Important Points to Note

Transcription

TEORI EKONOMI MIKRO
DOSEN:
DR. ARDITO BHINADI, SE., M.SI
JURUSAN ILMU EKONOMI, FAKULTAS EKONOMI,
UPN “VETERAN” YOGYAKARTA
2013
RANCANGAN PEMBELAJARAN SEMESTER ( RPS)
Program Studi /Jurusan
Matakuliah / Kode
SKS / Semester
Mata Kuliah Prasyarat
Dosen
:
:
:
:
:
EKONOMI PEMBANGUNAN/ILMU EKONOMI
TEORI EKONOMI MIKRO /
3 (tiga x 50 menit)/ II (dua)
Ekonomi Mikro Pengantar
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
1
(Satu)
2
(Dua)
3
(Tiga)
4
(Empat)
Kompetensi
Pokok/Sub-pokok
Bahasan
Model-Model
Ekonomi
Metoda
Pembelajaran
Ceramah dan
diskusi
Media
Pembelajaran
Papan tulis,
LCD, Laptop,
Metoda
Evaluasi
Pertanyaan
kuis/umpan
balik
Referensi
Mahasiswa
mampu
memahami
preferensi dan
utilitas
konsumen.
Preferensi dan
Utilitas
Mahasiswa
Presentasi,
Ceramah dan
diskusi
Papan tulis,
LCD, Laptop,
Pertanyaan
kuis/umpan
balik
Ch 3
Mahasiswa
mampu efek
substitusi dan
pendapatan.
Efek Substitusi dan
Pendapatan
Mahasiswa
Presentasi,
Ceramah dan
diskusi
Papan tulis,
LCD, Laptop,
Pertanyaan
kuis/umpan
balik
Ch 5
Mahasiswa
mampu
memahami
hubungan
permintaan antar
barang.
Hubungan
Permintaan Antar
Barang
Mahasiswa
Presentasi,
Ceramah dan
diskusi
Papan tulis,
LCD, Laptop,
Pertanyaan
kuis/umpan
balik
Ch 6
Mahasiswa
mampu
memahami
berbagai model
ekonomi.
Ch1
1
5
(Lima)
6
(Enam)
7
(Tujuh)
Pertemuan
Ke
8
(Delapan)
9
(Sembilan)
10
(Sepuluh)
11
(Sebelas)
12
(Dua Belas)
13
(Tiga Belas)
Mahasiswa
mampu
memahami
fungsi-fungsi
produksi
Fungsi-Fungsi
Produksi
Mahasiswa
Presentasi,
Ceramah dan
diskusi
Papan tulis,
LCD, Laptop,
Pertanyaan
kuis/umpan
balik
Ch 9
Mahasiswa
mampu
memahami
fungsi-fungsi
biaya.
Fungsi-Fungsi
Biaya.
Mahasiswa
Presentasi,
Ceramah dan
diskusi
Papan tulis,
LCD, Laptop,
Pertanyaan
kuis/umpan
balik
Ch 10
Mahasiwa
mampu
menghitung
maksimisasi
laba.
Maksimisasi Laba
Mahasiswa
Presentasi,
Ceramah dan
Diskusi
Papan tulis,
LCD, Laptop
Pertanyaan
umpan balik
Ch 11
Media
Pembelajaran
Papan tulis,
LCD, Laptop
Metoda
Evaluasi
Pertanyaan
umpan balik
Referensi
Kompetensi
Mahasiwa
mampu
memahami
model
persaingan
keseimbangan
parsial.
Mahasiwa
mampu
memahami
keseimbangan
umum dan
kesejahteraan.
Mahasiwa
mampu
memahami
monopoli.
Ujian Tengah Semester
Pokok/Sub-pokok
Metoda
Bahasan
Pembelajaran
Model Persaingan Diskusi dan
Keseimbangan
Kuis
Parsial
Ch 12
Keseimbangan
Umum dan
Kesejahteraan
Diskusi dan
Kuis
Papan tulis,
LCD, Laptop
Pertanyaan
umpan balik
Ch 13
Monopoli
Diskusi dan
Kuis
Papan tulis,
LCD, Laptop
Pertanyaan
umpan balik
Ch 14
Mahasiwa
mampu
memahami
persaingan tidak
sempurna.
Persaingan Tidak
Sempurna
Diskusi dan
Kuis
Papan tulis,
LCD, Laptop
Pertanyaan
umpan balik
Ch 15
Mahasiwa
mampu
memahami pasar
tenaga kerja
Pasar Tenaga Kerja
Diskusi dan
Kuis
Papan tulis,
LCD, Laptop
Pertanyaan
umpan balik
Ch 16
Mahasiwa
mampu
memahami
informasi
asimetris.
Asimetris Informasi
Diskusi dan
Kuis
Papan tulis,
LCD, Laptop
Pertanyaan
umpan balik
Ch 18
2
14
(Empat
Belas)
Mahasiwa
mampu
memahami
eksternalitas dan
barang publik.
Eksternalitas dan
Barang Publik
Diskusi dan
Kuis
Papan tulis,
LCD, Laptop
Pertanyaan
umpan balik
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
2.Ujian Akhir Semester
3.Partisipasi Kelas
4.Tugas-Tugas
=
=
=
=
30%
30%
20%
20%
3
Microeconomic Theory
Basic Principles and Extensions, 9e
Chapter 1
ECONOMIC MODELS
By
WALTER NICHOLSON
Slides prepared by
Linda Ghent
Eastern Illinois University
Copyright ©2005 by South-Western, a division of Thomson Learning. All rights reserved.
1
Theoretical Models
2
Verification of Economic Models
• Economists use models to describe
economic activities
• There are two general methods used to
verify economic models:
– direct approach
• While most economic models are
abstractions from reality, they provide
aid in understanding economic behavior
3
• establishes the validity of the model’s
assumptions
– indirect approach
• shows that the model correctly predicts realworld events
4
1
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?
Features of Economic Models
• Ceteris Paribus assumption
• Optimization assumption
• Distinction between positive and
normative analysis
5
Ceteris Paribus Assumption
• Ceteris Paribus means “other things the
same”
• Economic models attempt to explain
simple relationships
6
Optimization Assumptions
• Many economic models begin with the
assumption that economic actors are
rationally pursuing some goal
– focus on the effects of only a few forces at a
time
– other variables are assumed to be unchanged
during the period of study
7
– consumers seek to maximize their utility
– firms seek to maximize profits (or minimize
costs)
– government regulators seek to maximize
public welfare
8
2
Optimization Assumptions
Positive-Normative Distinction
• Optimization assumptions generate
precise, solvable models
• Positive economic theories seek to
explain the economic phenomena that
is observed
• Optimization models appear to be
perform fairly well in explaining reality
• Normative economic theories focus on
what “should” be done
9
The Economic Theory of Value
10
• The Founding of Modern Economics
• Early Economic Thought
– 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)
– “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”
• the value of an item meant its “value in use”
• the price of an item meant its “value in exchange”
11
12
3
• Labor Theory of Exchange Value
• The Marginalist Revolution
– 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
– 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
– producing diamonds requires more labor than
producing water
13
• Marshallian Supply-Demand Synthesis
14
Supply-Demand Equilibrium
Price
– 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
15
Equilibrium
QD = Qs
S
The supply curve has a positive
slope because marginal cost
rises as quantity increases
P*
D
Q*
The demand curve has a
negative slope because
the marginal value falls as
quantity increases
Quantity per period
16
4
• A more general model is
qD = 1000 - 100p
qS = -125 + 125p
qD = a + bp
Equilibrium  qD = qS
qS = c + dp
1000 - 100p = -125 + 125p
Equilibrium  qD = qS
225p = 1125
a + bp = c + dp
p* = 5
q* = 500
p* 
17
A shift in demand will lead to a new equilibrium:
ac
d b
18
Price
An increase in demand...
S
Q’D = 1450 - 100P
Q’D = 1450 - 100P = QS = -125 + 125P
225P = 1575
P* = 7
Q* = 750
…leads to a rise in the
equilibrium price and
quantity.
7
5
D’
D
500 750
19
Quantity per period
20
5
• General Equilibrium Models
• 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
– 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
21
A Production Possibility Frontier
Quantity of food
(weekly)
Opportunity cost of
clothing = 1/2 pound of food
10
9.5
Opportunity cost of
clothing = 2 pounds of food
4
2
3 4
12 13
22
• 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
Quantity of clothing
(weekly)
23
24
6
• Suppose that the production possibility
frontier can be represented by
dy 1
 4 x  2x
 (225  2x 2 )1/ 2  ( 4 x ) 

dx 2
2y
y
2x 2  y 2  225
• when x=5, y=13.2, the slope= -2(5)/13.2= -0.76
• To find the slope, we can solve for Y
• when x=10, y=5, the slope= -2(10)/5= -4
y  225  2x 2
• the slope rises as y rises
• If we differentiate
dy 1
 4 x  2x
 (225  2x 2 )1/ 2  ( 4 x ) 

dx 2
2y
y
25
• 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
27
26
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
28
7
Important Points to Note:
• Economics is the study of how scarce
resources are allocated among
alternative uses
• 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
– economists use simple models to
understand the process
29
30
• The supply-demand model is only a
partial-equilibrium model
• Testing the validity of a model is a
difficult task
– a general equilibrium model is needed to
look at many markets together
– are the model’s assumptions
reasonable?
– does the model explain real-world
events?
31
32
8
Axioms of Rational Choice
• Completeness
Chapter 3
– if A and B are any two situations, an
individual can always specify exactly one of
these possibilities:
PREFERENCES AND UTILITY
• A is preferred to B
• B is preferred to A
• A and B are equally attractive
1
2
• Transitivity
• Continuity
– 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
– 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
3
4
1
Utility
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)
• 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
5
6
Utility
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
• 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 orderpreserving transformation
– ceteris paribus assumption
7
8
2
Economic Goods
Indifference Curves
• In the utility function, the x’s are assumed
to be “goods”
• An indifference curve shows a set of
consumption bundles among which the
individual is indifferent
– more is preferred to less
Quantity of y
Quantity of y
Combinations (x1, y1) and (x2, y2)
provide the same level of utility
Preferred to x*, y*
?
y*
y1
?
Worse
than
x*, y*
y2
Quantity of x
Quantity of x
9
x*
U1
x1
x2
10
Marginal Rate of Substitution
Marginal Rate of Substitution
• The negative of the slope of the
indifference curve at any point is called
the marginal rate of substitution (MRS)
• MRS changes as x and y change
– reflects the individual’s willingness to trade y
for x
Quantity of y
Quantity of y
MRS  
dy
dx U U1
y1
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
y1
y2
y2
U1
Quantity of x
x1
x2
11
U1
Quantity of x
x1
x2
12
3
Indifference Curve Map
Transitivity
• Each point must have an indifference
curve through it
• Can any two of an individual’s indifference
curves intersect?
Quantity of y
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
Quantity of y
Increasing utility
U3
U2
C
U1 < U2 < U3
B
U2
A
But B is preferred to A
because B contains more
x and y than A
U1
U1
Quantity of x
Quantity of x
13
Convexity
14
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 y
• 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 y
This implies that “well-balanced” bundles are preferred
to bundles that are heavily weighted toward one
commodity
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
y1
(y1 + y2)/2
y*
y2
U1
Quantity of x
x*
15
U1
Quantity of x
x1
(x1 + x2)/2
x2
16
4
Utility and the MRS
Utility and the MRS
• Suppose an individual’s preferences for
hamburgers (y) and soft drinks (x) can
be represented by
MRS = -dy/dx = 100/x2
• Note that as x rises, MRS falls
– when x = 5, MRS = 4
– when x = 20, MRS = 0.25
utility  10  x  y
• Solving for y, we get
y = 100/x
• Solving for MRS = -dy/dx:
MRS = -dy/dx = 100/x2
17
Marginal Utility
18
Deriving the MRS
• Suppose that an individual has a utility
function of the form
• Therefore, we get:
utility = U(x,y)
dy
MRS  
dx
• The total differential of U is
U
U
dU 
dx 
dy
x
y
Uconstant
U
 x
U
y
• MRS is the ratio of the marginal utility of
x to the marginal utility of y
• Along any indifference curve, utility is
constant (dU = 0)
19
20
5
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
Convexity of Indifference
Curves
• Suppose that the utility function is
utility  x  y
• We can simplify the algebra by taking the
logarithm of this function
• this is independent of how utility is measured
– diminishing marginal utility depends on how
utility is measured
U*(x,y) = ln[U(x,y)] = 0.5 ln x + 0.5 ln y
• Thus, these two concepts are different
21
22
Curves
Curves
• If the utility function is
• Thus,
U(x,y) = x + xy + y
• There is no advantage to transforming
this utility function, so
U * 0.5
y
MRS  x  x 
U * 0.5 x
y
y
U
1 y
MRS  x 
U 1  x
y
23
24
6
Curves
Curves
• Thus,
utility  x 2  y 2
U *
2x x
MRS  x 

U * 2y y
y
• For this example, it is easier to use the
transformation
U*(x,y) = [U(x,y)]2 = x2 + y2
25
Examples of Utility Functions
• Cobb-Douglas Utility
utility = U(x,y) =
26
• Perfect Substitutes
utility = U(x,y) = x + y
xy
where  and  are positive constants
Quantity of y
The indifference curves will be linear.
The MRS will be constant along the
indifference curve.
– The relative sizes of  and  indicate the
relative importance of the goods
U3
U1
27
U2
Quantity of x
28
7
• CES Utility (Constant elasticity of
substitution)
• Perfect Complements
utility = U(x,y) = min (x, y)
utility = U(x,y) = x/ + y/
Quantity of y
The indifference curves will be
L-shaped. Only by choosing more
of the two goods together can utility
be increased.
when   0 and
utility = U(x,y) = ln x + ln y
when  = 0
– Perfect substitutes   = 1
– Cobb-Douglas   = 0
– Perfect complements   = -
U3
U2
U1
Quantity of x
29
• CES Utility (Constant elasticity of
substitution)
30
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
– The elasticity of substitution () is equal to
1/(1 - )
– 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 < /
• Perfect substitutes   = 
• Fixed proportions   = 0
31
32
8
Homothetic Preferences
Nonhomothetic Preferences
• For the general Cobb-Douglas function,
the MRS can be found as
• Some utility functions do not exhibit
homothetic preferences
utility = U(x,y) = x + ln y
U
x  1y   y
MRS  x 
 
U x  y 1  x
y
U
1
MRS  x   y
U
1
y
y
33
The Many-Good Case
The Many-Good Case
• Suppose utility is a function of n goods
given by
• We can find the MRS between any two
goods by setting dU = 0
utility = U(x1, x2,…, xn)
dU  0 
• The total differential of U is
dU 
34
U
U
dxi 
dx j
xi
x j
• Rearranging, we get
U
U
U
dx1 
dx2  ... 
dxn
x1
x2
xn
U
x i
MRS( x i for x j )  

U
dx i
x j
dx j
35
36
9
Multigood Indifference
Surfaces
Multigood Indifference
Surfaces
• We will define an indifference surface
as being the set of points in n
dimensions that satisfy the equation
• 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
U(x1,x2,…xn) = k
where k is any preassigned constant
37
38
• If individuals obey certain behavioral
postulates, they will be able to rank all
commodity bundles
• The negative of the slope of the
indifference curve measures the marginal
rate of substitution (MRS)
– the ranking can be represented by a utility
function
– in making choices, individuals will act as if
they were maximizing this function
– 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
• Utility functions for two goods can be
illustrated by an indifference curve map
– individuals prefer some balance in their
consumption choices
39
40
10
• 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
• 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 twogood cases to build intuition
• includes the other three as special cases
41
42
11
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:
Chapter 5
INCOME AND SUBSTITUTION
EFFECTS
x1* = d1(p1,p2,…,pn,I)
x2* = d2(p1,p2,…,pn,I)
1
•
•
•
xn* = dn(p1,p2,…,pn,I)
2
Homogeneity
Demand Functions
• If we were to double all prices and
income, the optimal quantities demanded
will not change
• If there are only two goods (x and y), we
can simplify the notation
x* = x(px,py,I)
– the budget constraint is unchanged
y* = y(px,py,I)
xi* = di(p1,p2,…,pn,I) = di(tp1,tp2,…,tpn,tI)
• Prices and income are exogenous
• Individual demand functions are
homogeneous of degree zero in all prices
and income
– the individual has no control over these
parameters
3
4
1
Homogeneity
Homogeneity
• With a Cobb-Douglas utility function
utility = U(x,y) =
• With a CES utility function
utility = U(x,y) = x0.5 + y0.5
the demand functions are
1
I
1
I
x* 

y* 

1  px / py px
1  py / px py
x0.3y0.7
the demand functions are
x* 
0 .3 I
px
y* 
0.7 I
py
• Note that a doubling of both prices and
income would leave x* and y*
unaffected
• Note that a doubling of both prices and
income would leave x* and y*
unaffected
5
6
Increase in Income
Changes in Income
• If both x and y increase as income rises,
x and y are normal goods
• 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
Quantity of y
As income rises, the individual chooses
to consume more x and y
B
C
A
U1
U3
U2
Quantity of x
7
8
2
Increase in Income
Normal and Inferior Goods
• If x decreases as income rises, x is an
inferior good
• A good xi for which xi/I  0 over some
range of income is a normal good in that
range
As income rises, the individual chooses
to consume less x and more y
Quantity of y
Note that the indifference
curves do not have to be
“oddly” shaped. The
assumption of a diminishing
MRS is obeyed.
C
B
U3
U2
A
• A good xi for which xi/I < 0 over some
range of income is an inferior good in
that range
U1
Quantity of x
9
10
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
• 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
– substitution effect
– income effect
– the income effect
11
12
3
Quantity of y
Suppose the consumer is maximizing
utility at point A.
Quantity of y
To isolate the substitution effect, we hold
“real” income constant but allow the
relative price of good x to change
If the price of good x falls, the consumer
will maximize utility at point B.
The substitution effect is the movement
from point A to point C
B
A
The individual substitutes
good x for good y
because it is now
relatively cheaper
C
A
U2
U1
U1
Quantity of x
Quantity of x
Substitution effect
Total increase in x
13
Quantity of y
A
Quantity of y
The income effect occurs because the
individual’s “real” income changes when
the price of good x changes
B
14
The income effect is the movement
from point C to point B
C
U2
U1
If x is a normal good,
the individual will buy
more because “real”
income increased
An increase in the price of good x means that
the budget constraint gets steeper
The substitution effect is the
movement from point A to point C
C
A
B
U1
The income effect is the
movement from point C
to point B
U2
Quantity of x
Quantity of x
Substitution effect
Income effect
Income effect
15
16
4
Price Changes for
Inferior Goods
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
17
• 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
18
Giffen’s Paradox
A Summary
• If the income effect of a price change is
strong enough, there could be a positive
relationship between price and quantity
demanded
• Utility maximization implies that (for normal
goods) a fall in price leads to an increase in
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
19
– 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
20
5
A Summary
A Summary
• Utility maximization implies that (for normal
goods) a rise in price leads to a decline in
quantity demanded
• Utility maximization implies that (for inferior
goods) no definite prediction can be made
for changes in price
– 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
21
The Individual’s Demand Curve
• An individual’s demand for x depends
on preferences, all prices, and income:
– 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
22
Quantity of y
As the price
of x falls...
px
…quantity of x
demanded rises.
x* = x(px,py,I)
px’
• It may be convenient to graph the
individual’s demand for x assuming that
income and the price of y (py) are held
constant
px’’
px’’’
U1
x1
23
I = px’ + py
x2
x3
I = px’’ + py
U2
U3
Quantity of x
I = px’’’ + py
x
x’
x’’
x’’’
Quantity of x
24
6
• 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
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
25
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
27
26
Demand Functions and Curves
• We discovered earlier that
x* 
0 .3 I
px
y* 
0.7 I
py
• If the individual’s income is $100, these
functions become
x* 
30
px
y* 
70
py
28
7
Demand Functions and Curves
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
• Any change in income will shift these
demand 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
29
30
• An alternative approach holds real income
(or utility) constant while examining
reactions to changes in px
• 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 twodimensional representation of the
compensated demand function
– 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
31
x* = xc(px,py,U)
32
8
Holding utility constant, as price falls...
Quantity of y
slope  
px
px '
py
slope  
…quantity demanded
rises.
px ' '
py
Compensated &
Uncompensated Demand
px
At px’’, the curves intersect because
the individual’s income is just sufficient
to attain utility level U2
px’
px’’
px’’
slope  
px ' ' '
py
x
px’’’
xc
xc
U2
x’
x’’
x’’’
x’
Quantity of x
x’’
x’’’
x’’
Quantity of x
Quantity of x
33
Compensated &
Compensated &
At prices above px2, income
compensation is positive because the
individual needs some help to remain
on U2
px
34
px
At prices below px2, income
compensation is negative to prevent an
increase in utility from a lower price
px’
px’’
px’’
px’’’
x
x
xc
x’
x*
xc
x***
Quantity of x
35
x’’’
Quantity of x
36
9
Compensated Demand
Functions
Compensated &
• 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
• 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
– the compensated demand curve reflects only
substitution effects
utility  V ( I, px , py ) 
I
0.5
x
2p py0.5
37
Compensated Demand
Functions
Compensated Demand
Functions
• To obtain the compensated demand
functions, we can solve the indirect
utility function for I and then substitute
into the Marshallian demand functions
x
Vpy0.5
px0.5
y
38
x
Vpy0.5
px0.5
y
Vpx0.5
py0.5
• Demand now depends on utility (V)
rather than income
• Increases in px reduce the amount of x
demanded
Vpx0.5
py0.5
– only a substitution effect
39
40
10
A Mathematical Examination
of a Change in Price
• Our goal is to examine how purchases of
good x change when px changes
• Instead, we will use an indirect approach
• Remember the expenditure function
x/px
minimum expenditure = E(px,py,U)
• 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
41
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 42
x x c
x E



px px
E px
xc (px,py,U) = x[px,py,E(px,py,U)]
• We can differentiate the compensated
demand function and get
• The first term is the slope of the
compensated demand curve
x
x
x E



px px
E px
c
x x c
x E



px px
E px
• Then, by definition
– the mathematical representation of the
substitution effect
43
44
11
The Slutsky Equation
• The substitution effect can be written as
x x c
x E



px px
E px
substituti on effect 
• The second term measures the way in
which changes in px affect the demand
for x through changes in purchasing
power
x c
x

px px
U constant
• The income effect can be written as
income effect  
– the mathematical representation of the
income effect
x E
x E

 

E px
I px
45
46
• The utility-maximization hypothesis
shows that the substitution and income
effects arising from a price change can be
represented by
• 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
x
 substituti on effect  income effect
px
x
x

px px
47
x
U constant
x
I
48
12
x
x

px px
x
U constant
x
I
x
x

px px
• The first term is the substitution effect
x
U constant
x
I
• The second term is the income effect
– always negative as long as MRS is
diminishing
– the slope of the compensated demand curve
must be negative
– 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
49
A Slutsky Decomposition
50
• We can demonstrate the decomposition
of a price effect using the Cobb-Douglas
example studied earlier
• The Marshallian demand function for
good x was
0 .5 I
x ( p x , py , I ) 
px
51
• The Hicksian (compensated) demand
function for good x was
x c ( px , py ,V ) 
Vpy0.5
px0.5
• The overall effect of a price change on
the demand for x is
x
 0 .5 I

px
px2
52
13
• This total effect is the sum of the two
effects that Slutsky identified
• The substitution effect is found by
differentiating the compensated demand
function
• We can substitute in for the indirect utility
function (V)
 0.5(0.5 Ipx0.5 py0.5 )py0.5
p
1.5
x

 0.25I
px2
0. 5
x c  0.5Vpy

px
p1x.5
53
• Calculation of the income effect is easier
income effect   x
 0.5I  0.5
x
0.25I
 


I
px2
 px  px
• Interestingly, the substitution and income
effects are exactly the same size
55
54
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)
ex ,px 
x / x
x px


px / px px x
56
14
Marshallian Demand
Elasticities
Price Elasticity of Demand
• The own price elasticity of demand is
always negative
• Income elasticity of demand (ex,I)
e x ,I 
x / x x I


I / I I x
– the only exception is Giffen’s paradox
• The size of the elasticity is important
• Cross-price elasticity of demand (ex,py)
ex ,py 
– if ex,px < -1, demand is elastic
– if ex,px > -1, demand is inelastic
– if ex,px = -1, demand is unit elastic
x / x
x py


py / py py x
57
Price Elasticity and Total
Spending
58
Price Elasticity and Total
Spending
• Total spending on x is equal to
( p x x )
x
 px 
 x  x[ex,px  1]
px
px
total spending =pxx
• Using elasticity, we can determine how
total spending changes when the price of
x changes
( p x x )
x
 px 
 x  x[ex,px  1]
px
px
59
• 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
60
15
Compensated Price Elasticities
• It is also useful to define elasticities
based on the compensated demand
function
• 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)
61
• The compensated own price elasticity of
demand (exc,px) is
exc,px 
x c / x c x c px


px / px px x c
• The compensated cross-price elasticity
of demand (exc,py) is
e
c
x ,py
x c / x c x c py



py / py py x c
62
• The relationship between Marshallian
and compensated price elasticities can
be shown using the Slutsky equation
px x
p x c px
x

 ex,px  xc 

x
x px
x px x
I
• If sx = pxx/I, then
ex,px  exc,px  sx ex,I
63
64
16
Homogeneity
• The Slutsky equation shows that the
compensated and uncompensated price
elasticities will be similar if
• Demand functions are homogeneous of
degree zero in all prices and income
• Euler’s theorem for homogenous
functions shows that
– the share of income devoted to x is small
– the income elasticity of x is small
0  px 
x
x
x
 py 
I
px
py
I
65
Homogeneity
66
Engel Aggregation
• Dividing by x, we get
• Engel’s law suggests that the income
elasticity of demand for food items is
less than one
0  ex,px  ex,py  ex,I
• Any proportional change in all prices
and income will leave the quantity of x
demanded unchanged
– this implies that the income elasticity of
demand for all nonfood items must be
greater than one
67
68
17
Engel Aggregation
Cournot Aggregation
• We can see this by differentiating the
budget constraint with respect to
income (treating prices as constant)
1  px 
1  px 
• 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
x
y
 py 
I
I
x xI
y yI

 py 

 s x e x , I  s y ey , I
I xI
I yI
69
Demand Elasticities
Cournot Aggregation
• The Cobb-Douglas utility function is
I
x
y
 0  px 
 x  py 
px
px
px
0  px 
70
U(x,y) = xy
(+=1)
• The demand functions for x and y are
x px x
p
y px y
   x  x  py 
 
px I x
I
px I y
x
0  s x ex,px  s x  sy ey ,px
I
px
y
I
py
s x ex,px  sy ey ,px  s x
71
72
18
Demand Elasticities
Demand Elasticities
• Calculating the elasticities, we get
ex ,px 
– homogeneity
x px
I p

  2  x  1
px x
p x  I 
 
 px 
e x , py
e x ,I
• We can also show
ex,px  ex,py  ex,I  1 0  1  0
– Engel aggregation
p
x py


 0 y  0
py x
x
x I 
I

 

1
I x px  I 
 
 px 
s x e x , I  s y ey , I    1    1      1
– Cournot aggregation
s x ex,px  sy ey ,px    ( 1)    0    s x
73
74
Demand Elasticities
Demand Elasticities
• We can also use the Slutsky equation to
derive the compensated price elasticity
• The CES utility function (with  = 2,
 = 5) is
U(x,y) = x0.5 + y0.5
exc,px  ex,px  sx ex,I  1 (1)    1  
• The compensated price elasticity
depends on how important other goods
(y) are in the utility function
75
• The demand functions for x and y are
x
I
px (1  px py1 )
y
I
py (1  px1py )
76
19
Demand Elasticities
Demand Elasticities
• We will use the “share elasticity” to
derive the own price elasticity
es x ,px 
• Thus, the share elasticity is given by
esx ,px 
s x px

 1 ex,px
px s x
• Therefore, if we let px = py
• In this case,
sx 
 py1
 px py1
s x px
px




px s x (1  px py1 )2 (1  px py1 )1 1  px py1
px x
1

I
1  px py1
ex,px  es x ,px  1 
1
 1  1.5
1 1
77
Demand Elasticities
78
Demand Elasticities
• Thus, the share elasticity is given by
• The CES utility function (with  = 0.5,
 = -1) is
es x ,px 
U(x,y) = -x -1 - y -1
• The share of good x is
sx 

px x
1

0.5 0.5
I
1  py px
0.5 py0.5 px1.5
s x px
px



0.5 0.5 2
0.5 0.5 1
px s x (1  py px ) (1  py px )
0.5 py0.5 px0.5
1  py0.5 px0.5
• Again, if we let px = py
79
ex,px  esx ,px  1 
0.5
 1  0.75
2
80
20
Consumer Surplus
Consumer Welfare
• One way to evaluate the welfare cost of a
price increase (from px0 to px1) would be
to compare the expenditures required to
achieve U0 under these two situations
• An important problem in welfare
economics is to devise a monetary
measure of the gains and losses that
individuals experience when prices
change
expenditure at px0 = E0 = E(px0,py,U0)
expenditure at px1 = E1 = E(px1,py,U0)
81
82
Consumer Welfare
Consumer Welfare
Suppose the consumer is maximizing
utility at point A.
Quantity of y
• In order to compensate for the price rise,
this person would require a
compensating variation (CV) of
If the price of good x rises, the consumer
will maximize utility at point B.
A
CV = E(px1,py,U0) - E(px0,py,U0)
B
U1
The consumer’s utility
falls from U1 to U2
U2
Quantity of x
83
84
21
Consumer Welfare
Consumer Welfare
The consumer could be compensated so
that he can afford to remain on U1
Quantity of y
CV is the amount that the
C
• The derivative of the expenditure function
with respect to px is the compensated
demand function
individual would need to be
compensated
A
E ( px , py ,U0 )
B
px
U1
 x c ( px , py ,U0 )
U2
Quantity of x
85
Consumer Welfare
86
px
Consumer Welfare
• The amount of CV required can be found
by integrating across a sequence of
small increments to price from px0 to px1
p1x
p1x
px0
px0
welfare loss
px1
CV   dE   x c ( px , py ,U0 )dpx
– this integral is the area to the left of the
compensated demand curve between px0
and px1
When the price rises from px0 to px1,
the consumer suffers a loss in welfare
px0
xc(px…U0)
x1
87
x0
Quantity of x
88
22
The Consumer Surplus
Concept
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)?
• 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
89
The Consumer Surplus
Concept
90
Consumer Welfare
px
• The area below the compensated
demand curve and above the market
price is called consumer surplus
When the price rises from px0 to px1, the actual
market reaction will be to move from A to C
The consumer’s utility falls from U0 to U1
px1
– the extra benefit the person receives by
being able to make market transactions at
the prevailing market price
px
C
A
0
x(px…)
xc(...U0)
xc(...U1)
x1
91
x0
Quantity of x
92
23
Consumer Welfare
px
px1
Consumer Welfare
Is the consumer’s loss in welfare
best described by area px1BApx0
[using xc(...U0)] or by area px1CDpx0
[using xc(...U1)]?
C
B
A
px0
D
xc(...U
0)
Is U0 or U1 the
appropriate utility
target?
px
px1
We can use the Marshallian demand
curve as a compromise
C
B
A
px0
D
x(px…)
xc(...U
0)
xc(...U1)
x1
x0
The area px1CApx0
falls between the
sizes of the welfare
losses defined by
xc(...U0) and
xc(...U1)
xc(...U1)
x1
Quantity of x
x0
Quantity of x
93
94
Welfare Loss from a Price
Increase
Consumer Surplus
• We will define consumer surplus as the
area below the Marshallian demand
curve and above price
• Suppose that the compensated demand
function for x is given by
– 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
95
x c ( px , py ,V ) 
Vpy0.5
px0.5
• The welfare cost of a price increase
from px = 1 to px = 4 is given by
4
CV  Vpy0.5 px0.5  2Vpy0.5 px0.5
1
px  4
p X 1
96
24
Increase
Welfare Loss from Price
Increase
• Suppose that we use the Marshallian
demand function instead
• If we assume that V = 2 and py = 2,
CV = 222(4)0.5 – 222(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 = 122(4)0.5 – 122(1)0.5 = 4
x( px , py , I )  0.5Ipx-1
• The welfare loss from a price increase
from px = 1 to px = 4 is given by
4
Loss   0.5 Ipx-1dpx  0.5 I ln px
97
Increase
1
px  4
px 1
98
Revealed Preference and
the Substitution Effect
• If income (I) is equal to 8,
• 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
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
99
100
25
• 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
101
Suppose that, when the budget constraint is
given by I1, A is chosen
Quantity of y
A must still be preferred to B when income
is I3 (because both A and B are available)
A
If B is chosen, the budget
constraint must be similar to
that given by I2 where A is not
available
B
I3
I1
I2
Quantity of x
102
Negativity of the
Substitution Effect
Negativity of the
Substitution Effect
• Suppose that an individual is indifferent
between two bundles: C and D
• Since the individual is indifferent between
C and D
– When C is chosen, D must cost at least as
much as C
pxCxC + pyCyC ≤ pxCxD + pyCyD
• Let pxC,pyC be the prices at which
bundle C is chosen
• Let pxD,pyD be the prices at which
bundle D is chosen
– When D is chosen, C must cost at least as
much as D
pxDxD + pyDyD ≤ pxDxC + pyDyC
103
104
26
Negativity of the
Substitution Effect
Negativity of the
Substitution Effect
• Suppose that only the price of x changes
(pyC = pyD)
• Rearranging, we get
pxC(xC
- xD) +
pyC(yC
-yD) ≤ 0
(pxC – pxD)(xC - xD) ≤ 0
pxD(xD - xC) + pyD(yD -yC) ≤ 0
• Adding these together, we get
(pxC – pxD)(xC - xD) + (pyC – pyD)(yC - yD) ≤ 0
• This implies that price and quantity move
in opposite direction when utility is held
constant
– the substitution effect is negative
105
106
Mathematical Generalization
Mathematical Generalization
• If, at prices pi0 bundle xi0 is chosen
instead of bundle xi1 (and bundle xi1 is
affordable), then
• Consequently, at prices that prevail
when bundle 1 is chosen (pi1), then
n
p x
i 1
0
i
n
p x
n
0
i
  pi0 xi1
i 1
1 0
i i
n
  pi1xi1
i 1
i 1
• Bundle 0 has been “revealed preferred”
to bundle 1
107
• Bundle 0 must be more expensive than
bundle 1
108
27
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
• 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
109
110
• When purchasing power changes
(income changes but prices remain the
same), budget constraints shift
• 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
– for normal goods, an increase in income
means that more is purchased
– for inferior goods, an increase in income
means that less is purchased
• no unambiguous prediction is possible
111
112
28
• A rise in the price of a good also
causes income and substitution effects
• The Marshallian demand curve
summarizes the total quantity of a good
demanded at each possible price
– for normal goods, less will be demanded
– for inferior goods, the net result is
ambiguous
– changes in price prompt movements
along the curve
– changes in income, prices of other goods,
or preferences may cause the demand
curve to shift
113
114
• 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)
115
• 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
116
29
• There are many relationships among
demand elasticities
• Welfare effects of price changes can
be measured by changing areas below
either compensated or ordinary
demand curves
– 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
– such changes affect the size of the
consumer surplus that individuals receive
by being able to make market transactions
117
118
• 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
119
30
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 twodimensional graphs
Chapter 6
DEMAND RELATIONSHIPS
AMONG GOODS
1
2
Gross Complements
Quantity of y
Gross Substitutes
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
y1
y0
U1
U0
Quantity of y
When the price of y falls, the substitution
effect may be so large that the consumer
purchases less x and more y
In this case, we call x and y gross
substitutes
y1
y0
x/py < 0
U1
x/py > 0
U0
x0 x1
x1
Quantity of x
3
x0
Quantity of x
4
1
Substitutes and Complements
A Mathematical Treatment
• The change in x caused by changes in py
can be shown by a Slutsky-type equation
x
x

py py
 y
U constant
substitution
effect (+)
• For the case of many goods, we can
generalize the Slutsky analysis
x
I
xi xi

p j p j
U constant
xi
I
for any i or j
income effect
(-) if x is normal
combined effect
(ambiguous)
 xj
5
Substitutes and Complements
• Two goods are substitutes if one good
may replace the other in use
– 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
6
Gross Substitutes and
Complements
• The concepts of gross substitutes and
complements include both substitution
and income effects
– examples: tea & coffee, butter & margarine
• Two goods are complements if they are
used together
– two goods are gross substitutes if
xi /pj > 0
– examples: coffee & cream, fish & chips
– two goods are gross complements if
xi /pj < 0
7
8
2
Asymmetry of the Gross
Definitions
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
• 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)
9
Definitions
10
Definitions
gives us the following first-order conditions:
L/x = 1/x - px = 0
• Inserting this into the budget constraint, we
can find the Marshallian demand for y
pyy = I – py
L/y = 1 - py = 0
– an increase in py causes a decline in spending
on y
L/ = I - pxx - pyy = 0
• Manipulating the first two equations, we get
pxx = py
11
• 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)
12
3
Net Substitutes and
Complements
Net Substitutes and
Complements
• The concepts of net substitutes and
complements focuses solely on substitution
effects
– two goods are net substitutes if
xi
p j
0
xi
p j
U constant
– two goods are net complements if
xi
p j
U constant
x j
pi
U constant
13
Even though x and y are gross
complements, they are net substitutes
y1
y0
U1
Since MRS is diminishing,
the own-price substitution
effect must be negative so
the cross-price substitution
effect must be positive
U0
x0 x1

0
U constant
Gross Complements
Quantity of y
• This definition looks only at the shape of
the indifference curve
• This definition is unambiguous because
the definitions are perfectly symmetric
14
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
Quantity of x
15
16
4
Goods
Goods
• To prove this, we can start with the
compensated demand function
• In elasticity terms, we get
eic1  eic2  ...  einc  0
xc(p1,…pn,V)
• Since the negativity of the substitution
effect implies that eiic  0, it must be the
case that
• Applying Euler’s theorem yields
p1 
xic
x c
x c
 p2  i  ...  pn i  0
p1
p2
pn
e
c
ij
0
j i
17
18
Composite Commodity Theorem
Composite Commodities
• 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)
• 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”
19
20
5
• Let p20…pn0 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
• The individual’s budget constraint is
I = p1x1 + p20x2 +…+ pn0xn = p1x1 + y
• If we assume that all of the prices p20…pn0
change by the same factor (t > 0) then the
budget constraint becomes
I = p1x1 + tp20x2 +…+ tpn0xn = p1x1 + ty
– changes in p1 or t induce substitution effects
y = p20x2 + p30x3 +…+ pn0xn
21
p20…pn0
• As long as
move together, we can
confine our examination of demand to
choices between buying x1 and
“everything else”
22
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
• The theorem makes no prediction about
how choices of x2…xn behave
– only focuses on total spending on x2…xn
23
24
6
Example: Composite
Commodity
Example: Composite
Commodity
• Suppose that an individual receives
utility from three goods:
utility  U ( x, y , z )  
1 1 1
 
x y z
• The Lagrangian technique can be used
to derive demand functions
– food (x)
– housing services (y), measured in
hundreds of square feet
– household operations (z), measured by
electricity use
x
I
p x  p x p y  p x pz
z
• Assume a CES utility function
y
I
py  py px  py pz
I
pz  pz px  pz py
25
Example: Composite
Commodity
26
Example: Composite
Commodity
• 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)
• 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
h = 4y + 1z
• The initial quantity of housing is the total
spent on housing (75)
27
28
7
Example: Composite
Commodity
Example: Composite
Commodity
• Now x can be shown as a function of I,
px, and ph
x
• 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
I
py  3 px ph
• If I = 100, px = 1, py = 4, and ph = 1, then
x* = 25 and spending on housing (h*) =
75
29
Example: Composite
Commodity
100
100

7
1 3 4
• Housing purchases would be given by
Ph h*  100 
100 600

7
7
30
Household Production Model
• 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
• 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
• This means that the amount of h that is
consumed can also be computed as
h* = 4y* + 1z* = 150/7
x* 
31
32
8
• Assume that there are three goods that
a person might want to purchase in the
market: x, y, and z
• The individual’s goal is to choose x,y,
and z so as to maximize utility
utility = U(a1,a2)
– these goods provide no direct utility
– these goods can be combined by the
individual to produce either of two homeproduced goods: a1 or a2
subject to the production functions
a1 = f1(x,y,z)
a2 = f2(x,y,z)
and a financial budget constraint
• the technology of this household production
can be represented by a production function
pxx + pyy + pzz = I
33
34
The Linear Attributes Model
• Two important insights from this general
model can be drawn
• 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
– 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
35
a1 = ax1x + ay1y + az1z
a2 = ax2x + ay2y + az2z
36
9
The ray 0x shows the combinations of a1 and a2
available from successively larger amounts of good x
a2
x
y
The ray 0y shows the combinations of
a1 and a2 available from successively
larger amounts of good y
z
• If the individual spends all of his or her
income on good x
x* = I/px
• That will yield
a1* = ax1x* = (ax1I)/px
The ray 0z shows the
combinations of a1 and
a2 available from
successively larger
amounts of good z
a2* = ax2x* = (ax2I)/px
a1
0
37
x* is the combination of a1 and a2 that would be
obtained if all income was spent on x
a2
38
All possible combinations from mixing the
three market goods are represented by
the shaded triangular area x*y*z*
a2
x
x
y* is the combination of a1 and a2 that
y
x*
would be obtained if all income was
spent on y
y
x*
y*
y*
z
z* is the combination of
a1 and a2 that would be
z
obtained if all income was
spent on z
Z*
0
z*
a1
0
39
a1
40
10
A utility-maximizing individual would never
consume positive quantities of all three
goods
a2
x
Individuals with a preference toward
a1 will have indifference curves similar
to U0 and will consume only y and z
y
U1
z
U0
0
Individuals with a preference
toward a0 will have
indifference curves similar
to U1 and will consume only
x and y
a1
41
• 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
42
• 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
• 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
43
44
11
• Focusing only on the substitution
effects from price changes does
provide a symmetric definition
• 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
– two goods are net substitutes if xi c/pj >
0 and net complements if xi c/pj < 0
– because xic /pj = xjc /pi, there is no
ambiguity
– Hicks’ second law of demand shows that
net substitutes are more prevalent
45
46
• 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
47
12
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)
Chapter 9
PRODUCTION FUNCTIONS
q = f(k,l)
1
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
marginal physical product of capital  MPk 
marginal physical product of labor  MPl 
q
 fk
k
q
 fl
l
3
2
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
MPk  2f
 2  fkk  f11  0
k
k
MPl  2f
 2  fll  f22  0
l
l
4
1
Diminishing Marginal
Productivity
Average Physical Product
• 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
• Labor productivity is often measured by
average productivity
APl 
output
q f (k, l )
 
labor input l
l
• Note that APl also depends on the
amount of capital employed
– we need to consider flk which is often > 0
5
A Two-Input Production
Function
Function
• Suppose the production function for
flyswatters can be represented by
q = f(k,l) = 600k
2l2
-
• The marginal productivity function is
MPl = q/l = 120,000l - 3000l2
k 3l3
which diminishes as l increases
• This implies that q has a maximum value:
• To construct MPl and APl, we must
assume a value for k
120,000l - 3000l2 = 0
40l = l2
l = 40
– let k = 10
• The production function becomes
q = 60,000l2 - 1000l3
6
7
• Labor input beyond l = 40 reduces output8
2
Function
Function
• To find average productivity, we hold
k=10 and solve
• In fact, when l = 30, both APl and MPl are
equal to 900,000
APl = q/l = 60,000l - 1000l2
• Thus, when APl is at its maximum, APl
and MPl are equal
• APl reaches its maximum where
APl/l = 60,000 - 2000l = 0
l = 30
9
10
Isoquant Map
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)
• Each isoquant represents a different level
of output
– output rises as we move northeast
k per period
q = 30
f(k,l) = q0
q = 20
11
l per period
12
3
Marginal Rate of Technical
Substitution (RTS)
• The slope of an isoquant shows the rate
at which l can be substituted for k
k per period
- slope = marginal rate of technical
substitution (RTS)
RTS > 0 and is diminishing for
kA
increasing inputs of labor
A
q = 20
l per period
lA
• 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
RTS (l for k ) 
B
kB
Marginal Rate of Technical
Substitution (RTS)
RTS and Marginal Productivities
• Take the total differential of the production
function:
f
f
dq   dl 
 dk  MPl  dl  MPk  dk
l
k
MPl  dl  MPk  dk

q q0
14
• 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
• Along an isoquant dq = 0, so
 dk
dl
q q0
13
lB
RTS (l for k ) 
 dk
dl
MPl
MPk
15
16
4
• Using the fact that dk/dl = -fl/fk along an
isoquant and Young’s theorem (fkl = flk)
• To show that isoquants are convex, we
would like to show that d(RTS)/dl < 0
• Since RTS = fl/fk
dRTS (fk2fll  2fk fl fkl  fl 2fkk )

dl
(fk )3
dRTS d (fl / fk )

dl
dl
dRTS [fk (fll  flk  dk / dl )  fl (fkl  fkk  dk / dl )]

dl
(fk )2
17
• 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
19
• 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
18
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
20
5
A Diminishing RTS
A Diminishing RTS
• Because
2
fll = 1200k -
• Cross differentiation of either of the
marginal productivity functions yields
6k 3l
fkk = 1200l 2 - 6kl 3
fkl = flk = 2400kl - 9k 2l 2
this production function exhibits
diminishing marginal productivities for
sufficiently large values of k and l
which is positive only for kl < 266
– fll and fkk < 0 if kl > 200
21
A Diminishing RTS
22
Returns to Scale
• 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
23
• 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
24
6
Returns to Scale
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
• 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
25
26
Constant Returns to Scale
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
27
• 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
28
k-1
7
• 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
• The production function will be
homothetic
• Geometrically, all of the isoquants are
radial expansions of one another
29
30
Returns to Scale
• Along a ray from the origin (constant k/l),
the RTS will be the same on all isoquants
• Returns to scale can be generalized to a
production function with n inputs
q = f(x1,x2,…,xn)
k per period
The isoquants are equally
spaced as output expands
• 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
q=3
q=2
q=1
l per period
31
32
8
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
%(k / l ) d (k / l ) RTS  ln( k / l )




%RTS dRTS k / l
 ln RTS
 is the ratio of these
k per period
proportional changes
 measures the
RTSA
A
• The value of  will always be positive
because k/l and RTS move in the same
direction
33
RTSB
(k/l)A
(k/l)B
B
curvature of the
isoquant
q = q0
l per period
34
• If  is high, the RTS will not change
much relative to k/l
• Generalizing the elasticity of substitution
to the many-input case raises several
complications
– the isoquant will be relatively flat
– 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
• 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
• Both RTS and k/l will change as we
move from point A to point B
35
36
9
The Linear Production Function
• Suppose that the production function is
The Linear Production Function
Capital and labor are perfect substitutes
q = f(k,l) = ak + bl
• This production function exhibits constant
returns to scale
k per period
RTS is constant as k/l changes
f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l)
=
slope = -b/a
• All isoquants are straight lines
– RTS is constant
–=
q2
q1
q3
l per period
37
Fixed Proportions
38
Fixed Proportions
No substitution between labor and capital
is possible
q = min (ak,bl) a,b > 0
k/l is fixed at b/a
k per period
• Capital and labor must always be used
in a fixed ratio
– the firm will always operate along a ray
where k/l is constant
=0
q3
q3/a
• Because k/l is constant,  = 0
q2
q1
39
l per period
q3/b
40
10
Cobb-Douglas Production
Function
Cobb-Douglas Production
Function
• The Cobb-Douglas production function is
linear in logarithms
q = f(k,l) = Akalb A,a,b > 0
• This production function can exhibit any
returns to scale
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
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
41
42
A Generalized Leontief
Production Function
CES Production Function
q = f(k,l) = [k + l] /   1,   0,  > 0
–  > 1  increasing returns to scale
q = f(k,l) = k + l + 2(kl)0.5
–  < 1  decreasing returns to scale
• Marginal productivities are
• For this production function
 = 1/(1-)
–  = 1  linear production function
–  = -  fixed proportions production function
–  = 0  Cobb-Douglas production function
43
fk = 1 + (k/l)-0.5
fl = 1 + (k/l)0.5
• Thus,
RTS 
fl
1  (k / l )0.5

fk 1  (k / l )0.5
44
11
Technical Progress
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
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
45
Technical Progress
46
Technical Progress
• Differentiating the production function
with respect to time we get
• Dividing by q gives us
dq / dt dA / dt f / k dk f / l dl





q
A
f (k, l ) dt f (k, l ) dt
dq dA
df (k, l )

 f (k, l )  A 
dt
dt
dt
dq / dt dA / dt f
k
dk / dt f
l
dl / dt




 

q
A
k f (k, l )
k
l f (k, l )
l
dq dA q
q  f dk f dl 

 

 
dt
dt A f (k, l)  k dt l dt 
47
48
12
Technical Progress
Technical Progress
• For any variable x, [(dx/dt)/x] is the
proportional growth rate in x
• Since
– denote this by Gx
• Then, we can write the equation in terms
of growth rates
Gq  GA 
f
k
f
l

 Gk  
 Gl
k f (k, l )
l f (k, l )
f
k
q k


  eq,k
k f (k, l ) k q
f
l
q l


  eq,l
l f (k, l ) l q
Gq  GA  eq,kGk  eq,lGl
49
Technical Progress in the
Cobb-Douglas Function
q = A(t)f(k,l) =
50
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-
51
• Taking logarithms and differentiating
with respect to t gives the growth
equation
 ln q  ln q q q / t



 Gq
t
q t
q
52
13
(ln A  t   ln k  (1   ) ln l )
t
 ln k
 ln l
 
 (1   ) 
   Gk  (1   )Gl
t
t
Gq 
• 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
53
54
• The entire production function can be
illustrated by an isoquant map
• Isoquants are usually assumed to be
convex
– the slope of an isoquant is the marginal
rate of technical substitution (RTS)
– they obey the assumption of a diminishing
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
• 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
55
56
14
• The elasticity of substitution ()
provides a measure of how easy it is to
substitute one input for another in
production
• 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
57
– a high  implies nearly straight isoquants
– a low  implies that isoquants are nearly
L-shaped
58
• 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
59
15
Definitions of Costs
• It is important to differentiate between
accounting cost and economic cost
Chapter 10
– the accountant’s view of cost stresses outof-pocket expenses, historical costs,
depreciation, and other bookkeeping
entries
– economists focus more on opportunity cost
COST FUNCTIONS
Copyright ©2005 by South-western, a division of Thomson learning. All rights reserved.
1
2
• Capital Costs
• Labor 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
– 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
• we will use v to denote the rental rate for capital
3
4
1
Economic Cost
• 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
• 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
• part of accounting profits would be considered as
entrepreneurial costs by economists
5
Two Simplifying Assumptions
• There are only two inputs
6
Economic Profits
• Total costs for the firm are given by
– homogeneous labor (l), measured in laborhours
– homogeneous capital (k), measured in
machine-hours
total costs = C = wl + vk
• Total revenue for the firm is given by
total revenue = pq = pf(k,l)
• entrepreneurial costs are included in capital costs
• Inputs are hired in perfectly competitive
markets
– firms are price takers in input markets
7
• Economic profits () are equal to
 = total revenue - total cost
 = pq - wl - vk
 = pf(k,l) - wl - vk
8
2
Cost-Minimizing Input Choices
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
• 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
– for now, we will assume that the firm has
already chosen its output level (q0) and
wants to minimize its costs
9
• Mathematically, we seek to minimize
total costs given q = f(k,l) = q0
• Setting up the Lagrangian:
10
• Dividing the first two conditions we get
w f / l

 RTS (l for k )
v f / k
L = wl + vk + [q0 - f(k,l)]
• The cost-minimizing firm should equate
the RTS for the two inputs to the ratio of
their prices
• First order conditions are
L/l = w - (f/l) = 0
L/k = v - (f/k) = 0
L/ = q0 - f(k,l) = 0
11
12
3
• Cross-multiplying, we get
• Note that this equation’s inverse is also
of interest
fk fl

v w
• For costs to be minimized, the marginal
productivity per dollar spent should be
the same for all inputs
w v
 
fl fk
• The Lagrangian multiplier shows how
much in extra costs would be incurred
by increasing the output constraint
slightly
13
14
Given output q0, we wish to find the least costly
point on the isoquant
The minimum cost of producing q0 is C2
k per period
k per period
C1
C3
Costs are represented by
parallel lines with a slope of w/v
This occurs at the
tangency between the
isoquant and the total cost
curve
C1
C3
C2
C2
C1 < C2 < C3
k*
q0
q0
l per period
15
l*
The optimal choice
is l*, k*
l per period
16
4
Contingent Demand for Inputs
• In Chapter 4, we considered an
individual’s expenditure-minimization
problem
• 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
– 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?
– it is based on the level of the firm’s output
17
The Firm’s Expansion Path
18
• The firm can determine the costminimizing 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 costminimizing choices
The expansion path is the locus of costminimizing tangencies
k per period
The curve shows
how inputs increase
as output increases
E
q1
– called the firm’s expansion path
q0
q00
19
l per period
20
5
Cost Minimization
• The expansion path does not have to be
a straight line
Cobb-Douglas:
– 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
q = kl
• The Lagrangian expression for cost
minimization of producing q0 is
L = vk + wl + (q0 - k  l )
21
22
Cost Minimization
Cost Minimization
• The first-order conditions for a minimum
are
• Dividing the first equation by the second
gives us
L/k = v - k -1l = 0
w k l 1  k

   RTS
v k  1l   l
L/l = w - k l -1 = 0
L/ = q0 - k  l  = 0
• This production function is homothetic
– the RTS depends only on the ratio of the two
inputs
– the expansion path is a straight line
23
24
6
Cost Minimization
Cost Minimization
CES:
• The first-order conditions for a minimum
are
q = (k  + l )/
L/k = v - (/)(k + l)(-)/()k-1 = 0
• The Lagrangian expression for cost
minimization of producing q0 is
L/l = w - (/)(k + l)(-)/()l-1 = 0
L/ = q0 - (k  + l )/ = 0
L = vk + wl + [q0 - (k  + l )/]
25
26
Cost Minimization
Total Cost Function
• Dividing the first equation by the second
gives us
• 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
w  1
 
v k 
 1
1
k 
 
l
1/ 
k 
 
l
C = C(v,w,q)
• This production function is also
homothetic
• As output (q) increases, total costs
increase
27
28
7
Marginal Cost Function
Average Cost Function
• The average cost function (AC) is found
by computing total costs per unit of
output
average cost  AC(v ,w, q ) 
• The marginal cost function (MC) is
found by computing the change in total
costs for a change in output produced
C(v ,w, q )
q
marginal cost  MC(v ,w, q ) 
C(v ,w, q )
q
29
30
Graphical Analysis of
Total Costs
Total Costs
• Suppose that k1 units of capital and l1
units of labor input are required to
produce one unit of output
Total
costs
With constant returns to scale, total costs
are proportional to output
AC = MC
C
C(q=1) = vk1 + wl1
Both AC and
MC will be
constant
• 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)
Output
31
32
8
Total Costs
Total Costs
• Suppose instead that total costs start
out as concave and then becomes
convex as output increases
Total
costs
C
Total costs rise
dramatically as
output increases
after diminishing
returns set in
– 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
Output
33
Total Costs
Average
and
marginal
costs
Shifts in Cost Curves
MC is the slope of the C curve
MC
AC
min AC
34
If AC > MC,
AC must be
falling
If AC < MC,
AC must be
rising
• 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
Output
35
36
9
Some Illustrative Cost
Functions
Functions
• Suppose we have a fixed proportions
technology such that
• Suppose we have a Cobb-Douglas
technology such that
q = f(k,l) = k l 
q = f(k,l) = min(ak,bl)
• Production will occur at the vertex of the
L-shaped isoquants (q = ak = bl)
• Cost minimization requires that
w  k
 
v  l
C(w,v,q) = vk + wl = v(q/a) + w(q/b)
v w 
C(w ,v , q )  a  
a b 
k
37
Functions
l q

 

• Now we can derive total costs as
C(v,w, q )  vk  wl  q1/ Bv  / w  / 
 /  
w
  /  
v
 /  
where
B  (  )  /  / 
• A similar method will yield
k q
1 /  

 

38
Functions
• If we substitute into the production
function and solve for l, we will get
1/  
 w
 l
 v
 /  
which is a constant that involves only
the parameters  and 
w  /  v  /  
39
40
10
Functions
Properties of Cost Functions
• Homogeneity
• Suppose we have a CES technology
such that
q = f(k,l) =
(k 
– cost functions are all homogeneous of
degree one in the input prices
+ l )/
• 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
• To derive the total cost, we would use
the same method and eventually get
C(v,w, q )  vk  wl  q1/  (v  / 1  w  / 1 )( 1) / 
C(v,w, q )  q1/  (v 1  w 1 )1/ 1
41
42
• Nondecreasing in q, v, and w
• Concave in input prices
– cost functions are derived from a costminimization process
• any decline in costs from an increase in one of
the function’s arguments would lead to a
contradiction
43
– 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
44
11
Concavity of Cost Function
At w1, the firm’s costs are C(v,w1,q1)
• Some of these properties carry over to
average and marginal costs
Costs
If the firm continues to
buy the same input mix
as w changes, its cost
function would be Cpseudo
Cpseudo
– homogeneity
– effects of v, w, and q are ambiguous
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)
C(v,w 1,q1)
w
w1
45
Input Substitution
Input Substitution
• Putting this in proportional terms as
• 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
k 
 
l
46
s
(k / l ) w / v  ln( k / l )


(w / v ) k / l  ln( w / v )
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
w 
 
v 
47
48
12
Partial Elasticity of Substitution
• The partial elasticity of substitution
between two inputs (xi and xj) with
prices wi and wj is given by
sij 
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
( x i / x j ) w j / w i
 ln( xi / x j )


(w j / w i ) xi / x j
 ln( w j / w i )
• 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
49
Technical Progress
50
Technical Progress
• Improvements in technology also lower
cost curves
• Suppose that total costs (with constant
returns to scale) are
• 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)
C0 = C0(q,v,w) = qC0(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)
51
52
13
Shifting the Cobb-Douglas
Cost Function
Cost Function
• The Cobb-Douglas cost function is
• If v = 3 and w = 12, the relationship is
C(v,w, q )  vk  wl  q1/ Bv  / w  / 
C(3,12, q )  2q 36  12q
where
B  (  )
  /     /  

• If we assume  =  = 0.5, the total cost
curve is greatly simplified:
– C = 480 to produce q =40
– AC = C/q = 12
– MC = C/q = 12
C(v,w, q )  vk  wl  2qv 0.5w 0.5
53
Cost Function
54
• If v = 3 and w = 27, the relationship is
• Contingent demand functions for all of
the firms inputs can be derived from the
cost function
C(3,27, q )  2q 81  18q
– C = 720 to produce q =40
– AC = C/q = 18
– MC = C/q = 18
– 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
55
56
14
• Suppose we have a fixed proportions
technology
• The cost function is
• For this cost function, contingent
demand functions are quite simple:
v w 
C(w ,v , q )  a  
a b 
k c (v ,w , q ) 
C(v ,w , q ) q

v
a
l c (v ,w , q ) 
C(v ,w , q ) q

w
b
57
58
• For this cost function, the derivation is
messier:
• Suppose we have a Cobb-Douglas
technology
• The cost function is
k c (v ,w , q ) 
C(v,w, q )  vk  wl  q1/ Bv  / w  / 

59
C


 q 1/  Bv  /  w  /  
v   

w 
 q 1/  B 

v 
 /  
60
15
l c (v ,w , q ) 

C


 q 1/  Bv  /  w  /  
w   

w 
 q 1/  B 

v 
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
  /  
• The contingent demands for inputs
depend on both inputs’ prices
q = f(k1,l)
61
62
Short-Run Total Costs
• Short-run total cost for the firm is
• Short-run costs are not minimal costs
for producing the various output levels
SC = vk1 + wl
– 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
• 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)
63
64
16
Short-Run Marginal and
Average Costs
k per period
Because capital is fixed at k1,
the firm cannot equate RTS
with the ratio of input prices
• The short-run average total cost (SAC)
function is
SAC = total costs/total output = SC/q
• The short-run marginal cost (SMC) function
is
k1
q2
SMC = change in SC/change in output = SC/q
q1
q0
l1
l2
l per period
l3
65
66
Relationship between ShortRun and Long-Run Costs
SC (k2)
Total
costs
SC (k1)
Costs
C
SMC (k0)
q0
q1
q2
Output
67
SAC (k0)
MC
AC
The long-run
C curve can
be derived by
varying the
level of k
SC (k0)
SMC (k1)
q0
q1
SAC (k1)
The geometric
relationship
between shortrun and long-run
AC and MC can
also be shown
Output
68
17
• 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
• 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
AC = MC = SAC = SMC
69
70
• The firm’s average cost (AC = C/q)
and marginal cost (MC = C/q) can
be derived directly from the total-cost
function
• 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
– if the total cost curve has a general cubic
shape, the AC and MC curves will be ushaped
• it also shows the relationship between output
level and total cost
• this relationship is summarized by the total
cost function, C(v,w,q)
71
72
18
• All cost curves are drawn on the
assumption that the input prices are
held constant
• Input demand functions can be derived
from the firm’s total-cost function
through partial differentiation
– 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
73
– these input demands will depend on the
quantity of output the firm chooses to
produce
• are called “contingent” demand functions
74
• 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, highercost input combinations than it would
choose if it were possible to vary all
inputs
75
19
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
Chapter 11
PROFIT MAXIMIZATION
– the nature of the contractual relationship
between the providers of inputs to a firm
may be quite complicated
1
2
Modeling Firms’ Behavior
Contractual Relationships
• Some contracts between providers of
inputs may be explicit
• Most economists treat the firm as a
single decision-making unit
– may specify hours, work details, or
compensation
– the decisions are made by a single
dictatorial manager who rationally pursues
some goal
• Other arrangements will be more
implicit in nature
• usually profit-maximization
– decision-making authority or sharing of
tasks
3
4
1
Profit Maximization
Profit Maximization
• A profit-maximizing firm chooses both
its inputs and its outputs with the sole
goal of achieving maximum economic
profits
• 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
– seeks to maximize the difference between
total revenue and total economic costs
5
Output Choice
6
Output Choice
• Total revenue for a firm is given by
• 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
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
d
dR dC
 ' (q ) 

0
dq
dq dq
(q) = R(q) – C(q) = p(q)q –C(q)
dR dC

dq dq
7
8
2
Output Choice
Second-Order Conditions
• To maximize economic profits, the firm
should choose the output for which
marginal revenue is equal to marginal
cost
• MR = MC is only a necessary condition
for profit maximization
• For sufficiency, it is also required that
d 2
d' (q )

0
2
dq q q *
dq q q *
dR dC
MR 

 MC
dq dq
• “marginal” profit must be decreasing at
the optimal level of q
9
Profit Maximization
Marginal Revenue
Profits are maximized when the slope of
the revenue function is equal to the slope of
the cost function
revenues & costs
10
C
R
The second-order
condition prevents us
from mistaking q0 as
a maximum
• 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
marginal revenue  MR(q ) 
q0
q*
dR d [ p(q )  q ]
dp

 pq
dq
dq
dq
output
11
12
3
Marginal Revenue
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
• 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
13
MR = dR/dq = -q/5 + 10
14
Marginal Revenue and
Elasticity
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
• 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
MR = MC
-q/5 + 10 = 4
q = 30
eq,p 
15
dq / q dq p


dp / p dp q
16
4
Elasticity
Elasticity
• This means that
MR  p 

 q dp 
q  dp
1 
  p1 
 p1  


dq
 p dq 
 eq,p 
– 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
eq,p < -1
MR > 0
eq,p = -1
MR = 0
eq,p > -1
MR < 0
• if the demand is infinitely elastic, eq,p = - and
marginal revenue will equal price
17
The Inverse Elasticity Rule
• Because MR = MC when the firm
maximizes profit, we can see that

1 
MC  p1 
 e 
q ,p 

18
p  MC
1

p
eq,p
p  MC
1

p
eq,p
• The gap between price and marginal
cost will fall as the demand curve facing
the firm becomes more elastic
19
• 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
20
5
Average Revenue Curve
Marginal 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
• 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
– shows the revenue per unit yielded by
alternative output choices
21
price
22
As output increases from 0 to q1, total
revenue increases so MR > 0
As output increases beyond q1, total
revenue decreases so MR < 0
• 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
p1
D (average revenue)
output
q1
MR
23
24
6
The Constant Elasticity Case
The Constant Elasticity Case
• We showed (in Chapter 5) that a
demand function of the form
• This means that
R = pq = kq(1+b)/b
q = apb
and
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
MR = dr/dq = [(1+b)/b]kq1/b = [(1+b)/b]p
• This implies that MR is proportional to
price
where k = (1/a)1/b
25
Short-Run Supply by a
Price-Taking Firm
price
Price-Taking Firm
price
SMC
SAC
p* = MR
q*
26
SMC
SAC
p* = MR
SAVC
SAVC
Maximum profit
occurs where
p = SMC
Since p > SAC,
profit > 0
output
q*
27
output
28
7
Price-Taking Firm
Price-Taking Firm
price
price
SMC
SMC
p**
SAC
p* = MR
SAC
p* = MR
SAVC
SAVC
If the price rises
to p**, the firm
will produce q**
and  > 0
q*
q**
If the price falls to
p***, the firm will
produce q***
output
Profit maximization
requires that p =
SMC and that SMC
is upward-sloping
p***
q***
q*
output
<0
29
Price-Taking Firm
30
Price-Taking Firm
• 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
• 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 profitmaximizing decision is to shut down and
produce no output
• the firm will produce no output if p < SAVC
31
32
8
Price-Taking Firm
price
Short-Run Supply
• Suppose that the firm’s short-run total cost
curve is
SMC
SC(v,w,q,k) = vk1 + wq1/k1-/
SAC
SAVC
The firm’s short-run
supply curve is the
SMC curve that is
above SAVC
where k1 is the level of capital held
constant in the short run
• Short-run marginal cost is
SMC(v ,w, q, k1 ) 
output
SC w (1 ) /   / 
 q
k1
q

33
Short-Run Supply
Short-Run Supply
• The price-taking firm will maximize profit
where p = SMC
SMC 
• SAVC < SMC for all values of  < 1
• Therefore, quantity supplied will be
– there is no price low enough that the firm will
want to shut down
 /(1 )
  /(1 )
1
k
p
• To find the firm’s shut-down price, we
need to solve for SAVC
SVC = wq1/k1-/
SAVC = SVC/q = wq(1-)/k1-/
w (1 ) /   / 
q
k1
p

w 
q   

34
 /(1 )
35
36
9
Profit Functions
Profit Functions
• A firm’s economic profit can be
expressed as a function of inputs
• A firm’s profit function shows its
maximal profits as a function of the
prices that the firm faces
 = pq - C(q) = pf(k,l) - vk - wl
• Only the variables k and l are under the
firm’s control
( p,v,w )  Max (k, l )  Max[ pf (k, l )  vk  wl ]
k ,l
k ,l
– the firm chooses levels of these inputs in
order to maximize profits
• treats p, v, and w as fixed parameters in its
decisions
37
Properties of the Profit
Function
38
Function
• Homogeneity
• Nondecreasing in output price
– 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
39
– 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
40
10
Function
Function
• Nonincreasing in input prices
• Convex in output 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
– 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
( p1,v ,w )  ( p2 ,v ,w )
 p  p2

  1
,v ,w 
2
 2

41
Producer Surplus in the
Short Run
Envelope Results
• We can apply the envelope theorem to
see how profits respond to changes in
output and input prices
• Because the profit function is
nondecreasing in output prices, we know
that if p2 > p1
( p,v ,w )
 q( p,v ,w )
p
(p2,…)  (p1,…)
• The welfare gain to the firm of this price
increase can be measured by
( p,v ,w )
 k ( p,v ,w )
v
( p,v ,w )
 l ( p,v ,w )
w
42
welfare gain = (p2,…) - (p1,…)
43
44
11
Short Run
Short Run
SMC
price
If the market price
is p1, the firm will
produce q1
p2
p1
If the market price
rises to p2, the firm
will produce q2
q1
output
q2
SMC
price
The firm’s profits
rise by the shaded
area
p2
p1
q1
q2
output
45
Short Run
Short Run
• Mathematically, we can use the
envelope theorem results
p2
p2
p1
p1
46
• 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
welfare gain   q( p)dp   ( / p)dp
 ( p2 ,...)  ( p1,...)
47
48
12
Short Run
Short Run
SMC
price
Suppose that the
firm’s shutdown
price is p0
p1
• The extra profits available from facing a
price of p1 are defined to be producer
surplus
p1
producer surplus  ( p1,...)  ( p0 ,...)   q( p)dp
p
0
p0
output
q1
49
Short Run
Short Run
SMC
price
Producer surplus
at a market price
of p1 is the
shaded area
p1
50
p
• 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
0
q1
output
51
52
13
Short Run
Profit Maximization and
Input Demand
• 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
53
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)
54
Input Demand
• The first-order conditions for a maximum
are
/k = p[f/k] – v = 0
/l = p[f/l] – w = 0
• These first-order conditions for profit
maximization also imply cost
minimization
– they imply that RTS = w/v
• 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
55
56
14
Input Demand
Input Demand Functions
• In principle, the first-order conditions can
be solved to yield input demand functions
• To ensure a true maximum, secondorder conditions require that
Capital Demand = k(p,v,w)
Labor Demand = l(p,v,w)
kk = fkk < 0
ll = fll < 0
• These demand functions are
unconditional
kk ll - kl2 = fkkfll – fkl2 > 0
– capital and labor must exhibit sufficiently
diminishing marginal productivities so that
marginal costs rise as output expands
– they implicitly allow the firm to adjust its
output to changing prices
57
Single-Input Case
Single-Input Case
• This reduces to
• We expect l/w  0
– diminishing marginal productivity of labor
1  p  fll 
• The first order condition for profit
maximization was
l
w
• Solving further, we get
l
1

w p  fll
/l = p[f/l] – w = 0
• Taking the total differential, we get
dw  p 
58
• Since fll  0, l/w  0
fl l

 dw
l w
59
60
15
Two-Input Case
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 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
• when k changes, the entire fl function changes
• But, even in this case, l/w  0
61
62
Substitution Effect
k per period
Output Effect
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
A decline in w will lower the firm’s MC
Price
MC
MC’
Consequently, the firm
will choose a new level
of output that is higher
P
q0
l per period
63
q0 q1
Output
64
16
Output Effect
Cross-Price Effects
Output will rise to q1
k per period
• No definite statement can be made
about how capital usage responds to a
wage change
Thus, the output effect
also implies a negative
relationship between l
and w
– 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
q1
q0
l per period
65
Substitution and Output
Effects
66
Substitution and Output
Effects
• We have two concepts of demand for
any input
• Differentiation with respect to w yields
– the conditional demand for labor,
– the unconditional demand for labor, l(p,v,w)
lc(v,w,q)
• At the profit-maximizing level of output
l( p,v ,w ) l c (v ,w, q ) l c (v ,w, q ) q



w
w
q
w
substitution
effect
lc(v,w,q) = l(p,v,w)
output
effect
total effect
67
68
17
• 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
• 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
69
70
• Marginal revenue and the price
elasticity of demand are related by the
formula
• 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)

1 
MR  p1 
 e 
q ,p 

– if price falls below minimum AVC, the
firm’s profit-maximizing choice is to shut
down and produce nothing
71
72
18
• The firm’s reactions to the various
prices it faces can be judged through
use of its profit function
• 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
– shows maximum profits for the firm given
the price of its output, the prices of its
inputs, and the production technology
73
74
• Short-run changes in market price
result in changes in the firm’s short-run
profitability
• Profit maximization provides a theory
of the firm’s derived demand for inputs
– 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
75
– 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
76
19
Market Demand
• Assume that there are only two goods
(x and y)
Chapter 12
– An individual’s demand for x is
THE PARTIAL EQUILIBRIUM
COMPETITIVE MODEL
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
Market demand for X   xi ( px , py , Ii )
i 1
1
2
Market Demand
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
To derive the market demand curve, we sum the
quantities demanded at every price
px
px
Individual 1’s
demand curve
Individual 2’s
demand curve
px
Market demand
curve
px*
x1
x1 *
X
x2
x
x2 *
x
X*
x
x1* + x2* = X*
3
4
1
Shifts in the Market
Demand Curve
Shifts in Market Demand
• Suppose that individual 1’s demand for
oranges is given by
x1 = 10 – 2px + 0.1I1 + 0.5py
• 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)
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
5
6
• If py rises to 6, the market demand curve
shifts outward to
X = 27 – 3px + 4 + 1 + 6 = 38 – 3px
• 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
– 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 + 4 + 1 + 4 = 36 – 3px
X = 27 – 3px + 3 + 1.5 + 4 = 35.5 – 3px
– note that X is a normal good for both buyers
7
8
2
Generalizations
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
• The market demand function for xi is the
sum of each individual’s demand for that
good
m
X i   xij ( p1,..., pn , I j )
j 1
• The market demand function depends on
the prices of all goods and the incomes
and preferences of all buyers
xij = xij(p1,…,pn, Ij)
9
Elasticity of Market Demand
• The price elasticity of market demand is
measured by
eQ,P 
10
Elasticity of Market Demand
• The cross-price elasticity of market
demand is measured by
QD (P, P ' , I ) P

P
QD
eQ,P 
QD (P, P ' , I ) P '

P '
QD
• The income elasticity of market demand is
measured by
• Market demand is characterized by
whether demand is elastic (eQ,P <-1) or
inelastic (0> eQ,P > -1)
eQ,I 
11
QD (P, P ' , I ) I

I
QD
12
3
Timing of the Supply Response
• In the analysis of competitive pricing, the
time period under consideration is
important
– very short run
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
• no supply response (quantity supplied is fixed)
• price will adjust to clear the market
– short run
• existing firms can alter their quantity supplied, but
no new firms can enter the industry
– the supply curve is a vertical line
– long run
• new firms may enter an industry
13
Pricing in the Very Short Run
Price
S
When quantity is fixed in the
very short run, price will rise
from P1 to P2 when the demand
rises from D to D’
P2
P1
14
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
D’
D
Q*
Quantity
15
16
4
Perfect Competition
Short-Run Market Supply
• A perfectly competitive industry is one
that obeys the following assumptions:
• The quantity of output supplied to the
entire market in the short run is the sum
of the quantities supplied by each firm
– 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
– 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
• its actions have no effect on the market price
– information is perfect
– transactions are costless
17
18
Short-Run Market Supply Curve
To derive the market supply curve, we sum the
quantities supplied at every price
P
Firm A’s
supply curve
P
sB
sA
P
Firm B’s
supply curve
Market supply
curve
S
Short-Run Market Supply
Function
• The short-run market supply function
shows total quantity supplied by each
firm to a market
n
Qs (P,v ,w )   qi (P,v ,w )
P1
i 1
q1A
quantity
q1B
quantity
Q1
Quantity
q1A + q1B = Q1
19
• Firms are assumed to face the same
market price and the same prices for
inputs
20
5
Short-Run Supply Elasticity
• The short-run supply elasticity describes
the responsiveness of quantity supplied
to changes in market price
eS,P 
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
% change in Q supplied QS P


% change in P
P QS
• Because price and quantity supplied are
positively related, eS,P > 0
(i = 1,2,…,100)
• This means that the market supply
function is given by
100
100
10P 1000P

3
i 1 3
Qs   qi  
i 1
21
A Short-Run Supply Function
22
Equilibrium Price
Determination
• In this case, computation of the
elasticity of supply shows that it is unit
elastic
• An equilibrium price is one at which
quantity demanded is equal to quantity
supplied
QS (P,v ,w ) P 1000
P



1
P
QS
3 1000P / 3
– neither suppliers nor demanders have an
incentive to alter their economic decisions
eS,P 
• An equilibrium price (P*) solves the
equation:
QD (P *, P ' , I )  QS (P *,v,w )
23
24
6
Equilibrium Price
Determination
Equilibrium Price
Determination
• The equilibrium price depends on many
exogenous factors
The interaction between
market demand and market
supply determines the
equilibrium price
Price
S
– changes in any of these factors will likely
result in a new equilibrium price
P1
D
Quantity
Q1
25
26
Market Reaction to a
Shift in Demand
Market Reaction to a
Shift in Demand
If many buyers experience
an increase in their demands,
the market demand curve
will shift to the right
Price
S
If the market price rises,
firms will increase their
level of output
Price
SMC
SAC
P2
P1
D’
Equilibrium price and
equilibrium quantity will
both rise
P2
P1
This is the short-run
supply response to an
increase in market price
D
Q1
Q2
Quantity
q1
27
q2
Quantity
28
7
Shifts in Supply and
Demand Curves
Shifts in Supply and
Demand Curves
• Demand curves shift because
• 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
– incomes change
– prices of substitutes or complements change
– preferences change
• Supply curves shift because
– input prices change
– technology changes
– number of producers change
29
30
Shifts in Supply
Small increase in price,
large drop in quantity
Price
Shifts in Demand
Large increase in price,
small drop in quantity
Price
S’
Large increase in price,
small rise in quantity
Price
S’
S
Small increase in price,
large rise in quantity
Price
S
S
S
P’
P’
P
P’
P’
P
P
P
D’
D
D
Q’
Q
Elastic Demand
Quantity
Q’ Q
Inelastic Demand
D’
D
Quantity
31
Q
Q’
Elastic Supply
D
Quantity
Q Q’
Inelastic Supply
Quantity
32
8
Changing Short-Run Equilibria
• Suppose that the market demand for
luxury beach towels is
• Suppose instead that the demand for
luxury towels rises to
QD = 10,000 – 500P
QD = 12,500 – 500P
• Solving for the new equilibrium, we find
and the short-run market supply is
QS = 1,000P/3
P* = $15
Q* = 5,000
• Setting these equal, we find
• Equilibrium price and quantity both rise
P* = $12
Q* = 4,000
33
• Suppose that the wage of towel cutters
rises so that the short-run market supply
becomes
34
Mathematical Model of
Supply and Demand
• Suppose that the demand function is
represented by
QD = D(P,)
QS = 800P/3
• Solving for the new equilibrium, we find
–  is a parameter that shifts the demand curve
• D/ = D can have any sign
P* = $13.04
Q* = 3,480
– D/P = DP < 0
• Equilibrium price rises and quantity falls
35
36
9
Supply and Demand
Supply and Demand
• The supply relationship can be shown as
QS = S(P,)
–  is a parameter that shifts the supply curve
• S/ = S can have any sign
• To analyze the comparative statics of
this model, we need to use the total
differentials of the supply and demand
functions:
dQD = DPdP + Dd
dQS = SPdP + Sd
– S/P = SP > 0
• Maintenance of equilibrium requires that
• Equilibrium requires that QD = QS
dQD = dQS
37
38
Supply and Demand
Supply and Demand
• Suppose that the demand parameter ()
changed while  remains constant
• The equilibrium condition requires that
• We can convert our analysis to elasticities
eP , 
DPdP + Dd = SPdP
D
P

 SP  DP
• Because SP - DP > 0, P/ will have the
same sign as D
39
eP , 
D
P 

 

 P SP  DP P
D

Q
P
(SP  DP ) 
Q

eQ,
eS,P  eQ,P
40
10
Long-Run Analysis
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
• 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
41
42
Long-Run Competitive
Equilibrium
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
43
• 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
44
11
Long-Run Competitive
Equilibrium
Long-Run Equilibrium:
Constant-Cost Case
• We will assume that all firms in an
industry have identical cost curves
• Assume that the entry of new firms in an
industry has no effect on the cost of
inputs
– no firm controls any special resources or
technology
– no matter how many firms enter or leave
an industry, a firm’s cost curves will remain
unchanged
• The equilibrium long-run position
requires that each firm earn zero
economic profit
• This is referred to as a constant-cost
industry
45
46
Constant-Cost Case
Constant-Cost Case
Suppose that market demand rises to D’
This is a long-run equilibrium for this industry
Price
SMC
P = MC = AC
Price
MC
Price
SMC
Price
MC
Market price rises to P2
S
S
AC
AC
P2
P1
P1
D’
D
q1
A Typical Firm
Quantity
Q1
Total Market
D
47
Quantity
q1
A Typical Firm
Quantity
Q1 Q2
48
Quantity
Total Market
12
Constant-Cost Case
Constant-Cost Case
In the short run, each firm increases output to q2
Price
SMC
Price
MC
In the long run, new firms will enter the industry
Economic profit > 0
Economic profit will return to 0
Price
SMC
Price
MC
S
S
AC
S’
AC
P2
P1
P1
D’
D’
D
q1
q2
Quantity
A Typical Firm
D
49
Quantity
Q1 Q2
Total Market
The long-run supply curve will be a horizontal line
(infinitely elastic) at p1
SMC
Q1
Q3
50
Quantity
Total Market
• Suppose that the total cost curve for a
typical firm in the bicycle industry is
Price
MC
Quantity
Infinitely Elastic Long-Run
Supply
Constant-Cost Case
Price
q1
A Typical Firm
S
S’
TC = q3 – 20q2 + 100q + 8,000
AC
• Demand for bicycles is given by
P1
LS
D’
QD = 2,500 – 3P
D
q1
A Typical Firm
Quantity
Q1
Q3
51
Quantity
52
Total Market
13
Infinitely Elastic Long-Run
Supply
Shape of the Long-Run
Supply Curve
• To find the long-run equilibrium for this
market, we must find the low point on the
typical firm’s average cost curve
• The zero-profit condition is the factor that
determines the shape of the long-run cost
curve
– where AC = MC
AC = q2 – 20q + 100 + 8,000/q
MC = 3q2 – 40q + 100
– this occurs where q = 20
– 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
• If q = 20, AC = MC = $500
– this will be the long-run equilibrium price
53
54
Increasing-Cost Industry
• The entry of new firms may cause the
average costs of all firms to rise
Suppose that we are in long-run equilibrium in this industry
Price
SMC
– 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
P = MC = AC
Price
MC
S
AC
P1
D
55
q1
Quantity
A Typical Firm (before entry)
Q1
56
Quantity
Total Market
14
Positive profits attract new firms and supply shifts out
Market price rises to P2 and firms increase output to q2
Price
SMC
MC
Entry of firms causes costs for each firm to rise
SMC’
Price
Price
MC’
Price
S
S
S’
AC’
AC
P2
P3
P1
P1
D’
D’
D
q1
q2
Quantity
D
57
Quantity
Q1 Q2
Total Market
Price
MC’
S
LS
p3
D’
D
A Typical Firm (after entry)
Quantity
Q3
58
Quantity
• permits the development of more efficient
transportation and communications networks
p1
Q1
Q3
– new firms may attract a larger pool of
trained labor
– entry of new firms may provide a “critical
mass” of industrialization
S’
AC’
q3
Q1
Total Market
• The entry of new firms may cause the
average costs of all firms to fall
The long-run supply curve will be upward-sloping
SMC’
Quantity
Decreasing-Cost Industry
Price
q3
A Typical Firm (after entry)
59
Quantity
60
Total Market
15
Decreasing-Cost Case
Suppose that we are in long-run equilibrium in this industry
Price
SMC
P = MC = AC
Price
MC
Market price rises to P2 and firms increase output to q2
Price
SMC
Price
MC
S
S
AC
AC
P2
P1
P1
D
q1
Quantity
D’
D
q1
q2
Quantity
61
Quantity
Q1
Total Market
62
Quantity
Q1 Q2
Total Market
Positive profits attract new firms and supply shifts out
The long-run industry supply curve will be downward-sloping
Entry of firms causes costs for each firm to fall
Price
SMC’
Price
Price
SMC’
S
MC’
Price
S
MC’
S’
S’
AC’
AC’
P1
P1
P3
D’
D
q1 q3
Quantity
Q1
Total Market
Q3
63
Quantity
P3
D
q1 q3
Quantity
Q1
D’
LS
64
Q3 Quantity
Total Market
16
Classification of Long-Run
Supply Curves
Classification of Long-Run
Supply Curves
• Constant Cost
• Decreasing Cost
– entry does not affect input costs
– the long-run supply curve is horizontal at
the long-run equilibrium price
– entry reduces input costs
– the long-run supply curve is negatively
sloped
• Increasing Cost
– entry increases inputs costs
– the long-run supply curve is positively
sloped
65
Long-Run Elasticity of Supply
• The long-run elasticity of supply (eLS,P)
records the proportionate change in longrun industry output to a proportionate
change in price
eLS ,P 
% change in Q QLS P


% change in P
P QLS
• eLS,P can be positive or negative
– the sign depends on whether the industry
exhibits increasing or decreasing costs
67
66
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
68
17
• 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*
• A shift in demand that changes the
equilibrium industry output to Q1 will
change the equilibrium number of firms to
69
n1 = Q1/q*
• The change in the number of firms is
Q1  Q0
q*
– completely determined by the extent of the
demand shift and the optimal output level for
70
the typical firm
n1  n0 
• The effect of a change in input prices is
more complicated
• The optimal level of output for each firm
may also be affected
• Therefore, the change in the number of
firms becomes
– we need to know how much minimum
average cost is affected
– we need to know how an increase in longrun equilibrium price will affect quantity
demanded
Q Q
n1  n0  *1  *0
q1 q0
71
72
18
Rising Input Costs and
Industry Structure
Rising Input Costs and
Industry Structure
• Suppose that the total cost curve for a
typical firm in the bicycle industry is
• At q = 22, MC = AC = $672 so the longrun equilibrium price will be $672
• If demand can be represented by
TC = q3 – 20q2 + 100q + 8,000
QD = 2,500 – 3P
and then rises to
then QD = 484
• This means that the industry will have
22 firms (484  22)
TC = q3 – 20q2 + 100q + 11,616
• The optimal scale of each firm rises
from 20 to 22 (where MC = AC)
73
Long Run
74
Long Run
• In the long-run, all profits are zero and
there are no fixed costs
• Short-run producer surplus represents
the return to a firm’s owners in excess
of what would be earned if output was
zero
– owners are indifferent about whether they
are in a particular market
– the sum of short-run profits and fixed costs
75
• they could earn identical returns on their
investments elsewhere
• Suppliers of inputs may not be indifferent
about the level of production in an
industry
76
19
Long Run
Long Run
• In the constant-cost case, input prices
are assumed to be independent of the
level of production
• 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
– 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
77
off
Ricardian Rent
– the area above the long-run supply curve
and below the market price
• this would equal zero in the case of constant
costs
78
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)
79
• 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
80
20
Ricardian Rent
Ricardian Rent
The owners of low-cost firms will earn positive profits
Price
MC
Price
The owners of the marginal firm will earn zero profit
Price
Price
MC
AC
AC
S
S
P*
P*
D
Quantity
q*
Low-Cost Firm
Q*
D
81
Quantity
q*
Total Market
Q*
Quantity
Marginal Firm
82
Quantity
Total Market
Ricardian Rent
Ricardian Rent
• Firms with higher costs (than the
marginal firm) will stay out of the market
Each point on the supply curve represents minimum
average cost for some firm
Price
For each firm, P – AC represents
profit per unit of output
– would incur losses at a price of P*
• Profits earned by intramarginal firms
can persist in the long run
Total long-run profits can be
computed by summing over all
units of output
S
– they reflect a return to a unique resource
P*
• The sum of these long-run profits
constitutes long-run producer surplus
D
Q*
83
Total Market
Quantity
84
21
Ricardian Rent
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
• It is the scarcity of low-cost inputs that
creates the possibility of Ricardian rent
• In industries with upward-sloping longrun supply curves, increases in output
not only raise firms’ costs but also
generate factor rents for inputs
– reflect the present value of all future profits
85
86
• 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
87
• 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
88
22
• 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
• 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
89
90
• 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
91
• 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
92
23
Perfectly Competitive
Price System
• We will assume that all markets are
perfectly competitive
Chapter 13
– there is some large number of homogeneous
goods in the economy
GENERAL EQUILIBRIUM AND
WELFARE
• both consumption goods and factors of
production
1
– each good has an equilibrium price
– there are no transaction or transportation
costs
– individuals and firms have perfect information
2
Assumptions of Perfect
Competition
Law of One Price
• A homogeneous good trades at the
same price no matter who buys it or
who sells it
• 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
– 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
• There are a large number of firms
producing each good
– each firm takes all prices as given and
attempts to maximize profits
• these actions would tend to equalize the price
of the good
3
4
1
Edgeworth Box Diagram
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
5
Labor for y
Capital
Oy in y
production
Capital for y
Total Capital
Ox
Labor in x production
Total Labor
6
• 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
Capital
for x

A
Capital
in x
production
– any point in the box represents a fully
employed allocation of the available
resources to x and y
Labor in y production
Labor for x
• 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
– they illustrate the actual production outcomes
7
8
2
Point A is inefficient because, by moving along y1, we can increase
x from x1 to x2 while holding y constant
Oy
• 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
Total Capital
y1
y2
• The efficient allocations will occur where
the isoquants are tangent to one another
A
Ox
9

x2
x1
10
Total Labor
We could also increase y from y1 to y2 while holding x constant
by moving along x1
Oy
At each efficient point, the RTS (of k for l) is equal in both
x and y production
Oy
y1
p4
y2
A

y2
Total Capital
Total Capital
y1
x2
p3
x4
y3
p2
y4
x1
x3
p1
x2
x1
Ox
Total Labor
11
Ox
Total Labor
12
3
Production Possibility Frontier
• The locus of efficient points shows the
maximum output of y that can be
produced for any level of x
Production Possibility Frontier
Each efficient point of production
becomes a point on the production
possibility frontier
Quantity of y
Ox
y4
y3
– we can use this information to construct a
production possibility frontier
p1
p2
• shows the alternative outputs of x and y that
can be produced with the fixed capital and
labor inputs that are employed efficiently
p4
y1
x1
13
The negative of the slope of
the production possibility
frontier is the rate of product
transformation (RPT)
p3
y2
x2
x3
x4 Oy
Quantity of x
14
Rate of Product Transformation
Rate of Product Transformation
• The rate of product transformation (RPT)
between two outputs is the negative of
the slope of the production possibility
frontier
• The rate of product transformation shows
how x can be technically traded for y
while continuing to keep the available
productive inputs efficiently employed
RPT (of x for y )   slope of production
RPT (of x for y )  
dy
(along OxOy )
dx
15
16
4
Shape of the Production
Possibility Frontier
• The production possibility frontier shown
earlier exhibited an increasing RPT
• Suppose that the costs of any output
combination are C(x,y)
– along the production possibility frontier,
C(x,y) is constant
– this concave shape will characterize most
production situations
• We can write the total differential of the
cost function as
• RPT is equal to the ratio of MCx to MCy
dC 
C
C
 dx 
 dy  0
x
y
17
• Rewriting, we get
RPT  
18
• As production of x rises and production
of y falls, the ratio of MCx to MCy rises
dy
C / x MCx
(along OxOy ) 

dx
C / y MCy
– 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
• The RPT is a measure of the relative
marginal costs of the two goods
– this could also occur if some inputs were
more suited for x production than for y
production
19
20
5
Opportunity Cost
• 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
• 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
21
22
Concavity of the Production
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
• Suppose that the production of x and y
depends only on labor and the production
functions are
x  f (lx )  lx0.5
y  f (ly )  ly0.5
• If labor supply is fixed at 100, then
lx + ly = 100
– this opportunity cost rises as more x is
produced
• The production possibility frontier is
23
x2 + y2 = 100
for x,y  0
24
6
Concavity of the Production
• The RPT can be calculated by taking the
total differential:
 dy  ( 2x ) x


dx
2y
y
2xdx  2ydy  0 or RPT 
• The slope of the production possibility
frontier increases as x output increases
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
– the frontier is concave
25
26
Determination of
Equilibrium Prices
Determination of
Equilibrium Prices
If the prices of x and y are px and py,
society’s budget constraint is C
Quantity of y
C
There is excess demand for x and
excess supply of y
Quantity of y
C
Output will be x1, y1
y1
The price of x will rise and
the price of y will fall
y1
Individuals will demand x1’, y1’
y1 ’
excess
supply
y1 ’
U3
U2
U1
x1
x1 ’
U3
C
U2
 px
slope 
py
Quantity of x
U1
27
x1 ’
x
1
C
slope 
 px
py
Quantity of x
28
excess demand
7
Determination of
Equilibrium Prices
The equilibrium prices will
be px* and py*
Quantity of y C*
C
The equilibrium output will
be x1* and y1*
y1
y1 *
y1 ’
U3
U2
U1
C*
x
x1 *
x1 ’
1
slope 
 px*
py*
– we would move in a clockwise direction
along the production possibility frontier
C
slope 
 px
py
Quantity of x
• 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
29
• Technical progress in the production of
good x will shift the production
possibility curve outward
30
Production of x
Technical progress in the production
of x will shift the production possibility
curve out
Quantity of y
– this will lower the relative price of x
– more x will be consumed
The relative price of x will fall
More x will be consumed
• if x is a normal good
U3
– the effect on y is ambiguous
U2
U1
31
x1 *
x2 *
Quantity of x
32
8
General Equilibrium Pricing
• Suppose that the production possibility
frontier can be represented by
• Profit-maximizing firms will equate RPT
and the ratio of px /py
x 2 + y 2 = 100
RPT 
• Suppose also that the community’s
preferences can be represented by
x px

y py
• Utility maximization requires that
U(x,y) = x0.5y0.5
MRS 
y px

x py
33
• Equilibrium requires that firms and
individuals face the same price ratio
RPT 
34
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
x px y

  MRS
y py x
or
x* = y*
– what effect would the elimination of these
tariffs have on factor prices?
35
36
9
Quantity of
manufactured
goods (y)
If the corn laws completely prevented
trade, output would be x0 and y0
The equilibrium prices will be
px* and py*
y0
Quantity of
manufactured
goods (y)
Removal of the corn laws will change
the prices to px’ and py’
Output will be x1’ and y1’
y1 ’
Individuals will demand x1 and y1
y0
y1
U2
U2
U1
slope 
U1
slope 
 px*
py*
Quantity of Grain (x)
x0
x1 ’
x0
x1
 px '
py '
37
Quantity of
manufactured
goods (y)
exports
of
goods
Grain imports will be x1 – x1’
These imports will be financed by
the export of manufactured goods
equal to y1’ – y1
y1 ’
y0
y1
U2
U1
slope 
x1 ’
x0
imports of grain
x1
38
 px '
py '
• 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
39
40
10
A repeal of the corn laws would result in a movement from p3 to
p1 where more y and less x is produced
Oy
y1
p4
Total Capital
y2
p3
– the relative price of capital will fall
– the relative price of labor will rise
x4
y3
p2
y4
x3
p1
x1
Ox
x2
Total Labor
• 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
41
• The repeal of the corn laws will be
harmful to capital owners and helpful to
laborers
42
Political Support for
Trade Policies
Existence of General
Equilibrium Prices
• 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
• Beginning with 19th century investigations
by Leon Walras, economists have
examined whether there exists a set of
prices that equilibrates all markets
simultaneously
– free trade policies will result in rising relative
wages for skilled workers and in falling
relative wages for unskilled workers
43
– if this set of prices exists, how can it be
found?
44
11
Equilibrium Prices
Equilibrium Prices
• Suppose that there are n goods in fixed
supply in this economy
• We will write this demand function as
dependent on the whole set of prices (P)
– 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
Di (P)
• Walras’ problem: Does there exist an
equilibrium set of prices such that
Di (P*) = Si
for all values of i ?
45
46
Excess Demand Functions
Excess Demand Functions
• The excess demand function for any
good i at any set of prices (P) is defined
to be
• Demand functions are homogeneous of
degree zero
EDi (P) = Di (P) – Si
• This means that the equilibrium
condition can be rewritten as
– this implies that we can only establish
equilibrium relative prices in a Walrasiantype model
• Walras also assumed that demand
functions are continuous
EDi (P*) = Di (P*) – Si = 0
– small changes in price lead to small changes
in quantity demanded
47
48
12
Walras’ Law
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
• 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
n
 P  ED (P )  0
i 1
i
i
49
50
Walras’ Proof of the Existence
of Equilibrium Prices
• The market equilibrium conditions
provide (n-1) independent equations in
(n-1) unknown relative prices
• 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’)
– 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
51
– in changing p2 from its initial position to p2’,
the price calculated for good 1 does not
52
need to remain an equilibrium price
13
• Using the provisional prices p1’ and p2’,
solve for p3’
• 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
– 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
53
Brouwer’s Fixed-Point Theorem
54
f (X)
• 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*
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]
1
Any continuous function must
cross the 45 line
f (X*)
This point of crossing is a
“fixed point” because f maps
this point (X*) into itself

45
0
55
X*
1
x
56
14
• A mapping is a rule that associates the
points in one set with points in another set
• 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
– 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 ndimensional 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
57
– closed (they contain their boundaries)
– bounded (none of their dimensions is infinitely
large)
– convex (they have no “holes” in them)
58
Proof of the Existence of
Equilibrium Prices
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
• These new prices will retain their original
relative values and will sum to 1
pi ' 
pi
• These new prices will sum to 1
n
p ' 1
i 1
n
p
i
i 1
pi ' pi

pj ' pj
59
i
60
15
Equilibrium Prices
Free Goods
• 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
• 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”
61
62
Mapping the Set of Prices
Into Itself
Free Goods
• The equilibrium conditions are
• In order to achieve equilibrium, prices of
goods in excess demand should be
raised, whereas those in excess supply
should have their prices lowered
EDi (P*) = 0 for pi* > 0
EDi (P*)  0 for pi* = 0
• Note that this set of equilibrium prices
continues to obey Walras’ law
63
64
16
Into Itself
Into Itself
• Two problems exist with this mapping
• First, nothing ensures that the prices will
be nonnegative
• We define the mapping F(P) for any
normalized set of prices (P), such that
the ith component of F(P) is given by
– the mapping must be redefined to be
F i(P) = pi + EDi (P)
F i(P) = Max [pi + EDi (P),0]
• The mapping performs the necessary
task of appropriately raising or lowering
prices
– the new prices defined by the mapping must
be positive or zero
65
66
Application of Brouwer’s
Theorem
Into Itself
• Second, the recalculated prices are not
necessarily normalized
• Thus, F satisfies the conditions of the
Brouwer fixed-point theorem
– they will not sum to 1
– it will be simple to normalize such that
– it is a continuous mapping of the set S into
itself
• There exists a point (P*) that is mapped
into itself
• For this point,
n
 F (P )  1
i
i 1
– we will assume that this normalization has
been done
67
pi* = Max [pi* + EDi (P*),0]
for all i
68
17
Application of Brouwer’s
Theorem
A General Equilibrium with
Three Goods
• This says that P* is an equilibrium set of
prices
• The economy of Oz is composed only of
three precious metals: (1) silver, (2)
gold, and (3) platinum
– for pi* > 0,
pi* = pi* + EDi (P*)
EDi (P*) = 0
– For pi* = 0,
– there are 10 (thousand) ounces of each
metal available
• The demands for gold and platinum are
pi* + EDi (P*)  0
EDi (P*)  0
D2  2
69
Three Goods

D3  
p
p2
 2 3  18
p1
p1
70
Three Goods
• This system of simultaneous equations
can be solved as
• Equilibrium in the gold and platinum
markets requires that demand equal
supply in both markets simultaneously
2
p2 p3

 11
p1 p1
p2/p1 = 2
p3/p1 = 3
• In equilibrium:
p2 p3

 11  10
p1 p1
– 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
p
p2
 2 3  18  10
p1
p1
71
72
18
Three Goods
Smith’s Invisible Hand
Hypothesis
• Because Walras’ law must hold, we know
• 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
p1ED1 = – p2ED2 – p3ED3
• Substituting the excess demand functions
for gold and silver and substituting, we get
p1ED1  2
pp
p2
p22 p2 p3

 p2  2 3  2 3  8 p3
p1
p1
p1
p1
ED1  2
p2 p
p
p22
 2 32  2  8 3
2
p1
p1 p1
p1
73
74
Smith’s Invisible Hand
Hypothesis
Pareto Efficiency
• 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
• 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
75
76
19
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
77
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
78
Single Firm
Single Firm
• 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
• 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:
x = f (kx, lx)
L = f (kx, lx) + [y’ – g (k’ - kx, l’ - lx)]
L/kx = fk + gk = 0
• If we assume full employment, the
production function for y is
y = g (ky, ly) = g (k’ - kx, l’ - lx)
L/lx = fl + gl = 0
79
L/ = y’ – g (k’ - kx, l’ - lx) = 0
80
20
Single Firm
Efficient Allocation of
Resources among Firms
• From the first two conditions, we can see
that
• Resources should be allocated to those
firms where they can be most efficiently
used
fk g k

fl
gl
– 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
RTSx (k for l) = RTSy (k for l)
81
82
• The allocational problem is to maximize
• Suppose that there are two firms
producing x and their production
functions are
x = f1(k1, l1) + f2(k2, l2)
subject to the constraints
k1 + k2 = k’
l1 + l2 = l’
f1(k1, l1)
f2(k2, l2)
• Assume that the total supplies of capital
and labor are k’ and l’
83
• Substituting, the maximization problem
becomes
x = f1(k1, l1) + f2(k’ - k1, l’ - l1)
84
21
• First-order conditions for a maximum
are
• These first-order conditions can be
rewritten as
x
f
f
f
f
 1  2  1  2 0
k1 k1 k1 k1 k 2
f1
f
 2
k1 k 2
x f1 f2 f1 f2




0
l1 l1 l1 l1 l2
85
Efficient Choice of Output
by Firms
f1 f2

l1 l2
• The marginal physical product of each
input should be equal across the two
firms
86
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
• The Lagrangian for this problem is
L = x1 + x2 + [y* - f1(x1) - f2(x2)]
and yields the first-order condition:
f1/x1 = f2/x2
yi = fi (xi ) for i=1,2
• The rate of product transformation
(RPT) should be the same for all firms
producing these goods
• The overall optimization problem is to
produce the maximum amount of x for
any given level of y (y*)
87
88
22
by Firms
by Firms
Firm A is relatively efficient at producing cars, while Firm B
is relatively efficient at producing trucks
If each firm was to specialize in its efficient product, total
output could be increased
Cars
Cars
RPT 
RPT 
2
1
100
Cars
1
1
100
Trucks
50
Firm A
Cars
RPT 
100
Trucks
50
Firm B
89
Theory of Comparative
Advantage
RPT 
2
1
1
1
100
Trucks
50
Firm A
Trucks
50
Firm B
90
Efficiency in Product Mix
• The theory of comparative advantage
was first proposed by Ricardo
• Technical efficiency is not a sufficient
condition for Pareto efficiency
– countries should specialize in producing
those goods of which they are relatively
more efficient producers
– 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
• 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
91
92
23
• The condition necessary to ensure that
the right goods are produced is
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
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
Any point on PP represents a
point of technical efficiency
P
Output of x
93
Output of y
Only one point on PP will maximize
Crusoe’s utility
At the point of
tangency, Crusoe’s
MRS will be equal to
the technical RPT
P
U3
U2
• 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
U1
P
94
T(x,y) = 0
Output of x
95
96
24
• Crusoe’s problem is to maximize his
utility subject to the production
constraint
• First-order conditions for an interior
maximum are
L U
T


0
x x
x
• Setting up the Lagrangian yields
L U
T


0
y y
y
L = U(x,y) + [T(x,y)]
97
98
Competitive Prices and
Efficiency
• Combining the first two, we get
U / x T / x

U / y T / y
or
MRS ( x for y )  
L
 T ( x, y )  0

dy
(along T )  RPT ( x for y )
dx
99
• 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
100
25
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”
• 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
101
102
• 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)
• If this is true for every firm, then with a
competitive labor market
pxfl = w
103
pxfl1 = w = pxfl2
fl1 = fl2
• Every firm that produces x has identical
marginal productivities of every input in
the production of x
104
26
• Recall that the RPT (of x for y) is equal
to MCx /MCy
• In perfect competition, each profitmaximizing 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
• 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
105
106
• 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
107
Output of y
x* and y* represent the efficient output mix
slope  
px*
py*
P
Only with a price ratio of
px*/py* will supply and
demand be in equilibrium
y*
U0
x*
P
Output of x
108
27
Departing from the
Competitive Assumptions
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
• The ability of competitive markets to
achieve efficiency may be impaired
because of
– imperfect competition
– externalities
– public goods
– imperfect information
109
110
Imperfect Competition
Externalities
• Imperfect competition includes all
situations in which economic agents
exert some market power in determining
market prices
• 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
– these agents will take these effects into
account in their decisions
• Market prices no longer carry the
informational content required to achieve
Pareto efficiency
111
– there is a divergence between private and
social marginal cost
112
28
Public Goods
Imperfect Information
• 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
• 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
– they are nonexclusive
• extra individuals cannot be precluded from
consuming the good
113
114
Distribution
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
• 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
115
116
29
Distribution
Distribution
OJ
UJ
1
• Any point within the Edgeworth box in
which the MRS for Smith is unequal to
that for Jones offers an opportunity for
Pareto improvements
UJ2
US4
UJ3
Total Y
US3
UJ4
– both can move to higher levels of utility
through trade
US2
US1
OS
Total X
117
Distribution
Contract Curve
OJ
• In an exchange economy, all efficient
allocations lie along a contract curve
UJ1
UJ2
– points off the curve are necessarily
inefficient
US4
UJ3
• individuals can be made better off by moving to
the curve
US3
UJ4
• Along the contract curve, individuals’
preferences are rivals
US2
A

OS
Any trade in this area is
an improvement over A
118
US1
119
– one may be made better off only by making
the other worse off
120
30
Exchange with Initial
Endowments
Contract Curve
OJ
UJ1
• Suppose that the two individuals
possess different quantities of the two
goods at the start
UJ2
US4
UJ3
– it is possible that the two individuals could
both benefit from trade if the initial
allocations were inefficient
US3
UJ4
US2
A

US1
Contract curve
OS
121
122
Endowments
Endowments
• 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
Suppose that A represents
the initial endowments
UJA
A

123
OJ
OS
USA
124
31
Endowments
Endowments
OJ
OJ
Only allocations between M1
and M2 will be acceptable to
both
Neither individual would be
willing to accept a lower level
of utility than A gives
UJA
UJA
M2

M1

A

A

USA
OS
125
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
127
USA
OS
126
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
128
32
• 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
• 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)
129
130
• Competitive prices will result in a
Pareto-efficient allocation of resources
– this is the First Theorem of Welfare
Economics
• Factors that will interfere with
competitive markets’ abilities to
achieve efficiency include
– market power
– externalities
– existence of public goods
– imperfect information
131
132
33
• 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
133
34
Monopoly
• A monopoly is a single supplier to a
market
• This firm may choose to produce at any
point on the market demand curve
Chapter 14
MODELS OF MONOPOLY
1
Barriers to Entry
2
Technical 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
• 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
– there are two general types of barriers to
entry
• technical barriers
• legal barriers
3
4
1
Technical Barriers to Entry
Legal Barriers to Entry
• Another technical basis of monopoly is
special knowledge of a low-cost
productive technique
• 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
– 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
5
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
6
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
7
8
2
Profit Maximization
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
MC
Price
The monopolist will maximize
profits where MR = MC
AC
P*
The firm will charge a price
of P*
C
Profits can be found in
the shaded rectangle
MR
Q*
D
Quantity
9
10
• The gap between a firm’s price and its
marginal cost is inversely related to the
price elasticity of demand facing the firm
• Two general conclusions about monopoly
pricing can be drawn:
P  MC
1

P
eQ,P
– 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
where eQ,P is the elasticity of demand
for the entire market
11
12
3
Monopoly Profits
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
• The size of monopoly profits in the long
run will depend on the relationship
between average costs and market
demand for the product
– 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
13
Monopoly Profits
No Monopoly Supply Curve
Price
Price
MC
MC
• With a fixed market demand curve, the
supply “curve” for a monopolist will only
be one point
AC
AC
P*=AC
P*
14
– the price-output combination where MR =
MC
C
MR
Q*
Positive profits
D
MR
Quantity
Q*
Zero profit
D
Quantity
15
• If the demand curve shifts, the marginal
revenue curve shifts and a new profitmaximizing output will be chosen
16
4
Monopoly with Linear Demand
• Suppose that the market for frisbees
has a linear demand curve of the form
• To maximize profits, the monopolist
chooses the output for which MR = MC
• We need to find total revenue
Q = 2,000 - 20P
or
TR = PQ = 100Q - Q2/20
P = 100 - Q/20
• Therefore, marginal revenue is
• The total costs of the frisbee producer
are given by
MR = 100 - Q/10
while marginal cost is
C(Q) = 0.05Q2 + 10,000
MC = 0.01Q
17
• Thus, MR = MC where
• To see that the inverse elasticity rule
holds, we can calculate the elasticity of
demand at the monopoly’s profitmaximizing level of output
100 - Q/10 = 0.01Q
Q* = 500
18
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
eQ,P 
19
Q P
 75 
  20
  3
P Q
 500 
20
5
Monopoly and Resource
Allocation
• The inverse elasticity rule specifies that
• To evaluate the allocational effect of a
monopoly, we will use a perfectly
competitive, constant-cost industry as a
basis of comparison
P  MC
1
1


P
eQ,P 3
• Since P* = 75 and MC = 50, this
relationship holds
– the industry’s long-run supply curve is
infinitely elastic with a price equal to both
marginal and average cost
21
22
Allocation
Price
Allocation
If this market was competitive, output would
be Q* and price would be P*
Under a monopoly, output would be Q**
and price would rise to P**
P**
MC=AC
P*
Price
Consumer surplus would fall
Producer surplus will rise
Consumer surplus falls by more
than producer surplus rises.
P**
MC=AC
P*
D
D
MR
Q**
There is a deadweight
loss from monopoly
MR
Q*
Quantity
Q**
23
Q*
Quantity
24
6
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:
• The competitive price in this market will
be
Q = Pe
where e is the price elasticity of demand
(e < -1)
Pc = c
and the monopoly price is given by
c
Pm 
1
1
e
25
26
• The consumer surplus associated with
any price (P0) can be computed as
• Therefore, under perfect competition


P0
P0
CSc  
CS   Q(P )dP   P edP
c e 1
e 1
and under monopoly
e 1
CS 
e 1

e 1
0
P
P

e 1P
e 1
0
27


 c 


 1 1 


e
CSm   
e 1
28
7
• Taking the ratio of these two surplus
measures yields
• Monopoly profits are given by


CSm  1 


CSc  1  1 


e



 c

m  PmQm  cQm  
 c Qm
 1 1



e


e 1
e
• If e = -2, this ratio is ½
– consumer surplus under monopoly is half
what it is under perfect competition
29
 c  



    c 
 c 
e

  

m  
 1 1   1 1 
 1 1 

 



e 
e
e


e 1

1
e
30
Monopoly and Product Quality
• To find the transfer from consumer
surplus into monopoly profits we can
divide monopoly profits by the competitive
consumer surplus
• The market power enjoyed by a monopoly
may be exercised along dimensions other
than the market price of its product




m
 e  1 1 


CSc  e  1  1 


e

e 1
• If e = -2, this ratio is ¼
 e 


 1 e 
– 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
e
31
32
8
• 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
• First-order conditions for a maximum are

P
 P (Q, X )  Q
 CQ  0
Q
Q
– MR = MC for output decisions
• If costs are given by C(Q,X), the
monopoly will choose Q and X to
maximize

P
Q
 CX  0
X
X
 = P(Q,X)Q - C(Q,X)
33
– Marginal revenue from increasing quality by
one unit is equal to the marginal cost of
34
making such an increase
• The level of product quality that will be
opted for under competitive conditions is
the one that maximizes net social welfare
• The difference between the quality choice
of a competitive industry and the
monopolist is:
Q*
SW   P(Q, X )dQ  C(Q, X )
0
• Maximizing with respect to X yields
Q*
SW
  PX (Q, X )dQ  C X  0
0
X
35
– 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
36
9
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
• 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
– profit-seeking middlemen will destroy any
discriminatory pricing scheme if possible
• price discrimination becomes possible if resale is
costly
38
37
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
Under perfect price discrimination, the monopolist
charges a different price to each buyer
Price
The first buyer pays P1 for Q1 units
P1
The second buyer pays P2 for Q2-Q1 units
P2
MC
– perfect or first-degree price discrimination
• extracts all consumer surplus
• no deadweight loss
D
The monopolist will
continue this way until the
marginal buyer is no
longer willing to pay the
good’s marginal cost
Quantity
39
Q1 Q2
40
Q2
10
• 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
41
• Therefore,
P = 100 - Q/20 = MC = 0.1Q
Q* = 666
• Total revenue and total costs will be
666
Q*
R   P (Q )dQ  100Q 
0
Q2
40 0
 55,511
c(Q)  0.05Q 2  10,000  32,178
• Profit is much larger (23,333 > 15,000) 42
Market Separation
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
• All the monopolist needs to know in this
case is the price elasticities of demand
for each market
– can follow a different pricing policy in each
market
– third-degree price discrimination
– set price according to the inverse elasticity
rule
• If the marginal cost is the same in all
markets,
Pi (1 
43
1
1
)  Pj (1  )
ei
ej
44
11
Market Separation
Market Separation
If two markets are separate, maximum profits occur by
setting different prices in the two markets
Price
1
)
ej
Pi

Pj (1  1 )
ei
(1 
The market with the less
elastic demand will be
charged the higher price
P1
P2
• The profit-maximizing price will be
higher in markets where demand is less
elastic
MC
MC
D
D
MR
Quantity in Market 1
MR
Q1*
0
Q2*
Quantity in Market 2
45
Third-Degree Price
Discrimination
46
Third-Degree Price
Discrimination
• Suppose that the demand curves in two
separated markets are given by
• Optimal choices and prices are
Q1 = 9
P1 = 15
Q1 = 24 – P1
Q2 = 6
P2 = 9
Q2 = 24 – 2P2
• Profits for the monopoly are
• Suppose that MC = 6
• Profit maximization requires that
 = (P1 - 6)Q1 + (P2 - 6)Q2 = 81 + 18 = 99
MR1 = 24 – 2Q1 = 6 = MR2 = 12 – Q2
47
48
12
Third-Degree Price
Discrimination
Third-Degree Price
Discrimination
• 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
DW 1 = 0.5(P1-MC)(18-Q1) = 0.5(15-6)(18-9) = 40.5
DW 2 = 0.5(P2-MC)(12-Q2) = 0.5(9-6)(12-6) = 9
49
Third-Degree Price
Discrimination
• If this monopoly was to pursue a singleprice 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
50
Two-Part Tariffs
• 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
• 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
51
52
13
Two-Part Tariffs
Two-Part Tariffs
• Because the average price paid by any
demander is
• 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
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)
– this might not be the most profitable
approach
– in general, optimal pricing schedules will
depend on a variety of contingencies
53
Two-Part Tariffs
Two-Part Tariffs
• Suppose there are two different buyers
with the demand functions
• With this marginal price, demander 2
obtains consumer surplus of 36
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
54
q2 = 12
55
– 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
56
14
Regulation of Monopoly
• 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 such as the utility,
communications, and transportation
industries are highly regulated in many
countries
– natural monopolies exhibit declining
average costs over a wide range of output
57
58
Because natural monopolies exhibit
decreasing costs, MC falls below AC
Price
An unregulated monopoly will
maximize profit at Q1 and P1
P1
C1
C2
AC
P2
MR
Q1
If regulators force the
monopoly to charge a
price of P2, the firm will
suffer a loss because
P2 < C2
MC
Q2 D
Price
Suppose that the regulatory commission allows the
monopoly to charge a price of P1 to some users
Other users are offered the lower price
of P2
The profits on the sales to highprice customers are enough to
cover the losses on the sales to
low-price customers
P1
C1
C2
AC
MC
P2
Quantity
Q1
59
Q2 D
Quantity
60
15
• 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
• Suppose that a regulated utility has a
production function of the form
q = f (k,l)
– 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
• The firm’s actual rate of return on
capital is defined as
s
pf (k, l )  wl
k
61
62
• If =0, regulation is ineffective and the
monopoly behaves like any profitmaximizing firm
• If =1, the Lagrangian reduces to
• 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
L = (s0 – v)k
subject to this constraint
which (assuming s0>v), will mean that
the monopoly will hire infinite amounts
of capital – an implausible result
L = pf (k,l) – wl – vk + [wl + s0k – pf (k,l)]
63
64
16
• Because s0>v and <1, this means that
• Therefore, 0<<1 and the first-order
conditions for a maximum are:
pfk < v
L
 pfl  w  (w  pfl )  0
l
• The firm will hire more capital than it
would under unregulated conditions
L
 pfk  v  (s0  pfk )  0
k
– it will also achieve a lower marginal
productivity of capital
L
 wl  s0  pf (k, l )  0

65
66
Dynamic Views of Monopoly
• Some economists have stressed the
beneficial role that monopoly profits can
play in the process of economic
development
• The most profitable level of output for
the monopolist is the one for which
cost
– 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
67
– at this output level, price will exceed
marginal cost
– the profitability of the monopolist will
depend on the relationship between price
and average cost
68
17
• Relative to perfect competition,
monopoly involves a loss of consumer
surplus for demanders
• Monopolies may opt for different levels
of quality than would perfectly
competitive firms
• Durable good monopolists may be
constrained by markets for used goods
– 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
69
70
• A monopoly may be able to increase its
profits further through price
discrimination – charging different
prices to different categories of buyers
• Governments often choose to regulate
natural monopolies (firms with
diminishing average costs over a broad
range of output levels)
– the ability of the monopoly to practice
price discrimination depends on its ability
to prevent arbitrage among buyers
– the type of regulatory mechanisms
adopted can affect the behavior of the
regulated firm
71
72
18
Pricing Under
Homogeneous Oligopoly
• We will assume that the market is
perfectly competitive on the demand side
Chapter 15
– there are many buyers, each of whom is a
price taker
TRADITIONAL MODELS OF
IMPERFECT COMPETITION
• We will assume that the good obeys the
law of one price
– this assumption will be relaxed when product
differentiation is discussed
1
Pricing Under
2
Pricing Under
• We will assume that there is a relatively
small number of identical firms (n)
• The inverse demand function for the
good shows the price that buyers are
willing to pay for any particular level of
industry output
– we will initially start with n fixed, but later
allow n to vary through entry and exit in
response to firms’ profitability
P = f(Q) = f(q1+q2+…+qn)
• The output of each firm is qi (i=1,…,n)
• Each firm’s goal is to maximize profits
– symmetry in costs across firms will usually
require that these outputs are equal
i = f(Q)qi –Ci(qi)
i = f(q1+q2+…qn)qi –Ci
3
4
1
Oligopoly Pricing Models
Oligopoly Pricing Models
• The Cournot model assumes that firm i
treats firm j’s output as fixed in its
decisions
• The quasi-competitive model assumes
price-taking behavior by all firms
– P is treated as fixed
– qj/qi = 0
• The cartel model assumes that firms
can collude perfectly in choosing
industry output and P
• The conjectural variations model
assumes that firm j’s output will respond
to variations in firm i’s output
– qj/qi  0
5
6
Quasi-Competitive Model
Quasi-Competitive Model
• Each firm is assumed to be a price taker
• The first-order condition for profitmaximization 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
Price
If each firm acts as a price taker, P = MCi
so QC output is produced and sold at a
price of PC
MC
PC
D
MR
QC
7
Quantity
8
2
Cartel Model
Cartel Model
• The assumption of price-taking behavior
may be inappropriate in oligopolistic
industries
• In this case, the cartel acts as a
multiplant monopoly and chooses qi for
each firm so as to maximize total
industry profits
– 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
 = PQ – [C1(q1) + C2(q2) +…+ Cn(qn)]
n
  f (q1  q2  ...  qn )[q1  q2  ...  qn ]   Ci (qi )
i 1
9
10
Cartel Model
Cartel Model
are that

P
 P  (q1  q2  ...  qn )
 MC(qi )  0
qi
qi
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
Price
PM
MC
MR(Q) = MCi(qi)
• At the profit-maximizing point, marginal
revenue will be equal to each firm’s
marginal cost
D
MR
QM
11
Quantity
12
3
Cartel Model
Cournot 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
• 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
13
Cournot Model
14
Cournot Model
• The first-order conditions for a profit
maximization are
• Each firm’s output will exceed the cartel
output
i
P
 P  qi
 MCi (qi )  0
qi
qi
– the firm-specific marginal revenue is larger
than the market-marginal revenue
• Each firm’s output will fall short of the
competitive output
• 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
15
– qi P/qi < 0
16
4
Cournot’s Natural Springs
Duopoly
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
• 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
17
Duopoly
18
Duopoly
• Because each firm has zero marginal
costs, the quasi-competitive solution
will result in a market price of zero
• The cartel solution to this problem can
be found by maximizing industry
revenue (and profits)
– total demand will be 120
– the division of output between the two
firms is indeterminate
 = PQ = 120Q - Q2
• The first-order condition is
/Q = 120 - 2Q = 0
• each firm has zero marginal cost over all
output ranges
19
20
5
Duopoly
Duopoly
• The profit-maximizing output, price, and
level of profit are
• The two firms’ revenues (and profits) are
given by
Q = 60
P = 60
1 = Pq1 = (120 - q1 - q2) q1 = 120q1 - q12 - q1q2
 = 3,600
• First-order conditions for a maximum are
2 = Pq2 = (120 - q1 - q2) q2 = 120q2 - q22 - q1q2
• The precise division of output and
profits is indeterminate
1
 120  2q1  q2  0
q1
2
 120  2q2  q1  0
q2
21
Duopoly
22
Duopoly
• These first-order equations are called
reaction functions
• We can solve the reaction functions
simultaneously to find that
– show how each firm reacts to the other’s
output level
q1 = q2 = 40
P = 120 - (q1 + q2) = 40
1 = 2 = Pq1 = Pq2 = 1,600
• In equilibrium, each firm must produce
what the other firm thinks it will
23
• Note that the Cournot equilibrium falls
between the quasi-competitive model
and the cartel model
24
6
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
• For each firm i, we are concerned with
the assumed value of qj /qi for ij
– 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
25
• The first-order condition for profit
maximization becomes
Price Leadership Model
• Suppose that the market is composed
of a single price leader (firm 1) and a
fringe of quasi-competitors
 P
i
P q j 
 P  qi 


  MCi (qi )  0
qi
 qi j  i q j qi 
– firms 2,…,n would be price takers
– firm 1 would have a more complex reaction
function, taking other firms’ actions into
account
• 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
26
27
28
7
We can derive the demand curve facing
the industry leader
D represents the market demand curve
Price
Price
SC
SC
SC represents the supply
decisions of all of the n-1 firms in
For a price of P1 or above, the
leader will sell nothing
P1
the competitive fringe
For a price of P2 or below, the
leader has the market to itself
P2
D
D
Quantity
Quantity
29
30
Between P2 and P1, the
demand for the leader (D’)
is constructed by
subtracting what the fringe
will supply from total
market demand
Price
SC
P1
PL
D’
P2
MC’
MR’
QL
D
The leader would then set
MR’ = MC’ and produce QL
at a price of PL
Price
SC
P1
D’
P2
MC’
QC
31
The competitive fringe will
produce QC and total
industry output will be QT
(= QC + QL)
PL
MR’
Quantity
Market price will then be PL
QL
QT
D
Quantity
32
8
Stackelberg Leadership Model
• 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
• 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
33
• There is the possibility of a different
type of strategic leadership
• Assume that firm 1 knows that firm 2
chooses q2 so that
34
• 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
q2 = (120 – q1)/2
1 = Pq1 = 120q1 – q12 – q1q2
• Firm 1 can now calculate the conjectural
variation
1/q1 = 120 – 2q1 – q1(q2/q1) – q2 = 0
q2/q1 = -1/2
35
1/q1 = 120 – (3/2)q1 – q2 = 0
36
9
Product Differentiation
• Solving this simultaneously with firm 2’s
reaction function, we get
• Firms often devote considerable
resources to differentiating their
products from those of their competitors
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
– quality and style variations
– warranties and guarantees
– special service features
– product advertising
37
38
• 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
39
• 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
40
10
• We will assume that there are n firms
competing in a particular product group
• 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
– 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
pi = g(qi,pj,zi,zj)
• Presumably, pi/qi  0, pi/pj  0,
pi/zi  0, and pi/zj  0
total costs = Ci (qi,zi)
41
• 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
i
p C
 pi  qi i  i  0
qi
qi qi
i
p C
 qi i  i  0
zi
zi zi
42
43
• 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
44
11
Spatial Differentiation
• Suppose we are examining the case of
ice cream stands located on a beach
• The demand curve facing any one firm
may shift often
– assume that demanders are located
uniformly along the beach
– 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
• 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
45
• A person located at point E will be
indifferent between stands A and B if
L
Ice cream stands are located at points A
and B along a linear beach of length L

A

E
46
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

B
Suppose that a person is standing at point E
47
48
12
• The coordinate of point E is
L
a
x
y
x
b
a+x+y+b=L

A

E

x
pB  pA
Lab x
c
x
1
p  pA 
L  a  b  B

2
c

y
1
p  pB 
L  a  b  A

2
c

B
49
1
p p  pA2
(L  a  b ) p A  A B
2
2c
B  pB (b  y ) 
1
p p  pB2
(L  a  b)pB  A B
2
2c
50
• Each firm will choose its price so as to
maximize profits
• Profits for the two firms are
 A  p A (a  x ) 
pB  pA  cy
c
 A 1
p
p
 (L  a  b )  B  A  0
pA 2
2c c
B 1
p
p
 (L  a  b )  A  B  0
pB 2
2c c
51
52
13
• These can be solved to yield:
L
ab

pA  c  L 

3 

a
ab

pB  c  L 

3 

x
y
b
Because A is somewhat more favorably located
than B, pA will exceed pB

• These prices depend on the precise
locations of the stands and will differ
from one another
A

E

B
53
54
Entry
• 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
• 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
– 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
– to the extent that entry is possible, long-run
profits are constrained
– if entry is completely costless, long-run
profits will be zero
• this encourages a similarity of products
55
56
14
Entry
Entry
• If firms are price takers:
• 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
– 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:
57
Entry
Monopolistic Competition
Firms will initially be maximizing
profits at q*. Since P > AC,  > 0
Price
MC
AC
P*
Since  > 0, firms will
enter and the demand
facing the firm will shift
left
Entry will end when  = 0
P’
d
mr’
q’
mr
q* qm
d’
– each firm will face a downward-sloping
demand curve
– entry may reduce profits to zero, but
production at minimum average cost is not
ensured
58
Firms will exhibit excess
capacity = qm - q’
• 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
Quantity
59
60
15
• To find the equilibrium n, we must
examine each firm’s profit-maximizing
choice of pi
• Because
• Each firm also faces a demand curve
for its product of the form:
qi  0.01(n  1)pi  0.01 p j 
j i
303
n
• 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
i = piqi – ci
the first-order condition for a maximum is
i
303
 0.02(n  1)pi  0.01 p j 
 0.04(n  1)  0
pi
n
j i
61
• This means that
pi 
0.5 p j
j i
n 1
• The equilibrium n is determined by the
zero-profit condition

303
2
0.02(n  1)n
pi qi  ci  0
• Applying the equilibrium condition that pi
= pj yields
pi 
62
30,300  303 4(303)
4(303)

9
n 2 (n  1)
n
n
30,300
4
(n  1)n
• P approaches MC (4) as n gets larger
• Substituting in the expression for pi, we
find that
63
n  101
64
16
• The final equilibrium is
• If each firm faces a similar demand
function, this equilibrium is sustainable
pi = pj = 7
qi = 3
– no firm would find it profitable to enter this
industry
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
65
• But what if a potential entrant adopted a
large-scale production plan?
Contestable Markets and
Industry Structure
Perfectly Contestable Market
• Several economists have challenged
that this zero-profit equilibrium is
sustainable in the long run
– the low average cost may give the potential
entrant considerable leeway in pricing so as
to tempt customers of existing firms to
66
switch allegiances
• A market is perfectly contestable if entry
and exit are absolutely free
– 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
67
– 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
68
17
This market would be unsustainable
in a perfectly contestable market
Price
MC
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
AC
P*
P’
d
mr’
q’
mr
q* q’
d’
• 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
Quantity
69
70
• 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
In a perfectly contestable market, equilibrium
requires that P = MC = AC
Price
AC1
AC2
AC3
The number of firms is
completely determined by
market demand (Q*) and
by the output level that
minimizes AC (q*)
AC4
P*
n = Q*/q*
– this number may be relatively small (unlike
the perfectly competitive case)
71
D
q*
2q*
3q*
Q*=4q*
Quantity
72
18
Barriers to Entry
Barriers to Entry
• If barriers to entry prevent free entry and
exit, the results of this model must be
modified
• The completely flexible type of hit-andrun behavior assumed in the contestable
markets theory may be subject to barriers
to entry
– 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
– some types of capital investments may not
be reversible
– demanders may not respond to price
differentials quickly
73
74
A Contestable Natural
Monopoly
Monopoly
• Suppose that the total cost of producing
electric power is given by
• If the producer behaves as a monopolist,
it will maximize profits by
C(Q) = 100Q + 8,000
– since AC declines over all output ranges,
MR = 200 - (2Q)/5 = MC = 100
Qm = 250
Pm = 150
this is a natural monopoly
• The demand for electricity is given by
m = R - C = 37,500 - 33,000 = 4,500
QD = 1,000 - 5P
75
• These profits will be tempting to would-be
entrants
76
19
Monopoly
Monopoly
• If there are no entry barriers, a potential
entrant can offer electricity customers a
lower price and still cover costs
• If electricity production is fully
contestable, the only price viable under
threat of potential entry is average cost
– this monopoly solution might not represent
a viable equilibrium
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
77
Monopoly
• 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
79
situation
78
• 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
80
20
• In markets with few firms, output and
price decisions are interdependent
• The Cournot model provides a
tractable approach to oligopoly
markets, but neglects important
strategic issues
– each firm must consider its rivals’
decisions
– modeling such interdependence is
difficult because of the need to consider
conjectural variations
81
82
• Product differentiation can be
analyzed in a standard profitmaximization framework
• Entry conditions are important
determinants of the long-run
sustainability of various market
equilibria
– with differentiated products, the law of
one price no longer holds and firms may
have somewhat more leeway in their
pricing decisions
– with perfect contestability, equilibria may
resemble perfectly competitive ones
even though there are relatively few
firms in the market
83
84
21
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
Chapter 16
LABOR MARKETS
– engaging in market work at a real wage
rate of w
– leisure (nonwork)
1
Allocation of Time
2
Allocation of Time
• Assume that an individual’s utility
depends on consumption (c) and hours
of leisure (h)
• Combining the two constraints, we get
utility = U(c,h)
• An individual has a “full income” of 24w
c = w(24 – h)
c + wh = 24w
• In seeking to maximize utility, the
individual is bound by two constraints
– may spend the full income either by
working (for real income and consumption)
or by not working (enjoying leisure)
l + h = 24
c = wl
• The opportunity cost of leisure is w
3
4
1
Utility Maximization
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)
• Dividing the two, we get
U / c
 w  MRS (h for c )
U / h
• 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
• The first-order conditions are
L/c = U/c -  = 0
– to be a true maximum, the MRS (of h for c)
must be diminishing
L/h = U/h -  = 0
5
Income and
Substitution Effects
6
Consumption
Income and
• 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
7
in opposite directions
B
C
A
U2
U1
The individual chooses
less leisure as a result
of the increase in w
Leisure
substitution effect > income effect
8
2
Consumption
Income and
A Mathematical Analysis
of Labor Supply
B
C
A
U1
U2
Leisure
The individual
chooses more
leisure as a result
of the increase in
w
substitution effect < income effect
• 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 laborleisure choice
9
A Mathematical Analysis
of Labor Supply
10
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
• 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)
– solving this minimization problem will yield
exactly the same solution as the utility
maximization problem
– hours worked will depend on both the
wage and the amount of nonlabor income
– since leisure is a normal good, l/n < 0
11
12
3
• A small change in w will change the
minimum expenditures required by
• This means that a labor supply
function can be calculated by partially
differentiating the expenditure function
E/w = -l
– this is the extent to which labor earnings
are increased by the wage change
– because utility is held constant, this
function should be interpreted as a
“compensated” (constant utility) labor
supply function
lc(w,U)
13
Slutsky Equation of
Labor Supply
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
l c
l
l E



w w E w
14
15
• Substituting for E/w, we get
l c
l
l
l
l

l

l
w w
E w
n
• Introducing a different notation for lc ,
and rearranging terms gives us the
Slutsky equation for labor supply:
l
l

w w
l
U U 0
l
n
16
4
Cobb-Douglas Labor Supply
• Suppose that utility is of the form
• The Lagrangian expression for utility
maximization is
U  c  h
L = ch + (w + n - wh - c)
• The budget constraint is
• First-order conditions are
c = wl + n
L/c = c-h -  = 0
and the time constraint is
l+h=1
– note that we have set maximum work time
to 1 hour for convenience
L/h = ch- - w = 0
L/ = w + n - wh - c = 0
17
18
• Dividing the first by the second yields
• Substitution into the full income
constraint yields
h
h
1


c (1   )c w
wh 
c = (w + n)
h = (w + n)/w
– the person spends  of his income on
consumption and  = 1- on leisure
– the labor supply function is
1 
c

l (w , n )  1  h  (1  ) 
19
n
w
20
5
• Note that if n = 0, the person will work
(1-) of each hour no matter what the
wage is
• 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
– the substitution and income effects of a
change in w offset each other and leave l
unaffected
21
22
CES Labor Supply
• 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
23
U (c, h ) 
c  h



• Budget share equations are given by
sc 
c
1

w  n (1  w  )
sh 
wh
1

w  n (1  w   )
– where  = /(-1)
24
6
Market Supply Curve for Labor
CES Labor Supply
To derive the market supply curve for labor, we sum
the quantities of labor offered at every wage
• Solving for leisure gives
h
w n
w  w 1 
w
Individual A’s
supply curve
w
sA
and
l(w, n )  1  h 
Individual B’s
supply curve
w
Total labor
supply curve
sB
S
w*
w 1   n
w  w 1 
lA*
l
lB*
l
l
l*
lA* + lB* = l*
25
26
Market Supply Curve for Labor
Note that at w0, individual B would choose to remain
out of the labor force
w
Individual A’s
supply curve
w
sA
Individual B’s
supply curve
w
Total labor
supply curve
sB
S
w0
l
l
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
l
As w rises, l rises for two reasons: increased hours
of work and increased labor force participation 27
28
7
Labor Market Equilibrium
real wage
Mandated Benefits
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
S
w*
D
l*
At any wage below w*, the quantity
of labor demanded will be greater
than the quantity of labor supplied
quantity of labor
29
• 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
30
Mandated Benefits
Mandated Benefits
• Suppose that, prior to the mandate, the
supply and demand for labor are
• Suppose that the government mandates
that all firms provide a benefit to their
workers that costs t per unit of labor
hired
lS = a + bw
lD = c – dw
• Setting lS = lD yields an equilibrium wage
of
w* = (c – a)/(b + d)
– 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
31
w+k
32
8
Mandated Benefits
Mandated Benefits
• Equilibrium in the labor market then
requires that
• If workers derive no value from the
mandated benefits (k = 0), the mandate
is just like a tax on employment
a + b(w + k) = c – d(w + t)
– similar results will occur as long as k < t
• This means that the net wage is
w ** 
• If k = t, the new wage falls precisely by
the amount of the cost and the
equilibrium level of employment does not
change
c  a bk  dt
bk  dt

w *
bd
bd
bd
33
Mandated Benefits
34
Wage Variation
• If k > t, the new wage falls by more than
the cost of the benefit and the
equilibrium level of employment rises
• 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
35
36
9
Wage Variation
Wage Variation
• Human Capital
• Compensating Differentials
– 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
37
– 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
38
Monopsony in the
Labor Market
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 marginal expense (ME) associated
with any input is the increase in total
costs of that input that results from hiring
one more unit
– the firm faces the entire market supply curve
– to increase its hiring of labor, the firm must
pay a higher wage
39
– if the firm faces an upward-sloping supply
curve for that input, the marginal expense will
exceed the market price of the input
40
10
Monopsony in the
Labor Market
Monopsony in the
Labor Market
• If the total cost of labor is wl, then
MEl 
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*)
Wage
MEl
wl
w
w l
l
l
S
• In the competitive case, w/l = 0 and
MEl = w
• If w/l > 0, MEl > w
w*
The wage paid by the
firm will also be lower
than the competitive
level (w*)
w1
D
41
Monopsonistic Hiring
l1
Labor
l*
42
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
• The firm’s wage bill is
wl = l2/50
• The marginal expense associated with
hiring miners is
MEl = wl/l = l/25
• If the coal mine is the only hirer of
miners in the local area, it faces a labor
supply curve of the form
• Setting MEl = MRPl, we find that the
optimal quantity of labor is 500 and the
optimal wage is $10
l = 50w
43
44
11
Labor Unions
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
• 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
– the union faces a demand curve for labor
– as the sole supplier, it can choose at which
point it will operate
– even if workers would benefit from the
union, they may choose to be “free riders”
• this point depends on the union’s goals
45
46
Labor Unions
Wage
Labor Unions
The union may wish to maximize the total
wage bill (wl).
This occurs where
MR = 0
S
Wage
economic rent of its employed members
This occurs where
S
w2
l1 workers will be
w1
l2 workers will be
hired and paid a
wage of w1
D
MR
l1
Labor
This choice will
create an excess
supply of labor
47
MR = S
hired and paid a
wage of w2
D
MR
l2
Labor
Again, this will
cause an excess
supply of labor
48
12
Labor Unions
Wage
Modeling a Union
employment of its members
This occurs where
D=S
S
l = 50w
l3 workers will be
w3
hired and paid a
wage of w3
• Assume that the monopsony has a
MRPL curve of the form
MRPl = 70 – 0.1l
D
• The monopsonist will choose to hire 500
workers at a wage of $10
MR
l3
• A monopsonistic hirer of coal miners
faces a supply curve of
Labor
49
Modeling a Union
50
A Union Bargaining Model
• If a union can establish control over
labor supply, other options become
possible
• 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
– competitive solution where l = 583 and
w = $11.66
– monopoly solution where l = 318 and
w = $38.20
51
52
13
• Assuming that l* solves the firm’s
problem, the union’s goal is to choose w
to maximize utility
• This two-stage game can be solved by
backward induction
• The firm’s second-stage problem is to
maximize its profits:
U(w,l) = U[w,l*(w)]
 = R(l) – wl
• The first-order condition for a maximum is
R’(l) = w
and the first-order condition for a
maximum is
U1 + U2l’ = 0
U1/U2 = l’
53
54
• 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
• 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
55
56
14
• An increase in the real wage rate
creates income and substitution
effects that operate in different
directions in affecting the quantity of
labor supplied
• 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
– this result can be summarized by a
Slutsky-type equation much like the
one already derived in consumer
theory
57
58
• Monopsony power by firms on the
demand side of the market will
reduce both the quantity of labor
hired and the real wage rate
• 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
– as in the monopoly case, there will be a
welfare loss
59
60
15
Properties of Information
• Information is not easy to define
Chapter 18
– 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 pricequantity characterization used in supply
and demand analysis
THE ECONOMICS OF
INFORMATION
1
Properties of Information
2
The Value of Information
• Studying information also becomes
difficult due to some technical properties
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
– it is durable and retains value after its use
– it can be nonrival and nonexclusive
• Better information can reduce uncertainty
and lead to better decisions and higher
utility
• in this manner it can be considered a public
good
3
4
1
• Assume an individual forms subjective
opinions about the probabilities of two
states of the world
• Assume that information can be
measured by the number of “messages”
(m) purchased
– “good times” (probability = g) and “bad
times” (probability = b)
– g and b will be functions of m
• Information is valuable because it helps
the individual revise his estimates of
these probabilities
5
6
• First-order conditions for a constrained
maximum are:
• The individual’s goal will be to maximize
E(U) = gU(W g) + bU(W b)
L
 gU ' (Wg )  pg  0
Wg
subject to
I = pgW g + pbW b + pmm
L
 bU ' (Wb )  pb  0
Wb
• We need to set up the Lagrangian
L = gU(W g) + bU(W b) + (I-pgW g-pbW b-pmm)
7
L
 I  pgWg  pbWb  pm m  0

8
2
• First-order conditions for a constrained
maximum are:
• 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 utilitymaximizing level of information to buy
dWg
dWb
L
 gU ' (Wg )
 bU ' (Wb )
m
dm
dm
d g
dWg
d b
 U (Wg )
 U (Wb )
 pg
dm
dm
dm
dWb
 pb
 pm  0
dm
9
10
Asymmetry of Information
Information and Insurance
• The level of information that a person buys
will depend on the price per unit
• Information costs may differ significantly
across individuals
• 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
– 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
11
– may also be able to take actions that
impact these probabilities
12
3
Moral Hazard
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
• Suppose a risk-averse individual faces
the risk of a loss (l) that will lower
wealth
– parking an insured car in an unsafe area
– choosing not to install a sprinkler system in
an insured home
– the probability of a loss is 
– this probability can be lowered by the
amount the person spends on preventive
measures (a)
13
Moral Hazard
14
Moral Hazard
• The first-order condition for a maximum is
• Wealth in the two states is given by
E


 U (W1 )  (1  )U ' (W1 )  U (W2 )  U ' (W2 )  0
a
a
a

U ' (W2 )  (1  )U ' (W1 )  [U (W2 )  U (W1 )]
a
W1 = W 0 - a
W2 = W 0 - a - l
• The individual chooses a to maximize
E(U) = E = (1-)U(W 1) + U(W 2)
15
– 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
16
4
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
• The person can maximize expected utility
by choosing x such that W 1 = W 2
W1 = W 0 - a - p
W2 = W 0 - a - p - l + x
• A fair premium would be equal to
p = x
E
 


 (1  )U ' (W1 )1  l   U (W1 )
a
a 
a

 


 U ' (W2 )1  l   U (W2 )
0
a 
a

17
• Since W 1 = W 2, this condition becomes
1  l
• The first-order condition is

a
– 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
19
at the optimal level
18
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
20
5
Moral Hazard
Adverse Selection
• In the simplest case, the insurer might set
a premium based on the average
probability of loss experienced by some
group of people
• Individuals may have different probabilities
of experiencing a loss
• If individuals know the probabilities more
accurately than insurers, insurance
markets may not function properly
– no variation in premiums allowed for specific
precautionary activities
• each individual would have an incentive to reduce
his level of precautionary activities
– it will be difficult for insurers to set premiums
based on accurate measures of expected loss
21
22
Adverse Selection
W2
Adverse Selection
W2
certainty line
Suppose that one person has a probability of loss
equal to H, while the other has a probability of loss
equal to l
certainty line
Assume that two individuals
have the same initial wealth
(W*) and each face a
potential loss of l
W *- l
G
W*- l
E
W*
F
W1
E
W*
23
Both individuals would
prefer to move to the
certainty line if premiums
are actuarially fair
W1
24
6
Adverse Selection
W2
Adverse Selection
The lines show the market opportunities for each
person to trade W 1 for W2 by buying fair insurance
certainty line
F
slope 
G
W*- l
E
slope 
 (1  H )
H
 (1  l )
l
The low-risk person will
maximize utility at point
F, while the high-risk
person will choose G
W1
W*
• 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
25
26
Adverse Selection
W2
Adverse Selection
One possible solution would be for the insurer to
offer premiums based on the average probability of
loss
W2
Point M is not an equilibrium because further trading
opportunities exist for low-risk individuals
certainty line
F
H
G
W*- l
M
E
W*
certainty line
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
W1
F
H
G
W*- l
M
UH
N
E
W*
27
UL
An insurance policy
such as N would be
unattractive to highrisk individuals, but
attractive to low-risk
individuals and
profitable for insurers
W1
28
7
Adverse Selection
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
Suppose that insurers offer policy G. High-risk
individuals will opt for full insurance.
W2
certainty line
F
G
W*- l
UH
E
W*
Insurers cannot offer
any policy that lies
above UH because
they cannot prevent
high-risk individuals
from taking advantage
of it
W1
29
Adverse Selection
Adverse Selection
The best policy that low-risk individuals can obtain is
one such as J
W2
certainty line
F
G
W*- l
J
UH
E
W*
30
The policies G and J
represent a
separating equilibrium
W1
31
• 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
32
8
The Principal-Agent
Relationship
Adverse Selection
• Market signals can be drawn from a
number of sources
• One important way in which asymmetric
information may affect the allocation of
resources is when one person hires
another person to make decisions
– 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
– patients hiring physicians
– investors hiring financial advisors
– car owners hiring mechanics
– stockholders hiring managers
33
34
The Principal-Agent
Relationship
The Principal-Agent
Relationship
• 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
• 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
35
36
9
The Principal-Agent
Relationship
Profits
The Principal-Agent
Relationship
If the manager is also the
owner of the firm, he will
maximize his utility at
profits of * and benefits of
b*
*
Profits
The owner-manager maximizes
profit because any other ownermanager will also want b* in
benefits
b* represents a true
cost of doing business
*
U1
U1
Owner’s constraint
b*
Benefits
b*
Benefits
37
The Principal-Agent
Relationship
38
The Principal-Agent
Relationship
• Suppose that the manager is not the
sole owner of the firm
• The new budget constraint continues to
include the point b*, *
– suppose there are two other owners who
play no role in operating the firm
– the manager could still make the same
decision that a sole owner could)
• $1 in benefits only costs the manager
$0.33 in profits
• For benefits greater than b*, the slope
of the budget constraint is only -1/3
– the other $0.67 is effectively paid by the
other owners in terms of reduced profits
39
40
10
The Principal-Agent
Relationship
The Principal-Agent
Relationship
Given the manager’s budget
constraint, he will maximize
utility at benefits of b**
Profits
Agent’s constraint
*
**
U2
Profits for the
firm will be ***
U1
***
b*
b**
Benefits
• 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
41
Using the Corporate Jet
42
• A firm owns a fleet of corporate jets
used mainly for business purposes
• Suppose that all would-be applicants
have the same utility function
– the firm has just fired a CEO for misusing
the corporate fleet
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
• The firm wants to structure a
management contract that provides
better incentives for cost control
43
44
11
• Because jet use is expensive,  = 800
(thousand) if j =0 and  = 162 if j =1
• 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
– 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
45
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 
47
46
The Owner-Manager
Relationship
• 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
48
12
The Owner-Manager
Relationship
The Owner-Manager
Relationship
• 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
• 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
– with this compensation package, the
manager’s income is maximized by setting
s(a)/a = /a = 0
49
50
Asymmetric Information
Hidden Action
• Models of the principal-agent relationship
have introduced asymmetric information
into this problem in two ways
• 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)
– 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”
51
(a) = ’(a) + u
where ’ represents expected profits
52
13
Hidden Action
Hidden Information
• Because owners observe only  and not
’, they can only use actual profits in their
compensation function
• 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
– 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
– will be adapted as new information becomes
available
53
54
• Information is valuable because it
permits individuals to increase the
expected utility of their decisions
• Information has a number of special
properties that suggest that
inefficiencies associated with
imperfect and asymmetric information
may be quite prevalent
– individuals might be willing to pay
something to acquire additional
information
– differing costs of acquisition
– some aspects of a public good
55
56
14
• The presence of asymmetric
information may affect a variety of
market outcomes, many of which are
illustrated in the context of insurance
theory
• 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
– insurers may have less information
about potential risks than do insurance
purchasers
57
58
• Informational asymmetries can also
lead to adverse selection in insurance
markets
• Asymmetric information may also
cause some (principal) economic
actors to hire others (agents) to make
decisions for them
– 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
59
– providing the correct incentives to the
agent is a difficult problem
60
15
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
Chapter 19
EXTERNALITIES AND
PUBLIC GOODS
– chemical manufacturers releasing toxic
fumes
– noise from airplanes
– motorists littering roadways
1
Interfirm Externalities
2
Beneficial 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
• The relationship between the two firms
can be beneficial
– two firms, one producing honey and the
other producing apples
y = f(k,l;x)
3
4
1
Externalities in Utility
Public Goods Externalities
• Externalities can also occur if the
activities of an economic agent directly
affect an individual’s utility
• Public goods are nonexclusive
– externalities can decrease or increase
utility
• It is also possible for someone’s utility to
be dependent on the utility of another
– 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
utility = US(x1,…,xn;UJ)
5
Externalities and Allocative
Inefficiency
6
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
• 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*
7
– can consume them or use them in
production
8
2
Inefficiency
Inefficiency
• 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
• 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
yo = g(xi,xo)
9
Inefficiency
10
Finding the Efficient Allocation
• The economic problem is to maximize
utility subject to the four constraints
listed earlier
• The quantities of each good in this
economy are constrained by the initial
stocks available and by the additional
production that takes place
L = U(xc,yc) + 1[f(yi) - xo] + 2[g(xi,xo) - yo]
xc + xi = xo + x*
+
3(xc + xi - xo - x*) + 4(yc + yi - yo - y*)
yc + yi = xo + y*
11
12
3
• The six first-order conditions are
• Taking the ratio of the first two, we find
MRS = U1/U2 = 3/4
L/xc = U1 + 3 = 0
• The third and sixth equation also imply
that
L/yc = U2 + 4 = 0
L/xi = 2g1 + 3 = 0
MRS = 3/4 = 2g1/2 = g1
L/yi = 1fy + 4 = 0
L/xo = -1 + 2g2 - 3 = 0
L/yo = -2 - 4 = 0
13
• Optimality in y production requires that
the individual’s MRS in consumption
equals the marginal productivity of x in
the production of y
14
• To achieve efficiency in x production,
we must also consider the externality
this production poses to y
• Combining the last three equations
gives
• This equation requires the individual’s
MRS to equal dy/dx obtained through x
production
MRS = 3/4 = (-1 + 2g2)/4 = -1/4 + 2g2/4
– 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
MRS = 1/fy - g2
15
16
4
Inefficiency of the
Competitive Allocation
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
• But the producer of x would choose y
input so that
Py = Pxfy
Px/Py = 1/fy
and the profit-maximizing producer of y
would choose x input according to
Px = Pyg1
17
Production Externalities
• This means that the producer of x would
disregard the externality that its
production poses for y and will
overproduce x
18
• The downstream firm has a similar
production function but its output may
be affected by chemicals that firm x
pours in the river
• Suppose that two newsprint producers
are located along a river
• The upstream firm has a production
function of the form
x = 2,000lx0.5
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
19
20
5
• 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
50  p 
x
 1,000lx0.5
lx
• lx* = 400
• If  = 0 (no externalities), ly* = 400
21
22
• 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
• If  = -0.1 and x0 = 38,000, firm y will
maximize profits by
50  p 
• 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
y
 1,000ly0.5 ( 40,000  38,000)0.1
ly
50  468ly0.5
x = 2,000(399)0.5 = 39,950
• Because of the externality, ly* = 87 and
y output will be 8,723
and output of y becomes
y = 2,000(88)0.5(1,950)-0.1 = 8,796
23
24
6
• If firm x was to hire one more worker, its
own output would rise to
• 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
x = 2,000(401)0.5 = 40,050
– the private marginal value product of the
401st worker is equal to the wage
25
• 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
26
Solutions to the
Externality Problem
Solutions to the
Externality Problem
Price
• The output of the externality-producing
activity is too high under a marketdetermined 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
MC’
Market equilibrium
will occur at p1, x1
S = MC
If there are external
costs in the
production of x,
social marginal costs
are represented by
p1
MC’
D
Quantity of x
27
x1
28
7
Solutions to the
Externality Problem
Price
MC’
S = MC
p2
tax
A Pigouvian Tax on Newsprint
A tax equal to these
additional marginal
costs will reduce
output to the socially
optimal level (x2)
The price paid for the
good (p2) now
reflects all costs
• 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
D
Quantity of x
x2
29
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
30
Taxation in the General
Equilibrium Model
• The optimal Pigouvian tax in our
general equilibrium model is to set
t = -pyg2
(1 - t)MPl = (1 - t)1,000(361)-0.5 = 50
– 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
t = 0.05
• Therefore, a 5 percent tax on the price
firm x receives would eliminate the
externality
31
32
8
Equilibrium Model
Equilibrium Model
• With the optimal tax, firm x now faces a
net price of (px - t) and will choose y
input according to
• The Pigouvian tax scheme requires that
regulators have enough information to
set the tax properly
py = (px - t)fy
• The resulting allocation of resources will
achieve
– in this case, they would need to know firm
y’s production function
MRS = px/py = (1/fy) + t/py = (1/fy) - g2
33
34
Pollution Rights
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
• 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
– x’s choice to purchase these rights is
identical to its output choice
y = pyg(xi,xo) + rxo
35
36
9
Pollution Rights
The Coase Theorem
• 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
• The key feature of the pollution rights
equilibrium is that the rights are welldefined and tradable with zero
transactions costs
• The initial assignment of rights is
irrelevant
– subsequent trading will always achieve the
same, efficient equilibrium
37
The Coase Theorem
38
The Coase Theorem
• Suppose that firm x is initially given xT
rights to produce (and to pollute)
• Profits for firm y are given by
y = pyg(xi,xo) - r(xT - xo)
– it can choose to use these for its own
production or it may sell some to firm y
• Profit maximization in this case will lead
to precisely the same solution as in the
case where firm y was assigned the
rights
• Profits for firm x are given by
x = pxxo + r(xT - xo) = (px - r)xo + rxT
x = (px - r)f(yi) + rxT
39
40
10
The Coase Theorem
Attributes of Public Goods
• The independence of initial rights
assignment is usually referred to as the
Coase Theorem
• 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
– 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
41
42
• A good is nonrival if consumption of
additional units of the good involves
zero social marginal costs of production
• Some examples of these types of goods
include:
Exclusive
Yes
Rival
No
43
Yes
Hot dogs,
cars,
houses
Bridges,
swimming
pools
No
Fishing
grounds,
clean air
National
defense,
mosquito
control
44
11
Public Goods and
Resource Allocation
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
45
Public Goods and
Resource Allocation
• 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
47
• 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 + ysB)
46
Public Goods and
Resource Allocation
• The necessary conditions for efficient
resource allocation consist of choosing
the levels of ysA and ysB 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* - ysB) - K]
48
12
Public Goods and
Resource Allocation
Public Goods and
Resource Allocation
are
L/ysA = U1Af’ - U2A + U1Bf’ = 0
L/ysB = U1Af’ - U2B + U1Bf’ = 0
• We can now derive the optimality
condition for the production of x
• From the initial first-order condition we
know that
U1A/U2A + U1B/U2B = 1/f’
• Comparing the two equations, we find
MRSA + MRSB = 1/f’
U2B = U2A
49
• The MRS must reflect all consumers
because all will get the same benefits
50
Failure of a
Competitive Market
Failure of a
Competitive Market
• Production of x and y in competitive
markets will fail to achieve this allocation
• For public goods, the value of producing
one more unit is the sum of each
consumer’s valuation of that output
– 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
– individual demand curves should be added
vertically rather than horizontally
• Thus, the usual market demand curve
will not reflect the full marginal valuation
• it would provide too little incentive to produce x
51
52
13
Inefficiency of a
Nash Equilibrium
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
• The first-order condition for a maximum
is
choose sA to maximize UA[f(sA + sB),yA - sA]
U1Af’ - U2A = 0
U1A/U2A = 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
53
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
54
(for i=1,2)
• Assume each roommate has $300 to
spend and that px = $100 and py = $0.20
55
• 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
56
14
• If person 1 believes that person 2 will
not buy any paintings, he could choose
to purchase one and receive utility of
• We can show that this solution is
inefficient by calculating each person’s
MRS
U1(x,y1) = 11/3(1,000)2/3 = 100
MRSi 
while person 2’s utility will be
U i / x
y
 i
U i / y i 2 x
• At the allocations described,
U2(x,y2) = 11/3(1,500)2/3 = 131
• Person 2 has gained from his free-riding
position
MRS1 = 1,000/2 = 500
MRS2 = 1,500/2 = 750
57
58
• Since MRS1 + MRS2 = 1,250, the
roommates would be willing to sacrifice
1,250 granola bars to have one additional
painting
• To calculate the efficient level of x, we
must set the sum of each person’s MRS
equal to the price ratio
– an additional painting would only cost them
500 granola bars
– too few paintings are bought
MRS1  MRS2 
y1 y 2 y1  y 2 px 100




2x 2x
2x
py 0.20
• This means that
y1 + y2 = 1,000x
59
60
15
Lindahl Pricing of
Public Goods
• Substituting into the budget constraint,
we get
• Swedish economist E. Lindahl
suggested that individuals might be
willing to be taxed for public goods if they
knew that others were being taxed
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
61
game
Lindahl Pricing of
Public Goods
62
Lindahl Pricing of
Public Goods
• 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)]
– 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
• The first-order condition is given by
U1A - AU2B(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’
64
16
Lindahl Pricing of
Public Goods
Shortcomings of the
Lindahl Solution
• An equilibrium would occur when
A+B = 1
• The incentive to be a free rider is very
strong
– 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’
– 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
65
66
• Externalities may cause a
misallocation of resources because of
a divergence between private and
social marginal cost
• If transactions costs are small, private
bargaining among the parties
affected by an externality may bring
social and private costs into line
– traditional solutions to this divergence
includes mergers among the affected
parties and adoption of suitable
Pigouvian taxes or subsidies
– the proof that resources will be
efficiently allocated under such
circumstances is sometimes called the
Coase theorem
67
68
17
• Public goods provide benefits to
individuals on a nonexclusive basis no one can be prevented from
consuming such goods
• Private markets will tend to
underallocate resources to public
goods because no single buyer can
appropriate all of the benefits that
such goods provide
– such goods are usually nonrival in that
the marginal cost of serving another
user is zero
69
70
• 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
71
18

Important Points to Note

Transcription

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