

As output increases from 0 to q1, total

revenue increases so MR > 0

As output increases beyond q1, total

revenue decreases so MR < 0

24


• When the demand curve shifts, its

associated marginal revenue curve

shifts as well

– a marginal revenue curve cannot be

calculated without referring to a specific

demand curve

7

25

The Constant Elasticity Case

• We showed (in Chapter 5) that a

demand function of the form

q = apb

has a constant price elasticity of

demand equal to b

• Solving this equation for p, we get

p = (1/a)1/bq1/b = kq1/b where k = (1/a)1/b

26

The Constant Elasticity Case

• This means that

R = pq = kq(1+b)/b

and

MR = dr/dq = [(1+b)/b]kq1/b = [(1+b)/b]p

• This implies that MR is proportional to

price

27

Short-Run Supply by a Price-Taking Firm

output

price SMC

SAC

SAVC

p* = MR

q*

Maximum profit

occurs where

p = SMC

28


output

price SMC

SAC

SAVC

p* = MR

q*

Since p > SAC,

profit > 0

8

29


output

price SMC

SAC

SAVC

p* = MR

q*

If the price rises

to p**, the firm

will produce q**

and > 0

q**

p**

30


output

price SMC

SAC

SAVC

p* = MR

q*

If the price falls to

p***, the firm will

produce q***

q***

p*** Profit maximization

requires that p =

SMC and that SMC

is upward-sloping

< 0

31


• The positively-sloped portion of the

short-run marginal cost curve is the

short-run supply curve for a price-taking

firm

– it shows how much the firm will produce at

every possible market price

– firms will only operate in the short run as

long as total revenue covers variable cost

• the firm will produce no output if p < SAVC 32


• Thus, the price-taking firm’s short-run

supply curve is the positively-sloped

portion of the firm’s short-run marginal

cost curve above the point of minimum

average variable cost

– for prices below this level, the firm’s profit-

maximizing decision is to shut down and

produce no output

9

33


output

price SMC

SAC

SAVC

The firm’s short-run

supply curve is the

SMC curve that is

above SAVC

34

Short-Run Supply

• Suppose that the firm’s short-run total cost

curve is

SC(v,w,q,k) = vk1 + wq1/k1-/

where k1 is the level of capital held

constant in the short run

• Short-run marginal cost is

/

1

/)1(

1),,,( kqw

q

SCkqwvSMC

35

Short-Run Supply

• The price-taking firm will maximize profit

where p = SMC

pkq

wSMC

/

1

/)1(

• Therefore, quantity supplied will be

)1/()1/(

1

)1/(

pk

wq

36

Short-Run Supply

• To find the firm’s shut-down price, we

need to solve for SAVC

SVC = wq1/k1-/

SAVC = SVC/q = wq(1-)/k1-/

• SAVC < SMC for all values of < 1

– there is no price low enough that the firm will

want to shut down

10

37

Profit Functions

• A firm’s economic profit can be

expressed as a function of inputs

= pq - C(q) = pf(k,l) - vk - wl

• Only the variables k and l are under the

firm’s control

– the firm chooses levels of these inputs in

order to maximize profits

• treats p, v, and w as fixed parameters in its

decisions

38

Profit Functions

• A firm’s profit function shows its

maximal profits as a function of the

prices that the firm faces

]),([),(),,(,,

lllll

wvkkpfMaxkMaxwvpkk

39

Properties of the Profit Function

• Homogeneity

– the profit function is homogeneous of

degree one in all prices

• with pure inflation, a firm will not change its

production plans and its level of profits will keep

up with that inflation

40


• Nondecreasing in output price

– a firm could always respond to a rise in the

price of its output by not changing its input

or output plans

• profits must rise

11

41


• Nonincreasing in input prices

– if the firm responded to an increase in an

input price by not changing the level of that

input, its costs would rise

• profits would fall

42


• Convex in output prices

– the profits obtainable by averaging those

from two different output prices will be at

least as large as those obtainable from the

average of the two prices

wv

ppwvpwvp,,

22

),,(),,( 2121

43

Envelope Results

• We can apply the envelope theorem to

see how profits respond to changes in

output and input prices

),,(),,(

wvpqp

wvp

),,(),,(

wvpkv

wvp

),,(),,(

wvpw

wvpl

44

Producer Surplus in the Short Run

• Because the profit function is

nondecreasing in output prices, we know

that if p2 > p1

(p2,…) (p1,…)

• The welfare gain to the firm of this price

increase can be measured by

welfare gain = (p2,…) - (p1,…)

12

45


output

price SMC

p1

q1

If the market price

is p1, the firm will

produce q1

If the market price

rises to p2, the firm

will produce q2

p2

q2

46

The firm’s profits

rise by the shaded

area


output

price SMC

p1

q1

p2

q2

47


• Mathematically, we can use the

envelope theorem results

,...)(,...)(

)/()( gain welfare

12

2

1

2

1

pp

dppdppqp

p

p

p

48


• We can measure how much the firm

values the right to produce at the

prevailing price relative to a situation

where it would produce no output

13

49


output

price SMC

p1

q1

Suppose that the

firm’s shutdown

price is p0

p

0

50


• The extra profits available from facing a

price of p1 are defined to be producer

surplus

1

0

)(,...)(,...)( surplus producer 01

p

p

dppqpp

51

Producer surplus

at a market price

of p1 is the

shaded area


output

price SMC

p1

q1

p

0

52


• Producer surplus is the extra return that

producers make by making transactions

at the market price over and above what

they would earn if nothing was

produced

– the area below the market price and above

the supply curve

14

53


• Because the firm produces no output at

the shutdown price, (p0,…) = -vk1

– profits at the shutdown price are equal to the

firm’s fixed costs

• This implies that

producer surplus = (p1,…) - (p0,…)

= (p1,…) – (-vk1) = (p1,…) + vk1

– producer surplus is equal to current profits

plus short-run fixed costs 54

Profit Maximization and Input Demand

• A firm’s output is determined by the

amount of inputs it chooses to employ

– the relationship between inputs and

outputs is summarized by the production

function

• A firm’s economic profit can also be

expressed as a function of inputs

(k,l) = pq –C(q) = pf(k,l) – (vk + wl)

55


• The first-order conditions for a maximum

are

/k = p[f/k] – v = 0

/l = p[f/l] – w = 0

• A profit-maximizing firm should hire any

input up to the point at which its marginal

contribution to revenues is equal to the

marginal cost of hiring the input 56


• These first-order conditions for profit

maximization also imply cost

minimization

– they imply that RTS = w/v

15

57


• To ensure a true maximum, second-

order conditions require that

kk = fkk < 0

ll = fll < 0

kk ll - kl2 = fkkfll – fkl

2 > 0

– capital and labor must exhibit sufficiently

diminishing marginal productivities so that

marginal costs rise as output expands 58

Input Demand Functions

• In principle, the first-order conditions can

be solved to yield input demand functions

Capital Demand = k(p,v,w)

Labor Demand = l(p,v,w)

• These demand functions are

unconditional

– they implicitly allow the firm to adjust its

output to changing prices

59

Single-Input Case

• We expect l/w 0

– diminishing marginal productivity of labor

• The first order condition for profit

maximization was

/l = p[f/l] – w = 0

• Taking the total differential, we get

dww

fpdw

l

l

l

60

Single-Input Case

• This reduces to

wfp

lll1

• Solving further, we get

ll

l

fpw

1

• Since fll 0, l/w 0

16

61

Two-Input Case

• For the case of two (or more inputs), the

story is more complex

– if there is a decrease in w, there will not

only be a change in l but also a change in

k as a new cost-minimizing combination of

inputs is chosen

• when k changes, the entire fl function changes

• But, even in this case, l/w 0

62

Two-Input Case

• When w falls, two effects occur

– substitution effect

• if output is held constant, there will be a

tendency for the firm to want to substitute l for k

in the production process

– output effect

• a change in w will shift the firm’s expansion

path

• the firm’s cost curves will shift and a different

output level will be chosen

63

Substitution Effect

q0

l per period

k per period

If output is held constant at q0 and w

falls, the firm will substitute l for k in

the production process

Because of diminishing

RTS along an isoquant,

the substitution effect will

always be negative

64

Output Effect

Output

Price

A decline in w will lower the firm’s MC

MC

MC’

Consequently, the firm

will choose a new level

of output that is higher

P

q0 q1

17

65

Output Effect

q0

l per period

k per period Thus, the output effect

also implies a negative

relationship between l

and w

Output will rise to q1

q1

66

Cross-Price Effects

• No definite statement can be made

about how capital usage responds to a

wage change

– a fall in the wage will lead the firm to

substitute away from capital

– the output effect will cause more capital to

be demanded as the firm expands

production

67

Substitution and Output Effects

• We have two concepts of demand for

any input

– the conditional demand for labor, lc(v,w,q)

– the unconditional demand for labor, l(p,v,w)

• At the profit-maximizing level of output

lc(v,w,q) = l(p,v,w)

68

Substitution and Output Effects

• Differentiation with respect to w yields

w

q

q

qwv

w

qwv

w

wvp cc

),,(),,(),,( lll

substitution

effect

output

effect

total effect

18

69


• In order to maximize profits, the firm

should choose to produce that output

level for which the marginal revenue is

equal to the marginal cost

70


• If a firm is a price taker, its output

decisions do not affect the price of its

output

– marginal revenue is equal to price

• If the firm faces a downward-sloping

demand for its output, marginal

revenue will be less than price

71


• Marginal revenue and the price

elasticity of demand are related by the

formula

pqepMR

,

11

72


• The supply curve for a price-taking,

profit-maximizing firm is given by the

positively sloped portion of its marginal

cost curve above the point of minimum

average variable cost (AVC)

– if price falls below minimum AVC, the

firm’s profit-maximizing choice is to shut

down and produce nothing

19

73


• The firm’s reactions to the various

prices it faces can be judged through

use of its profit function

– shows maximum profits for the firm given

the price of its output, the prices of its

inputs, and the production technology

74


• The firm’s profit function yields

particularly useful envelope results

– differentiation with respect to market price

yields the supply function

– differentiation with respect to any input

price yields the (inverse of) the demand

function for that input

75


• Short-run changes in market price

result in changes in the firm’s short-run

profitability

– these can be measured graphically by

changes in the size of producer surplus

– the profit function can also be used to

calculate changes in producer surplus

76


• Profit maximization provides a theory

of the firm’s derived demand for inputs

– the firm will hire any input up to the point

at which the value of its marginal product

is just equal to its per-unit market price

– increases in the price of an input will

induce substitution and output effects that

cause the firm to reduce hiring of that

input

1

1

Chapter 12

THE PARTIAL EQUILIBRIUM

COMPETITIVE MODEL


Market Demand

• Assume that there are only two goods

(x and y)

– An individual’s demand for x is

Quantity of x demanded = x(px,py,I)

– If we use i to reflect each individual in the

market, then the market demand curve is

n

i

iyxi ppxX1

),,( for demand Market I

3

Market Demand

• To construct the market demand curve,

PX is allowed to vary while Py and the

income of each individual are held

constant

• If each individual’s demand for x is

downward sloping, the market demand

curve will also be downward sloping

4

Market Demand

x x x

px px px

x1* x2*

px*

To derive the market demand curve, we sum the

quantities demanded at every price

x1

Individual 1’s

demand curve

x2

Individual 2’s

demand curve

Market demand

curve

X*

X

x1* + x2* = X*

2

5

Shifts in the Market Demand Curve

• The market demand summarizes the

ceteris paribus relationship between X

and px

– changes in px result in movements along the

curve (change in quantity demanded)

– changes in other determinants of the

demand for X cause the demand curve to

shift to a new position (change in demand) 6

Shifts in Market Demand

• Suppose that individual 1’s demand for

oranges is given by

x1 = 10 – 2px + 0.1I1 + 0.5py

and individual 2’s demand is

x2 = 17 – px + 0.05I2 + 0.5py

• The market demand curve is

X = x1 + x2 = 27 – 3px + 0.1I1 + 0.05I2 + py

7


• To graph the demand curve, we must

assume values for py, I1, and I2

• If py = 4, I1 = 40, and I2 = 20, the market

demand curve becomes

X = 27 – 3px + 4 + 1 + 4 = 36 – 3px

8


• If py rises to 6, the market demand curve

shifts outward to

X = 27 – 3px + 4 + 1 + 6 = 38 – 3px

– note that X and Y are substitutes

• If I1 fell to 30 while I2 rose to 30, the

market demand would shift inward to

X = 27 – 3px + 3 + 1.5 + 4 = 35.5 – 3px

– note that X is a normal good for both buyers

3

9

Generalizations

• Suppose that there are n goods (xi, i = 1,n)

with prices pi, i = 1,n.

• Assume that there are m individuals in the

economy

• The j th’s demand for the i th good will

depend on all prices and on Ij

xij = xij(p1,…,pn, Ij)

10

Generalizations

• The market demand function for xi is the

sum of each individual’s demand for that

good

),,...,( 1

1

jn

m

j

iji ppxX I

• The market demand function depends on

the prices of all goods and the incomes

and preferences of all buyers

11

Elasticity of Market Demand

• The price elasticity of market demand is

measured by

D

DPQ

Q

P

P

PPQe

),',(,

I

• Market demand is characterized by

whether demand is elastic (eQ,P <-1) or

inelastic (0> eQ,P > -1)

12

Elasticity of Market Demand

• The cross-price elasticity of market

demand is measured by

D

DPQ

Q

P

P

PPQe

'

'

),',(,

I

• The income elasticity of market demand is

measured by

D

DQ

Q

PPQe

I

I

II

),',(,

4

13

Timing of the Supply Response • In the analysis of competitive pricing, the

time period under consideration is

important

– very short run

• no supply response (quantity supplied is fixed)

– short run

• existing firms can alter their quantity supplied, but

no new firms can enter the industry

– long run

• new firms may enter an industry 14

Pricing in the Very Short Run

• In the very short run (or the market

period), there is no supply response to

changing market conditions

– price acts only as a device to ration demand

• price will adjust to clear the market

– the supply curve is a vertical line

15

Pricing in the Very Short Run

Quantity

Price

S

D

Q*

P1

D’

P2

When quantity is fixed in the

very short run, price will rise

from P1 to P2 when the demand

rises from D to D’

16

Short-Run Price Determination

• The number of firms in an industry is

fixed

• These firms are able to adjust the

quantity they are producing

– they can do this by altering the levels of the

variable inputs they employ

5

17

Perfect Competition

• A perfectly competitive industry is one

that obeys the following assumptions:

– there are a large number of firms, each

producing the same homogeneous product

– each firm attempts to maximize profits

– each firm is a price taker

• its actions have no effect on the market price

– information is perfect

– transactions are costless 18

Short-Run Market Supply

• The quantity of output supplied to the

entire market in the short run is the sum

of the quantities supplied by each firm

– the amount supplied by each firm depends

on price

• The short-run market supply curve will

be upward-sloping because each firm’s

short-run supply curve has a positive

slope

19

Short-Run Market Supply Curve

quantity Quantity quantity

P P P

q1A q1

B

P1

To derive the market supply curve, we sum the

quantities supplied at every price

sA

Firm A’s

supply curve sB Firm B’s

supply curve

Market supply

curve

Q1

S

q1A + q1

B = Q1

20

Short-Run Market Supply Function

• The short-run market supply function

shows total quantity supplied by each

firm to a market

n

i

is wvPqwvPQ1

),,(),,(

• Firms are assumed to face the same

market price and the same prices for

inputs

6

21

Short-Run Supply Elasticity

• The short-run supply elasticity describes

the responsiveness of quantity supplied

to changes in market price

S

SPS

Q

P

P

Q

P

Qe

in change %

supplied in change %,

• Because price and quantity supplied are

positively related, eS,P > 0

22

A Short-Run Supply Function

• Suppose that there are 100 identical

firms each with the following short-run

supply curve

qi (P,v,w) = 10P/3 (i = 1,2,…,100)

• This means that the market supply

function is given by

3

1000

3

10100

1

100

1

PPqQ

i i

is

23

A Short-Run Supply Function

• In this case, computation of the

elasticity of supply shows that it is unit

elastic

1

3/10003

1000),,(,

P

P

Q

P

P

wvPQe

S

SPS

24

Equilibrium Price Determination

• An equilibrium price is one at which

quantity demanded is equal to quantity

supplied

– neither suppliers nor demanders have an

incentive to alter their economic decisions

• An equilibrium price (P*) solves the

equation:

),*,(),'*,( wvPQPPQ SD I

7

25


• The equilibrium price depends on many

exogenous factors

– changes in any of these factors will likely

result in a new equilibrium price

26


Quantity

Price

S

D

Q1

P1

The interaction between

market demand and market

supply determines the

equilibrium price

27

Market Reaction to a Shift in Demand

Quantity

Price

S

D

Q1

P1

Q2

P2 Equilibrium price and

equilibrium quantity will

both rise

If many buyers experience

an increase in their demands,

the market demand curve

will shift to the right

D’

28

Market Reaction to a Shift in Demand

Quantity

Price

SMC

q1

P1

This is the short-run

supply response to an

increase in market price

q2

P2

If the market price rises,

firms will increase their

level of output

SAC

8

29

Shifts in Supply and Demand Curves

• Demand curves shift because

– incomes change

– prices of substitutes or complements change

– preferences change

• Supply curves shift because

– input prices change

– technology changes

– number of producers change 30

Shifts in Supply and Demand Curves

• When either a supply curve or a

demand curve shift, equilibrium price

and quantity will change

• The relative magnitudes of these

changes depends on the shapes of the

supply and demand curves

31

Shifts in Supply

Quantity Quantity

Price Price S

S’

S

S’

D

D

P

P

Q

P’

Q’

P’

Q Q’

Elastic Demand Inelastic Demand

Small increase in price,

large drop in quantity

Large increase in price,

small drop in quantity

32

Shifts in Demand

Quantity Quantity

Price Price

S

S

D D

P P

Q

P’

Q’

P’

Q Q’

Elastic Supply Inelastic Supply

Small increase in price,

large rise in quantity

Large increase in price,

small rise in quantity

D’ D’

9

33

Changing Short-Run Equilibria

• Suppose that the market demand for

luxury beach towels is

QD = 10,000 – 500P

and the short-run market supply is

QS = 1,000P/3

• Setting these equal, we find

P* = $12

Q* = 4,000 34


• Suppose instead that the demand for

luxury towels rises to

QD = 12,500 – 500P

• Solving for the new equilibrium, we find

P* = $15

Q* = 5,000

• Equilibrium price and quantity both rise

35


• Suppose that the wage of towel cutters

rises so that the short-run market supply

becomes

QS = 800P/3

• Solving for the new equilibrium, we find

P* = $13.04

Q* = 3,480

• Equilibrium price rises and quantity falls 36

Mathematical Model of Supply and Demand

• Suppose that the demand function is

represented by

QD = D(P,)

– is a parameter that shifts the demand curve

• D/ = D can have any sign

– D/P = DP < 0

10

37


• The supply relationship can be shown as

QS = S(P,)

– is a parameter that shifts the supply curve

• S/ = S can have any sign

– S/P = SP > 0

• Equilibrium requires that QD = QS

38


• To analyze the comparative statics of

this model, we need to use the total

differentials of the supply and demand

functions:

dQD = DPdP + Dd

dQS = SPdP + Sd

• Maintenance of equilibrium requires that

dQD = dQS

39


• Suppose that the demand parameter ()

changed while remains constant

• The equilibrium condition requires that

DPdP + Dd = SPdP

PP DS

DP

• Because SP - DP > 0, P/ will have the

same sign as D 40


• We can convert our analysis to elasticities

PDS

D

P

Pe

PP

P

,

PQPS

Q

PP

Pee

e

Q

PDS

QD

e,,

,

,

)(

11

41

Long-Run Analysis

• In the long run, a firm may adapt all of its

inputs to fit market conditions

– profit-maximization for a price-taking firm

implies that price is equal to long-run MC

• Firms can also enter and exit an industry

in the long run

– perfect competition assumes that there are

no special costs of entering or exiting an

industry 42

Long-Run Analysis

• New firms will be lured into any market

for which economic profits are greater

than zero

– entry of firms will cause the short-run

industry supply curve to shift outward

– market price and profits will fall

– the process will continue until economic

profits are zero

43

Long-Run Analysis

• Existing firms will leave any industry for

which economic profits are negative

– exit of firms will cause the short-run industry

supply curve to shift inward

– market price will rise and losses will fall

– the process will continue until economic

profits are zero

44

Long-Run Competitive Equilibrium

• A perfectly competitive industry is in

long-run equilibrium if there are no

incentives for profit-maximizing firms to

enter or to leave the industry

– this will occur when the number of firms is

such that P = MC = AC and each firm

operates at minimum AC

12

45

Long-Run Competitive Equilibrium

• We will assume that all firms in an

industry have identical cost curves

– no firm controls any special resources or

technology

• The equilibrium long-run position

requires that each firm earn zero

economic profit

46

Long-Run Equilibrium: Constant-Cost Case

• Assume that the entry of new firms in an

industry has no effect on the cost of

inputs

– no matter how many firms enter or leave

an industry, a firm’s cost curves will remain

unchanged

• This is referred to as a constant-cost

industry

47


A Typical Firm Total Market Quantity Quantity

SMC MC

AC

S

D

q1

P1

Q1

This is a long-run equilibrium for this industry

P = MC = AC Price Price

48


A Typical Firm Total Market

q1 Quantity Quantity

SMC MC

AC

S

D

P1

Q1

P2

Market price rises to P2

Q2

Suppose that market demand rises to D’

D’

Price Price

13

49




SMC MC

AC

S

D

P1

Q1

D’

P2

Economic profit > 0

Q2

In the short run, each firm increases output to q2

q2

Price Price

50




SMC MC

AC

S

D

P1

Q1

D’

Economic profit will return to 0

Q3

In the long run, new firms will enter the industry

S’

Price Price

51




SMC MC

AC

S

D

P1

Q1

D’

Q3

S’

The long-run supply curve will be a horizontal line

(infinitely elastic) at p1

LS

Price Price

52

Infinitely Elastic Long-Run Supply

• Suppose that the total cost curve for a

typical firm in the bicycle industry is

TC = q3 – 20q2 + 100q + 8,000

• Demand for bicycles is given by

QD = 2,500 – 3P

14

53

Infinitely Elastic Long-Run Supply

• To find the long-run equilibrium for this

market, we must find the low point on the

typical firm’s average cost curve

– where AC = MC

AC = q2 – 20q + 100 + 8,000/q

MC = 3q2 – 40q + 100

– this occurs where q = 20

• If q = 20, AC = MC = $500

– this will be the long-run equilibrium price 54

Shape of the Long-Run Supply Curve

• The zero-profit condition is the factor that

determines the shape of the long-run cost

curve

– if average costs are constant as firms enter,

long-run supply will be horizontal

– if average costs rise as firms enter, long-run

supply will have an upward slope

– if average costs fall as firms enter, long-run

supply will be negatively sloped

55

Long-Run Equilibrium: Increasing-Cost Industry

• The entry of new firms may cause the

average costs of all firms to rise

– prices of scarce inputs may rise

– new firms may impose “external” costs on

existing firms

– new firms may increase the demand for

tax-financed services

56


A Typical Firm (before entry) Total Market


SMC MC

AC

S

D

P1

Q1

Suppose that we are in long-run equilibrium in this industry


15

57




SMC MC

AC

S

D

P1

Q1


D’

P2

Market price rises to P2 and firms increase output to q2

Q2 q2

Price Price

58


A Typical Firm (after entry) Total Market Quantity Quantity

SMC’ MC’

AC’

S

D

P1

Q1

D’

q3

P3

Entry of firms causes costs for each firm to rise

Q3

Positive profits attract new firms and supply shifts out

S’

Price Price

59


A Typical Firm (after entry) Total Market


SMC’ MC’

AC’

S

D

p1

Q1

D’

p3

Q3

S’

The long-run supply curve will be upward-sloping

LS

Price Price

60

Long-Run Equilibrium: Decreasing-Cost Industry

• The entry of new firms may cause the

average costs of all firms to fall

– new firms may attract a larger pool of

trained labor

– entry of new firms may provide a “critical

mass” of industrialization

• permits the development of more efficient

transportation and communications networks

16

61

Long-Run Equilibrium: Decreasing-Cost Case



SMC MC

AC

S

D

P1

Q1

Suppose that we are in long-run equilibrium in this industry


62




SMC MC

AC

S

D

P1

Q1


D’

P2

Market price rises to P2 and firms increase output to q2

Q2 q2

Price Price

63




SMC’ MC’

AC’

S

D

P1

Q1

D’ P3

Entry of firms causes costs for each firm to fall

Q3 q3

Positive profits attract new firms and supply shifts out

S’

Price Price

64




SMC’ MC’

AC’

S

D

P1

Q1

The long-run industry supply curve will be downward-sloping

D’ P3

Q3 q3

S’

LS

Price Price

17

65

Classification of Long-Run Supply Curves

• Constant Cost

– entry does not affect input costs

– the long-run supply curve is horizontal at

the long-run equilibrium price

• Increasing Cost

– entry increases inputs costs

– the long-run supply curve is positively

sloped 66

Classification of Long-Run Supply Curves

• Decreasing Cost

– entry reduces input costs

– the long-run supply curve is negatively

sloped

67

Long-Run Elasticity of Supply

• The long-run elasticity of supply (eLS,P)

records the proportionate change in long-

run industry output to a proportionate

change in price

LS

LSPLS

Q

P

P

Q

P

Qe

in change %

in change %,

• eLS,P can be positive or negative

– the sign depends on whether the industry

exhibits increasing or decreasing costs 68

Comparative Statics Analysis of Long-Run Equilibrium

• Comparative statics analysis of long-run

equilibria can be conducted using

estimates of long-run elasticities of

supply and demand

• Remember that, in the long run, the

number of firms in the industry will vary

from one long-run equilibrium to another

18

69


• Assume that we are examining a

constant-cost industry

• Suppose that the initial long-run

equilibrium industry output is Q0 and the

typical firm’s output is q* (where AC is

minimized)

• The equilibrium number of firms in the

industry (n0) is Q0/q*

70


• A shift in demand that changes the

equilibrium industry output to Q1 will

change the equilibrium number of firms to

n1 = Q1/q*

• The change in the number of firms is

*q

QQnn 01

01

– completely determined by the extent of the

demand shift and the optimal output level for

the typical firm

71


• The effect of a change in input prices is

more complicated

– we need to know how much minimum

average cost is affected

– we need to know how an increase in long-

run equilibrium price will affect quantity

demanded

72


• The optimal level of output for each firm

may also be affected

• Therefore, the change in the number of

firms becomes

**

0

0

1

101

q

Q

q

Qnn

19

73

Rising Input Costs and Industry Structure

• Suppose that the total cost curve for a

typical firm in the bicycle industry is

TC = q3 – 20q2 + 100q + 8,000

and then rises to

TC = q3 – 20q2 + 100q + 11,616

• The optimal scale of each firm rises

from 20 to 22 (where MC = AC) 74

Rising Input Costs and Industry Structure

• At q = 22, MC = AC = $672 so the long-

run equilibrium price will be $672

• If demand can be represented by

QD = 2,500 – 3P

then QD = 484

• This means that the industry will have

22 firms (484 22)

75

Producer Surplus in the Long Run

• Short-run producer surplus represents

the return to a firm’s owners in excess

of what would be earned if output was

zero

– the sum of short-run profits and fixed costs

76


• In the long-run, all profits are zero and

there are no fixed costs

– owners are indifferent about whether they

are in a particular market

• they could earn identical returns on their

investments elsewhere

• Suppliers of inputs may not be indifferent

about the level of production in an

industry

20

77


• In the constant-cost case, input prices

are assumed to be independent of the

level of production

– inputs can earn the same amount in

alternative occupations

• In the increasing-cost case, entry will bid

up some input prices

– suppliers of these inputs will be made better

off 78


• Long-run producer surplus represents

the additional returns to the inputs in an

industry in excess of what these inputs

would earn if industry output was zero

– the area above the long-run supply curve

and below the market price

• this would equal zero in the case of constant

costs

79

Ricardian Rent

• Long-run producer surplus can be most

easily illustrated with a situation first

described by economist David Ricardo

– assume that there are many parcels of land

on which a particular crop may be grown

• the land ranges from very fertile land (low costs

of production) to very poor, dry land (high costs

of production)

80

Ricardian Rent

• At low prices only the best land is used

• Higher prices lead to an increase in

output through the use of higher-cost

land

– the long-run supply curve is upward-sloping

because of the increased costs of using less

fertile land

21

81

Ricardian Rent

Low-Cost Firm Total Market q* Quantity Quantity

MC

AC

S

D

P*

Q*

The owners of low-cost firms will earn positive profits

Price Price

82

Ricardian Rent

Marginal Firm Total Market q* Quantity Quantity

MC

AC

S

D

P*

Q*

The owners of the marginal firm will earn zero profit

Price Price

83

Ricardian Rent

• Firms with higher costs (than the

marginal firm) will stay out of the market

– would incur losses at a price of P*

• Profits earned by intramarginal firms

can persist in the long run

– they reflect a return to a unique resource

• The sum of these long-run profits

constitutes long-run producer surplus

84

Ricardian Rent

For each firm, P – AC represents

profit per unit of output

Total Market Quantity

S

D

P*

Q*

Each point on the supply curve represents minimum

average cost for some firm

Total long-run profits can be

computed by summing over all

units of output

Price

22

85

Ricardian Rent

• The long-run profits for the low-cost firms

will often be reflected in the prices of the

unique resources owned by those firms

– the more fertile the land is, the higher its

price

• Thus, profits are said to be capitalized

inputs’ prices

– reflect the present value of all future profits

86

Ricardian Rent

• It is the scarcity of low-cost inputs that

creates the possibility of Ricardian rent

• In industries with upward-sloping long-

run supply curves, increases in output

not only raise firms’ costs but also

generate factor rents for inputs

87


• In the short run, equilibrium prices are

established by the intersection of what

demanders are willing to pay (as reflected

by the demand curve) and what firms are

willing to produce (as reflected by the

short-run supply curve)

– these prices are treated as fixed in both

demanders’ and suppliers’ decision-making

processes

88


• A shift in either demand or supply will

cause the equilibrium price to change

– the extent of such a change will depend on

the slopes of the various curves

• Firms may earn positive profits in the

short run

– because fixed costs must always be paid,

firms will choose a positive output as long as

revenues exceed variable costs

23

89


• In the long run, the number of firms is

variable in response to profit opportunities

– the assumption of free entry and exit implies

that firms in a competitive industry will earn

zero economic profits in the long run (P = AC)

– because firms also seek maximum profits, the

equality P = AC = MC implies that firms will

operate at the low points of their long-run

average cost curves

90


• The shape of the long-run supply curve

depends on how entry and exit affect

firms’ input costs

– in the constant-cost case, input prices do not

change and the long-run supply curve is

horizontal

– if entry raises input costs, the long-run supply

curve will have a positive slope

91


• Changes in long-run market equilibrium

will also change the number of firms

– precise predictions about the extent of these

changes is made difficult by the possibility

that the minimum average cost level of

output may be affected by changes in input

costs or by technical progress

92


• If changes in the long-run equilibrium in a

market change the prices of inputs to that

market, the welfare of the suppliers of

these inputs will be affected

– such changes can be measured by changes

in the value of long-run producer surplus

1

1

Chapter 13

GENERAL EQUILIBRIUM AND

WELFARE


Perfectly Competitive Price System

• We will assume that all markets are

perfectly competitive

– there is some large number of homogeneous

goods in the economy • both consumption goods and factors of

production

– each good has an equilibrium price

– there are no transaction or transportation

costs

– individuals and firms have perfect information

3

Law of One Price • A homogeneous good trades at the

same price no matter who buys it or

who sells it

– if one good traded at two different prices,

demanders would rush to buy the good

where it was cheaper and firms would try

to sell their output where the price was

higher

• these actions would tend to equalize the price

of the good 4

Assumptions of Perfect Competition

• There are a large number of people

buying any one good

– each person takes all prices as given and

seeks to maximize utility given his budget

constraint

• There are a large number of firms

producing each good

– each firm takes all prices as given and

attempts to maximize profits

2

5

General Equilibrium

• Assume that there are only two goods, x

and y

• All individuals are assumed to have

identical preferences

– represented by an indifference map

• The production possibility curve can be

used to show how outputs and inputs are

related

6

Edgeworth Box Diagram • Construction of the production possibility

curve for x and y starts with the

assumption that the amounts of k and l

are fixed

• An Edgeworth box shows every possible

way the existing k and l might be used to

produce x and y

– any point in the box represents a fully

employed allocation of the available

resources to x and y

7

Edgeworth Box Diagram

Ox

Oy

Total Labor

To

tal C

ap

ital

A

Cap

ital

for

x

Cap

ital fo

r y

Labor for y Labor for x Capital

in y

production

Capital

in x

production

Labor in y production

Labor in x production 8


• Many of the allocations in the Edgeworth

box are technically inefficient

– it is possible to produce more x and more y by

shifting capital and labor around

• We will assume that competitive markets

will not exhibit inefficient input choices

• We want to find the efficient allocations

– they illustrate the actual production outcomes

3

9


• We will use isoquant maps for the two

goods

– the isoquant map for good x uses Ox as the

origin

– the isoquant map for good y uses Oy as the

origin

• The efficient allocations will occur where

the isoquants are tangent to one another

10


Ox

Oy

Total Labor

To

tal C

ap

ital

x2

x1

y1

y2

A

Point A is inefficient because, by moving along y1, we can increase

x from x1 to x2 while holding y constant

11


Ox

Oy

Total Labor

To

tal C

ap

ital

x2

x1

y1

y2

A

We could also increase y from y1 to y2 while holding x constant

by moving along x1

12


Ox

Oy

Total Labor

To

tal C

ap

ital

At each efficient point, the RTS (of k for l) is equal in both

x and y production

x2 x1

x4

x3

y1

y2

y3

y4

p4

p3

p2

p1

4

13

Production Possibility Frontier

• The locus of efficient points shows the

maximum output of y that can be

produced for any level of x

– we can use this information to construct a

production possibility frontier

• shows the alternative outputs of x and y that

can be produced with the fixed capital and

labor inputs that are employed efficiently

14

Production Possibility Frontier

Quantity of x

Quantity of y

p4

p3

p2

p1

y1

y2

y3

y4

x1 x2 x3 x4

Ox

Oy

Each efficient point of production

becomes a point on the production

possibility frontier

The negative of the slope of

the production possibility

frontier is the rate of product

transformation (RPT)

15

Rate of Product Transformation

• The rate of product transformation (RPT)

between two outputs is the negative of

the slope of the production possibility

frontier

frontiery possibilit

production of slope ) for (of yxRPT

) (along ) for (of yxOOdx

dyyxRPT

16

Rate of Product Transformation

• The rate of product transformation shows

how x can be technically traded for y

while continuing to keep the available

productive inputs efficiently employed

5

17

Shape of the Production Possibility Frontier

• The production possibility frontier shown

earlier exhibited an increasing RPT

– this concave shape will characterize most

production situations

• RPT is equal to the ratio of MCx to MCy

18


• Suppose that the costs of any output

combination are C(x,y)

– along the production possibility frontier,

C(x,y) is constant

• We can write the total differential of the

cost function as

0

dy

y

Cdx

x

CdC

19


• Rewriting, we get

y

xyx

MC

MC

yC

xCOO

dx

dyRPT

/

/) (along

• The RPT is a measure of the relative

marginal costs of the two goods

20


• As production of x rises and production

of y falls, the ratio of MCx to MCy rises

– this occurs if both goods are produced

under diminishing returns

• increasing the production of x raises MCx, while

reducing the production of y lowers MCy

– this could also occur if some inputs were

more suited for x production than for y

production

6

21


• But we have assumed that inputs are

homogeneous

• We need an explanation that allows

homogeneous inputs and constant

returns to scale

• The production possibility frontier will be

concave if goods x and y use inputs in

different proportions 22

Opportunity Cost

• The production possibility frontier

demonstrates that there are many

possible efficient combinations of two

goods

• Producing more of one good

necessitates lowering the production of

the other good

– this is what economists mean by opportunity

cost

23

Opportunity Cost

• The opportunity cost of one more unit of

x is the reduction in y that this entails

• Thus, the opportunity cost is best

measured as the RPT (of x for y) at the

prevailing point on the production

possibility frontier

– this opportunity cost rises as more x is

produced

24

Concavity of the Production Possibility Frontier

• Suppose that the production of x and y

depends only on labor and the production

functions are

5.0)( xxfx ll 5.0)( yyfy ll

• If labor supply is fixed at 100, then

lx + ly = 100

• The production possibility frontier is

x2 + y2 = 100 for x,y 0

7

25

Concavity of the Production Possibility Frontier

• The RPT can be calculated by taking the

total differential:

y

x

y

x

dx

dyRPTydyxdx

2

)2( or 022

• The slope of the production possibility frontier increases as x output increases

– the frontier is concave

26

Determination of Equilibrium Prices

• We can use the production possibility

frontier along with a set of indifference

curves to show how equilibrium prices

are determined

– the indifference curves represent

individuals’ preferences for the two goods

27


Quantity of x

Quantity of y

U1

U2

U3

y1

x1

Output will be x1, y1

If the prices of x and y are px and py,

society’s budget constraint is C

y

x

p

p slope

C

C

Individuals will demand x1’, y1’

x1’

y1’

28


Quantity of x

Quantity of y

y1

x

1

U1

U2

U3

y

x

p

p slope

C

C

The price of x will rise and

the price of y will fall

x1’

y1’

There is excess demand for x and

excess supply of y

excess

supply

excess demand

8

29


Quantity of x

Quantity of y

y1

x

1

U1

U2

U3

y

x

p

p slope

C

C

x1’

y1’

The equilibrium output will

be x1* and y1* y1*

x1*

The equilibrium prices will

be px* and py*

C*

C*

*

* slope

y

x

p

p

30

Comparative Statics Analysis

• The equilibrium price ratio will tend to

persist until either preferences or

production technologies change

• If preferences were to shift toward good

x, px /py would rise and more x and less

y would be produced

– we would move in a clockwise direction

along the production possibility frontier

31

Comparative Statics Analysis

• Technical progress in the production of

good x will shift the production

possibility curve outward

– this will lower the relative price of x

– more x will be consumed

• if x is a normal good

– the effect on y is ambiguous

32

Technical Progress in the Production of x

Quantity of x

Quantity of y

U1

U2

U3

x1*

The relative price of x will fall

More x will be consumed

x2*

Technical progress in the production

of x will shift the production possibility

curve out

9

33

General Equilibrium Pricing

• Suppose that the production possibility

frontier can be represented by

x 2 + y 2 = 100

• Suppose also that the community’s

preferences can be represented by

U(x,y) = x0.5y0.5

34


• Profit-maximizing firms will equate RPT

and the ratio of px /py

y

x

p

p

y

xRPT

• Utility maximization requires that

y

x

p

p

x

yMRS

35


• Equilibrium requires that firms and

individuals face the same price ratio

MRSx

y

p

p

y

xRPT

y

x

or

x* = y*

36

The Corn Laws Debate

• High tariffs on grain imports were

imposed by the British government after

the Napoleonic wars

• Economists debated the effects of these

“corn laws” between 1829 and 1845

– what effect would the elimination of these

tariffs have on factor prices?

10

37


Quantity of Grain (x)

Quantity of

manufactured

goods (y)

U1

U2

x0

If the corn laws completely prevented

trade, output would be x0 and y0

y0

The equilibrium prices will be

px* and py*

*

* slope

y

x

p

p

38



Quantity of

manufactured

goods (y)

x0

U1

U2

y0

Removal of the corn laws will change

the prices to px’ and py’

'

' slope

y

x

p

p

Output will be x1’ and y1’

x1’

y1’

y1

x1

Individuals will demand x1 and y1

39



Quantity of

manufactured

goods (y)

y1

x0

U1

U2

x1

y0

'

' slope

y

x

p

p

x1’

y1’

Grain imports will be x1 – x1’

imports of grain

These imports will be financed by

the export of manufactured goods

equal to y1’ – y1 exports

of

goods

40


• We can use an Edgeworth box diagram

to see the effects of tariff reduction on

the use of labor and capital

• If the corn laws were repealed, there

would be an increase in the production

of manufactured goods and a decline in

the production of grain

11

41


Ox

Oy

Total Labor

To

tal C

ap

ital

A repeal of the corn laws would result in a movement from p3 to

p1 where more y and less x is produced

x2 x1

x4

x3

y1

y2

y3

y4

p4

p3

p2

p1

42


• If we assume that grain production is

relatively capital intensive, the movement

from p3 to p1 causes the ratio of k to l to

rise in both industries

– the relative price of capital will fall

– the relative price of labor will rise

• The repeal of the corn laws will be

harmful to capital owners and helpful to

laborers

43

Political Support for Trade Policies

• Trade policies may affect the relative

incomes of various factors of production

• In the United States, exports tend to be

intensive in their use of skilled labor

whereas imports tend to be intensive in

their use of unskilled labor

– free trade policies will result in rising relative

wages for skilled workers and in falling

relative wages for unskilled workers 44

Existence of General Equilibrium Prices

• Beginning with 19th century investigations

by Leon Walras, economists have

examined whether there exists a set of

prices that equilibrates all markets

simultaneously

– if this set of prices exists, how can it be

found?

12

45


• Suppose that there are n goods in fixed

supply in this economy

– let Si (i =1,…,n) be the total supply of good i

available

– let pi (i =1,…n) be the price of good i

• The total demand for good i depends on

all prices

Di (p1,…,pn) for i =1,…,n 46


• We will write this demand function as

dependent on the whole set of prices (P)

Di (P)

• Walras’ problem: Does there exist an

equilibrium set of prices such that

Di (P*) = Si

for all values of i ?

47

Excess Demand Functions

• The excess demand function for any

good i at any set of prices (P) is defined

to be

EDi (P) = Di (P) – Si

• This means that the equilibrium

condition can be rewritten as

EDi (P*) = Di (P*) – Si = 0

48

Excess Demand Functions

• Demand functions are homogeneous of

degree zero

– this implies that we can only establish

equilibrium relative prices in a Walrasian-

type model

• Walras also assumed that demand

functions are continuous

– small changes in price lead to small changes

in quantity demanded

13

49

Walras’ Law

• A final observation that Walras made

was that the n excess demand equations

are not independent of one another

• Walras’ law shows that the total value of

excess demand is zero at any set of

prices

n

i

ii PEDP1

0)(

50

Walras’ Law

• Walras’ law holds for any set of prices

(not just equilibrium prices)

• There can be neither excess demand for

all goods together nor excess supply

51

Walras’ Proof of the Existence of Equilibrium Prices

• The market equilibrium conditions

provide (n-1) independent equations in

(n-1) unknown relative prices

– can we solve the system for an equilibrium

condition?

• the equations are not necessarily linear

• all prices must be nonnegative

• To attack these difficulties, Walras set up

a complicated proof

52


• Start with an arbitrary set of prices

• Holding the other n-1 prices constant,

find the equilibrium price for good 1 (p1’)

• Holding p1’ and the other n-2 prices

constant, solve for the equilibrium price

of good 2 (p2’)

– in changing p2 from its initial position to p2’,

the price calculated for good 1 does not

need to remain an equilibrium price

14

53


• Using the provisional prices p1’ and p2’,

solve for p3’

– proceed in this way until an entire set of

provisional relative prices has been found

• In the 2nd iteration of Walras’ proof,

p2’,…,pn’ are held constant while a new

equilibrium price is calculated for good 1

– proceed in this way until an entire new set of

prices is found

54


• The importance of Walras’ proof is its

ability to demonstrate the simultaneous

nature of the problem of finding

equilibrium prices

• Because it is cumbersome, it is not

generally used today

• More recent work uses some relatively

simple tools from advanced mathematics

55

Brouwer’s Fixed-Point Theorem

• Any continuous mapping [F(X)] of a

closed, bounded, convex set into itself

has at least one fixed point (X*) such

that F(X*) = X*

56


x

f (X)

1

1

0

45

Any continuous function must

cross the 45 line

Suppose that f(X) is a continuous function defined

on the interval [0,1] and that f(X) takes on the

values also on the interval [0,1]

This point of crossing is a

“fixed point” because f maps

this point (X*) into itself

X*

f (X*)

15

57


• A mapping is a rule that associates the

points in one set with points in another set

– let X be a point for which a mapping (F) is

defined

• the mapping associates X with some point Y = F(X)

– if a mapping is defined over a subset of n-

dimensional space (S), and if every point in S

is associated (by the rule F) with some other

point in S, the mapping is said to map S into

itself 58


• A mapping is continuous if points that are

“close” to each other are mapped into other

points that are “close” to each other

• The Brouwer fixed-point theorem considers

mappings defined on certain kinds of sets

– closed (they contain their boundaries)

– bounded (none of their dimensions is infinitely

large)

– convex (they have no “holes” in them)

59

Proof of the Existence of Equilibrium Prices

• Because only relative prices matter, it is

convenient to assume that prices have

been defined so that the sum of all prices

is equal to 1

• Thus, for any arbitrary set of prices

(p1,…,pn), we can use normalized prices

of the form

n

i

i

ii

p

pp

1

'

60


• These new prices will retain their original

relative values and will sum to 1

1'1

n

i

ip

j

i

j

i

p

p

p

p

'

'

• These new prices will sum to 1

16

61


• We will assume that the feasible set of

prices (S) is composed of all

nonnegative numbers that sum to 1

– S is the set to which we will apply Brouwer’s

theorem

– S is closed, bounded, and convex

– we will need to define a continuous mapping

of S into itself

62

Free Goods

• Equilibrium does not really require that

excess demand be zero for every market

• Goods may exist for which the markets

are in equilibrium where supply exceeds

demand (negative excess demand)

– it is necessary for the prices of these goods

to be equal to zero

– “free goods”

63

Free Goods

• The equilibrium conditions are

EDi (P*) = 0 for pi* > 0

EDi (P*) 0 for pi* = 0

• Note that this set of equilibrium prices

continues to obey Walras’ law

64

Mapping the Set of Prices Into Itself

• In order to achieve equilibrium, prices of

goods in excess demand should be

raised, whereas those in excess supply

should have their prices lowered

17

65


• We define the mapping F(P) for any

normalized set of prices (P), such that

the ith component of F(P) is given by

F i(P) = pi + EDi (P)

• The mapping performs the necessary

task of appropriately raising or lowering

prices

66


• Two problems exist with this mapping

• First, nothing ensures that the prices will

be nonnegative

– the mapping must be redefined to be

F i(P) = Max [pi + EDi (P),0]

– the new prices defined by the mapping must

be positive or zero

67


• Second, the recalculated prices are not

necessarily normalized

– they will not sum to 1

– it will be simple to normalize such that

n

i

i PF1

1)(

– we will assume that this normalization has

been done 68

Application of Brouwer’s Theorem

• Thus, F satisfies the conditions of the

Brouwer fixed-point theorem

– it is a continuous mapping of the set S into

itself

• There exists a point (P*) that is mapped

into itself

• For this point,

pi* = Max [pi* + EDi (P*),0] for all i

18

69

Application of Brouwer’s Theorem

• This says that P* is an equilibrium set of prices

– for pi* > 0,

pi* = pi* + EDi (P*)

EDi (P*) = 0

– For pi* = 0,

pi* + EDi (P*) 0

EDi (P*) 0

70

A General Equilibrium with Three Goods

• The economy of Oz is composed only of

three precious metals: (1) silver, (2)

gold, and (3) platinum

– there are 10 (thousand) ounces of each

metal available

• The demands for gold and platinum are

1121

3

1

22

p

p

p

pD 182

1

3

1

23

p

p

p

pD

71


• Equilibrium in the gold and platinum

markets requires that demand equal

supply in both markets simultaneously

101121

3

1

2 p

p

p

p

101821

3

1

2 p

p

p

p

72


• This system of simultaneous equations can be solved as

p2/p1 = 2 p3/p1 = 3

• In equilibrium:

– gold will have a price twice that of silver

– platinum will have a price three times that of silver

– the price of platinum will be 1.5 times that of gold

19

73


• Because Walras’ law must hold, we know

p1ED1 = – p2ED2 – p3ED3

• Substituting the excess demand functions

for gold and silver and substituting, we get

3

1

23

1

322

1

32

1

22

11 822 pp

p

p

ppp

p

pp

p

pEDp

1

3

1

2

21

23

21

22

1 822p

p

p

p

p

p

p

pED

74

Smith’s Invisible Hand Hypothesis

• Adam Smith believed that the

competitive market system provided a

powerful “invisible hand” that ensured

resources would find their way to where

they were most valued

• Reliance on the economic self-interest

of individuals and firms would result in a

desirable social outcome

75

Smith’s Invisible Hand Hypothesis

• Smith’s insights gave rise to modern

welfare economics

• The “First Theorem of Welfare

Economics” suggests that there is an

exact correspondence between the

efficient allocation of resources and the

competitive pricing of these resources

76

Pareto Efficiency

• An allocation of resources is Pareto

efficient if it is not possible (through

further reallocations) to make one person

better off without making someone else

worse off

• The Pareto definition identifies allocations

as being “inefficient” if unambiguous

improvements are possible

20

77

Efficiency in Production

• An allocation of resources is efficient in

production (or “technically efficient”) if no

further reallocation would permit more of

one good to be produced without

necessarily reducing the output of some

other good

• Technical efficiency is a precondition for

Pareto efficiency but does not guarantee

Pareto efficiency 78

Efficient Choice of Inputs for a Single Firm

• A single firm with fixed inputs of labor

and capital will have allocated these

resources efficiently if they are fully

employed and if the RTS between

capital and labor is the same for every

output the firm produces

79


• Assume that the firm produces two

goods (x and y) and that the available

levels of capital and labor are k’ and l’

• The production function for x is given by

x = f (kx, lx)

• If we assume full employment, the

production function for y is

y = g (ky, ly) = g (k’ - kx, l’ - lx)

80


• Technical efficiency requires that x

output be as large as possible for any

value of y (y’)

• Setting up the Lagrangian and solving for

the first-order conditions:

L = f (kx, lx) + [y’ – g (k’ - kx, l’ - lx)]

L/kx = fk + gk = 0

L/lx = fl + gl = 0

L/ = y’ – g (k’ - kx, l’ - lx) = 0

21

81


• From the first two conditions, we can see

that

ll g

g

f

f kk


RTSx (k for l) = RTSy (k for l)

82

Efficient Allocation of Resources among Firms

• Resources should be allocated to those

firms where they can be most efficiently

used

– the marginal physical product of any

resource in the production of a particular

good should be the same across all firms

that produce the good

83


• Suppose that there are two firms

producing x and their production

functions are

f1(k1, l1)

f2(k2, l2)

• Assume that the total supplies of capital

and labor are k’ and l’

84


• The allocational problem is to maximize

x = f1(k1, l1) + f2(k2, l2)

subject to the constraints

k1 + k2 = k’

l1 + l2 = l’

• Substituting, the maximization problem

becomes

x = f1(k1, l1) + f2(k’ - k1, l’ - l1)

22

85


• First-order conditions for a maximum

are

02

2

1

1

1

2

1

1

1

k

f

k

f

k

f

k

f

k

x

02

2

1

1

1

2

1

1

1

lllll

ffffx

86


• These first-order conditions can be

rewritten as

2

2

1

1

k

f

k

f

2

2

1

1

ll

ff

• The marginal physical product of each

input should be equal across the two

firms

87

Efficient Choice of Output by Firms

• Suppose that there are two outputs (x

and y) each produced by two firms

• The production possibility frontiers for

these two firms are

yi = fi (xi ) for i=1,2

• The overall optimization problem is to

produce the maximum amount of x for

any given level of y (y*) 88


• The Lagrangian for this problem is

L = x1 + x2 + [y* - f1(x1) - f2(x2)]

and yields the first-order condition:

f1/x1 = f2/x2

• The rate of product transformation

(RPT) should be the same for all firms

producing these goods

23

89


Trucks Trucks

Cars Cars

Firm A Firm B

50 50

100 100 1

2RPT 1

1RPT

Firm A is relatively efficient at producing cars, while Firm B

is relatively efficient at producing trucks

90


Trucks Trucks

Cars Cars

Firm A Firm B

50 50

100 100 1

2RPT 1

1RPT

If each firm was to specialize in its efficient product, total

output could be increased

91

Theory of Comparative Advantage

• The theory of comparative advantage

was first proposed by Ricardo

– countries should specialize in producing

those goods of which they are relatively

more efficient producers

• these countries should then trade with the rest

of the world to obtain needed commodities

– if countries do specialize this way, total

world production will be greater 92

Efficiency in Product Mix

• Technical efficiency is not a sufficient

condition for Pareto efficiency

– demand must also be brought into the

picture

• In order to ensure Pareto efficiency, we

must be able to tie individual’s

preferences and production possibilities

together

24

93


• The condition necessary to ensure that

the right goods are produced is

MRS = RPT

– the psychological rate of trade-off between

the two goods in people’s preferences must

be equal to the rate at which they can be

traded off in production

94


Output of x

Output of y Suppose that we have a one-person (Robinson

Crusoe) economy and PP represents the

combinations of x and y that can be produced

P

P

Any point on PP represents a

point of technical efficiency

95


Output of x

Output of y

P

P At the point of

tangency, Crusoe’s

MRS will be equal to

the technical RPT

Only one point on PP will maximize

Crusoe’s utility

U1

U2

U3

96


• Assume that there are only two goods

(x and y) and one individual in society

(Robinson Crusoe)

• Crusoe’s utility function is

U = U(x,y)

• The production possibility frontier is

T(x,y) = 0

25

97


• Crusoe’s problem is to maximize his

utility subject to the production

constraint

• Setting up the Lagrangian yields

L = U(x,y) + [T(x,y)]

98


• First-order conditions for an interior

maximum are

0

x

T

x

U

x

L

0

y

T

y

U

y

L

0),(

yxT

L

99


• Combining the first two, we get

yT

xT

yU

xU

/

/

/

/

or

) for ( ) (along ) for ( yxRPTTdx

dyyxMRS

100

Competitive Prices and Efficiency

• Attaining a Pareto efficient allocation of

resources requires that the rate of

trade-off between any two goods be the

same for all economic agents

• In a perfectly competitive economy, the

ratio of the prices of the two goods

provides the common rate of trade-off to

which all agents will adjust

26

101

Competitive Prices and Efficiency

• Because all agents face the same

prices, all trade-off rates will be

equalized and an efficient allocation will

be achieved

• This is the “First Theorem of Welfare

Economics”

102


• In minimizing costs, a firm will equate

the RTS between any two inputs (k and

l) to the ratio of their competitive prices

(w/v)

– this is true for all outputs the firm produces

– RTS will be equal across all outputs

103


• A profit-maximizing firm will hire

additional units of an input (l) up to the

point at which its marginal contribution

to revenues is equal to the marginal

cost of hiring the input (w)

pxfl = w

104


• If this is true for every firm, then with a

competitive labor market

pxfl1 = w = pxfl

2

fl1 = fl

2

• Every firm that produces x has identical

marginal productivities of every input in

the production of x

27

105


• Recall that the RPT (of x for y) is equal

to MCx /MCy

• In perfect competition, each profit-

maximizing firm will produce the output

level for which marginal cost is equal to

price

• Since px = MCx and py = MCy for every

firm, RTS = MCx /MCy = px /py

106


• Thus, the profit-maximizing decisions

of many firms can achieve technical

efficiency in production without any

central direction

• Competitive market prices act as

signals to unify the multitude of

decisions that firms make into one

coherent, efficient pattern

107


• The price ratios quoted to consumers

are the same ratios the market presents

to firms

• This implies that the MRS shared by all

individuals will be equal to the RPT

shared by all the firms

• An efficient mix of goods will therefore

be produced

108


Output of x

Output of y

P

P

U0

x* and y* represent the efficient output mix

x*

y*

Only with a price ratio of

px*/py* will supply and

demand be in equilibrium

*

* slope

y

x

p

p

28

109

Laissez-Faire Policies

• The correspondence between

competitive equilibrium and Pareto

efficiency provides some support for the

laissez-faire position taken by many

economists

– government intervention may only result in

a loss of Pareto efficiency

110

Departing from the Competitive Assumptions

• The ability of competitive markets to

achieve efficiency may be impaired

because of

– imperfect competition

– externalities

– public goods

– imperfect information

111

Imperfect Competition • Imperfect competition includes all

situations in which economic agents

exert some market power in determining

market prices

– these agents will take these effects into

account in their decisions

• Market prices no longer carry the

informational content required to achieve

Pareto efficiency 112

Externalities • An externality occurs when there are

interactions among firms and individuals

that are not adequately reflected in

market prices

• With externalities, market prices no

longer reflect all of a good’s costs of

production

– there is a divergence between private and

social marginal cost

29

113

Public Goods

• Public goods have two properties that

make them unsuitable for production in

markets

– they are nonrival

• additional people can consume the benefits of

these goods at zero cost

– they are nonexclusive

• extra individuals cannot be precluded from

consuming the good

114

Imperfect Information

• If economic actors are uncertain about

prices or if markets cannot reach

equilibrium, there is no reason to expect

that the efficiency property of

competitive pricing will be retained

115

Distribution

• Although the First Theorem of Welfare

Economics ensures that competitive

markets will achieve efficient allocations,

there are no guarantees that these

allocations will exhibit desirable

distributions of welfare among individuals

116

Distribution

• Assume that there are only two people

in society (Smith and Jones)

• The quantities of two goods (x and y) to

be distributed among these two people

are fixed in supply

• We can use an Edgeworth box diagram

to show all possible allocations of these

goods between Smith and Jones

30

117

Distribution OJ

OS

Total Y

Total X

UJ4

UJ3

UJ2

UJ1

US4

US3

US2

US1

118

Distribution

• Any point within the Edgeworth box in

which the MRS for Smith is unequal to

that for Jones offers an opportunity for

Pareto improvements

– both can move to higher levels of utility

through trade

119

Distribution OJ

OS

UJ4

UJ3

UJ2

UJ1

US4

US3

US2

US1

A

Any trade in this area is

an improvement over A 120

Contract Curve

• In an exchange economy, all efficient

allocations lie along a contract curve

– points off the curve are necessarily

inefficient

• individuals can be made better off by moving to

the curve

• Along the contract curve, individuals’

preferences are rivals

– one may be made better off only by making

the other worse off

31

121

Contract Curve OJ

OS

UJ4

UJ3

UJ2

UJ1

US4

US3

US2

US1

A

Contract curve

122

Exchange with Initial Endowments

• Suppose that the two individuals

possess different quantities of the two

goods at the start

– it is possible that the two individuals could

both benefit from trade if the initial

allocations were inefficient

123


• Neither person would engage in a trade

that would leave him worse off

• Only a portion of the contract curve

shows allocations that may result from

voluntary exchange

124


OJ

OS

UJA

USA

A

Suppose that A represents

the initial endowments

32

125


OJ

OS

UJA

USA

A

Neither individual would be

willing to accept a lower level

of utility than A gives

126


OJ

OS

UJA

USA

A

Only allocations between M1

and M2 will be acceptable to

both

M1

M2

127

The Distributional Dilemma

• If the initial endowments are skewed in

favor of some economic actors, the

Pareto efficient allocations promised by

the competitive price system will also

tend to favor those actors

– voluntary transactions cannot overcome

large differences in initial endowments

– some sort of transfers will be needed to

attain more equal results 128

The Distributional Dilemma

• These thoughts lead to the “Second

Theorem of Welfare Economics”

– any desired distribution of welfare among

individuals in an economy can be achieved

in an efficient manner through competitive

pricing if initial endowments are adjusted

appropriately

33

129


• Preferences and production

technologies provide the building

blocks upon which all general

equilibrium models are based

– one particularly simple version of such a

model uses individual preferences for two

goods together with a concave production

possibility frontier for those two goods

130


• Competitive markets can establish

equilibrium prices by making marginal

adjustments in prices in response to

information about the demand and

supply for individual goods

– Walras’ law ties markets together so that

such a solution is assured (in most cases)

131


• Competitive prices will result in a

Pareto-efficient allocation of resources

– this is the First Theorem of Welfare

Economics

132


• Factors that will interfere with

competitive markets’ abilities to

achieve efficiency include

– market power

– externalities

– existence of public goods

– imperfect information

34

133


• Competitive markets need not yield

equitable distributions of resources,

especially when initial endowments are

very skewed

– in theory any desired distribution can be

attained through competitive markets

accompanied by lump-sum transfers

• there are many practical problems in

implementing such transfers

1

1

Chapter 14

MODELS OF MONOPOLY


Monopoly

• A monopoly is a single supplier to a

market

• This firm may choose to produce at any

point on the market demand curve

3

Barriers to Entry

• The reason a monopoly exists is that

other firms find it unprofitable or

impossible to enter the market

• Barriers to entry are the source of all

monopoly power

– there are two general types of barriers to

entry

• technical barriers

• legal barriers

4

Technical Barriers to Entry

• The production of a good may exhibit

decreasing marginal and average costs

over a wide range of output levels

– in this situation, relatively large-scale firms

are low-cost producers

• firms may find it profitable to drive others out of

the industry by cutting prices

• this situation is known as natural monopoly

• once the monopoly is established, entry of new

firms will be difficult

2

5

Technical Barriers to Entry

• Another technical basis of monopoly is

special knowledge of a low-cost

productive technique

– it may be difficult to keep this knowledge

out of the hands of other firms

• Ownership of unique resources may

also be a lasting basis for maintaining a

monopoly

6

Legal Barriers to Entry

• Many pure monopolies are created as a

matter of law

– with a patent, the basic technology for a

product is assigned to one firm

– the government may also award a firm an

exclusive franchise to serve a market

7

Creation of Barriers to Entry

• Some barriers to entry result from actions

taken by the firm

– research and development of new products

or technologies

– purchase of unique resources

– lobbying efforts to gain monopoly power

• The attempt by a monopolist to erect

barriers to entry may involve real

resource costs 8

Profit Maximization

• To maximize profits, a monopolist will

choose to produce that output level for

which marginal revenue is equal to

marginal cost

– marginal revenue is less than price because

the monopolist faces a downward-sloping

demand curve

• he must lower its price on all units to be sold if it

is to generate the extra demand for this unit

3

9

Profit Maximization

• Since MR = MC at the profit-maximizing

output and P > MR for a monopolist, the

monopolist will set a price greater than

marginal cost

10

C

Profits can be found in

the shaded rectangle

Profit Maximization

AC

MC

D MR

Quantity

Price

Q*

The monopolist will maximize

profits where MR = MC

P* The firm will charge a price

of P*

11


• The gap between a firm’s price and its

marginal cost is inversely related to the

price elasticity of demand facing the firm

PQeP

MCP

,

1

where eQ,P is the elasticity of demand

for the entire market

12


• Two general conclusions about monopoly

pricing can be drawn:

– a monopoly will choose to operate only in

regions where the market demand curve is

elastic

• eQ,P < -1

– the firm’s “markup” over marginal cost

depends inversely on the elasticity of market

demand

4

13

Monopoly Profits

• Monopoly profits will be positive as long

as P > AC

• Monopoly profits can continue into the

long run because entry is not possible

– some economists refer to the profits that a

monopoly earns in the long run as

monopoly rents

• the return to the factor that forms the basis of

the monopoly

14

Monopoly Profits

• The size of monopoly profits in the long

run will depend on the relationship

between average costs and market

demand for the product

15

Monopoly Profits

Quantity

Price MC

AC

MR D

Quantity

Price MC

AC

MR D

Positive profits Zero profit

P* P*=AC

C

Q* Q*

16

No Monopoly Supply Curve

• With a fixed market demand curve, the

supply “curve” for a monopolist will only

be one point

– the price-output combination where MR = MC

• If the demand curve shifts, the marginal

revenue curve shifts and a new profit-

maximizing output will be chosen

5

17

Monopoly with Linear Demand

• Suppose that the market for frisbees

has a linear demand curve of the form

Q = 2,000 - 20P

or

P = 100 - Q/20

• The total costs of the frisbee producer

are given by

C(Q) = 0.05Q2 + 10,000

18


• To maximize profits, the monopolist

chooses the output for which MR = MC

• We need to find total revenue

TR = PQ = 100Q - Q2/20

• Therefore, marginal revenue is

MR = 100 - Q/10

while marginal cost is

MC = 0.01Q

19


• Thus, MR = MC where

100 - Q/10 = 0.01Q

Q* = 500 P* = 75

• At the profit-maximizing output,

C(Q) = 0.05(500)2 + 10,000 = 22,500

AC = 22,500/500 = 45

= (P* - AC)Q = (75 - 45)500 = 15,000

20


• To see that the inverse elasticity rule

holds, we can calculate the elasticity of

demand at the monopoly’s profit-

maximizing level of output

3500

7520,

Q

P

P

Qe PQ

6

21


• The inverse elasticity rule specifies that

3

11

,

PQeP

MCP

• Since P* = 75 and MC = 50, this

relationship holds

22

Monopoly and Resource Allocation

• To evaluate the allocational effect of a

monopoly, we will use a perfectly

competitive, constant-cost industry as a

basis of comparison

– the industry’s long-run supply curve is

infinitely elastic with a price equal to both

marginal and average cost

23


Quantity

Price

MC=AC

D

MR

If this market was competitive, output would

be Q* and price would be P*

Q*

P*

Under a monopoly, output would be Q**

and price would rise to P**

Q**

P**

24

Consumer surplus would fall

Producer surplus will rise

There is a deadweight

loss from monopoly


Quantity

Price

MC=AC

D

MR

Q* Q**

P*

P** Consumer surplus falls by more

than producer surplus rises.

7

25

Welfare Losses and Elasticity

• Assume that the constant marginal (and

average) costs for a monopolist are

given by c and that the compensated

demand curve has a constant elasticity:

Q = Pe

where e is the price elasticity of demand

(e < -1)

26


• The competitive price in this market will

be

Pc = c

and the monopoly price is given by

e

cPm 1

1

27


• The consumer surplus associated with

any price (P0) can be computed as

00

)(P

e

PdPPdPPQCS

11

1

0

1

0

e

P

e

PCS

e

P

e

28


• Therefore, under perfect competition

1

1

e

cCS

e

c

1

11

1

e

e

c

CS

e

m

and under monopoly

8

29


• Taking the ratio of these two surplus

measures yields 1

11

1

e

c

m

e

CS

CS

• If e = -2, this ratio is ½

– consumer surplus under monopoly is half

what it is under perfect competition 30


• Monopoly profits are given by

mmmmm Qc

e

ccQQP

1

1

e

e

c

e

c

e

e

cee

m

1

11

11

11

1

31


• To find the transfer from consumer

surplus into monopoly profits we can

divide monopoly profits by the competitive

consumer surplus

e

e

c

m

e

e

e

e

e

CS

111

11

1

• If e = -2, this ratio is ¼ 32

Monopoly and Product Quality

• The market power enjoyed by a monopoly

may be exercised along dimensions other

than the market price of its product

– type, quality, or diversity of goods

• Whether a monopoly will produce a

higher-quality or lower-quality good than

would be produced under competition

depends on demand and the firm’s costs

9

33


• Suppose that consumers’ willingness to

pay for quality (X) is given by the inverse

demand function P(Q,X) where

P/Q < 0 and P/X > 0

• If costs are given by C(Q,X), the

monopoly will choose Q and X to

maximize

= P(Q,X)Q - C(Q,X)

34


• First-order conditions for a maximum are

0),(

QC

Q

PQXQP

Q

0

XC

X

PQ

X

– MR = MC for output decisions

– Marginal revenue from increasing quality by

one unit is equal to the marginal cost of

making such an increase

35


• The level of product quality that will be

opted for under competitive conditions is

the one that maximizes net social welfare

*

0),(),(

Q

XQCdQXQPSW

• Maximizing with respect to X yields

*

00),(

Q

XX CdQXQPX

SW

36


• The difference between the quality choice

of a competitive industry and the

monopolist is:

– the monopolist looks at the marginal

valuation of one more unit of quality

assuming that Q is at its profit-maximizing

level

– the competitve industry looks at the marginal

value of quality averaged across all output

levels

10

37


• Even if a monopoly and a perfectly

competitive industry chose the same

output level, they might opt for diffferent

quality levels

– each is concerned with a different margin

in its decision making

38

Price Discrimination

• A monopoly engages in price

discrimination if it is able to sell otherwise

identical units of output at different prices

• Whether a price discrimination strategy is

feasible depends on the inability of

buyers to practice arbitrage

– profit-seeking middlemen will destroy any

discriminatory pricing scheme if possible

• price discrimination becomes possible if resale is

costly

39

Perfect Price Discrimination

• If each buyer can be separately

identified by the monopolist, it may be

possible to charge each buyer the

maximum price he would be willing to

pay for the good

– perfect or first-degree price discrimination

• extracts all consumer surplus

• no deadweight loss

40

The monopolist will

continue this way until the

marginal buyer is no

longer willing to pay the

good’s marginal cost

Q1

P1

The first buyer pays P1 for Q1 units

Q2

Q2

P2 The second buyer pays P2 for Q2-Q1 units


Quantity

Price

D

Under perfect price discrimination, the monopolist

charges a different price to each buyer

MC

11

41


• Recall the example of the frisbee

manufacturer

• If this monopolist wishes to practice

perfect price discrimination, he will want

to produce the quantity for which the

marginal buyer pays a price exactly

equal to the marginal cost

42


• Therefore,

P = 100 - Q/20 = MC = 0.1Q

Q* = 666

• Total revenue and total costs will be

511,5540

100)(

666

0

*

0

2

Q Q

QdQQPR

178,32000,1005.0)( 2 QQc

• Profit is much larger (23,333 > 15,000)

43

Market Separation

• Perfect price discrimination requires the

monopolist to know the demand function

for each potential buyer

• A less stringent requirement would be to

assume that the monopoly can separate its

buyers into a few identifiable markets

– can follow a different pricing policy in each

market

– third-degree price discrimination 44

Market Separation

• All the monopolist needs to know in this

case is the price elasticities of demand

for each market

– set price according to the inverse elasticity

rule

• If the marginal cost is the same in all

markets,

)1

1()1

1(j

j

i

ie

Pe

P

12

45

Market Separation


)1

1(

)1

1(

i

j

j

i

e

e

P

P

• The profit-maximizing price will be

higher in markets where demand is less

elastic

46

Market Separation

Quantity in Market 2 Quantity in Market 1

Price

0

D D MR MR

MC MC

Q2*

P2

Q1*

P1

If two markets are separate, maximum profits occur by

setting different prices in the two markets

The market with the less

elastic demand will be

charged the higher price

47

Third-Degree Price Discrimination

• Suppose that the demand curves in two

separated markets are given by

Q1 = 24 – P1

Q2 = 24 – 2P2

• Suppose that MC = 6

• Profit maximization requires that

MR1 = 24 – 2Q1 = 6 = MR2 = 12 – Q2

48


• Optimal choices and prices are

Q1 = 9 P1 = 15

Q2 = 6 P2 = 9

• Profits for the monopoly are

= (P1 - 6)Q1 + (P2 - 6)Q2 = 81 + 18 = 99

13

49


• The allocational impact of this policy can be

evaluated by calculating the deadweight

losses in the two markets

– the competitive output would be 18 in market 1

and 12 in market 2

DW1 = 0.5(P1-MC)(18-Q1) = 0.5(15-6)(18-9) = 40.5

DW2 = 0.5(P2-MC)(12-Q2) = 0.5(9-6)(12-6) = 9

50


• If this monopoly was to pursue a single-

price policy, it would use the demand

function Q = Q1 + Q2 = 48 – 3P

• So marginal revenue would be

MR = 16 – 2Q/3

• Profit-maximization occurs where

Q = 15 P = 11

51


• The deadweight loss is smaller with one

price than with two:

DW = 0.5(P-MC)(30-Q) = 0.5(11-6)(15) = 37.5

52

Two-Part Tariffs

• A linear two-part tariff occurs when

buyers must pay a fixed fee for the right

to consume a good and a uniform price

for each unit consumed

T(q) = a + pq

• The monopolist’s goal is to choose a

and p to maximize profits, given the

demand for the product

14

53

Two-Part Tariffs

• Because the average price paid by any

demander is

p’ = T/q = a/q + p

this tariff is only feasible if those who

pay low average prices (those for whom

q is large) cannot resell the good to

those who must pay high average

prices (those for whom q is small)

54

Two-Part Tariffs

• One feasible approach for profit

maximization would be for the firm to set

p = MC and then set a equal to the

consumer surplus of the least eager

buyer

– this might not be the most profitable

approach

– in general, optimal pricing schedules will

depend on a variety of contingencies

55

Two-Part Tariffs

• Suppose there are two different buyers

with the demand functions

q1 = 24 - p1

q2 = 24 - 2p2

• If MC = 6, one way for the monopolist to

implement a two-part tariff would be to

set p1 = p2 = MC = 6

q1 = 18 q2 = 12

56

Two-Part Tariffs

• With this marginal price, demander 2

obtains consumer surplus of 36

– this would be the maximum entry fee that

can be charged without causing this buyer

to leave the market

• This means that the two-part tariff in this

case would be

T(q) = 36 + 6q

15

57

Regulation of Monopoly

• Natural monopolies such as the utility,

communications, and transportation

industries are highly regulated in many

countries

58


• Many economists believe that it is

important for the prices of regulated

monopolies to reflect marginal costs of

production accurately

• An enforced policy of marginal cost

pricing will cause a natural monopoly to

operate at a loss

– natural monopolies exhibit declining

average costs over a wide range of output

59


Quantity

Price

D

MR

AC

MC

Because natural monopolies exhibit

decreasing costs, MC falls below AC

C1

P1

Q1

An unregulated monopoly will

maximize profit at Q1 and P1

C2

P2

Q2

If regulators force the

monopoly to charge a

price of P2, the firm will

suffer a loss because

P2 < C2

60

cover the losses on the sales to

low-price customers

The profits on the sales to high-

price customers are enough to


Quantity

Price

D

AC

MC

Suppose that the regulatory commission allows the

monopoly to charge a price of P1 to some users

P1

Q1

C1

Other users are offered the lower price

of P2

P2

Q2

C2

16

61


• Another approach followed in many

regulatory situations is to allow the

monopoly to charge a price above

marginal cost that is sufficient to earn a

“fair” rate of return on investment

– if this rate of return is greater than that

which would occur in a competitive market,

there is an incentive to use relatively more

capital than would truly minimize costs

62


• Suppose that a regulated utility has a

production function of the form

q = f (k,l)

• The firm’s actual rate of return on

capital is defined as

k

wkpfs

ll

),(

63


• Suppose that s is constrained by

regulation to be equal to s0, then the

firm’s problem is to maximize profits

= pf (k,l) – wl – vk

subject to this constraint


L = pf (k,l) – wl – vk + [wl + s0k – pf (k,l)]

64


• If =0, regulation is ineffective and the

monopoly behaves like any profit-

maximizing firm

• If =1, the Lagrangian reduces to

L = (s0 – v)k

which (assuming s0>v), will mean that

the monopoly will hire infinite amounts

of capital – an implausible result

17

65


• Therefore, 0<<1 and the first-order

conditions for a maximum are:

0)(

ll

lpfwwpf

L

0)( 0

kk pfsvpf

k

L

0),(0

ll kpfsw

L

66


• Because s0>v and <1, this means that

pfk < v

• The firm will hire more capital than it

would under unregulated conditions

– it will also achieve a lower marginal

productivity of capital

67

Dynamic Views of Monopoly

• Some economists have stressed the

beneficial role that monopoly profits can

play in the process of economic

development

– these profits provide funds that can be

invested in research and development

– the possibility of attaining or maintaining a

monopoly position provides an incentive to

keep one step ahead of potential competitors 68


• The most profitable level of output for

the monopolist is the one for which


cost

– at this output level, price will exceed

marginal cost

– the profitability of the monopolist will

depend on the relationship between price

and average cost

18

69


• Relative to perfect competition,

monopoly involves a loss of consumer

surplus for demanders

– some of this is transferred into monopoly

profits, whereas some of the loss in

consumer surplus represents a

deadweight loss of overall economic

welfare

– it is a sign of Pareto inefficiency

70


• Monopolies may opt for different levels

of quality than would perfectly

competitive firms

• Durable good monopolists may be

constrained by markets for used goods

71


• A monopoly may be able to increase its

profits further through price

discrimination – charging different

prices to different categories of buyers

– the ability of the monopoly to practice

price discrimination depends on its ability

to prevent arbitrage among buyers

72


• Governments often choose to regulate

natural monopolies (firms with

diminishing average costs over a broad

range of output levels)

– the type of regulatory mechanisms

adopted can affect the behavior of the

regulated firm

1

1

Chapter 15

TRADITIONAL MODELS OF

IMPERFECT COMPETITION


Pricing Under Homogeneous Oligopoly

• We will assume that the market is

perfectly competitive on the demand side

– there are many buyers, each of whom is a

price taker

• We will assume that the good obeys the

law of one price

– this assumption will be relaxed when product

differentiation is discussed

3


• We will assume that there is a relatively

small number of identical firms (n)

– we will initially start with n fixed, but later

allow n to vary through entry and exit in

response to firms’ profitability

• The output of each firm is qi (i=1,…,n)

– symmetry in costs across firms will usually

require that these outputs are equal

4


• The inverse demand function for the

good shows the price that buyers are

willing to pay for any particular level of

industry output

P = f(Q) = f(q1+q2+…+qn)

• Each firm’s goal is to maximize profits

i = f(Q)qi –Ci(qi)

i = f(q1+q2+…qn)qi –Ci

2

5

Oligopoly Pricing Models

• The quasi-competitive model assumes

price-taking behavior by all firms

– P is treated as fixed

• The cartel model assumes that firms

can collude perfectly in choosing

industry output and P

6

Oligopoly Pricing Models

• The Cournot model assumes that firm i treats firm j’s output as fixed in its

decisions

– qj/qi = 0

• The conjectural variations model

assumes that firm j’s output will respond

to variations in firm i’s output

– qj/qi 0

7

Quasi-Competitive Model

• Each firm is assumed to be a price taker

• The first-order condition for profit-

maximization is

i /qi = P – (Ci /qi) = 0

P = MCi (qi) (i=1,…,n)

• Along with market demand, these n

supply equations will ensure that this

market ends up at the short-run

competitive solution 8

Quasi-Competitive Model

Quantity

Price

MC

D

MR

QC

PC

If each firm acts as a price taker, P = MCi

so QC output is produced and sold at a

price of PC

3

9

Cartel Model

• The assumption of price-taking behavior

may be inappropriate in oligopolistic

industries

– each firm can recognize that its output

decision will affect price

• An alternative assumption would be that

firms act as a group and coordinate their

decisions so as to achieve monopoly

profits 10

Cartel Model

• In this case, the cartel acts as a

multiplant monopoly and chooses qi for

each firm so as to maximize total

industry profits

= PQ – [C1(q1) + C2(q2) +…+ Cn(qn)]

n

i

iinn qCqqqqqqf1

2121 )(]...)[...(

11

Cartel Model


are that

0)()...( 21

i

i

n

i

qMCq

PqqqP

q


MR(Q) = MCi(qi)

• At the profit-maximizing point, marginal

revenue will be equal to each firm’s

marginal cost 12

Cartel Model

Quantity

Price

MC

D

MR

QM

PM

If the firms form a group and act as a

monopoly, MR = MCi so QM output is

produced and sold at a price of PM

4

13

Cartel Model

• There are three problems with the cartel

solution

– these monopolistic decisions may be illegal

– it requires that the directors of the cartel

know the market demand function and

each firm’s marginal cost function

– the solution may be unstable

• each firm has an incentive to expand output

because P > MCi

14

Cournot Model

• Each firm recognizes that its own

decisions about qi affect price

– P/qi 0

• However, each firm believes that its

decisions do not affect those of any

other firm

– qj /qi = 0 for all j i

15

Cournot Model

• The first-order conditions for a profit

maximization are

0)(

ii

i

i

i

i qMCq

PqP

q

• The firm maximizes profit where MRi = MCi

– the firm assumes that changes in qi affect

its total revenue only through their direct

effect on market price

16

Cournot Model

• Each firm’s output will exceed the cartel

output

– the firm-specific marginal revenue is larger

than the market-marginal revenue

• Each firm’s output will fall short of the

competitive output

– qi P/qi < 0

5

17

Cournot Model

• Price will exceed marginal cost, but

industry profits will be lower than in the

cartel model

• The greater the number of firms in the

industry, the closer the equilibrium point

will be to the competitive result

18

Cournot’s Natural Springs Duopoly

• Assume that there are two owners of

natural springs

– each firm has no production costs

– each firm has to decide how much water

to supply to the market

• The demand for spring water is given

by the linear demand function

Q = q1 + q2 = 120 - P

19


• Because each firm has zero marginal

costs, the quasi-competitive solution

will result in a market price of zero

– total demand will be 120

– the division of output between the two

firms is indeterminate

• each firm has zero marginal cost over all

output ranges

20


• The cartel solution to this problem can

be found by maximizing industry

revenue (and profits)

= PQ = 120Q - Q2

• The first-order condition is

/Q = 120 - 2Q = 0

6

21


• The profit-maximizing output, price, and

level of profit are

Q = 60

P = 60

= 3,600

• The precise division of output and

profits is indeterminate

22


• The two firms’ revenues (and profits) are

given by

1 = Pq1 = (120 - q1 - q2) q1 = 120q1 - q12 - q1q2

2 = Pq2 = (120 - q1 - q2) q2 = 120q2 - q22 - q1q2

• First-order conditions for a maximum are

02120 21

1

1

qq

q02120 12

2

2

qq

q

23


• These first-order equations are called

reaction functions

– show how each firm reacts to the other’s

output level

• In equilibrium, each firm must produce

what the other firm thinks it will

24


• We can solve the reaction functions

simultaneously to find that

q1 = q2 = 40

P = 120 - (q1 + q2) = 40

1 = 2 = Pq1 = Pq2 = 1,600

• Note that the Cournot equilibrium falls

between the quasi-competitive model

and the cartel model

7

25

Conjectural Variations Model

• In markets with only a few firms, we can

expect there to be strategic interaction

among firms

• One way to build strategic concerns into

our model is to consider the

assumptions that might be made by one

firm about the other firm’s behavior

26


• For each firm i, we are concerned with

the assumed value of qj /qi for ij

– because the value will be speculative,

models based on various assumptions

about its value are termed conjectural

variations models

• they are concerned with firm i’s conjectures

about firm j’s output variations

27


• The first-order condition for profit

maximization becomes

0)(

ii

ij i

j

ji

i

i

i qMCq

q

q

P

q

PqP

q

• The firm must consider how its output

decisions will affect price in two ways

– directly

– indirectly through its effect on the output

decisions of other firms 28

Price Leadership Model

• Suppose that the market is composed

of a single price leader (firm 1) and a

fringe of quasi-competitors

– firms 2,…,n would be price takers

– firm 1 would have a more complex reaction

function, taking other firms’ actions into

account

8

29


Quantity

Price

D

D represents the market demand curve

SC

SC represents the supply

decisions of all of the n-1 firms in

the competitive fringe

30


Quantity

Price

D

SC

We can derive the demand curve facing

the industry leader

For a price of P1 or above, the

leader will sell nothing P1

P2

For a price of P2 or below, the

leader has the market to itself

31


Quantity

Price

D

SC

P1

P2

Between P2 and P1, the

demand for the leader (D’) is constructed by

subtracting what the fringe

will supply from total

market demand

D’

MR’

MC’

The leader would then set

MR’ = MC’ and produce QL

at a price of PL

PL

QL 32


Quantity

Price

D

SC

P1

P2

Market price will then be PL

D’

MR’

MC’

PL

QL

The competitive fringe will

produce QC and total

industry output will be QT

(= QC + QL)

QC QT

9

33


• This model does not explain how the

price leader is chosen or what happens

if a member of the fringe decides to

challenge the leader

• The model does illustrate one tractable

example of the conjectural variations

model that may explain pricing behavior

in some instances

34

Stackelberg Leadership Model

• The assumption of a constant marginal

cost makes the price leadership model

inappropriate for Cournot’s natural

spring problem

– the competitive fringe would take the entire

market by pricing at marginal cost (= 0)

– there would be no room left in the market

for the price leader

35


• There is the possibility of a different

type of strategic leadership

• Assume that firm 1 knows that firm 2

chooses q2 so that

q2 = (120 – q1)/2

• Firm 1 can now calculate the conjectural

variation

q2/q1 = -1/2 36


• This means that firm 2 reduces its output

by ½ unit for each unit increase in q1

• Firm 1’s profit-maximization problem can

be rewritten as

1 = Pq1 = 120q1 – q12 – q1q2

1/q1 = 120 – 2q1 – q1(q2/q1) – q2 = 0

1/q1 = 120 – (3/2)q1 – q2 = 0

10

37


• Solving this simultaneously with firm 2’s

reaction function, we get

q1 = 60

q2 = 30

P = 120 – (q1 + q2) = 30

1 = Pq1 = 1,800

2 = Pq2 = 900

• Again, there is no theory on how the

leader is chosen 38

Product Differentiation

• Firms often devote considerable

resources to differentiating their

products from those of their competitors

– quality and style variations

– warranties and guarantees

– special service features

– product advertising

39


• The law of one price may not hold,

because demanders may now have

preferences about which suppliers to

purchase the product from

– there are now many closely related, but not

identical, products to choose from

• We must be careful about which

products we assume are in the same

market 40


• The output of a set of firms constitute a

product group if the substitutability in

demand among the products (as

measured by the cross-price elasticity) is

very high relative to the substitutability

between those firms’ outputs and other

goods generally

11

41


• We will assume that there are n firms

competing in a particular product group

– each firm can choose the amount it spends

on attempting to differentiate its product

from its competitors (zi)

• The firm’s costs are now given by

total costs = Ci (qi,zi)

42


• Because there are n firms competing in

the product group, we must allow for

different market prices for each (p1,...,pn)

• The demand facing the ith firm is

pi = g(qi,pj,zi,zj)

• Presumably, pi/qi 0, pi/pj 0,

pi/zi 0, and pi/zj 0

43


• The ith firm’s profits are given by

i = piqi –Ci(qi,zi)

• In the simple case where zj/qi, zj/zi,

pj/qi, and pj/zi are all equal to zero,

the first-order conditions for a maximum

are

0

i

i

i

iii

i

i

q

C

q

pqp

q

0

i

i

i

ii

i

i

z

C

z

pq

z 44


• At the profit-maximizing level of output,


cost

• Additional differentiation activities should

be pursued up to the point at which the

additional revenues they generate are

equal to their marginal costs

12

45


• The demand curve facing any one firm

may shift often

– it depends on the prices and product

differentiation activities of its competitors

• The firm must make some assumptions

in order to make its decisions

• The firm must realize that its own actions

may influence its competitors’ actions

46

Spatial Differentiation

• Suppose we are examining the case of

ice cream stands located on a beach

– assume that demanders are located

uniformly along the beach

• one at each unit of beach

• each buyer purchases exactly one ice cream

cone per period

– ice cream cones are costless to produce but

carrying them back to one’s place on the

beach results in a cost of c per unit traveled

47


A B

L

Ice cream stands are located at points A

and B along a linear beach of length L

Suppose that a person is standing at point E

E

48


• A person located at point E will be

indifferent between stands A and B if

pA + cx = pB + cy

where pA and pB are the prices charged

by each stand, x is the distance from E

to A, and y is the distance from E to B

13

49


A B

L

E

a + x + y + b = L

x y a b

50


• The coordinate of point E is

c

cyppx AB

xbaLc

ppx AB

c

ppbaLx AB

2

1

c

ppbaLy BA

2

1

51


• Profits for the two firms are

c

ppppbaLxap ABA

AAA2

)(2

1)(

2

c

ppppbaLybp BBA

BBB2

)(2

1)(

2

52


• Each firm will choose its price so as to

maximize profits

0

2)(

2

1

c

p

c

pbaL

pAB

A

A

02

)(2

1

c

p

c

pbaL

pBA

B

B

14

53


• These can be solved to yield:

3

baLcpA

3

baLcpB

• These prices depend on the precise

locations of the stands and will differ

from one another

54


A B

L

E

Because A is somewhat more favorably located

than B, pA will exceed pB

x y a b

55


• If we allow the ice cream stands to

change their locations at zero cost,

each stand has an incentive to move to

the center of the beach

– any stand that opts for an off-center

position is subject to its rival moving

between it and the center and taking a

larger share of the market

• this encourages a similarity of products

56

Entry

• In perfect competition, the possibility of

entry ensures that firms will earn zero

profit in the long run

• These conditions continue to operate

under oligopoly

– to the extent that entry is possible, long-run

profits are constrained

– if entry is completely costless, long-run

profits will be zero

15

57

Entry

• Whether firms in an oligopolistic

industry with free entry will be directed

to the point of minimum average cost

depends on the nature of the demand

facing them

58

Entry

• If firms are price takers:

– P = MR = MC for profit maximization, P = AC for zero profits, so production takes

place at MC = AC

• If firms have some control over price:

– each firm will face a downward-sloping

demand curve

– entry may reduce profits to zero, but

production at minimum average cost is not

ensured

59

Entry

Quantity

Price

d

mr

MC

AC

q’

P’

q*

P*

Firms will initially be maximizing

profits at q*. Since P > AC, > 0

d’ mr’

Since > 0, firms will

enter and the demand

facing the firm will shift

left

Entry will end when = 0

Firms will exhibit excess

capacity = qm - q’

qm

60

Monopolistic Competition

• The zero-profit equilibrium model just

shown was developed by Chamberlin

who termed it monopolistic competition

– each firm produces a slightly differentiated

product and entry is costless

• Suppose that there are n firms in a

market and that each firm has the total

cost schedule

ci = 9 + 4qi

16

61


• Each firm also faces a demand curve

for its product of the form:

ij

jiin

ppnq303

01.0)1(01.0

• We will define an equilibrium for this

industry to be a situation in which prices

must be equal

– pi = pj for all i and j

62


• To find the equilibrium n, we must

examine each firm’s profit-maximizing

choice of pi

• Because

i = piqi – ci

the first-order condition for a maximum is

ij

ji

i

i nn

ppnp

0)1(04.0303

01.0)1(02.0

63


• This means that

2)1(02.0

303

1

5.0

nnn

p

pij

j

i

• Applying the equilibrium condition that pi

= pj yields

4)1(

300,30

nnpi

• P approaches MC (4) as n gets larger 64


• The equilibrium n is determined by the

zero-profit condition

• Substituting in the expression for pi, we

find that

0 iii cqp

nnnn

)303(49

)303(4

)1(

303300,302

101n

17

65


• The final equilibrium is

pi = pj = 7

qi = 3

i = 0

• In this equilibrium, each firm has pi = ACi, but pi > MCi = 4

• Because ACi = 4 + 9/qi, each firm has

diminishing AC throughout all output

ranges 66


• If each firm faces a similar demand

function, this equilibrium is sustainable

– no firm would find it profitable to enter this

industry

• But what if a potential entrant adopted a

large-scale production plan?

– the low average cost may give the potential

entrant considerable leeway in pricing so as

to tempt customers of existing firms to

switch allegiances

67

Contestable Markets and Industry Structure

• Several economists have challenged

that this zero-profit equilibrium is

sustainable in the long run

– the model ignores the effects of potential entry on market equilibrium by focusing

only on actual entrants

– need to distinguish between competition in

the market and competition for the market

68

Perfectly Contestable Market

• A market is perfectly contestable if entry

and exit are absolutely free

– no outside potential competitor can enter

by cutting price and still make a profit

• if such profit opportunities existed, potential

entrants would take advantage of them

18

69


Quantity

Price

d

mr

MC

AC

d’ mr’

q’

P’

q*

P*

This market would be unsustainable

in a perfectly contestable market

q’

Because P > MC, a

potential entrant can take

one zero-profit firm’s

market away and

encroach a bit on other

firms’ markets where, at

the margin, profits are

attainable

70


• Therefore, to be perfectly contestable,

the market must be such that firms earn

zero profits and price at marginal costs

– firms will produce at minimum average cost

– P = AC = MC

• Perfect contestability guides market

equilibrium to a competitive-type result

71


• If we let q* represent the output level for

which average costs are minimized and

Q* represent the total market demand

when price equals average cost, then

the equilibrium number of firms in the

industry is given by

n = Q*/q*

– this number may be relatively small (unlike

the perfectly competitive case) 72


Quantity

Price

D

AC1 AC2 AC3 AC4

q* 2q* 3q* Q*=4q*

P*

In a perfectly contestable market, equilibrium

requires that P = MC = AC

The number of firms is

completely determined by

market demand (Q*) and

by the output level that

minimizes AC (q*)

19

73

Barriers to Entry

• If barriers to entry prevent free entry and

exit, the results of this model must be

modified

– barriers to entry can be the same as those

that lead to monopolies or can be the result

of some of the features of oligopolistic

markets

• product differentiation

• strategic pricing decisions

74

Barriers to Entry

• The completely flexible type of hit-and-

run behavior assumed in the contestable

markets theory may be subject to barriers

to entry

– some types of capital investments may not

be reversible

– demanders may not respond to price

differentials quickly

75

A Contestable Natural Monopoly

• Suppose that the total cost of producing

electric power is given by

C(Q) = 100Q + 8,000

– since AC declines over all output ranges,

this is a natural monopoly

• The demand for electricity is given by

QD = 1,000 - 5P

76


• If the producer behaves as a monopolist,

it will maximize profits by

MR = 200 - (2Q)/5 = MC = 100

Qm = 250

Pm = 150

m = R - C = 37,500 - 33,000 = 4,500

• These profits will be tempting to would-be

entrants

20

77


• If there are no entry barriers, a potential

entrant can offer electricity customers a

lower price and still cover costs

– this monopoly solution might not represent

a viable equilibrium

78


• If electricity production is fully

contestable, the only price viable under

threat of potential entry is average cost

Q = 1,000 - 5P = 1,000 – 5(AC)

Q = 1,000 - 5[100 + (8,000/Q)]

Q2 - 500Q + 40,000 = 0

(Q - 400)(Q - 100) = 0

79


• Only Q = 400 is a sustainable entry

deterrent

• Under contestability, the market equilibrium

is Qc = 400

Pc = 120

• Contestability increased consumer welfare

from what it was under the monopoly

situation 80


• Markets with few firms offer potential

profits through the formation of a

monopoly cartel

– such cartels may, however, be unstable

and costly to maintain because each

member has an incentive to chisel on

price

21

81


• In markets with few firms, output and

price decisions are interdependent

– each firm must consider its rivals’

decisions

– modeling such interdependence is

difficult because of the need to consider

conjectural variations

82


• The Cournot model provides a

tractable approach to oligopoly

markets, but neglects important

strategic issues

83


• Product differentiation can be

analyzed in a standard profit-

maximization framework

– with differentiated products, the law of

one price no longer holds and firms may

have somewhat more leeway in their

pricing decisions

84


• Entry conditions are important

determinants of the long-run

sustainability of various market

equilibria

– with perfect contestability, equilibria may

resemble perfectly competitive ones

even though there are relatively few

firms in the market

1

1

Chapter 16

LABOR MARKETS


Allocation of Time

• Individuals must decide how to allocate

the fixed amount of time they have

• We will initially assume that there are

only two uses of an individual’s time

– engaging in market work at a real wage

rate of w

– leisure (nonwork)

3

Allocation of Time

• Assume that an individual’s utility

depends on consumption (c) and hours

of leisure (h)

utility = U(c,h)

• In seeking to maximize utility, the

individual is bound by two constraints

l + h = 24

c = wl

4

Allocation of Time

• Combining the two constraints, we get

c = w(24 – h)

c + wh = 24w

• An individual has a “full income” of 24w

– may spend the full income either by

working (for real income and consumption)

or by not working (enjoying leisure)

• The opportunity cost of leisure is w

2

5

Utility Maximization

• The individual’s problem is to maximize

utility subject to the full income constraint

• Setting up the Lagrangian

L = U(c,h) + (24w – c – wh)

• The first-order conditions are

L/c = U/c - = 0

L/h = U/h - = 0

6

Utility Maximization

• Dividing the two, we get

) for ( /

/chMRSw

hU

cU

• To maximize utility, the individual should

choose to work that number of hours for

which the MRS (of h for c) is equal to w

– to be a true maximum, the MRS (of h for c)

must be diminishing

7

Income and Substitution Effects

• Both a substitution effect and an income

effect occur when w changes

– when w rises, the price of leisure becomes

higher and the individual will choose less

leisure

– because leisure is a normal good, an

increase in w leads to an increase in leisure

• The income and substitution effects move

in opposite directions 8


U1

U2

Leisure

Consumption

A

B

C



The individual chooses

less leisure as a result

of the increase in w



substitution effect > income effect

3

9


U1 U2

Leisure

Consumption

A

B C



The individual

chooses more

leisure as a result

of the increase in

w



substitution effect < income effect 10

A Mathematical Analysis of Labor Supply

• We will start by amending the budget

constraint to allow for the possibility of

nonlabor income

c = wl + n

• Maximization of utility subject to this

constraint yields identical results

– as long as n is unaffected by the labor-

leisure choice

11

A Mathematical Analysis of Labor Supply

• The only effect of introducing nonlabor

income is that the budget constraint

shifts out (or in) in a parallel fashion

• We can now write the individual’s labor

supply function as l(w,n)

– hours worked will depend on both the

wage and the amount of nonlabor income

– since leisure is a normal good, l/n < 0 12

Dual Statement of the Problem

• The dual problem can be phrased as

choosing levels of c and h so that the

amount of expenditure (E = c – wl)

required to obtain a given utility level

(U0) is as small as possible

– solving this minimization problem will yield

exactly the same solution as the utility

maximization problem

4

13


• A small change in w will change the

minimum expenditures required by

E/w = -l

– this is the extent to which labor earnings

are increased by the wage change

14


• This means that a labor supply

function can be calculated by partially

differentiating the expenditure function

– because utility is held constant, this

function should be interpreted as a

“compensated” (constant utility) labor

supply function

lc(w,U)

15

Slutsky Equation of Labor Supply

• The expenditures being minimized in the

dual expenditure-minimization problem

play the role of nonlabor income in the

primary utility-maximization problem

lc(w,U) = l[w,E(w,U)] = l(w,N)

• Partial differentiation of both sides with

respect to w gives us

w

E

Eww

c

lll16

Slutsky Equation of Labor Supply

• Substituting for E/w, we get

nwEww

c

ll

lll

ll

• Introducing a different notation for lc ,

and rearranging terms gives us the

Slutsky equation for labor supply:

nww UU

ll

ll

0

5

17

Cobb-Douglas Labor Supply

• Suppose that utility is of the form

hcU

• The budget constraint is

c = wl + n

and the time constraint is

l + h = 1

– note that we have set maximum work time

to 1 hour for convenience 18


• The Lagrangian expression for utility

maximization is

L = ch + (w + n - wh - c)

• First-order conditions are

L/c = c-h - = 0

L/h = ch- - w = 0

L/ = w + n - wh - c = 0

19


• Dividing the first by the second yields

wc

h

c

h 1

)1(

cwh

1

20


• Substitution into the full income

constraint yields

c = (w + n)

h = (w + n)/w

– the person spends of his income on

consumption and = 1- on leisure

– the labor supply function is

w

nhnw

)1(1),(l

6

21


• Note that if n = 0, the person will work

(1-) of each hour no matter what the

wage is

– the substitution and income effects of a

change in w offset each other and leave l

unaffected

22


• If n > 0, l/w > 0

– the individual will always choose to spend

n on leisure

– Since leisure costs w per hour, an increase

in w means that less leisure can be bought

with n

23


• Note that l/n < 0

– an increase in nonlabor income allows this

person to buy more leisure

• income transfer programs are likely to reduce

labor supply

• lump-sum taxes will increase labor supply

24

CES Labor Supply


hchcU ),(

• Budget share equations are given by

)1(

1

wnw

csc

)1(

1

wnw

whsh

– where = /(-1)

7

25

CES Labor Supply

• Solving for leisure gives

1ww

nwh

and

1

1

1),(ww

nwhnwl

26

Market Supply Curve for Labor

l l l

w w w sA

sB

To derive the market supply curve for labor, we sum

the quantities of labor offered at every wage

Individual A’s

supply curve Individual B’s

supply curve Total labor

supply curve S

lA* lB*

w*

l*

lA* + lB* = l*

27

Market Supply Curve for Labor

l l l

w w w sA

sB

Note that at w0, individual B would choose to remain

out of the labor force

Individual A’s

supply curve Individual B’s

supply curve Total labor

supply curve S

w0

As w rises, l rises for two reasons: increased hours

of work and increased labor force participation 28

Labor Market Equilibrium

• Equilibrium in the labor market is

established through the interactions of

individuals’ labor supply decisions with

firms’ decisions about how much labor

to hire

8

29

Labor Market Equilibrium

real wage

quantity of labor

S

D

w*

l*

At w*, the quantity of labor demanded is

equal to the quantity of labor supplied

At any wage above w*, the quantity

of labor demanded will be less

than the quantity of labor supplied

At any wage below w*, the quantity

of labor demanded will be greater

than the quantity of labor supplied

30

Mandated Benefits

• A number of new laws have mandated

that employers provide special benefits

to their workers

– health insurance

– paid time off

– minimum severance packages

• The effects of these mandates depend

on how much the employee values the

benefit

31

Mandated Benefits

• Suppose that, prior to the mandate, the

supply and demand for labor are

lS = a + bw

lD = c – dw

• Setting lS = lD yields an equilibrium wage

of

w* = (c – a)/(b + d)

32

Mandated Benefits

• Suppose that the government mandates

that all firms provide a benefit to their

workers that costs t per unit of labor

hired

– unit labor costs become w + t

• Suppose also that the benefit has a

value of k per unit supplied

– the net return from employment rises to

w + k

9

33

Mandated Benefits

• Equilibrium in the labor market then

requires that

a + b(w + k) = c – d(w + t)

• This means that the net wage is

db

dtbkw

db

dtbk

db

acw

***

34

Mandated Benefits

• If workers derive no value from the

mandated benefits (k = 0), the mandate

is just like a tax on employment

– similar results will occur as long as k < t

• If k = t, the new wage falls precisely by

the amount of the cost and the

equilibrium level of employment does not

change

35

Mandated Benefits

• If k > t, the new wage falls by more than

the cost of the benefit and the

equilibrium level of employment rises

36

Wage Variation

• It is impossible to explain the variation

in wages across workers with the tools

developed so far

– we must consider the heterogeneity that

exists across workers and the types of jobs

they take

10

37

Wage Variation

• Human Capital

– differences in human capital translate into

differences in worker productivities

– workers with greater productivities would be

expected to earn higher wages

– while the investment in human capital is

similar to that in physical capital, there are

two differences

• investments are sunk costs

• opportunity costs are related to past investments 38

Wage Variation

• Compensating Differentials

– individuals prefer some jobs to others

– desirable job characteristics may make a

person willing to take a job that pays less

than others

– jobs that are unpleasant or dangerous will

require higher wages to attract workers

– these differences in wages are termed

compensating differentials

39

Monopsony in the Labor Market

• In many situations, the supply curve for

an input (l) is not perfectly elastic

• We will examine the polar case of

monopsony, where the firm is the single

buyer of the input in question

– the firm faces the entire market supply curve

– to increase its hiring of labor, the firm must

pay a higher wage 40


• The marginal expense (ME) associated

with any input is the increase in total

costs of that input that results from hiring

one more unit

– if the firm faces an upward-sloping supply

curve for that input, the marginal expense will

exceed the market price of the input

11

41


• If the total cost of labor is wl, then

ll

l

ll

ww

wME

• In the competitive case, w/l = 0 and

MEl = w

• If w/l > 0, MEl > w

42


Labor

Wage

S

MEl

D

l1

Note that the quantity of

labor demanded by this

firm falls short of the

level that would be hired

in a competitive labor

market (l*)

l*

w1

w* The wage paid by the

firm will also be lower

than the competitive

level (w*)

43

Monopsonistic Hiring

• Suppose that a coal mine’s workers can

dig 2 tons per hour and coal sells for

$10 per ton

– this implies that MRPl = $20 per hour

• If the coal mine is the only hirer of

miners in the local area, it faces a labor

supply curve of the form

l = 50w 44

Monopsonistic Hiring

• The firm’s wage bill is

wl = l2/50

• The marginal expense associated with

hiring miners is

MEl = wl/l = l/25

• Setting MEl = MRPl, we find that the

optimal quantity of labor is 500 and the

optimal wage is $10

12

45

Labor Unions

• If association with a union was wholly

voluntary, we can assume that every

member derives a positive benefit

• With compulsory membership, we

cannot make the same claim

– even if workers would benefit from the

union, they may choose to be “free riders”

46

Labor Unions

• We will assume that the goals of the

union are representative of the goals of

its members

• In some ways, we can use a monopoly

model to examine unions

– the union faces a demand curve for labor

– as the sole supplier, it can choose at which

point it will operate

• this point depends on the union’s goals

47

Labor Unions

Labor

Wage

D

MR

S

The union may wish to maximize the total

wage bill (wl). This occurs where

MR = 0

l1 workers will be

hired and paid a

wage of w1

l1

w1

This choice will

create an excess

supply of labor

48

Labor Unions

Labor

Wage

D

MR

S


economic rent of its employed members

This occurs where

MR = S

l2 workers will be

hired and paid a

wage of w2

l2

w2

Again, this will

cause an excess

supply of labor

13

49

Labor Unions

Labor

Wage

D

MR

S


employment of its members

This occurs where

D = S

l3 workers will be

hired and paid a

wage of w3

l3

w3

50

Modeling a Union

• A monopsonistic hirer of coal miners

faces a supply curve of

l = 50w

• Assume that the monopsony has a

MRPL curve of the form

MRPl = 70 – 0.1l

• The monopsonist will choose to hire 500

workers at a wage of $10

51

Modeling a Union

• If a union can establish control over

labor supply, other options become

possible

– competitive solution where l = 583 and

w = $11.66

– monopoly solution where l = 318 and

w = $38.20

52

A Union Bargaining Model

• Suppose a firm and a union engage in a

two-stage game

– first stage: union sets the wage rate its

workers will accept

– second stage: firm chooses its employment

level

14

53


• This two-stage game can be solved by

backward induction

• The firm’s second-stage problem is to

maximize its profits:

= R(l) – wl

• The first-order condition for a maximum is

R’(l) = w

54


• Assuming that l* solves the firm’s

problem, the union’s goal is to choose w

to maximize utility

U(w,l) = U[w,l*(w)]

and the first-order condition for a

maximum is

U1 + U2l’ = 0

U1/U2 = l’

55


• This implies that the union should choose

w so that its MRS is equal to the slope of

the firm’s labor demand function

• The result from this game is a Nash

equilibrium

56


• A utility-maximizing individual will

choose to supply an amount of labor at

which the MRS of leisure for

consumption is equal to the real wage

rate

15

57


• An increase in the real wage rate

creates income and substitution

effects that operate in different

directions in affecting the quantity of

labor supplied

– this result can be summarized by a

Slutsky-type equation much like the

one already derived in consumer

theory

58


• A competitive labor market will

establish an equilibrium real wage

rate at which the quantity of labor

supplied by individuals is equal to the

quantity demanded by firms

59


• Monopsony power by firms on the

demand side of the market will

reduce both the quantity of labor

hired and the real wage rate

– as in the monopoly case, there will be a

welfare loss

60


• Labor unions can be treated

analytically as monopoly suppliers of

labor

– the nature of labor market equilibrium in

the presence of unions will depend

importantly on the goals the union

chooses to pursue

1

1

Chapter 18

THE ECONOMICS OF

INFORMATION


Properties of Information

• Information is not easy to define

– it is difficult to measure the quantity of

information obtainable from different

actions

– there are too many forms of useful

information to permit the standard price-

quantity characterization used in supply

and demand analysis

3

Properties of Information

• Studying information also becomes

difficult due to some technical properties

of information

– it is durable and retains value after its use

– it can be nonrival and nonexclusive

• in this manner it can be considered a public

good

4

The Value of Information

• In many respects, lack of information

does represent a problem involving

uncertainty for a decision maker

– the individual may not know exactly what the

consequences of a particular action will be

• Better information can reduce uncertainty

and lead to better decisions and higher

utility

2

5


• Assume an individual forms subjective

opinions about the probabilities of two

states of the world

– “good times” (probability = g) and “bad

times” (probability = b)

• Information is valuable because it helps

the individual revise his estimates of

these probabilities

6


• Assume that information can be

measured by the number of “messages”

(m) purchased

– g and b will be functions of m

7


• The individual’s goal will be to maximize

E(U) = gU(Wg) + bU(Wb)

subject to

I = pgWg + pbWb + pmm

• We need to set up the Lagrangian

L = gU(Wg) + bU(Wb) + (I-pgWg-pbWb-pmm)

8


• First-order conditions for a constrained

maximum are:

0)('

ggg

g

pWUW

L

0)('

bbb

b

pWUW

L

0

mpWpWp mbbggI

L

3

9


• First-order conditions for a constrained

maximum are:

0

)()(

)(')('

mb

b

g

gb

b

g

g

bbb

g

gg

pdm

dWp

dm

dWp

dm

dWU

dm

dWU

dm

dWWU

dm

dWWU

m

L

10


• The first two equations show that the

individual will maximize utility at a point

where the subjective ratio of expected

marginal utilities is equal to the price

ratio (pg /pb)

• The last equation shows the utility-

maximizing level of information to buy

11

Asymmetry of Information

• The level of information that a person buys

will depend on the price per unit

• Information costs may differ significantly

across individuals

– some may possess specific skills for acquiring

information

– some may have experience that is relevant

– some may have made different former

investments in information services 12

Information and Insurance

• There are a number of information

asymmetries in the market for insurance

• Buyers are often in a better position to

know the likelihood of uncertain events

– may also be able to take actions that

impact these probabilities

4

13

Moral Hazard

• Moral hazard is the effect of insurance

coverage on individuals’ decisions to

take activities that may change the

likelihood or size of losses

– parking an insured car in an unsafe area

– choosing not to install a sprinkler system in

an insured home

14

Moral Hazard

• Suppose a risk-averse individual faces

the risk of a loss (l) that will lower

wealth

– the probability of a loss is

– this probability can be lowered by the

amount the person spends on preventive

measures (a)

15

Moral Hazard

• Wealth in the two states is given by

W1 = W0 - a

W2 = W0 - a - l

• The individual chooses a to maximize

E(U) = E = (1-)U(W1) + U(W2)

16

Moral Hazard

• The first-order condition for a maximum is

0)(')()(')1()( 2211

WU

aWUWU

aWU

a

E

aWUWUWUWU

)]()([)(')1()(' 1212

– the optimal point is where the expected

marginal utility cost from spending one

additional dollar on prevention is equal to the

reduction in the expected value of the utility loss

that may be encountered in bad times

5

17

Behavior with Insurance and Perfect Monitoring

• Suppose that the individual may purchase

insurance (premium = p) that pays x if a

loss occurs

• Wealth in each state becomes

W1 = W0 - a - p

W2 = W0 - a - p - l + x

• A fair premium would be equal to

p = x 18


• The person can maximize expected utility

by choosing x such that W1 = W2

• The first-order condition is

0)(1)('

)(1)(')1(

22

11

aWU

aWU

aWU

aWU

a

E

l

l

19


• Since W1 = W2, this condition becomes

a

l1

– at the utility maximizing choice, the marginal

cost of an extra unit of prevention should

equal the marginal reduction in the expected

loss provided by the extra spending

– with full insurance and actuarially fair

premiums, precautionary purchases still occur

at the optimal level 20

Moral Hazard

• So far, we have assumed that insurance

providers know the probability of a loss

and can charge the actuarially fair premium

– this is doubtful when individuals can undertake

precautionary activities

– the insurance provider would have to

constantly monitor each person’s activities to

determine the correct probability of loss

6

21

Moral Hazard

• In the simplest case, the insurer might set

a premium based on the average

probability of loss experienced by some

group of people

– no variation in premiums allowed for specific

precautionary activities

• each individual would have an incentive to reduce

his level of precautionary activities

22

Adverse Selection

• Individuals may have different probabilities

of experiencing a loss

• If individuals know the probabilities more

accurately than insurers, insurance

markets may not function properly

– it will be difficult for insurers to set premiums

based on accurate measures of expected loss

23

Adverse Selection

certainty line

W1

W2

W *

W *- l

Assume that two individuals

have the same initial wealth

(W*) and each face a

potential loss of l E

24

Adverse Selection

certainty line

W1

W2

W *

W * - l

Suppose that one person has a probability of loss

equal to H, while the other has a probability of loss

equal to l

E

F

G

Both individuals would

prefer to move to the

certainty line if premiums

are actuarially fair

7

25

Adverse Selection

certainty line

W1

W2

W *

W * - l

The lines show the market opportunities for each

person to trade W1 for W2 by buying fair insurance

E

F

G l

l

)1(slope

H

Hslope

)(1

The low-risk person will

maximize utility at point

F, while the high-risk

person will choose G

26

Adverse Selection

• If insurers have imperfect information

about which individuals fall into low- and

high-risk categories, this solution is

unstable

– point F provides more wealth in both states

– high-risk individuals will want to buy

insurance that is intended for low-risk

individuals

– insurers will lose money on each policy sold

27

Adverse Selection

certainty line

W1

W2

W *

W * - l E

F

G

One possible solution would be for the insurer to

offer premiums based on the average probability of

loss

H Since EH does not

accurately reflect the true

probabilities of each buyer,

they may not fully insure

and may choose a point

such as M

M

28

Point M is not an equilibrium because further trading

opportunities exist for low-risk individuals

UH

UL

Adverse Selection

certainty line

W1

W2

W *

W * - l E

F

G

H M

An insurance policy

such as N would be

unattractive to high-

risk individuals, but

attractive to low-risk

individuals and

profitable for insurers

N

8

29

Adverse Selection

• If a market has asymmetric information,

the equilibria must be separated in

some way

– high-risk individuals must have an

incentive to purchase one type of

insurance, while low-risk purchase another

30

Adverse Selection

certainty line

W1

W2

W *

W * - l E

F

G

Suppose that insurers offer policy G. High-risk

individuals will opt for full insurance.

UH

Insurers cannot offer

any policy that lies

above UH because

they cannot prevent

high-risk individuals

from taking advantage

of it

31

Adverse Selection

certainty line

W1

W2

W *

W * - l E

F

G UH

The policies G and J

represent a

separating equilibrium

The best policy that low-risk individuals can obtain is

one such as J

J

32

Adverse Selection

• Low-risk individuals could try to signal

insurers their true probabilities of loss

– insurers must be able to determine if the

signals are believable

– insurers may be able to infer accurate

probabilities by observing their clients’

market behavior

– the separating equilibrium identifies an

individual’s risk category

9

33

Adverse Selection

• Market signals can be drawn from a

number of sources

– the economic behavior must accurately

reflect risk categories

– the costs to individuals of taking the

signaling action must be related to the

probability of loss

34

The Principal-Agent Relationship

• One important way in which asymmetric

information may affect the allocation of

resources is when one person hires

another person to make decisions

– patients hiring physicians

– investors hiring financial advisors

– car owners hiring mechanics

– stockholders hiring managers

35


• In each of these cases, a person with less

information (the principal) is hiring a more

informed person (the agent) to make

decisions that will directly affect the

principal’s own well-being

36


• Assume that we can show a graph of the

owner’s (or manager’s) preferences in

terms of profits and various benefits (such

as fancy offices or use of the corporate

jet)

• The owner’s budget constraint will have a

slope of -1

– each $1 of benefits reduces profit by $1

10

37


Benefits

Profits

Owner’s constraint

U1

b*

*

If the manager is also the owner of the firm, he will maximize his utility at profits of * and benefits of b*

38


Benefits

Profits


U1

b*

*

The owner-manager maximizes

profit because any other owner-

manager will also want b* in

benefits

b* represents a true

cost of doing business

39


• Suppose that the manager is not the

sole owner of the firm

– suppose there are two other owners who

play no role in operating the firm

• $1 in benefits only costs the manager

$0.33 in profits

– the other $0.67 is effectively paid by the

other owners in terms of reduced profits

40


• The new budget constraint continues to

include the point b*, *

– the manager could still make the same

decision that a sole owner could)

• For benefits greater than b*, the slope

of the budget constraint is only -1/3

11

41


Benefits

Profits


U1

b*

*

U2

Given the manager’s budget

constraint, he will maximize

utility at benefits of b**

**

b**

Agent’s constraint

***

Profits for the

firm will be ***

42


• The firm’s owners are harmed by having

to rely on an agency relationship with

the firm’s manager

• The smaller the fraction of the firm that

is owned by the manager, the greater

the distortions that will be induced by

this relationship

43

Using the Corporate Jet

• A firm owns a fleet of corporate jets

used mainly for business purposes

– the firm has just fired a CEO for misusing

the corporate fleet

• The firm wants to structure a

management contract that provides

better incentives for cost control

44


• Suppose that all would-be applicants

have the same utility function

U(s,j) = 0.1s0.5 + j

where s is salary and j is jet use (0 or 1)

• All applicants have job offers from other

firms promising them a utility level of at

most 2.0

12

45


• Because jet use is expensive, = 800

(thousand) if j =0 and = 162 if j =1

– the directors will be willing to pay the new

CEO up to 638 providing that they can

guarantee that he will not use the

corporate jet for personal use

– a salary of more than 400 will just be

sufficient to get a potential candidate to

accept the job without jet usage 46


• If the directors find it difficult to monitor

the CEO’s jet usage, this could mean

that the firm ends up with < 0

• The owner’s may therefore want to

create a contract where the

compensation of the new CEO is tied to

profit

47

The Owner-Manager Relationship

• Suppose that the gross profits of the firm

depend on some specific action that a

hired manager might take (a)

net profits = ’ = (a) – s[(a)]

• Both gross and net profits are maximized

when /a = 0

– the owners’ problem is to design a salary

structure that provides an incentive for the

manager to choose a that maximizes 48


• The owners face two issues

– they must know the agent’s utility function

which depends on net income (IM)

IM = s[(a)] = c(a) = c0

• where c(a) represents the cost to the manager of

undertaking a

– they must design the compensation system

so that the agent is willing to take the job

• this requires that IM 0

13

49


• One option would be to pay no

compensation unless the manager

chooses a* and to pay an amount equal to

c(a*) + c0 if a* is chosen

• Another possible scheme is s(a) = (a) – f, where f = (a) – c(a*) – c0

– with this compensation package, the

manager’s income is maximized by setting

s(a)/a = /a = 0 50


• The manager will choose a* and receive

an income that just covers costs

IM = s(a*) – c(a*) – c0 = (a*) – f – c(a*) – c0 = 0

• This compensation plan makes the agent

the “residual claimant” to the firm’s profits

51

Asymmetric Information

• Models of the principal-agent relationship

have introduced asymmetric information

into this problem in two ways

– it is assumed that a manager’s action is not

directly observed and cannot be perfectly

inferred from the firm’s profits

• referred to as “hidden action”

– the agent-manager’s objective function is not

directly observed

• referred to as “hidden information” 52

Hidden Action

• The primary reason that the manager’s

action may be hidden is that profits

depend on random factors that cannot be

observed by the firm’s owner

• Suppose that profits depend on both the

manager’s action and on a random

variable (u)

(a) = ’(a) + u

where ’ represents expected profits

14

53

Hidden Action

• Because owners observe only and not

’, they can only use actual profits in their

compensation function

– a risk averse manager will be concerned that

actual profits will turn out badly and may

decline the job

• The owner might need to design a

compensation scheme that allows for

profit-sharing 54

Hidden Information

• When the principal does not know the

incentive structure of the agent, the

incentive scheme must be designed

using some initial assumptions about the

agent’s motivation

– will be adapted as new information becomes

available

55


• Information is valuable because it

permits individuals to increase the

expected utility of their decisions

– individuals might be willing to pay

something to acquire additional

information

56


• Information has a number of special

properties that suggest that

inefficiencies associated with

imperfect and asymmetric information

may be quite prevalent

– differing costs of acquisition

– some aspects of a public good

15

57


• The presence of asymmetric

information may affect a variety of

market outcomes, many of which are

illustrated in the context of insurance

theory

– insurers may have less information

about potential risks than do insurance

purchasers

58


• If insurers are unable to monitor the

behavior of insured individuals

accurately, moral hazard may arise

– being insured will affect the willingness to

make precautionary expenditures

– such behavioral effects can arise in any

contractual situation in which monitoring

costs are high

59


• Informational asymmetries can also

lead to adverse selection in insurance

markets

– the resulting equilibria may often be

inefficient because low-risk individuals will

be worse off than in the full information

case

– market signaling may be able to reduce

these inefficiencies

60


• Asymmetric information may also

cause some (principal) economic

actors to hire others (agents) to make

decisions for them

– providing the correct incentives to the

agent is a difficult problem

1

1

Chapter 19

EXTERNALITIES AND PUBLIC GOODS


Externality

• An externality occurs whenever the

activities of one economic agent affect

the activities of another economic agent

in ways that are not reflected in market

transactions

– chemical manufacturers releasing toxic

fumes

– noise from airplanes

– motorists littering roadways

3

Interfirm Externalities

• Consider two firms, one producing good

x and the other producing good y

• The production of x will have an external

effect on the production of y if the output

of y depends not only on the level of

inputs chosen by the firm but on the level

at which x is produced

y = f(k,l;x)

4

Beneficial Externalities

• The relationship between the two firms

can be beneficial

– two firms, one producing honey and the

other producing apples

2

5

Externalities in Utility

• Externalities can also occur if the

activities of an economic agent directly

affect an individual’s utility

– externalities can decrease or increase

utility

• It is also possible for someone’s utility to

be dependent on the utility of another

utility = US(x1,…,xn;UJ)

6

Public Goods Externalities

• Public goods are nonexclusive

– once they are produced, they provide

benefits to an entire group

– it is impossible to restrict these benefits to

the specific groups of individuals who pay

for them

7

Externalities and Allocative Inefficiency

• Externalities lead to inefficient

allocations of resources because

market prices do not accurately reflect

the additional costs imposed on or the

benefits provided to third parties

• We can show this by using a general

equilibrium model with only one

individual 8


• Suppose that the individual’s utility

function is given by

utility = U(xc,yc)

where xc and yc are the levels of x and y

consumed

• The individual has initial stocks of x* and

y*

– can consume them or use them in

production

3

9


• Assume that good x is produced using

only good y according to

xo = f(yi)

• Assume that the output of good y

depends on both the amount of x used in

the production process and the amount

of x produced

yo = g(xi,xo) 10


• For example, y could be produced

downriver from x and thus firm y must

cope with any pollution that production of

x creates

• This implies that g1 > 0 and g2 < 0

11


• The quantities of each good in this

economy are constrained by the initial

stocks available and by the additional

production that takes place

xc + xi = xo + x*

yc + yi = xo + y*

12

Finding the Efficient Allocation

• The economic problem is to maximize

utility subject to the four constraints

listed earlier


L = U(xc,yc) + 1[f(yi) - xo] + 2[g(xi,xo) - yo] +

3(xc + xi - xo - x*) + 4(yc + yi - yo - y*)

4

13


• The six first-order conditions are

L/xc = U1 + 3 = 0

L/yc = U2 + 4 = 0

L/xi = 2g1 + 3 = 0

L/yi = 1fy + 4 = 0

L/xo = -1 + 2g2 - 3 = 0

L/yo = -2 - 4 = 0 14


• Taking the ratio of the first two, we find

MRS = U1/U2 = 3/4

• The third and sixth equation also imply

that

MRS = 3/4 = 2g1/2 = g1

• Optimality in y production requires that

the individual’s MRS in consumption

equals the marginal productivity of x in

the production of y

15


• To achieve efficiency in x production,

we must also consider the externality

this production poses to y

• Combining the last three equations

gives

MRS = 3/4 = (-1 + 2g2)/4 = -1/4 + 2g2/4

MRS = 1/fy - g2

16


• This equation requires the individual’s

MRS to equal dy/dx obtained through x

production

– 1/fy represents the reciprocal of the

marginal productivity of y in x production

– g2 represents the negative impact that

added x production has on y output

• allows us to consider the externality from x production

5

17

Inefficiency of the Competitive Allocation

• Reliance on competitive pricing will result

in an inefficient allocation of resources

• A utility-maximizing individual will opt for

MRS = Px/Py

and the profit-maximizing producer of y

would choose x input according to

Px = Pyg1

18

Inefficiency of the Competitive Allocation

• But the producer of x would choose y

input so that

Py = Pxfy

Px/Py = 1/fy

• This means that the producer of x would

disregard the externality that its

production poses for y and will

overproduce x

19

Production Externalities

• Suppose that two newsprint producers

are located along a river

• The upstream firm has a production

function of the form

x = 2,000lx0.5

20


• The downstream firm has a similar

production function but its output may

be affected by chemicals that firm x pours in the river

y = 2,000ly0.5(x - x0)

(for x > x0)

y = 2,000ly0.5 (for x x0)

where x0 represents the river’s natural

capacity for pollutants

6

21

Production Externalities • Assuming that newsprint sells for $1 per

foot and workers earn $50 per day, firm

x will maximize profits by setting this

wage equal to the labor’s marginal

product 5.0000,150

x

x

xp l

l

• lx* = 400

• If = 0 (no externalities), ly* = 400 22


• When firm x does have a negative

externality ( < 0), its profit-maximizing

decision will be unaffected (lx* = 400

and x* = 40,000)

• But the marginal product of labor will be

lower in firm y because of the externality

23


• If = -0.1 and x0 = 38,000, firm y will

maximize profits by

1.05.0 )000,38000,40(000,150

y

y

yp l

l

5.046850 yl

• Because of the externality, ly* = 87 and

y output will be 8,723

24


• Suppose that these two firms merge

and the manager must now decide how

to allocate the combined workforce

• If one worker is transferred from x to y,

output of x becomes

x = 2,000(399)0.5 = 39,950

and output of y becomes

y = 2,000(88)0.5(1,950)-0.1 = 8,796

7

25


• Total output increased with no change

in total labor input

• The earlier market-based allocation

was inefficient because firm x did not

take into account the effect of its hiring

decisions on firm y

26


• If firm x was to hire one more worker, its

own output would rise to

x = 2,000(401)0.5 = 40,050

– the private marginal value product of the

401st worker is equal to the wage

• But, increasing the output of x causes

the output of y to fall (by about 21 units)

• The social marginal value product of the

additional worker is only $29

27

Solutions to the Externality Problem

• The output of the externality-producing

activity is too high under a market-

determined equilibrium

• Incentive-based solutions to the

externality problem originated with

Pigou, who suggested that the most

direct solution would be to tax the

externality-creating entity 28


Quantity of x

Price

S = MC

D

x1

p1

Market equilibrium

will occur at p1, x1

If there are external

costs in the

production of x,

social marginal costs

are represented by

MC’

MC’

8

29


Quantity of x

Price

S = MC

MC’

D

x2

tax

A tax equal to these

additional marginal

costs will reduce

output to the socially

optimal level (x2) p2

The price paid for the

good (p2) now

reflects all costs

30

A Pigouvian Tax on Newsprint

• A suitably chosen tax on firm x can

cause it to reduce its hiring to a level at

which the externality vanishes

• Because the river can handle pollutants

with an output of x = 38,000, we might

consider a tax that encourages the firm

to produce at that level

31

A Pigouvian Tax on Newsprint

• Output of x will be 38,000 if lx = 361

• Thus, we can calculate t from the labor

demand condition

(1 - t)MPl = (1 - t)1,000(361)-0.5 = 50

t = 0.05

• Therefore, a 5 percent tax on the price

firm x receives would eliminate the

externality 32

Taxation in the General Equilibrium Model

• The optimal Pigouvian tax in our

general equilibrium model is to set

t = -pyg2

– the per-unit tax on x should reflect the

marginal harm that x does in reducing y

output, valued at the price of good y

9

33


• With the optimal tax, firm x now faces a

net price of (px - t) and will choose y

input according to

py = (px - t)fy

• The resulting allocation of resources will

achieve

MRS = px/py = (1/fy) + t/py = (1/fy) - g2

34


• The Pigouvian tax scheme requires that

regulators have enough information to

set the tax properly

– in this case, they would need to know firm

y’s production function

35

Pollution Rights

• An innovation that would mitigate the

informational requirements involved with

Pigouvian taxation is the creation of a

market for “pollution rights”

• Suppose that firm x must purchase from

firm y the rights to pollute the river they

share

– x’s choice to purchase these rights is

identical to its output choice 36

Pollution Rights

• The net revenue that x receives per unit

is given by px - r, where r is the payment

the firm must make to firm y for each

unit of x it produces

• Firm y must decide how many rights to

sell firm x by choosing x output to

maximize its profits

y = pyg(xi,xo) + rxo

10

37

Pollution Rights

• The first-order condition for a maximum

is y/xo = pyg2 + r = 0

r = -pyg2

• The equilibrium solution is identical to

that for the Pigouvian tax

– from firm x’s point of view, it makes no

difference whether it pays the fee to the

government or to firm y 38

The Coase Theorem

• The key feature of the pollution rights

equilibrium is that the rights are well-

defined and tradable with zero

transactions costs

• The initial assignment of rights is

irrelevant

– subsequent trading will always achieve the

same, efficient equilibrium

39

The Coase Theorem

• Suppose that firm x is initially given xT

rights to produce (and to pollute)

– it can choose to use these for its own

production or it may sell some to firm y

• Profits for firm x are given by

x = pxxo + r(xT - xo) = (px - r)xo + rxT

x = (px - r)f(yi) + rxT

40

The Coase Theorem

• Profits for firm y are given by

y = pyg(xi,xo) - r(xT - xo)

• Profit maximization in this case will lead

to precisely the same solution as in the

case where firm y was assigned the

rights

11

41

The Coase Theorem

• The independence of initial rights

assignment is usually referred to as the

Coase Theorem

– in the absence of impediments to making

bargains, all mutually beneficial

transactions will be completed

– if transactions costs are involved or if

information is asymmetric, initial rights

assignments will matter

42

Attributes of Public Goods

• A good is exclusive if it is relatively easy

to exclude individuals from benefiting

from the good once it is produced

• A good is nonexclusive if it is

impossible, or very costly, to exclude

individuals from benefiting from the

good

43


• A good is nonrival if consumption of

additional units of the good involves

zero social marginal costs of production

44


Exclusive

Yes No

Yes Hot dogs,

cars, houses

Fishing grounds, clean air

Rival

No Bridges,

swimming pools

National defense, mosquito control

• Some examples of these types of goods

include:

12

45

Public Good

• A good is a pure public good if, once

produced, no one can be excluded from

benefiting from its availability and if the

good is nonrival -- the marginal cost of

an additional consumer is zero

46

Public Goods and Resource Allocation

• We will use a simple general equilibrium

model with two individuals (A and B)

• There are only two goods

– good y is an ordinary private good

• each person begins with an allocation (yA* and

yB*)

– good x is a public good that is produced

using y

x = f(ysA + ys

B)

47


• Resulting utilities for these individuals are

UA[x,(yA* - ysA)]

UB[x,(yB* - ysB)]

• The level of x enters identically into each

person’s utility curve

– it is nonexclusive and nonrival

• each person’s consumption is unrelated to what

he contributes to production

• each consumes the total amount produced 48


• The necessary conditions for efficient

resource allocation consist of choosing

the levels of ysA and ys

B that maximize

one person’s (A’s) utility for any given

level of the other’s (B’s) utility

• The Lagrangian expression is

L = UA(x, yA* - ysA) + [UB(x, yB* - ys

B) - K]

13

49



are

L/ysA = U1

Af’ - U2A + U1

Bf’ = 0

L/ysB = U1

Af’ - U2B + U1

Bf’ = 0

• Comparing the two equations, we find

U2B = U2

A

50


• We can now derive the optimality

condition for the production of x

• From the initial first-order condition we

know that

U1A/U2

A + U1B/U2

B = 1/f’

MRSA + MRSB = 1/f’

• The MRS must reflect all consumers

because all will get the same benefits

51

Failure of a Competitive Market

• Production of x and y in competitive

markets will fail to achieve this allocation

– with perfectly competitive prices px and py,

each individual will equate his MRS to px/py

– the producer will also set 1/f’ equal to px/py

to maximize profits

– the price ratio px/py will be too low

• it would provide too little incentive to produce x

52

Failure of a Competitive Market

• For public goods, the value of producing

one more unit is the sum of each

consumer’s valuation of that output

– individual demand curves should be added

vertically rather than horizontally

• Thus, the usual market demand curve

will not reflect the full marginal valuation

14

53

Inefficiency of a Nash Equilibrium

• Suppose that individual A is thinking

about contributing sA of his initial y endowment to the production of x

• The utility maximization problem for A is

then

choose sA to maximize UA[f(sA + sB),yA - sA]

54

Inefficiency of a Nash Equilibrium

• The first-order condition for a maximum

is

U1Af’ - U2

A = 0

U1A/U2

A = MRSA = 1/f’

• Because a similar argument can be

applied to B, the efficiency condition will

fail to be achieved

– each person considers only his own benefit

55

The Roommates’ Dilemma

• Suppose two roommates with identical

preferences derive utility from the number

of paintings hung on their walls (x) and the

number of granola bars they eat (y) with a

utility function of

Ui(x,yi) = x1/3yi2/3 (for i=1,2)

• Assume each roommate has $300 to

spend and that px = $100 and py = $0.20

56


• We know from our earlier analysis of

Cobb-Douglas utility functions that if each

individual lived alone, he would spend 1/3

of his income on paintings (x = 1) and 2/3

on granola bars (y = 1,000)

• When the roommates live together, each

must consider what the other will do

– if each assumed the other would buy

paintings, x = 0 and utility = 0

15

57


• If person 1 believes that person 2 will

not buy any paintings, he could choose

to purchase one and receive utility of

U1(x,y1) = 11/3(1,000)2/3 = 100

while person 2’s utility will be

U2(x,y2) = 11/3(1,500)2/3 = 131

• Person 2 has gained from his free-riding

position 58


• We can show that this solution is

inefficient by calculating each person’s

MRS

x

y

yU

xUMRS i

ii

ii

2/

/

• At the allocations described,

MRS1 = 1,000/2 = 500

MRS2 = 1,500/2 = 750

59


• Since MRS1 + MRS2 = 1,250, the

roommates would be willing to sacrifice

1,250 granola bars to have one additional

painting

– an additional painting would only cost them

500 granola bars

– too few paintings are bought

60


• To calculate the efficient level of x, we

must set the sum of each person’s MRS

equal to the price ratio

20.0

100

2222121

21

y

x

p

p

x

yy

x

y

x

yMRSMRS

• This means that

y1 + y2 = 1,000x

16

61


• Substituting into the budget constraint,

we get

0.20(y1 + y2) + 100x = 600

x = 2

y1 + y2 = 2,000

• The allocation of the cost of the

paintings depends on how each

roommate plays the strategic financing

game 62

Lindahl Pricing of Public Goods

• Swedish economist E. Lindahl

suggested that individuals might be

willing to be taxed for public goods if they

knew that others were being taxed

– Lindahl assumed that each individual would

be presented by the government with the

proportion of a public good’s cost he was

expected to pay and then reply with the

level of public good he would prefer

63


• Suppose that individual A would be

quoted a specific percentage (A) and

asked the level of a public good (x) he

would want given the knowledge that this

fraction of total cost would have to be

paid

• The person would choose the level of x

which maximizes

utility = UA[x,yA*- Af -1(x)] 64


• The first-order condition is given by

U1A - AU2

B(1/f’)=0

MRSA = A/f’

• Faced by the same choice, individual B

would opt for the level of x which satisfies

MRSB = B/f’

17

65


• An equilibrium would occur when

A+B = 1

– the level of public goods expenditure

favored by the two individuals precisely

generates enough tax contributions to pay

for it

MRSA + MRSB = (A + B)/f’ = 1/f’

66

Shortcomings of the Lindahl Solution

• The incentive to be a free rider is very

strong

– this makes it difficult to envision how the

information necessary to compute

equilibrium Lindahl shares might be

computed

• individuals have a clear incentive to understate

their true preferences

67


• Externalities may cause a

misallocation of resources because of

a divergence between private and

social marginal cost

– traditional solutions to this divergence

includes mergers among the affected

parties and adoption of suitable

Pigouvian taxes or subsidies

68


• If transactions costs are small, private

bargaining among the parties

affected by an externality may bring

social and private costs into line

– the proof that resources will be

efficiently allocated under such

circumstances is sometimes called the

Coase theorem

18

69


• Public goods provide benefits to

individuals on a nonexclusive basis -

no one can be prevented from

consuming such goods

– such goods are usually nonrival in that

the marginal cost of serving another

user is zero

70


• Private markets will tend to

underallocate resources to public

goods because no single buyer can

appropriate all of the benefits that

such goods provide

71


• A Lindahl optimal tax-sharing scheme

can result in an efficient allocation of

resources to the production of public

goods

– computation of these tax shares

requires substantial information that

individuals have incentives to hide

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