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LECTURE NOTES for
INTERMEDIATE MICROECONOMICS
Contents
1. CONSUMER’S THEORY ............................................................................................. 4
1.1 Budget constraint.......................................................................................................... 4
1.2 Preferences ................................................................................................................... 7
1.3 Utility ........................................................................................................................... 8
1.4 Consumer’s choice ..................................................................................................... 10
1.5 Comparative statics of Marshallian demand .............................................................. 12
1.6 Expenditure minimization problem and duality in consumption ............................... 15
1.7 Slutsky decomposition and Slutsky equation ............................................................. 17
1.8 Alternative approach to consumer’s theory: revealed preferences ............................ 19
1.9 Slutsky substitution effect .......................................................................................... 20
1.10 Measuring changes in consumer’s welfare .............................................................. 22
1.11 Price indices (optional)............................................................................................. 28
1.12 Sample exercise with solution .................................................................................. 30
2. CHOICE UNDER IN-KIND INCOME ....................................................................... 31
2.1 In-kind income ........................................................................................................... 31
2.2 Consumption-leisure model (Individual labour supply) ............................................ 33
2.3 Consumption choices over time: intertemporal choice .............................................. 35
2.4 Production and consumption over time (optional) ..................................................... 39
2.5 Applications of NPV rule: exhaustible resources (optional) ...................................... 41
2.6 Sample exercise with solution .................................................................................... 43
3. CHOICE UNDER UNCERTAINTY ........................................................................... 45
3.1 Gambles and contingent commodities ....................................................................... 45
3.2 Expected utility .......................................................................................................... 50
3.3 Willingness to pay to avoid risk ................................................................................. 52
Application 1. Obtaining additional information ............................................................. 53
Application 2. Demand for Insurance .............................................................................. 54
3.4 The Arrow–Pratt measure of risk aversion ................................................................ 57
3.5 Reducing risk via diversification................................................................................ 58
3.6 Sample exercise with solution .................................................................................... 59
4. GAME THEORY ......................................................................................................... 62
4.1 Simultaneous-move games ......................................................................................... 62
4.2 Sequential-move or extensive-form games ................................................................ 66
4.3 Repeated games .......................................................................................................... 70
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4.4 Sample exercise with solution .................................................................................... 73
5. THE FIRM ................................................................................................................... 75
5.1 Modeling the firm’s technological opportunities ....................................................... 75
5.2 Profit maximization and Cost minimization .............................................................. 77
5.3 Cost minimization ...................................................................................................... 77
5.4 Profit maximization in case of perfect competition ................................................... 85
5.5 Sample exercise with solution .................................................................................... 86
6. PERFECT COMPETITION ......................................................................................... 89
6.1 Perfect competition .................................................................................................... 89
6.2 Equilibrium and efficiency ......................................................................................... 93
Application 1: per unit tax analysis .................................................................................. 96
Application 2: price ceiling .............................................................................................. 98
Application 3: price support program .............................................................................. 98
6.3 Sample exercise with solution .................................................................................... 99
7. GENERAL EQUILIBRIUM AND WELFARE ECONOMICS ................................ 101
7.1 General equilibrium in exchange economy .............................................................. 101
7.2 Pareto optimum in exchange economy .................................................................... 105
7.3 Welfare theorems for exchange economy ................................................................ 108
7.4 Production economy ................................................................................................. 110
7.5 Pareto efficiency in economy with production ........................................................ 114
7.6 Sample exercise with solution .................................................................................. 118
8. MONOPOLY ............................................................................................................. 120
8.1 Pure monopoly ......................................................................................................... 120
8.2 Sources of monopoly and regulatory responses: ...................................................... 122
8.3 Monopolistic price discrimination ........................................................................... 126
8.4 Sample problem with solution.................................................................................. 136
9. OLIGOPOLY ............................................................................................................. 138
9.2 The Stackelberg model ............................................................................................. 140
9.3 Price-setting oligopolists: Bertrand model with homogenous good ........................ 142
9.4 Price leadership or Dominant firm model (optional) ............................................... 143
9.5 Repeated interactions ............................................................................................... 144
9.6 Bertrand model with differentiated goods ................................................................ 147
9.7 Sample exercise with solution .................................................................................. 149
10. FACTOR MARKETS .............................................................................................. 151
10.1 Demand for factors ................................................................................................. 151
10.2 The supply of factors and competitive equilibrium ............................................... 155
10.3 Monopsony and monopoly in factor markets ......................................................... 156
11. ASYMMETRIC INFORMATION .......................................................................... 159
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11.1 Types of asymmetric information problems .......................................................... 159
11.2 Adverse selection and the market for lemons ........................................................ 159
11.3 Adverse selection at insurance market ................................................................... 162
11.4 Private and Government Response to Adverse Selection Problem........................ 165
11.5 Spence model of job market signaling ................................................................... 166
11.6 Screening ................................................................................................................ 168
11.7 Sample exercise with solution ................................................................................ 177
12. EXTERNALITIES AND PUBLIC GOODS ........................................................... 181
12.1 Simple Model of Consumption Externalities ......................................................... 181
12.3 Government Regulation ......................................................................................... 185
12.4 Efficient Provision of Public Good ........................................................................ 186
12.5 Private Provision of Public Good ........................................................................... 187
12.6 The Commons Problem .......................................................................................... 189
12.7 Sample exercise with solution ................................................................................ 191
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1. CONSUMER’S THEORY
Economists assume that consumers choose the best bundle of goods they can afford.
Thus we have to describe more precisely what we mean by the “best” and what we mean by
“can afford”. We start with the concept of affordable bundle.
1.1 Budget constraint
Bundle is a vector of commodities. In an economy with N commodities bundle is described by
vector x  x1 , x 2 ,, x N  , where x i stays for the quantity of good i . Each commodity can
be consumed only in nonnegative amount, i.e. x i  0 .
Key assumption. Consumer is a price taker, i.e. price per unit of a commodity is not affected
by the number of units purchased.
Denote per unit price of commodity i by pi , then p   p1 , p2 ,, p N  is a price vector. If
consumer
purchases
bundle
x
at
prices
given
by
p,
then
he
spends
p1 x1  p2 x 2    p N x N or simply px .
Suppose that consumer has money income M , then he can afford all bundles that cost no
more than M .
We call the set of all affordable consumption bundles at prices p   p1 , p2 ,, p N  and
income M the budget set of the consumer. Budget set contains bundles x  0 that satisfy the
following constraint
p1 x1  p2 x 2    p N x N  M ,
which is called budget constraint.
In case of two goods we can illustrate budget set graphically using budget line.
x2
M
p2
Budget line slope  
p1
p2
Budget set
0
M
p1
x1
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Budget line is the set of bundles that cost exactly
x 2  ( M  p1 x1 ) / p2 , then the slope of budget line is:
M:
p1 x1  p2 x 2  M . As
dx 2
p
 1 .
dx1
p2
Note:
1)
vertical and horizontal intercepts represent bundles in which only one of the
commodities is consumed;
2)
the slope is equal to negative of the price ratio and reflects the opportunity cost of
consuming good 1 (in order to consume more of good 1 you have to give up some
consumption of good 2).
Change in the price of good 1 results in rotation of budget line.
x2
~
p1  p1
M
p2

~
p1
p2

p1
p2
M
~
p1
0
M
p1
x1
Change in income brings a parallel shift of budget line.
~
MM
x2
M
p2

p1
p2
~
M
p2
0
~
M
p1
M
p1
x1
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Non-linear budget constraints

Quantity discount
x2
100
4
M  100, p2  1
p1  4, x1  15 , and
p1  2 for each additional unit
2
0

15
35
x1
In-kind transfer of commodity 1, equal to x1
x2
x1
M / p2
M
p1
M
 x1
p1
x1
The numeriare
Budget set is not affected if all prices and income change proportionally. We can divide all
p
M
~
prices and income by the price of the second good: 1 x1  x 2 
or ~
p1 x1  x 2  M . In
p2
p2
this case we use second good as a numeriare.
2-good case
There are more than 2 commodities, but we can treat good 2 as a composite commodity with
price equal 1. Then good 2 represents the amount of all other (than good 1) commodities that
agent can purchase by spending $1. Notation: AOG
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1.2 Preferences
Preferences are defined over bundles, not goods.
Bundle is a vector of commodities: x  x1 , x 2 ,, x N 
Notation:
 weak preference relationship ‘bundle x is at least as good as bundle y’: x 
y
 strict preference relationship: x  y  x 
x
 y but not y 

indifference relationship x ~ y  x 
 y and y 
x
Assumptions

Completeness: consumer can compare any two bundles and tell, which one he/she prefers
or whether he/she is indifferent between them: for any two bundles x and y we have that
x
 y or y 
 x or both.

Transitivity: for any three bundles x, y, z such that x 
 y and y 
 z we have x 
z
The set of all bundles that are indifferent to a given bundle is called indifference curve:
  
IC x   x : x ~ x 

Implication of transitivity: different indifferent curves do not intersect (prove!).

Non-satiation: more is better
x2
These bundles are
x
preferred to ~
~
x
~
x is preferred
to these
bundles
x1
With non-satiation assumption, indifference curves cannot slope upward.
We will always assume completeness, transitivity and non-satiation if the opposite is not
postulated explicitly. In the second year course we also used the assumption of diminishing
MRS but we are not going to use it as a default assumption this year.
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Marginal rate of substitution of good 1 for good 2 MRS12 (i.e. putting of good 2 in place of
good 1) is the maximum amount of good 2 a person is willing to give up to obtain one
additional unit of good 1 or (in the limit) it is the rate at which commodity 2 must decrease as
commodity 1 increases by an infinitesimal amount to keep the individual on the same
indifference curve.
Suppose that indifference curve (IC) is described by the function x 2  f x1  . Then MRS 12 is
equal to the absolute value of the slope of IC
x 2
x1 0 x
1
MRS 12 x    lim
  lim
x
x1 0
f x1  x1   f x1 
  f x1 x  .
x1
x
Implication of diminishing MRS: convex to the origin indifference curves.
x2
x 2  f x1 
Diminishing
MRS12
x
MRS12 x 
x1
Examples of preferences
Perfect substitutes: goods that can be substituted for each other at a constant rate
Perfect complements: goods that have to be consumed in fixed proportions
“Bads” (violates the assumption of non-satiation)
1.3 Utility
Utility function is a function that assigns a number to every possible consumption bundle such
that more preferred bundles get assigned larger numbers than less preferred bundles and vice
versa, larger numbers are assigned to more preferred bundles.
Denote this function by u .
Note: utility function allows ranking bundles by their amount of utility but it does not allow
precise comparisons of how various bundles are valued relative to each other. Such function
is called ordinal as it only orders the bundles.
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As a result of its ordinal nature utility function is not unique. We can multiply utility by some
positive number and get different utility function that represents the same preferences.
In fact, we can use any positive monotonic transformation. This can be justified in the
following way. Suppose that u is a utility function that represents preferences of some
consumer and gu  is some arbitrary increasing function. Let us demonstrate that
v   g u   represents the same preferences as u . Consider two arbitrary consumption
bundles x and y . As u is a utility function that represents preferences of considered
consumer then x 
 y iff ux   uy . Since gu  is increasing then ux   uy implies that
gux   guy , which means that vx   vy .
Now, let us go in the opposite direction. Suppose that vx   gux   guy  vy . Is it
possible that ux   uy ? If ux   uy then gux   guy as gu  is increasing, and
we get a contradiction. Thus ux   uy iff vx   vy , that is v  gu represents the
same preferences as u if gu  is increasing.
Construction of utility function by assigning numbers to ICs
x2
ue
x
u ~x e
~
x
ux e
0
45
ux
u ~x
x1
Examples of utility functions
Perfect substitutes: ux1 , x 2   x1  x 2
Perfect complements: ux1 , x 2   minx1 , x 2 
Commodity 2 is a “Bad”: ux1 , x 2   x1  x 2
Cobb-Douglas utility function: ux1 , x 2   x1 x 2
Quasiliniar utility function: ux1 , x 2   x1  x 2
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How to calculate MRS
Marginal utility of commodity i - change in total utility due to an increase in consumption of
u
this commodity by (an infinitely small) additional unit: MU i 
.
x i
This concept allows calculating MRS as a ratio of marginal utilities. We start with an
indifference curve ux1 , x 2   u and take a full differential
u1 x1 , x 2 dx1  u2 x1 , x 2 dx 2  du  0 .
Rearranging we get an expression for MRS:
MRS 12 x  
dx 2
dx1

u
ux  / x1 MU 1

.
ux  / x 2 MU 2
Question. Check that MRS for gu is the same as MRS for u if g  0 .
Comment on MRS and convexity
Diminishing MRS (under nonsatiation) implies convex to the origin indifference curves.
Definition: S is a convex set if for any a and b from S a convex combination a  1   b
is also in the set S for any 0    1.
Preferences are convex if for any level of utility u the set of weakly preferred bundles
x : ux   u is convex (see the left panel). The right panel illustrates the case of nonconvex
preferences.
x2
x2
x1
x1
1.4 Consumer’s choice
Utility maximization problem
m ax ux1 ,  , x N 
xi 0
p1 x1  p2 x 2    p N x N  M
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Solution of this problem is called x  p, M  - Marshallian (ordinary) demand and the value of
the problem v p, M   ux  p, M  is called indirect utility function.
Note: non-satiation implies that income is always exhausted, i.e. budget constraint is satisfied
as equality (prove it!).
To solve the problem we setup a Lagrangean:
ℒ  ux1, , x N   M  p1 x1  p2 x 2   pN x N 
FOCs for interior solution:
 ℒ/xi = ux  x i  pi  0 
ux  x i  pi
 ℒ/ = M  p1 x1  p2 x 2    pN x N  0
 MRS ij 
ux  x i pi
p

 i
ux  x j p j p j
Graphical solution for N=2
x2
Interior optimum
 p1 x1  p2 x 2  M ,


 MRS 12 x  p1 / p2
 

 x1  0, x 2  0.
Increase in
utility
x
 

0
x1
Question. Consider any bundle with
x1  0, x 2  0
on budget line such that
MRS 12  p1 / p2 . Explain, why this bundle is not optimal.
Other interpretation of interior optimum condition: marginal utility per dollar spent has to be
the same for all goods:
 
 
MU 1 x 
MU 2 x 

.
p1
p2
FOCs are not sufficient to guarantee a maximum: bundles A and B both satisfy the FOC but
only bundle B gives the maximum utility while A violates second order condition (SOC).
Thus we should check SOC. If non-satiated preferences satisfy diminishing MRS then SOC
will hold automatically.
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x2
B
A
x1
Corner solutions.
x2
x
x2

 
MRS 12 x 
 
MRS 12 x 
0
p1
p2
x1
 
(a) x1  0, x 2  0 и MRS 12 x  
0
p1
p2
p1
p2
x
x1
 
(b) x1  0, x 2  0 и MRS 12 x  
p1
p2
1.5 Comparative statics of Marshallian demand
Comparative statics - comparison of two equilibria.
Marshallian (or ordinary) demand functions x1  p1 , p2 , M  and x1  p1 , p2 , M  we derive
from utility maximization problem.
Income change
Normal good - a good for which an increase in income increases consumption ceteris paribus.
Inferior good - a good for which an increase in income decreases consumption ceteris paribus.
Neutral good - a good for which an increase in income does not affect consumption ceteris
paribus.
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Income elasticity of demand- the percentage change in quantity demanded with respect to a
percentage change in income:  XM  lim
M 0
X  M
X / X
 M
M / M
X
.
For normal goods  XM  0 and for inferior goods  XM  0 .
Income
consumption
curve
x2
x1
0
Own price changes: derivation of individual’s demand curve
x2
M
p2
Price-consumption
curve

0
p3
 1
p2
p2
 1
p2

p11
p2
p10
p2
x1
p1
p13
p12
p11
p10
Demand curve
for good 1
x1
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Ordinary good - a good for which an increase in its price decreases consumption ceteris
paribus (individual demand curve is downward sloping)
Giffen good - a good for which an increase in its price results in an increase in consumption
ceteris paribus (individual demand curve is upward sloping)
x2
Giffen good
0
x1
Price elasticity of demand - the percentage change in quantity demanded of good X with
respect to a percentage change in its price:  Xp  lim
p 0
x
x
X p x  px
X / X

px / px
X
.
X
X
For ordinary good  pX  0 and for Giffen good  pX  0 .
Cross price changes (impact of a change in the price of one good on the quantity demanded
of another good)
Substitutes (goods that satisfy similar wants): an increase in the price of one good leads to an
increase in the quantity demanded of the other good.
Complements (goods that tend to be used together): an increase in the price of one good leads
to a decrease in the quantity demanded of the other good.
Unrelated goods: an increase in the price of one good has no impact on the quantity
demanded of other good.
Cross price elasticity of demand - the percentage change in quantity demanded of good X
with respect to a percentage change in price of good Y :  Xp  lim
p 0
x
Y
X pY  pY
X / X

pY / pY
X
.
X
X
X
Substitutes  pY  0 , complements  pY  0 , unrelated goods  pY  0 .
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1.6 Expenditure minimization problem and duality in consumption
Alternative objective of the consumer: attain a given level of satisfaction in a cheapest way.
Expenditure minimization problem:
m in p1 x1  p2 x 2
x i 0
u x 1 , x 2   u
Solving this problem we get compensated (or Hicksian) demand x  p, u  . In some textbooks
compensated demand is denoted by h p, u  . Plugging it into the objective function we obtain
the corresponding expenditure function e p, u   px p, u  .
Graphical derivation of compensated demand.



Start with identifying the set of bundles that give desired utility, i.e. draw the desired
IC
Illustrate iso-expenditure lines: p1 x1  p2 x 2 =const
Find the point that lies on IC and on the lowest iso-expenditure line:


u x1 , x2  u ,
for interior solution ( x1  0, x2  0 ) we have 
 MRS12 x   p1 / p2
 
Note: compensated demand curve reflects only SE, while ordinary demand curve shows both
SE and IE.
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p1
p1
Negative IE
Positive IE
~
p
~
p
x1  p, u
x1  p, M 
x1  p, u
0
x1  p, M 
x1
(а) normal good
0
x1
(b) inferior (but not Giffen) good
Duality in consumption
1. From utility-maximization (UMP) to expenditure minimization (EMP)
Let x  solves UMP under  p, M  and u   ux    v p, M  .
Consider EMP under p and u  .
If x  solves UMP under  p, M  then x  solves EMP under p, u   and ep, u    M .
2. From expenditure minimization (EMP) to utility-maximization (UMP)
Now, let us start with expenditure minimization problem. Suppose that x  solves EMP under
 p, u  . Let us fix the consumer’s income M  e p, u  and consider utility maximization
under  p, M  . Then x  solves UMP under  p, M  and ux    u .
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Summary of duality results:
x p, M   x  p, v p, M 
(1)
M  e p, v p, M 
(2)
x  p, u  x  p, e p, u
(3)
u  v p, e p, u
(4)
1.7 Slutsky decomposition and Slutsky equation
Price change
Substitution effect
the effect of a price change
on quantity demanded due
exclusively to the fact that its
relative price has changed
Income effect
the effect of a price change on
quantity
demanded
due
exclusively to the fact that the
consumer’s real income has
changed
Under Hickes approach constant real income means that consumer can attain the same
indifference curve, i.e. keeps utility constant.
Slutsky decomposition: x i  x iSE  x iIE  x ih  x i0   x i  x ih 
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x i x iSE x iIE


.
Relative changes:
p1 p1
p1
Sign of Hicksian SE:
x1SE
 0.
p1
Derivation of Slutsky equation
Consider the change in quantity of good i demanded due to the change of good j price.
Differentiate (3) with respect to p j :
x i  p, u  x i  p, M  x i  p, M  e p, u 



.
p j
p j
M
p j
Expenditure will increase proportionally to the quantity of the good consumed (This result is
e p, u 
 x j  p, u 
known as Shephard’s lemma and follows from the envelope theorem):
p j
and due to (3) x j  p, u  x j  p, e p, u  x j  p, M  , where M  e p, u .
Plug back and rearrange:
x i  p, M  x i  p, u 
x  p, M 

 x j  p, M  i
.
p j
p j 
M









IE
SE
Own price version of Slutsky equation:
Own SE can never be positive:
x i  p, M  x i  p, u 
x  p, M 

 x i  p, M  i
pi
p
M
i
 
IE
own SE
x i  p, u 
 0 . Prove it!
pi
Substitution effect is zero in case of kinked IC (for example, in case of Leontieff preferences)
Question: illustrate Slutsky decomposition for the case of Leontieff preferences.
Sign of IE depends on the nature of the good (normal or inferior).
Summary: impact of an increase in the price of a good
Type of good
SE
IE
TE=SE+IE
Normal



Inferior but not Giffen

+

Giffen

+
+
Question: illustrate Slutsky decomposition for the case of Giffen good
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1.8 Alternative approach to consumer’s theory: revealed preferences
Idea: if two bundles x and ~
x are affordable but only x is chosen, then x is revealed
~
preferred to x . Thus, if bundle x  x1, x2  is chosen under  p1 , p2 , M  then it is revealed
preferred to any bundle from the budget set x  0 : p1x1  p2 x2  M
x2
The weak axiom of revealed preference (WARP)
If bundle x is revealed preferred to ~x and the two bundles are not the same, then it cannot
happen that ~x is revealed preferred to x .
x2
M
p2
~
M
~
p2
0
Violation of WARP

~
x~
p, M

x  p, M 
M
p1
~
M
~
p1
x1
~
x  x , then ~
Check WARP: if p1 ~
x1  p2 ~
x 2  M and ~
p1 x1  ~
p2 x 2  M for any  p, M  and
~
~
p, M .


Application of revealed preferences
Replacement of per unit subsidy on good 1 by the lump sum subsidy of equivalent money cost
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x2
M  sx1
p2

x p, M  sx1
M
p2

1
sx
p2
0

x
M
p1
M
p1  s
x1
Question: prove that the vertical red segment gives the cost of subsidy in terms of good 2.
1.9 Slutsky substitution effect
Under Slutsky approach constant real income means that consumer can afford the initial
bundle. Thus we pivot the budget line around the original choice.
Slutsky decomposition with Slutsky SE: x i  x iSE  x iIE  x icomp  x i0   x i  x icomp 
Slutsky substitution effect is derived under the assumption that real income stays constant in a
sense that the old optimal bundle is still affordable under new prices: M comp  px 0 or in our
example M comp  M  p1 x10 .
Observed response= x i  x i  x i0
Compensated response - the change in quantity demanded resulting from changing the price
while simultaneously compensating the individual with income= x iSE  x icomp  x i0 
x i x iSE x iIE


.
Relative changes:
p1 p1
p1
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x2
M comp
p2
M
p2
x comp
x
x

0
p1
p2

0
p10
p2
x1
x
IE
1
x
SE
1
Sign of SE
Own substitution effect is always non-positive
x1SE
 0 (from revealed preferences).
p1
Sign of IE
x iIE x i  x icomp

.
p1
p1
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Note: M  M  M comp  p1 x10 or p1  
IE
x iIE
M
0 x i


x
. As
.
Plug
back:
1
p1
M
x10
x10  0 , then
x iIE
x iIE
 0 if
 0 , i.e. if good i is inferior,
p1
M
x iIE
x iIE
 0 if
 0 , i.e. if good i is normal
p1
M
x iIE
x iIE
 0 if
 0 , i.e. if good i is neutral with respect to income
p1
M
Own price Slutsky decomposition
x1 x1SE
x IE

 x10 1 . If p  0 , then Hicksian SE = Slutsky SE.
p1 p1
M
1.10 Measuring changes in consumer’s welfare
Demand curve as a marginal valuation schedule
Consider a quasi-linear utility function ux1 , x 2   vx1   x 2 , where v  0 and v  0 ,
v0  0 . Let p2  1 . In case of interior solution MRS 12 x  
v1 p1
or vx1   p1 .

1
1
$
Inverse demand
function
Ordinary (Marshallian) consumer surplus (CS) – the difference between what the consumer
is willing to pay and what he has to pay.
 
 
 
CS gross x10   p1 x1 dx1   vx1 dx1  v x10  v0  v x10 as v0  0 .
x10
x10
0
0
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$
Inverse demand
function
 
 
 


CS x10  CS gross x10  p10 x10  v x10  p10 x10  u x10 , M  p10 x10  M as x 2  M  p1 x1 .
Note: the derived equality between CS and utility adjusted for the income is correct for the
quasilinear preferences, do not apply it if preferences are different.
CS is calculated as the area below Marshallian inverse demand function above the market
  p x   p dx or as an area to the left from Marshallian demand
CS x    x  p dp , where p is the cut-off price at which the quantity
price CS x10  
x10
0
function
0
1
1
0
1
1
1
p
p10
1
1
1
demanded becomes equal to zero.
Application of CS
Suppose the price of good 1 goes up from p10 to p1 . What is the resulting change in
consumer’s welfare? It equals to the change in consumer’s surplus
 



CS  CS x1   CS x10  ux1 , M  p1 x1   M  u x10 , M  p10 x10  M 


 ux1 , M  p1 x1   u x10 , M  p10 x10 .
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$
CS after price
increase
Initial CS
Reduction in CS due to
the price increase
Problems with the concept of CS
In the presence of income effect CS is only an approximate measure of consumer welfare.
Reason: demand curve shows the relationship b/w price and quantity demanded holding other
things fixed, including money income. But with fixed money income the value that an
individual puts on an additional unit of a good may depend on the amount that he has already
spent on previous units of the good. As a result, price is not identical to consumer’s marginal
valuation of the associated unit of output.
Other measures of consumer’s welfare
Compensating variation (CV) - the change in money income just necessary to offset the
change in utility induced by the price change
By definition vp0 , M   u0  v p, M  CV  . From duality: M  CV  ep, u0  and



 

M  e p 0 , u 0 , which implies CV  e p, u0  e p0 , u0 .
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CV is measured at the new prices.
Equivalent variation (EV) - the change in money income that is equivalent in its effect on the
individual’s utility to a change in the price of a commodity.


M  e p, u , which implies EV  e p, u  ep , u.

By definition: v p, M   u  v p0 , M  EV . From duality: M  EV  e p0 , u

and
0
EV is measured at the initial prices.
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Relationships b/w EV, CV,  CS
EV and CV can be represented as areas bound by the compensated demand curves.
To measure CV we need a schedule that shows how the quantity demanded varies with price,
assuming that as price changes, the consumer’s money income is adjusted to keep him at
initial level of utility. This is a compensated demand curve that reflects only Hicks SE.
Inverse compensated demand function also can be interpreted as marginal valuation: it gives
marginal valuation for arbitrary preferences (not only quasi-linear one) as marginal valuation
of each dollar is not affected by the price.
Let’s prove that EV and CV can be represented as areas bound by the compensated demand
curves. We start with writing down Shephard’s lemma for good 1 and initial utility level:
e( p, u 0 )
 x1 ( p, u 0 ) .
p1
By integrating this function with respect to good 1 price from p10 to p1 , we can find
p1
 x1 ( p1 , p 2 , u )dp1 
0
p10
p1
e( p1 , p 2 , u0 )
dp1  e( p1 , p2 , u0 )  e( p10 , p2 , u0 )  CV .
0

p
1
p
1
Thus compensating variation can be represented as the area under corresponding compensated
demand curve x1( p1 , p 2 , u0 ) between initial and new prices.
Similarly, from Shephard’s lemma with new utility level we get expression for
equivalent variation:
p1
 x ( p ,p
1
p10
1
2
, u)dp1 
p1
e( p1 , p 2 , u)
dp1  e( p1 , p2 , u)  e( p10 , p2 , u)  EV

p
1
p0

1
Thus equivalent variation is represented by the area under another compensated demand curve
that corresponds to the new level of utility x1 ( p1 , p 2 , u) .
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EV
CS
CV
Finally, we can compare the three measures for the change in consumer’s welfare, resulting
from the considered price increase:
EV  CS  CV ,
where CS - is the corresponding change in consumer surplus.
We can explain this result intuitively. Consider the relationship between CV and CS . Price
increase reduces the purchasing power of income (real income), which in its turn leads to a
fall in demand for normal good. Due to the negative income effect Marshallian demand (that
includes income effect) lies to the left from corresponding compensated demand curve (that
reflects only substitution effect) for any price above p10 . As a result the area bounded by
Marshallian demand is smaller than area bounded by compensated demand curve x1( p, u0 ) .
Implication: Marshallian CS is only an approximation of the true CS measured as an area
below the compensated demand curve.
Question. Why Marshallian CS is widely used?
If price of normal good goes up then EV< CS <CV.
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If price of inferior good goes up then EV> CS >CV.
For the case of neutral good EV= CS =CV as Marshallian and compensated demand curves
coincide.
1.11 Price indices (optional)
Ideally, the changes in the cost of living would be measured by the change in money income
that is necessary for the consumer to achieve the same level of utility in the given year as in
the base year (index based on CV)
Then, if the consumer’s money income increases more (less) then this measure of the cost of
living we can infer that he is better (worse) off.
As compensated demand is not observable this ideal measure of cost of living (ICLI) cannot
be used.
Instead we use some approximations: Laspeyras price index (LPI) and Paashe price index
(PPI).
LPI - the ratio of the sum of given year prices weighted by the base year quantities to the sum
N
of base year prices weighted by the base year quantities: LPI t 
p x
0
i
p
0
i
t
i
i 1
N
i 1
.
0
i
x
x2
px 0 / p20
u LPI
u0
M LPI
p 20
M ICLI
p 20

x p, u 0
M 0 / p20


x  x p ,u
0
x
0
0


p1
p 20

0
p10
p 20
x1
Using base year quantities to weight the prices of goods in a different year, LPI does not
allow for the fact that a consumer tends to substitute away from goods that become relatively
expensive. It implies that an individual whose income is indexed in accordance with LPI can
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purchase base year bundle, i.e. he can never be worse off. Moreover he could be better off by
substituting away from relatively expensive goods. Thus agent might be better off even if his
income increases slightly less then LPI.
Conclusion: LPI overstates increases in the true cost of living.
Paashe price index (PPI) - the ratio of the sum of given year prices weighted by the given year
quantities to the sum of base year prices weighted by the given year quantities:
N
PPI t 
p x
t
i
t
i
p
0
i
t
i
i 1
N
i 1
.
x
PPI understates increases in the true cost of living.
PPI gives a minimum estimate of the increase in the TCL since it assumes (erroneously) that
N
had the consumer received in the base year an amount of income equal to
p
i 1
0
i
x it he would
choose given-year bundle. Instead the consumer would tend to buy relatively more of the
commodities which in base year were cheaper than in given year. This implies that for an
individual whose income is indexed in accordance with PPI, current year bundle was
affordable in the base year. Thus he is never better off. Moreover an agent could be worse off
even if his income rises a bit more than PPI.
x2
px  px  p, u 

p2
p2

p0 x
p2
p 0 x p 0 , u
p2

u0
u
p1
p2

x0

x

x p 0 , u

p10
p2
u PPI
0
x1
What determines the magnitudes of the errors in the LPI and PPI?

the extent to which relative prices change,

the extent to which the consumer substitutes b/w the commodities when relative prices
do change,

individual versus representative agents (average bundle) index.
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1.12 Sample exercise with solution
Bob uses his monthly income (M) to pay for water services and all other goods (represented
by a composite commodity). The price of the water services is p per m3, and the price of
composite commodity is 1. Bob’s preferences are represented by differentiable utility
function.
The local water company cannot cover its cost and considers two options to solve the
problem. It could raise the price by 10%. In this case Bob’s utility level is reduced from u 0 to
u 1 . Alternatively, the water company may keep per unit price constant but in addition
introduce fixed per month charge that results for Bob exactly in the same utility loss.
Which scheme brings more revenue to the water company? Which scheme results in greater
water conservation? Provide graphical and analytical solution.
Solution
Graphical solution
Let x stays for water consumption and y -for AOG. Revenue of water company is given by
the sum of revenue from sales (price multiplied by quantity) and fixed charge. As his income
is the same, then water expenditure equals M  y . Graphically we compare



TR1  p1 x p1 , M  M  y p1 , M





TR2  F  p 0 x p 0 , M  F  M  y p 0 , M  F .
and
From the graph we get TR1  TR 2 and x  x  x  x . Thus the second scheme brings
more revenue but the first scheme provides greater water conservation.
0
1
0
2
y
BC1
TR1
1
u0
u
BC2
TR2
BC-initial
x1
x2 x0
x
Algebraic solution
As bundles x 1 , y1  and x 2 , y 2  provide the same utility, then the change in quantity
demanded is due to Hicksian SE only. We know that own SE is nonpositive (proof to be
provided at class). As relative price goes up when we proceed from x 2 to x 1 and ICs are
smooth (due to differentiability of utility function) then x SE  x 1  x 2  0 . Thus x 1  x 2 ,
which means that water conservation is higher under the first scheme.
Due to nonsatiation with lower consumption of x we can have the same utility only with
increased consumption of y : ux 1 , y1   ux 2 , y 2  and x1  x 2 implies y1  y 2 . Thus
TR1  M  y1  M  y 2  TR2 .
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2. CHOICE UNDER IN-KIND INCOME
2.1 In-kind income
Let us suppose that instead of money income consumer is endowed with commodities bundle:
x1 , x 2  , i.e. he owns
x 1 of the first good and x2 units of the second one. This bundle is
called initial endowment and it is always on the budget line as it can be consumed without
market trade.
If he sells this bundle at market prices, he gets money income M  p1 , p2   p1 x1  p2 x 2 .
Note: now income depends on prices.
x2
x2 
p1 x1
p2
Agent sells good
1
Initial
endowment
x2
Agent buys good
1
p1
p2
x1
0
x1
p x
x1  2 2
p1
The effect of price change on budget set in case of in-kind income
x2
First good price
increase
Become
affordable
x2 
p1 x1
p2
x2
Become
unaffordable
p1
p2
0
x1
p1
p2
p x
x1  2 2
p1
x1
Utility maximization problem with in-kind income
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m ax ux1 , x 2 ,, x N 
xi 0
N
N
p x p x .
i 1
i
i
i 1
i
i
If x  - solution of UMP under price vector p , then

we say that consumer is a net buyer of good i if x i  x i ,

consumer is said to be a net seller of good i if x i  x i .
As income depends on prices, the income effect is different. On the one hand, the monetary
income changes as the same endowment will generate higher monetary income under
increased prices. On the other hand, if we fix the monetary income, its purchasing power is
reduced.
Slutsky equation with in-kind income
Differentiate demand for good i with respect to p j :
dx i  p, px 
x i  p, M 
x i  p, M   px 



.
dp j
p j
M
p j
Using Slutsky equation for fixed monetary income and rearranging we get
dx i  p, px 
x i  p, u 
x  p, M 
x i  p, M 

xj  i

 xj 
dp j
p j
M
M

x i  p, u 
x  p, M 
 x j  x j  i
p
M

j
 
Total IE
SE
Own-price version:
dx i  p, px 
x i  p, u 
x  p, M 

 x i  x i  i
.
dpi
pi
M
Conclusion: in case of in-kind income the sign of IE depends on the type of the good
(normal/inferior) and on the type of the agent (net buyer/net seller). The following table
summarizes the signs of income effect and the overall effect of the own price change taking
into account that own substitution effect can never be positive.
Good type
Agent type
Normal good
Inferior good
Net buyer
Net seller
Net buyer
Net seller
Income effect

+
+

Total effect

+/
+/

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2.2 Consumption-leisure model (Individual labour supply)
One of the applications of the model with in-kind income deals with labour supply. Assume
that we have only two commodities: leisure l  and aggregate consumption c  . Initial
endowment is given by  T , C  . Time endowment is divided between leisure l  and labour
L  :
l  L  T . Denoting the prices by w - wage rate (price of leisure) and p -price of
consumption good we get the following budget set:
pc  wl  pC  wT and 0  l  L , c  0 .
Assuming that preferences are represented by utility function uc, l  that increases in both c
and l , we can find the consumer’s choice from the following utility maximization problem:
max uc, l 
s.t. pc  wl  pC  wT
0  l  T, c  0 .
Interior solution:
MRS lc  w / p and pc  wl  pC  wT .
Reservation wage: w / p  MRS lc (l  T , c  C )
c
Increase in utility
C
wT
p
 w/ p
c
C
L
l
0
l
T
Analysis of the wage rate increase
dl  w, c  wT 
dw

l comp
l
l comp
l
 T  l 

L
.
w
M
w
M
SE
IE
SE: leisure becomes more expensive, thus SE reduces leisure and increases consumption of
aggregate commodity
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IE: Agent is a net seller of labour (he can never be a net buyer). Selling labour time at higher
price raises his real income.
c
 w / p
c0
 w0 / p
c
l IE
0
T
l SE
l
If leisure is normal, then it goes up due to increase in income and IE>0.
If leisure is inferior, then it goes down and IE<0.
Leisure
Normal good
Inferior good
Substitution effect


Income effect
+

 (if SE dominates)
+ (if IE dominates)

Total effect
Derivation of individual labour supply  p  1
c
Labour supply
w2
 w2
 w1
w1
 w0
w0
c
L
0
0
L1
L2
(а)
T
l
L0 L1 L2
L
(b)
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Possibility of backward bending labour supply (if leisure is normal)
c
w
 w2
Labour supply
 w1
IE dominates
w2
w1
SE dominates
 w0
w0
c
L0
0
(а)
L1
L2
T
l
L0 L2 L1
L
(b)
Question. Explain why an increase in the basic wage rate per hour offered to a worker may
decrease the number of hours she wishes to work while an overtime premium offered to the
same worker may increase the number of hours she wishes to work?
2.3 Consumption choices over time: intertemporal choice
Let us assume that there are two periods: current period ( t  0 ) and future period ( t  1 ).
Individual gets income of Y 0 in current period and Y1 in future period. This bundle
corresponds to his endowment point (the bundle of present and future consumption that can be
consumed without market trade). Assume that individual can borrow and lend at the same
market interest rate r .
If individual consumes less than he earns in the current period, then the difference
Y0  c 0  is saved and in the next period the agent gets additional income equal to
Y0  c0 1  r  . Thus his future consumption equals
c1  Y1  Y0  c 0 1  r  . If currently the
agent wants to consume more than he earns, then he has to borrow c 0  Y0  and in the future
he will repay the debt together with interest payments, thus his future consumption equals
c1  Y1  c 0  Y0 1  r   Y1  Y0  c 0 1  r  . It means that irrespective of whether agent
borrows or lends, his budget constraint is
c1  Y1  Y0  c 0 1  r  .
If we open the brackets and put consumption in the LHS, then the budget constraint can be
rewritten as
c 0 1  r   c1  Y0 1  r   Y1 .
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In the LHS we have the future value of the life-time consumption and in the RHS- the future
value of the life-time income. By dividing both sides by 1  r  the budget constraint can be
stated in terms of present values
c0 
c1
Y
 Y0  1 .
1 r
1 r
This budget constraint states that present value of lifetime consumption has to be equal to the
present value of endowment.
Graphically this intertemporal budget constraint can be represented by a straight line that
goes through endowment point and has a slope of  1  r  .
c1
Saves in
period 0
Borrows
in period 0
Assume that agent derives utility from consumption in both periods uc 0 , c1  . Let more
consumption in either period be preferred to less, so that utility increases as we move further
from the origin. If we assume diminishing MRS then we get convex indifference curves.
The MRS between current and future consumption reveals the intensity of individual’s
preferences for consumption in different periods of time. If we write down MRS 01 1   ,
then  is the rate of time preference.
A person is said to be impatient if when consumption levels are the same in both periods  is
positive, meaning that person is willing to forego more than $1 of future consumption to
increase current consumption by $1.
A person is said to be patient if when consumption levels are the same in both periods  is
negative, meaning that person is willing to forego less than $1 of future consumption to
increase current consumption by $1.
Intertemporal utility-maximization problem:
m ax uc0 , c1 
c 0  0 ,c1  0
c0 
c1
Y
 Y0  1
1 r
1 r
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In case of interior solution the point of tangency of budget constraint with IC indicates the
optimal consumption bundle. If c 0  Y0 , then agent is called a net lender (saver), if c 0  Y0 ,
then agent is called a net borrower.
c1
Net lender
1+r
c0
Conclusion: if financial market are perfect (agents can lend and borrow at the same interest
rate), then consumption decision is determined by the present value of life-time income, not
the income in current or future period alone.
Comparative statics
In this model interest rate plays a role of price.
An increase in the interest rate brings two effects: substitution effect and income effect. Due
to substitution effect current consumption falls as it becomes relatively more expensive. The
sign of income effect depends on whether we deal with net lender or net borrower (as
consumption stands for aggregate commodity it is treated as a normal good in each period).
For net borrower an increase in the interest rate decreases wealth and results in a fall in
current consumption. So for net borrower both effects move in the same direction and current
consumption definitely falls.
An increase in the interest rate increases the wealth of net lender and under given prices
results in an increase in current consumption. So for net lender current consumption falls and
saving increases if substitution effect dominates and current consumption rises together with
fall in saving when income effect dominates.
We can get the same results from the analysis of Slutsky equation. Income effect is
proportional to the amount saved (S), that is why it may become dominant if S is large
enough:
c 0 c 0comp
c

 (Y 0  c 0 ) 0 .
r 
r 
M


SE
IE
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Agent type
Net borrower
Net lender
Substitution effect

-
Income (wealth) effect

+
Total change in c 0

+/
Change in
borrowing/lending
 Borrowing=
 Lending=
= c0  Y0   c0  0
 0 if SE dominates
= Y 0  c 0   c 0 
 0 if IE dominates
Conclusions. Individual demand for borrowing is downward sloping, so does the aggregate
demand. Individual supply of lending could be backward bending (upward sloping under low
saving).
Lending-borrowing equilibrium.
Note: below we assume that backward bending part of individual supply disappears in process
of aggregation.
r
Supply of
lending
r
0
Demand
for
borrowing
B   L
Lending,
borrowing
More than 2 periods and bonds pricing
Consider a bond that pays a fixed coupon amount x each period (starting from the next period)
until a maturity date T and at T the face value F is paid.
If we denote the discount factor by   1/(1  r ) then the price of the bond i
P  x  2 x    T 1x  T F .
Let S  1    2    T 1  T then S     2    T  T 1  S 1  T 1 which gives
S
1  T 1
.
1 
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The price of a consol (perpetuity that never matures) we get as a special case, where F=0 and
1
1/(1  r )
x
T=. Thus the price is PCONSOL  x   2 x    T x    x
x
 .
1 
1  1/(1  r ) r
The price of a bond has an inverse relationship with the rate of interest.
2.4 Production and consumption over time (optional)
Suppose that investment (productive) opportunities are available but consumer has no access
to the financial market. Investment opportunities are described by the PPC.
PPC
In equilibrium ct  Qt and I   S  .
If PPC is given by Q1  F Y0  I  , then the consumption and production decision is given by
the solution of the problem
max uc 0 , c1 
s.t . Q1  F I , c 0  Q0  Y 0  I , c1  Y1  Q1
.
Suppose that investment (productive) opportunities are available and in addition consumer
can borrow and lend at the same market interest rate.
Consumption
Production
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c1
 W0 . The level of
1 r
wealth is the intercept of budget line with the horizontal axes. The highest attainable budget
line is tangent to the PPC. The corresponding highest level of wealth equals W0 . Then
Each budget line is associated with a specific level of wealth: c 0 
consumer chooses the best bundle under given level of wealth.
Investment financing.
Net lender I  Y 0  c 0   Q0  c 0  , net borrower I  Y 0  c 0   c 0  Q0 




 
 




own saving
lending
own saving
borrowing
On aggregate lending=borrowing, which implies that aggregate saving equals aggregate
investment.
Separation of consumption and production decisions
m ax uc 0 , c1 
c1
Y  Q1
 Q0  1
1 r
1 r
Q0  Y 0  I , Q1  F I 
s.t . c 0 
Note that production decision does not depend on consumers preferences.
FOC: F I    1  r .
Investment are chosen to maximize the consumers wealth, i.e.
Y  Q1 
Y  F I  
Y
F I  



maxW0  max Q0  1
  max Y 0  I  1
  Y 0  1  max  I 
.
1 r 
1 r 
1 r
1 r 



Separation theorem
If markets for intertemporal claims are perfect, individuals can separate investment decisions
(aimed at maximizing wealth) and consumption decision (dependent on consumer’s time
preferences).
Present value rule
Due to separation theorem production decision can be delegated to managers. Managers that
maximize the wealth of the firm will be making the correct investment decisions for all the
owners individually regardless of the possibly differing time-preferences of the owners.
Net present value of investment project in two-period model: NPV I    I 
F I 
.
1 r
If we have T periods and Rt I  is the net income in period t , then
NPV I   R0 I  
R1 I  R2 I 
R I 

   T 1 T 1
2
1  r 1  r 
1  r 
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Discrete case. If the number of investment projects is finite and these projects are mutually
exclusive, then the project with the maximum possible present value should be chosen (given
that it is positive).
If projects are not mutually exclusive then all projects with positive NPVs should be adopted.
2.5 Applications of NPV rule: exhaustible resources (optional)
Consider exhaustible resources industry (minerals or fossil fuels). The stock of the resource
Q is constant. The owner should decide how much of the stock to extract and sell in each
period. Consider two-period model. Assume that demand function is stable over time and
P q  - diminishing in q . Let extraction costs per unit of resource be constant and equal to c
in any period.
Competitive industry.
In case of competitive industry any firm is a price taker. Let pt stay for current price and
pt 1 - for future price.
If
pt 1  c
 pt  c , then it is profitable to postpone extraction and sell in future period
1 r
If
pt 1  c
 pt  c , then it is profitable to sell now
1 r
If
pt 1  c
 pt  c , then it is profitable to sell in both periods
1 r
Thus in competitive equilibrium extraction takes place in both periods if and only if
 pt 1  c    pt  c   r , i.e. price minus extraction costs (marginal
pt 1  c
 pt  c or
pt  c
1 r
profit) rises at the rate of interest.
The result can be explained intuitively. If marginal profit increases less than market interest
rate, then it is profitable to extract and sell the resource today and put money at bank deposit.
If the opposite is true, it is optimal to postpone extraction. In any case the profit maximizing
company has an incentive to change its production decision, which implies that currently
there is excess supply (if extraction today increases) or excess demand (if extraction is
postponed), which is not compatible with equilibrium. So, in equilibrium marginal profit
should grow at the rate equal to market interest rate.
The result can be derived formally from the profit maximization of representative firm:
m ax
q0 ,q1 0
 p0 q0  TC (q0 )   p1q1  TC (q1 ) /(1  r )
s.t . q0  q1  Q
FOC for interior solution implies:
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p0  TC (q0 )   p1  TC (q1 )/(1  r ) or
 p1  c    p0  c   r .
p0  c
This rule is known as Hotelling rule.
As extraction costs are constant, this implies that price will increase over time while
extraction falls (due to declining demand).
Graphical solution for linear demand functions.
Monopolistic industry
As monopolist is a price maker, marginal profit equals to MR  c , thus he will extract in both
MRt 1  c
periods if
 MRt  c , which implies that marginal revenue less marginal
1 r
MR1  c   MR0  c   r
extraction costs increases at market interest rate:
.
MR0  c
Analytical derivation of the Hotelling rule for the monopolistic industry.
m ax TR q0   TC (q0 )  TR q1   TC (q1 ) /(1  r )
q0 ,q1 0
s.t . q0  q1  Q
FOC for interior solution implies:
TRq0   TC (q0 )  TRq1   TC (q1 )/(1  r ) or
MRq1   c   MRq0   c   r
.
MRq0   c
Graphical solution and comparison with competitive case (linear demand functions and zero
marginal extraction costs).
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Graphical analysis suggests that monopolist would be more conservative: he would extract
less today and more tomorrow. As a result the prices for a monopolized industry would be
initially higher and become lower at later dates.
price
monopoly
Competitive
industry
time
extraction
monopoly
Competitive
industry
time
2.6 Sample exercise with solution
Explain, why an increase in the basic wage rate per hour offered to a worker may decrease the
number of hours she wishes to work while an overtime premium offered to the same worker
may increase the number of hours she wishes to work?
Solution
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Under overtime premium the initial bundle is just affordable and we observe a pivot of BL
around the previous choice. It means that Slutsky IE=0 and SE works in opposite direction to
the price change: with increased overtime payment leisure is more expensive, thus person will
reduce his consumption of leisure (moves from A to B). Thus, he will work more.
Under an increase in the basic wage rate the budget line becomes steeper and individual’s real
income increases as he can generate higher income supplying the same amount of labour.
Thus in addition to the SE that reduces leisure we observe income effect that increases
demand for leisure (in this case leisure must be a normal good) and it might happen that IE
dominates the SE so that total demand for leisure increases and, as a result, labour supply
decreases as it is demonstrated below.
c
w  / p
B
C
c0
w0 / p
A
IE
c
Lpremium
SE
0
leisure
T
LWage _ increase
l0
L0
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3. CHOICE UNDER UNCERTAINTY
3.1 Gambles and contingent commodities
A state of the world is the outcome of uncertain situation.
Contingent commodity is the amount of consumption, the level of which depends on the state
of the world occurring.
Flipping coin game
Suppose for each dollar you bet in a flipping coin game, you win  (and get your bet back) if
a heads comes up and lose your bet when a tail comes up.
States of the world: state 1- tails comes up and state 2- heads comes up.
Contingent commodities: consumption if tails comes up (denote by c1 ) and consumption if
heads comes up ( c 2 ).
Endowment point - consumption bundle of contingent commodities that is available when you
make no trades with the market.
In case of flipping coin game initially person has income of w in either state of the world.
Budget constraint for contingent commodities shows how much of each contingent
commodity you can have in each state of the world.
Let us denote the bet by z and assume that bet can never be negative, then we have the
following system that describes the budget constraint:
c1  w  z

c 2  w  z
0  z  w

If we solve this system with respect to z , then we get the budget constraint of the form:
c 2  w  w  c1  or c1  c 2  w1   where 0  c1  w .
w1   
endowment
1
c1
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Note that after the state of the world is determined, the person will consume only one
contingent commodity that corresponds to the state of the world that takes place.
Budget constraint does not extend to the right from endowment point as it was assumed that
agent is not allowed to select the other side of the original bet. That is, we do not allow
individual to make a bet in which he wins $1 if tails comes up and lose $ if heads comes up.
If person is allowed to take both sides of the gamble, then his budget constraint will be a
straight line that goes through initial endowment with a slope of   .
endowment
1
Budget constraint when both sides of the bet can be taken
Generalisation
Suppose that the terms of the gamble are such that consumption changes by x 1 in the first
state of the world (if tail appears) and by x 2 in the second state of the world (if heads
appears). Then we have the following system that describes the budget constraint:
c1  w  x1 z

c 2  w  x 2 z
If we solve this system with respect to z , then we get the budget constraint of the form:
c2  w 
Then the slope of the budget line equals
x2
w  c1  .
x1
dc 2 x 2

.
dc1 x1
Fair odds line
Each state of the world s can occur with some probability p s , where 0  ps  1 and
S
p
s 1
s
 1.
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Expected value of the gamble is the weighed sum of outcomes, where weights are equal to
S
probabilities: EV x    p s x s .
s 1
EV of the gain in original flipping coin game is  $1 p1    p2  $1 p1    1  p1  . If
coin is symmetric then p1  0.5 and EV gamble = 0.5  1 .
A gamble with zero expected monetary gain is called a fair gamble.
If expected monetary gain is different from zero, then such a gamble is said to be unfair.
Gamble with positive expected gain is said to be favourable; gamble with negative expected
gain is called unfaivourable.
The fair odds line is a budget constraint reflecting the opportunities presented by an
actuarially fair gamble (odds - the ratio of the probabilities of the two events).
With two states of the world fair gamble satisfies the condition x1  p1  x 2  p2  0 , which
implies that
x2
dc 2 x 2
p
p
  1   1 . As slope of budget line is

, the absolute value of
x1
dc1 x1
p2
1  p1
the slope of fair odds line equals to the ratio of the probabilities
dc 2 x 2
p

 1 .
dc1 x1
p2
Going back to our example of symmetric flipping coin we can illustrate the fair odds line as a
p
1/ 2
 1 . If   1 , the initial gamble is favourable
straight line with the slope of  1  
p2
1/ 2
and fair odds line would be flatter then budget line.
Expected
gain>0
Expected
gain<0
Fair odds line
c1
Note that expected value of consumption remains constant along fair odds line:
EV c   c1  p1  c 2  p2  w  x1 z   p1  w  x 2 z   p2  w  zx1  p1  x 2  p2   w .
Preferences
Three types of attitude toward risk can be distinguished.
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A person is said to be risk averse if he prefers a certain prospect with a particular expected
value to an uncertain prospect with the same expected value.
A person is said to be risk neutral if he is indifferent between a certain prospect with a
particular expected value and an uncertain prospect with the same expected value.
A person is said to be risk loving if he prefers an uncertain prospect with a particular expected
value to a certain prospect with the same expected value.
In order to illustrate certain prospects we will draw certainty line – the locus of all possible
certain consumption levels (i.e. the line c 2  c1 ).
Indifference curves for risk neutral agent are given by straight lines parallel to the fair odds
line. Reason: any uncertain prospect for a risk neutral agent is equivalent to certain bundle
with the same EV. Note that along FOL expected value of consumption is constant, thus all
these points lie on the same indifference curve. As more is better, agent becomes better off
while moving along certainty line further from the origin.
Increase in utility
FOL
To illustrate indifference curves for a risk averse agent, let us take two points on fair odds
line: certainty point A and some uncertain prospect B.
FOL
FOL
(1)
(2)
As both bundles (A and B) have the same expected value of consumption but A is certain,
then by definition risk averse agent would prefer certain bundle A to any uncertain prospect
like B that gives the same EV of consumption. It means that all points on FOL would give
lower utility than A. In other words, A belongs to indifference curve that lies further from the
origin. As a result ICs cannot be bowed outward as in diagram (1). Otherwise point B would
bring higher utility than A, which contradicts to risk aversion.
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ICs cannot cross fair odds line at certainty line as in diagram (2). Otherwise risky prospect
(D) would be equivalent to the certain one (A) with the same EV.
Thus the IC of risk averse agent satisfies the following properties:
 absolute value of the slope (MRS) at certainty points is equal to the ratio of probabilities
(absolute value of the slope of FOL);
 ICs are bowed in.
c2
FOL
c1
Exercise. Show that indifference curves of a risk lover are bowed out and at certainty points
have slope that is the same as the slope of FOL.
Optimal bet: the case of risk averse agent
By definition, a risk-averse agent will never participate in fair game (i.e. will make zero bet)
as his initial endowment lies on certainty line and is preferred to any risky prospect that
belongs to FOL.
Optimal choice
FOL=Budget line
Fair game
If game is favourable (this is the case if   1 ), then risk averse agent will take some risk and
optimal bet would be positive as risk is compensated by positive expected gain.
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Budget line
bet
FOL
Unfair favourable game
If game is unfavourable (this is the case if   1 ), then risk averse agent will make zero bet as
expected consumption at any point on budget line is less than at initial endowment and, in
addition, the endowment point is certain.
Optimal choice
Budget
line
FOL
Unfair unfavourable game
3.2 Expected utility
In presence of uncertainty utility depends on the quantities of contingent commodities and
corresponding probabilities. In principle, probabilities can enter utility function in quite
complex ways. Under some additional requirements on preferences utility function takes the
S
form which is linear in probabilities: U c1 , c 2 ,, c S ; p1 , p2 ,, p S    ps uc s  . A utility
s 1
function that takes this form is called a von Neumann-Morgenstern utility function or expected
utility function (EUF).
It is not entirely an ordinal function as only positive affine transformations are allowed:
aU  b , a  0 .
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EUF and attitude toward risk
Risk-averse person: U c1 , c 2 ; p,1  p  puc1   1  puc 2   u pc1  1  pc 2  for any
p  0, 1 and c1  c 2 . This is Jensen inequality which implies that uc  is strictly concave. It
implies that the marginal utility of a risk averse agent is decreasing in wealth.
u
Risk averse agent
uc2 
uc1 
c
0
Risk-neutral person: U c1 , c 2 ; p,1  p  puc1   1  puc 2   u pc1  1  pc 2  for any
p  0, 1 , which implies that uc  is linear.
u
Risk neutral agent
uc2 
uc1 
0
c1
c1
c
Risk-loving person: U  c1 , c2 ; p, 1  p   pu  c1   1  p  u  c2   u  pc1  1  p  c2  for
any p  0, 1 and c1  c 2 . This is a Jensen inequality which implies that utility function of a
risk-loving person uc  is strictly convex so that the marginal utility is an increasing function
of wealth.
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3.3 Willingness to pay to avoid risk
Certainty equivalent (CE) of a gamble – the certain wealth that would make an agent
indifferent b/w accepting the gamble and accepting the certain wealth:
uCE   puc1   1  puc2 
Let us show that for risk-averse agent certainty equivalent is less than the expected value of a
gamble. By definition of risk aversion: uCE   puc1   1  puc2   u(EV ) . Since uс  is
increasing then CE  EV .
It means that a risk averse person is ready to pay a risk premium EV  CV  0 to avoid the
risk, that is, to exchange a gamble for its expected value.
u
uc2 
uc1 
Risk
premium
c1
0
c
Example
Consider risk averse individual with initial wealth W . With probability p she can incur
losses of L , 0  L  W . Individual is offered to purchase full insurance that will compensate
all the loss in case of accident.
What is the maximum premium that this agent is willing to pay for this insurance? The
maximum premium should make this person indifferent between purchasing insurance and
staying at initial endowment. The expected utility at initial endowment is
EU NO _ INS  puW  L   1  puW  while with full insurance under premium R his utility
equals EU INS  uW  R . Thus the maximum premium could be derived from the following
equation
uW  RMAX   puW  L   1  puW  .
Note that W  RMAX  CE , which implies that this premium is given by the difference
between initial wealth and certainty equivalent:
RMAX  W  CE .
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u
c
Note: insurance premium  risk premium since insurance premium is calculated as a change
in wealth, not the change in expected wealth.
u
c
What is the minimum premium that a risk-neutral insurance company is willing to accept? For
a risk neutral agent expected utility is given by expected wealth. Thus, insurance company
will offer insurance iff the resulting expected profit is nonnegative:   R  pL  0 . Under
minimum premium   0 , which implies
RMIN  pL  W  EV .
Application 1. Obtaining additional information
Mary has a utility function EU economist  50  120 / c , where c is her consumption, measured
in thousands of dollars. If Mary becomes an economist, she will make 30 thousand per year
for certain. If she becomes a pediatrician, she will make $60 thousand if there is a baby boom
and $12 thousand otherwise. The probability of a baby boom is p=0.5.
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economist
pediatrician
boom
p= ½
30
no
p= ½
30
boom
p= ½
60  u(60)=50-120/60=48
no
p= ½
12 u(12)=50-120/12=40
By comparing utilities, we can find that Mary prefers to become an economist as
EU economist  46  44  EU pediatrician .
Now suppose a consulting firm has prepared demographic projections that indicate which
event will occur. Will she purchase the projection at price of $6000?
economist
30-6=24
pediatrician
60-6=54
boom
p= ½
economist
30-6=24
no
p= ½
pediatrician
12-6=6
As the maximum utility achieved without demographic projection was only 46 Mary would
be willing to purchase this projection.
Application 2. Demand for Insurance
Re-consider an example with insurance but now assume that any amount of insurance (full or
partial) might be purchased at insurance premium r per dollar of insurance coverage.
Insurance is actuarially fair if EV  $1 p  $r  0 or r  p .
If r  p , then insurance is unfair. The case of unfair favourable insurance ( r  p ) is quite
unrealistic as insurance company incurs loss under this price.
Thus we will consider only fair and unfavourable insurance.
Two states of the world: “loss” and “no loss”.
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Two contingent commodities: c L - consumption in case of loss, c NL - consumption in case of
no loss.
Denote by x the quantity of insurance and assume that over-insurance is not allowed (you
cannot insure more than L), then
c L  w  L  rx  x

c NL  w  rx
0  x  L

If we solve this system with respect to x , then we get the budget constraint of the form:
c NL  w 
r
w  L  c L  where w  L  c L  w  rL .
1 r
The slope of budget line is  r /1  r  .
endowment
W
budget line
W-L
The slope of fair odds line equals 
p
.
1 p
In case of fair insurance budget constraint coincides with the fair odds line and as a result
utility is maximised at certainty point, where indifference curve is tangent to budget line,
which means that person will purchase full insurance: c L  w  L  x  1  r   c NL  w  rx 
or x   L .
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Optimal choice
Fair insurance
Thus any risk averse agent under fair insurance purchases full insurance (insures all the loss).
In case of unfair unfavourable insurance ( r  p ) budget line is steeper than fair odds line. As
a result risk averse person will never purchase full insurance. He will either purchase partial
insurance ( 0  x   L ) or no insurance at all.
Optimal choice
Optimal choice
Unfavourable insurance
These results could be also derived algebraically if we setup and solve the expected utility
maximization problem:
maxpuW  L  X  rX   1  puW  rX 
X 0
As EU is strictly concave because of the risk-aversion then the FOC is both necessary and
sufficient.
FOC:




1  r p u W  L  X  1  r   r 1  p u W  rX   0

 0 if

X  0
Let us start with the fair insurance r  p and rearrange
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



u W  L  X  1  p   u W  pX   0

 0 if

X  0
Claim 1. X   0 . Suppose that this is not the case and X   0 . Then uW  L   uW   0
due to diminishing marginal utility. But this inequality violates FOC.
Claim 2. X   L .
Since X   0 FOC could be restated as uW  L  X  1  p  uW  pX    0 . Due to
diminishing marginal utility this requires equality of the levels of wealth
W  L  X  1  p  W  pX  , which implies X   L .
Now we proceed to unfair unfavourable insurance r  p . In this case it might happen that no
insurance is purchased:


EU  X   0  1  r p uW  L   r1  puW   0 .
If X   0 then 1  r p uW  L  X  1  r   r 1  puW  rX  . Rearranging we get:


u W  L  X  1  r  r 1  p 

1  r p
u W  rX 


1  r   1  p and 1  r p  r1  p . Thus we obtain
uW  L  X  1  r  r 1  p 

 1 , that is uW  L  X  1  r   uW  rX   . As u is

1  r p
uW  rX 
diminishing then W  L  X  1  r   W  rX  and we get that only partial insurance can be
As
rp
then
purchased X   L .
3.4 The Arrow–Pratt measure of risk aversion
How do we measure the degree of risk aversion?
Intuition: the degree of risk aversion should reflect the curvature of utility function, which
means that we should use the second derivative: u  . For risk-averse agent the second
derivative is negative and it is more convenient to put minus sign before the derivative. But
utility function is not unique: u and au  b, a  0 represent the same preferences. Solution:
divide by the first derivative.
Arrow–Pratt coefficient of absolute risk aversion:   
u
u
Result: the larger the Arrow-Pratt measure of risk aversion, the smaller gambles an individual
will take.
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This coefficient may change with wealth.
Example 1: u  lnW 
u  1/ W
and
u  1/ W 2 imply that coefficient of risk aversion   
u 1

u W
decreases as individual’s wealth goes up.
Conclusion: with an increase in wealth this person will take higher risks (buy less insurance
or invest more in risky asset).
Example 2: u  e aW , a  0
u  ae a
and
u  a 2 e a imply that coefficient of risk aversion   
u 
a
u
doesn’t change with an increase in wealth.
Conclusion: with an increase in wealth this person will take the same risks (buy the same
amount of insurance or invest the same sum in risky asset). This function is called CARA
(constant-absolute-risk-aversion) utility function.
3.5 Reducing risk via diversification
Diversification - the spreading of risk among several alternatives rather than choosing one
Example 1. Consider two companies A and B and assume that their performance depends on
the weather (see the table below).
A
B
Sunny (p=1/2)
100
20
Rainy (p=1/2)
20
100
If he invests in A or B only then his expected utility is EU A  EU B  u20  u100/ 2 .
u
EU0.5A+0.5B
EUA =EUB
20
60
100
c
Suppose that he invests 50% in A and 50% in B, then he gets (20+120)/2=60 in each state and
utility goes up: EU 0.5 A0.5 B  u(60)  u20  u100/ 2 .
Conclusion: If returns are perfectly negatively correlated diversification eliminates the risk
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Example 2. Assume that the shares of two companies (A and B) bring 100 or 20 with equal
probabilities but now returns are independent.
If a risk-averse agent invests in A or B only, he gets EU A  EU B  u20  u100/ 2 .
If he invests 50% in A and 50% in B then the following outcomes may be observed
Prob
1/4
Wealth
1/4
(20+20)/2=20
1/4
(20+100)/2=60
1/4
(100+20)/2=60
(100+100)/2=100
1
1
The resulting expected utility is EU 0.5 A  0.5 B  u(60)  u20  u100 .
2
4
Compare it with the initial one:

1   20  100  1
EU 0.5 A  0.5 B  EU A  u
  u20  u100  0 .
2 
2
 2

Graphical analysis for independent returns
We will proceed in two steps.
Step 1: EU A 
1
u20  u100
2
u
u(60)
EU0.5A+0.5B
EUA =EUB
20
Step 2:
100
60
c
1
1
EU A  u60 .
2
2
3.6 Sample exercise with solution
Susan
has
preferences,
represented
by the


12 
12
  1  p  5 
EU (c1 , c 2 , p)  p 5 
1  c1 
1  c2


following
expected
utility function

 , where c i stays for wealth in the state of the

world i i  1, 2 and is the probability of state 1. Suppose that Susan has $7 and in addition
owns a risky investment project that brings -$2 with probability 1/3 and $4 with probability
2/3.
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(a) Find the minimum sum (Xmin) at which this individual is willing to sell this project?
(b) Compare Xmin with the expected value of this project and explain the result.
(c) Produce graphical solution for part (b) using contingent commodities diagram.
(d) Dan is a risk-averse person with a smooth elementary utility function. He has the same
initial wealth and the same risky project as Susan. Assume that he was offered to sell the
project fully or partially at the price of $1 and he decided to sell 50% of the project. Is it true
that he will sell greater share of the project if the probability of loss goes up?
Solution
(a) Suzan will sell iff her utility does not diminish. Xmin is the sum that makes Suzan
indifferent between selling and keeping the project. Thus, in both cases EU should be the
1
same. EU project   5 
3
EU sell  5 
12  2 
12  1
2
3  2  4 11

  5 
  5  2  5  1 
1 7  2  3  1 7  4  3
3
3
3
12
11
36  32
12
4

,
 1.
 , X min 
1  7  X min
3 8  X min 3
4
(b) EV project  1   2  2  4  2  X min  1
3
3
Explanation.
If agent gets EVproject then he has the same expected consumption but if the project is sold
then she has no risk. By definition for a risk averse person (verify that Susan is risk averse)
this option is strictly preferred to the one with risk under the same EV, thus he will sell the
project even if the price is a bit lower.
(c) Contingent commodities graph with comments
11
E
-1/2
7+EV
7+Xmin
7
EV
Xmin
W
5
Fair odds lines
7+Xmin7+EV
c1- consumption if project is not successful
c2- consumption if project is successful
E- initial endowment if the person owns the project
W-initial wealth (without project)
Slope of FOL=-(1/3): (2/3)=-1/2
(d) Dan is a risk-averse person with a smooth elementary utility function. He has the same
initial wealth and the same risky project as Susan. Assume that he was offered to sell the
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project fully or partially at the price of $1 and he decided to sell 50% of the project. Is it true
that he will sell greater share of the project if the probability of loss goes up?
True. Optimal investment can be derived from
max pu7  21        1  pu7  41      
0, 1
FOC for interior solution: pu5  3  1  pu11  3 (*)
Let us prove that >0. Assume 0 then c1  3  0  uc1   0 as MU is
diminishing for a risk-averse person. Since p  0 then puc1  increases. c 2  3  0
 uc 2   0 as MU for a risk-averse person
Since 1  p  0 then 1  puc 2  decreases
AS the LHS of (*)  while the RHS  then they cannot be equal and (*) is violated. This
contradiction proves that >0.
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4. GAME THEORY
4.1 Simultaneous-move games
A situation of strategic interactions is called a game.
To specify a simultaneous-move game we have to identify

players(the decision makers): i  1, 2,, n ;

the set of strategies of each player: S1 , S2 ,, Sn
(strategy - complete description of the plan of actions of the player);

payoffs for each strategy profile: ui s1 , s2 ,, sn  , i  1, 2,, n
Simultaneous-move game can be represented in a matrix form (known as the normal form).
Strategy ~
s is a dominant strategy for player i if it performs better than any other strategy of
i
player i no matter what others are playing:
u ~
s , s   u s , s
i
i
i
i
i
i
 for all
si  ~
si and all s i .
If each player has a dominant strategy, then we get a dominant strategy equilibrium.
Example. Prisoners’ Dilemma.
Two prisoners who were partners in a crime were being questioned in separate rooms. Each
can either confess to the crime (and thereby implicate the other) or deny his participation in
the crime.
If only one prisoner confessed, then he would go for free and the other would spend 6 months
in prison. If both denied, then each would spend one month in prison (time required for
investigation) and if both confessed, then each would spend 3 months in a prison.
Note that each player would be better off by choosing strategy ‘confess’ whatever is the
strategy chosen by the other as 0 > -1 and -3 > -6. Thus in Prisoners Dilemma each player
would choose strategy confess, that is (‘confess’, ‘confess’) is a dominant strategy
equilibrium. Note, that the outcome of this equilibrium is inefficient as both agents would be
better off by playing (Deny, Deny). But (Deny, Deny) outcome cannot be achieved as players
are unable to cooperate.
Player B
Player A
Deny
Confess
Deny
-1,-1
-6, 0
Confess
0, -6
-3, -3
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Iterative elimination of dominated strategies (IEDS)
Even if a player does not have a dominant strategy, he might have one or more dominated
strategies.
Strategy si is a strictly dominated strategy for i if it yields a lower payoff compared to
another strategy (say s i ) irrespective of what others are playing:
ui si , si   ui si , si  for all s i
Rational agent will never use a dominated strategy, thus we can eliminate it.
If all strategies except one for each player can be eliminated by iteratively eliminating strictly
dominated strategies, the game is said to be dominance solvable.
Example. Consider the following simultaneous-move game with two players.
Player 2
Player 1
A2
B2
C2
a1
2,2
4,2
0,4
b1
4,0
6,8
2,2
c1
6,4
4,0
0,6
Step 1: eliminate a1 as it is dominated by b1 (2 < 4, 4 < 6, 0 < 2) and eliminate A2 as it is
dominated by C2 (2 < 4, 0 < 2, 4 < 6).
Step 2: Remaining game
Player 2
Player 1
B2
C2
b1
6,8
2,2
c1
4,0
0,6
Eliminate c1 as it is dominated by b1 (4<6, 0<2).
Step 3: Remaining game
Player 2
Player 1
b1
B2
C2
6,8
2,2
Eliminate C2 as it is dominated by B2 (2 < 8).
Equilibrium obtained by IEDS: (b1, B2)
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Nash equilibrium
In some games IEDS do not produce a unique outcome. To solve the games of this sort we
should increase our requirements to the rationality of the players. This bring us to the concept
of Nash equilibrium.
A strategy profile s1 , s2 ,, sn  is a Nash equilibrium (NE) if for each player i his strategy




si performs at least as well as any other strategy given si : ui si , si  ui si , s i for all
si  S i .
A dominant strategy equilibrium is also NE but NE does not require dominance.
Greater scope of NE comes at a cost: it places greater rationality requirements on players:
each player must correctly anticipate the strategies that the other players are going to play.
Example
Player B
Left
Right
Top
2, 1
0, 0
Bottom
0, 0
1, 1
Player A
In this game “Left” is best response for “Top” as 1>0 and “Top” is best response for “Left” as
2>0. Thus (Top, Left) is Nash equilibrium.
Similarly “Right” is best response for “Bottom” as 1>0 and “Bottom” is best response for
“Right” as 1>0. Thus (Bottom, Right) is another Nash equilibrium.
Some games do not have pure strategy NE.
Example: Matching coin game
Player 2
Player 1
Heads
Tails
Heads
+1, -1
-1, +1
Tails
-1, +1
+1, -1
Mixed strategy Nash equilibrium
A mixed strategy is a probability distribution over the set of pure strategies of the player.
Example: play ‘Heads’ with probability p=1/3 and ‘Tails’ with probability 1-p=2/3.
NE definition is the same: profile of mixed strategies that constitute a mutual best response
(BR).
Example. Derivation of mixed-strategy NE for Matching coin game.
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Denote the mixed strategy of agent 1 by  p, 1  p and the strategy of agent 2 – by q, 1  q  .
Derivation of BR for agent 1:
EU1  p1 q  1 (1  q)  1  p 1 q  1 (1  q)  1  2q  2 p2q  1  max
p0, 1
q  1/ 2
1,

BR1  0, 1, q  1/ 2
0,
q  1/ 2

Derivation of BR for agent 2:
EU 2  q 1 p  1 (1  p)  1  q1 p  1 (1  p)  2 p  1  2q1  2 p  max
p0, 1
p  1/ 2
1,

BR2  0, 1, p  1/ 2
0,
p  1/ 2

q
BR2
1
BR1
1/2
NE
0
1
1/2
p
Nash Equilibrium (NE): p  1/ 2, q  1/ 2
Derivation of mixed-strategy NE. Useful property
If in some NE two strategies (A and B) are played by agent i with positive probabilities they
should bring the same utility to this agent: ui siA , si   ui siB , si .
If utilities are different ui siA , si   ui siB , si  then we can increase expected utility by
reallocating the weight of low-utility strategy to the high-utility one










EU i  pui siA , si  1  pui siB , si  pui siA , si  1  pui siA , si  ui siA , s i .
It means that the initial strategy of player i doesn’t bring the highest possible expected utility
under given strategies of other players and so this strategy profile violates one of the
requirements of NE.
Let us demonstrate the application of this property for the Matching coin game.
Let us take the first player and calculate his expected utility for each pure strategy:
u1 H1   q  1(1  q)  2q  1 and u1 T1   q  1(1  q)  1  2q . This player will play both
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positive probabilities iff he is indifferent: u1 H1   u1 T1  , that is 2q  1  1  2q , which
happens if q  1/ 2 .
Thus in mixed NE (with 2 players) each player's chosen probability distribution must make
the other player indifferent between the strategies he is randomizing over.
Existence of NE
Existence theorem (Nash 1951). Every game with a finite number of players and finite
strategy sets has at least one NE (in mixed or pure strategies).
Existence result makes NE the main solution concept for simultaneous-move games. Proof is
based on fixed point theorems (take Game Theory Course).
Most of the models in Economics deal with continuous (rather than discrete) set of strategies:
Bertrand, Cournot. Many of the results are still applicable to continuous case.
4.2 Sequential-move or extensive-form games
Up until now we have been looking at games in which all players move simultaneously. But
in many situations one player gets to move first and the other players responds. For sequential
games we need to draw a game tree to depict the sequence of actions.
We start with games with perfect information, where each player can observe the moves of
players who act before them.
In extensive form games we differentiate between actions and strategies [strategy indicates a
complete plan of actions of the player].
Example:
1
L
R
2
2
T
(0,3)
B
(1,0)
T
(4,1)
B
(2,2)
Player 1 has two strategies: {L, R }
Player 2 has two actions {T, B} but four strategies: {TLTR, BLBR, BLTR, TLBR}, where the
lower index is used to indicate the choice of player 1.
Let us proceed from extensive to a normal form.
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Player 2
Player 1
TLTR,
BLBR
TLBR
BLTR
L
0, 3
1, 0
0, 3
1, 0
R
4, 1
2, 2
2, 2
4, 1
Sequential-move games with imperfect information
In imperfect information game at least one player is not perfectly informed about some of the
(or all of the) previous moves.
Reconsider the last example, assuming that now player 2 does not observe the action of player
1. To illustrate this lack of information we combine two decision nodes in one information set
(dotted line).
1
R
L
2
2
T
(0,3)
B
T
(1,0)
(4,1)
B
(2,2)
The strategy set of player 1 stays the same: { L, R } but the strategy set of player 2 is different
{ T, B }. As a result the normal form game looks like this:
Player 2
Player 1
L
T
0, 3
B
1, 0
R
4, 1
2, 2
Non-credible threats in NE and subgame perfection
Threat is credible if it would be in the self-interest of the player to carry out the threat if called
upon to do so.
The following example demonstrates that NE may involve non-credible threats
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Example
1
In
Out
2
Fight
(-1, 0)
No
(0,4)
(1, 2)
We can proceed to the normal form game:
In
Firm 1
Out
Firm 2
Fight
No
-1, 0
1, 2
0, 4
0, 4
There are two pure strategy NE: {In, No} and {Out, Fight}. The last one {Out, Fight}
includes non-credible threat as it is not profitable for firm 2 to fight if firm 1 enters as 0 < 2.
To solve this problem we need a refinement of NE concept in sequential games, that is, we
should impose additional conditions.
Subgame perfect Nash equilibrium (SPNE)
A subgame is a part of a game that starts from a node which is a singleton and includes all
successors of that node [you cannot cut an information set so that only part of it belongs to a
subgame!]
A strategy combination is SPNE if it induces a NE in every subgame.
Note: the whole game is also a subgame  SPNE is a NE but not every NE is SPNE
In finite games we can derive SPNE using backward induction.
In our example with two firms there are just two subgames: the game itself and the part that
starts from the decision node, where firm 1 goes in. If firm 1 plays ‘in’, the best choice for
firm 2 is to play ’no’ as 2>1. Anticipating that firm 2 will respond by playing ‘no’, firm 1
chooses ‘in’ as it gives 1 instead of 0. Thus only {In, No} constitutes SPNE.
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1
In
Out
2
(0,4)
No
Fight
(-1, 0)
(1, 2)
SPNE under imperfect information
Now, let us consider another example of sequential game, where information is imperfect.
1
In
Out
1
A
C
(3, 1)
2
B
C
D
(2, 4)
D
(0, -1) (-1, 0) (1, 3)
There are two subgames: the game itself and the part that starts from the decision node, where
firm 1 plays ‘In’. Let us consider this subgame and produce a normal form game for it.
Firm 2
Firm 1
A
C
3, 1
D
0, -1
B
-1, 0
1, 3
There are two pure strategy NE in this subgame: {A, C} and {B, D}.
If firm 1 plays ‘In’ and in the following subgame firms play {A, C} then firm 1 gets 3, which
is more than it would get staying out (3>2). Thus ‘In’ is firm’s 1 best response for {A, C} and
the strategy profile {(In, A), C} constitutes SPNE.
If firm 1 plays ‘In’ and in the subgame that follows firms play{B, D}then firm 1 gets 1, which
is less than 2. Thus ‘Out’ is a best response for {B, D} and we get one more SPNE: {(Out, B),
D}.
Thus in this game there are two pure strategy SPNE: {(In, A), C} and {(Out, B), D}.
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4.3 Repeated games
In a repeated game the same stage game is played several or infinite number of times. In
infinitely repeated game we discount the payoffs: the payoff of player i is given by

ui a0   ui a1    2 ui a2      t ui at  . In finitely repeated games there is no
t 0
discounting.
Finitely repeated game.
Let us suppose that the both players know that the same (for example Bertrand) game will be
played N times ( N is finite). What will the outcome be?
As the game is dynamic, we look for the perfect Nash equilibrium using backward induction.
Consider the last round. As this round is the final one and everybody knows it, then there is
no incentive for cooperation and every player will choose the static game Nash equilibrium
strategies by charging price equal to MC. Now consider what will happen on round N  1 . As
at the last round there will be no cooperation there is no incentive to cooperate at this round as
well.
If one cooperates by charging the monopolist price the rival will find optimal to cheat by
charging lower price and getting all the market. Each player has an incentive to deviate and as
a result the only equilibrium at this subgame is given by a static Nash equilibrium, where each
firm charges price equal to MC. The same logic proves that there would be no cooperation at
each round and the only perfect equilibrium corresponds to prices equal to marginal cost.
The result is not surprising as players cooperate only if there is a punishment for cheating.
With finitely repeated game at the last round cheating cannot be punished and this creates
incentive for deviation at each round.
In finitely repeated game with one NE this NE is played at each stage.
Infinitely repeated game.
If the game is infinitely repeated, then the last round does not exist and as a result backward
induction cannot be used.
With infinite number of rounds a deviation at any point of time could be punished in the
future. Trigger strategy – strategy in repeated game, where the player stops cooperating in
order to punish other player’s break with cooperation
Example. Infinitely repeated advertising game
Firm 2
Firm 1
A
A
2, 2
N
5, 1
N
1, 5
4, 4
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The only NE in one-shot game is {A, A}. We can note that this game is similar to Prisoners
dilemma: ‘A’ is a dominant strategy for each player but the payoffs of both firms could be
higher if they stop advertising.
Consider the following trigger strategy of player i (i=1,2):

start by playing N (cooperate at t=0)

in period t > 1:
 play N if (N, N) was played at t-1
 play A otherwise.
Let us find out, under which value of discount factor neither of the firms will deviate from
cooperative strategy. Assume that one of the firms deviate at moment t. Then we can calculate
the present value of the payoff starting from t and compare it with the payoff in the absence of
deviation.
Payoff in case of deviation at t (starting from t): 5    2   2  2  5  2
Payoff if tacit cooperation is sustained: 4    4   2  4   

.
1 
4
.
1 
4

. Solving this inequality we can get
 5 2
1 
1 
that if   1/ 3 (players are patient enough) cooperation can be sustained in infinitely repeated
advertising game.
There is no incentive for deviation if:
Note that the threat of punishment is credible. The payoff of the firm that implements
punishment (starting at t) is 1    2   2  2, while its payoff in the absence of punishment
is 1   1   2 1  . Thus punishment is in the self-interest of the firm as it gives higher
payoff: 1    2   2  2  1   1   2 1. As the payoffs are the same at every period it
is enough to look at one period only: 2>1.
Credibility is achieved due to the nature of the punishment: each firm uses Nash-reversion
strategy.
4
for each player. To compare it with payoffs in one1 
shot game we proceed to normalized payoff by multiplying by (1-).
The resulting equilibrium payoff is
Thus we demonstrated that (normalized) payoff (4, 4) can be sustained as SPNE in the
infinitely repeated game if  is high enough. But this is not the only possible payoff that could
be sustained in SPNE. There are other SPNE as well. For example, playing {A, A} at every
round is also SPNE. Thus we can sustain (2, 2).
The result about the set of payoffs that can be sustained as equilibrium outcome is known as
the `folk theorem'.
To identify all the (normalized) payoffs that could be sustained in the infinitely repeated
advertising game considered above, we will produce a graph.
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First of all, we will find all feasible payoffs by illustrating the convex combinations of
possible payoff profiles of the stage game (indicated by blue). Then we look only at those
payoffs from this set that are individually rational.
Individual rationality means that the equilibrium payoff of each player must be at least as
large as the minmax payoff of that player as otherwise a player who gets less than the minmax
payoff has incentive to deviate by playing the minmax strategy.
Player i’s minmax payoff of player i (denote it by *i ) is the lowest payoff in the stage game
that the rival can impose on him through his choice of a strategy s−i, given that player i
chooses his own strategy to maximize his own payoff: *i  min max ui  si , si  .
s i
si
Since we deal with Prisoners’ dilemma type game, where every player has a dominant
strategy, then maximum payoff is always provided by this trategy and, as, a result the minmax
payoff for each player coincides with the payoff that he gets in the dominant strategies
equilibrium (in our case it is 2 for each player).
2
5
4
2
1
1
2
4
5
1
Folk Theorem.
Any feasible individually rational pair of normalized payoffs (1, 2) can be supported as
SPNE for a high enough  ( close to 1).
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4.4 Sample exercise with solution
Answer the following questions.
(a) Consider the following extensive form game.
1
T
1
L
B
2
R
2
C
D
C
(0, 0)
(3, 1)
D
E
F
(5, 4)
(2, 5)
(4, 0)
(1, 2)
(i) Identify all pure strategies of player 2
(ii) Identify all subgames and find all pure strategy SPNE
(b) ’In any finite sequential game with perfect information if no player has the same payoffs
at any two distinct terminal nodes then the SPNE is unique’. Is this true or false? Explain.
(Note: you should take into account both pure and mixed strategies)
Solution
(ai) Pure strategies of player 2: CE, CF, DE, DF
(aii) Three subgames should be identified
Subgame 1
1
1
L
Subgame 3
T
B
2
R
2
C
(3, 1)
D
(0, 0)
E
C
(1, 2)
D
(5, 4)
F Subgame 2
(2, 5)
(4, 0)
Backward induction must be used to derive SPNE.
Subgame 1 NE: {L, C} (C is dominant strategy for player 2 as 1>0 and 2>0; L is BR for C as
3>1)
Subgame 2 NE: F as 5>4
Subgame 3: T (if agent 1 plays T then his payoff is 3; if he plays B the his payoff is 2. Thus
he is better off by playing T)
SPNE: {T L, C F}
(b) True. In finite game we find SPNE via backward induction by identifying NE in every
subgame.
In the last subgame the player who moves has a finite number of actions that result in
different payoffs u1 , u2 , , un . Moreover, due to the perfect information the payoffs at this
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subgame do not depend on the other players’ choice in this subgame as this is the only player
that moves in the considered subgame.
WLG we can assume that u1  u2 
 un . He is looking for p1 , p2 ,
, pn (where each
probability is b/w 0 and 1 and the sum=1) that max EU, where EU  p1 u1  p2u2 
 pn un
. If at least one probability different from pn is positive then this is not the Max EU as
EU  p1 u1  p2u2 
 pn un  p1un  p2un 
 pn un  un .
Thus max exists and it is unique. It means that the player that moves should never randomize
and should choose the action that generates the highest payoff at this stage (i.e pn  1 ) and
this action is unique.
Then we go backward and deal with the player that moves before last. He also has finite
number of actions and as he perfectly anticipates the action chosen by the last player every
action generates some particular payoff and the payoffs are different. Thus we apply the result
derived above that demonstrates that this agent should also play pure strategy in this subgame
and choose the action that generates the highest payoff. This action exists and is unique. We
continue backward and at every step identify unique action that generates the highest payoff.
The combination of these actions forms the resulting unique SPNE.
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5. THE FIRM
5.1 Modeling the firm’s technological opportunities
Production function is the relationship between the quantities of various inputs used per
period of time and the maximum quantity of output that can be produced per period of time:
Q  f z1 , z 2 ,, z n  .
We are going to concentrate on the case with two inputs: capital (K) and labour (L).
Examples of production functions (   0,   0 ):
 Fixed proportions or Leontieff technology minL, K  ,
 Linear technology L  K ,
 Cobb-Douglas production function AL K 
In case of two inputs we can represent production function graphically in terms of isoquants.
Isoquant is the locus of all the (technically efficient) combinations of inputs for producing a
given level of output.
Question: illustrate isoquants for the Leontieff, linear and Cobb-Douglas production
functions.
Properties of the production functions.
Marginal physical product of input i - the extra amount of output that can be produced when
the firm uses additional unit of this input, holding the levels of other inputs constant:
f
.
MPi 
z i
A technology exhibits decreasing/increasing/constant returns to factor when the marginal
physical product of an input falls/rises/stays constant as the amount of the input used
increases.
Question: characterize the three considered technologies (Leontieff, linear and CobbDouglas) with respect to returns to each factor of production
Marginal rate of technical substitution of labour for capital MRTS LK measures the rate at
which the firm has to substitute one input (L) for another (K) in order to keep output (Q)
dK
constant: MRTS LK  
. Thus MRTS equals to the absolute value of the slope of an
dL Q const
isoquant.
Definition suggests that MRTS can be calculated as the ratio of the marginal products:
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dQ  f K dK  f L dL  0 , which implies MRTS LK  
dK
dL

Q  const
f L MPL

.
f K MPK
If MRTS LK is some positive constant, then the factors are perfect substitutes (linear
technology)
If MRTS LK is 0, then the factors are perfect complements (Leontieff. Note: MRTS is not
defined at kink)
If MRTS LK isn’t constant, then the factors are imperfect substitutes (Cobb-Douglas).
In case of Cobb-Douglas technology we observe diminishing MRTS while moving along the
isoquant.
0
Most of the production functions used in empirical analyses are homothetic functions.
(Note: function f is homothetic if f  hgz  , where gz  is homogeneous of degree 1 and
h is monotone). For homothetic production functions the slopes of the isoquants are
preserved along every ray through the origin, i.e. MRTS remains the same for any given K/L
ratio whatever the level of output.
Returns to scale
If all inputs are changed by the same proportion this is referred to as a change in the scale of
production.
Decreasing returns to scale (DRS): a proportional increase in all inputs leads to less than
proportionate increase in output: f L, K   f L, K  for all L, K  and   1 .
Constant returns to scale (CRS): a proportional increase in all inputs leads to proportionate
increase in output: f L, K   f L, K  for all L, K  and   0 .
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Increasing returns to scale (IRS): a proportional increase in all inputs leads to greater than
proportionate increase in output: f L, K   f L, K  for all L, K  and   1 .
Returns to scale for homogenous production functions
Note: production function is homogeneous of degree t if f L, K   t f L, K  for all
L, K  and   0 .
If t  1 , then we have CRS technology,
If t  1 , then we have IRS technology,
If t  1 , then we have DRS technology.
5.2 Profit maximization and Cost minimization
Profit maximization problem
m ax  pQ  wL  rK 
K 0, L 0
f K , L   Q
Solution: demand for factors of production L p, w, r  , K  p, w, r  and supply of output
Q p, w, r  .
This problem can be divided into 2 sub-problems:
1)
Cost minimization problem
TC w, r , Q   m in wL  rK 
K 0, L 0
f K , L   Q
Solving this problem we obtain conditional demand for factors of production Lw, r, Q  ,
K w, r, Q  . Plugging the solution into the objective function we get the firm’s cost function:
TC w, r, Q  wLw, r, Q  rK w, r, Q .
2) profit maximization with respect to output max  pQ  TC w, r, Q  .
Q 0
5.3 Cost minimization
Cost minimization in the long run
In the long-run all factors are variable
TC LR w, r , Q   m in wL  rK 
K 0, L 0
F K , L   Q
Graphical solution of cost-minimization problem
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isocost lines
Isocost line - a line representing all input combinations that have the same cost for the firm.
Equation of isocost: wL  rK  const
At interior solution the slope of isoquant is equal to the slope of isocost:
MRTS LK L , K    w / r . This condition implies that if the firm uses both factors in
equilibrium, then they bring the same marginal product per dollar spent:




MPL L , K 
MPK L , K 
.

w
r
Thus if the firm uses both factors, then the corresponding quantities demanded can be found
from the following system






F L , K  Q




 MRTS LK L , K  w / r

This system could be derived algebraically. Lagrangean: ℒ  wL  rK  Q  f ( K , L)
FOCs for interior solution:
ℒ/L = w  FL  0

ℒ/K = r  FK  0
w  FL

r  FK
ℒ/ = Q  F ( K , L)  0
From the first two conditions we get MRTS LK 
FL w
 .
FK
r
Let Lw, r, Q  and K w, r, Q  be the solutions of long run cost minimization problem. These
demand functions are called conditional demands for factors of production since the
quantities depend on the level of output produced.
We can look for the optimum input combinations in production for different levels of output;
the resulting curve is known as expansion path. Expansion path allows for given factor prices
to get the relationship between output and the long run cost.
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Expansion
path
0
Exercise. Illustrate expansion path for the homothetic production function.
Analytically we get the long run cost function by plugging conditional demand functions into
the objective function: TC LR w, r, Q   wLw, r, Q   rK w, r, Q  .
Properties of long run costs
Long run marginal cost ( MC LR ) - the change in long-run total cost due to the production of
additional unit of output: MC LR 
TC LR
.
Q
Long run average cost ( AC LR ) - the long-run total cost divided by the number of units
produced: AC LR 
TC LR
.
Q
When long run average costs fall as output rises, costs are said to exhibit economies of scale.
When long run average costs rise with the output level, costs are said to exhibit diseconomies
of scale.
Due to homotheticity, K/L ratio under CRS is not affected by the level of output. As a result,
with CRS technology to increase output by  times we increase the employment of each
factor by  times. Then cost of production also increases by  times, while average costs
stay constant:
AC
LR
TC LR Q  TC LR Q  TC LR Q 
Q  


 AC LR Q  .
Q
Q
Q
That is, in case of CRS we have neither economies, nor diseconomies of scale.
NOTE: As AC=const, then TC LR  cQ , where c - cost of producing of one unit of output. It
implies that AC LR  MC LR  c .
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$
CRS
technology
With IRS we can produce Q by increasing the factors’ employment in a smaller proportion,
which implies that total cost also increase by less than  times1. As a result AC falls and we
deal with economies of scale:
AC LR Q  
TC LR Q  TC LR Q  TC LR Q 


 AC LR Q  .
Q
Q
Q
$
IRS
technology
In case of DRS under homothetic production function in order to increase output by  times,
we increase the factors’ employment in a greater proportion, which implies that total cost also
increases by more than  times. As a result AC goes up:
AC
LR
TC LR Q  TC LR Q  TC LR Q 
Q  


 AC LR Q  .
Q
Q
Q
$
1
DRS
technology
Note if production function isn’t homothetic then proportional factor increase does not necessarily correspond
to cost minimizing bundle but this implies that cost minimizing bundle might cost even less.
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That is, in case of DRS we have diseconomies of scale.
Relationships between LRAC and LRMC
TC Q 
MC Q 
 lim
 MC 0 .
Q

0
Q
1

If TC(0)=0, then AC 0  MC 0 . Proof: AC 0  lim

If AC reaches minimum at Q   0, then AC Q    MC Q  ,
Q 0
Proof. If AC reaches minimum at Q   0, then AC Q    0 , which implies

 TC Q   Q  TC Q   TC Q  MC Q   AC Q 
or
AC Q   

0
 
Q2
Q2
 Q 
AC Q    MC Q  .

If AC diminishes over some range of outputs, then AC Q   MC Q  for all Q from
considered range,
Proof. If AC diminishes over some range of Q , then AC Q   0 for each Q from given
range.
This
implies
that

 TC Q   Q  TC Q   TC Q  MC Q   AC Q 
AC Q   

 0 or AC Q  MCQ .
 
Q2
Q2
 Q 

If AC increases over some range of outputs, then AC Q   MC Q  for all Q from
considered range.
Proof. If AC increases over some range of Q , then AC Q   0 for each Q from given
range. This implies that

 TC Q   Q  TC Q   TC Q  MC Q   AC Q 

AC Q   

 0 or AC Q   MC Q  .
 
Q2
Q2
 Q 
Exercise. Suppose that AC and MC are U-shaped and TC(0)=0. Sketch AC and MC curves on
the same graph.
Cost minimization in the short-run
In the short-run, capital is fixed and we choose only one variable factor - labour:
TC SR w, r , Q, K   m in wL  r K
L 0
F K , L   Q ,
KK
This problem can be restated as
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TC SR w, r, Q, K   m in wL  r K
L 0
F K , L   Q
If F K ,0  0 , then labour employment is given by the condition F K , L   Q . Solving this
equation, we get short run demand for labour L  LQ, K  . Plugging into the objective
function we get short-run total cost as a sum of variable cost (VC) and fixed cost (FC):
TC SR w, r, Q, K   wLQ, K  



variable cost
r
K
.
fixed cost
Properties of the short-run cost
Short-run marginal cost ( MC SR ) - the change in short-run total cost due to the production of
additional unit of output: MC SR 
NOTE: As TC
SR
TC SR
.
Q
 VC Q   FC , then MC
SR
TC SR VC Q   FC  VC Q 



.
Q
Q
Q
Shape of the short-run MC is determined by return to variable factor.
MC SR 
VC Q  wLQ 
LQ 
w
.

w

Q
Q
Q
MPL
Conclusion:
 MC=const if MPL  const ,
 MC increases if MPL diminishes in Q,
 MC decreases if MPL increases in Q,
Short-run average cost ( AC SR ) - the short-run total cost divided by the number of unites
TC SR VC Q   FC

 AVC Q   AFC , where AVC denotes average
Q
Q
VC Q 
FC
variable cost: AVC Q  
and AFC denotes average fixed cost AFC 
.
Q
Q
produced: AC SR 
As FC  const , then AFC is diminishing function of output
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The shape of AVC depends on return to variable factor.


 VC Q    wLQ   QL Q   wLQ  w 
LQ   w  1
1



AVC Q   

L
Q


  
 w

  

2
Q
Q  Q  MPL APL
Q
 Q   Q 
As we can see dynamics of AVC depends on the relationship between average and marginal
products of labor.
 If MPL  APL , then AVC Q   0 , i.e. AVC is constant.
 If MPL  APL , then AVC Q   0 , i.e. AVC is increasing.
 If MPL  APL , then AVC Q   0 , i.e. AVC is diminishing.
Suppose that F K ,0  0 , i.e. labour is essential factor.
The first case takes place if marginal product is constant.
0
The second case takes place if MPL is diminishing.
0
The last case takes place if MPL is increasing.
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0
MPL
AVC
AFC
AC SR
constant
constant
decreasing
decreasing
diminishing
increasing
decreasing
decreasing at small Q
increasing at large Q
increasing
decreasing
decreasing
decreasing
NOTE: U-shaped short-run AC appears if MPL is diminishing
Relationships between AVC and short-run MC (the same as for long-run AC and MC)

If VC(0)=0, then AVC 0  MC 0 .

If AVC reaches minimum at Q   0, then AVC Q    MC Q   ,

If AVC diminishes over some range of outputs, then AVC Q   MC Q  for all Q from
considered range,

If AVC increases over some range of outputs, then AVC Q   MC Q  for all Q from
considered range.
Exercise. Suppose that AVC, AC and MC are U-shaped and TC(0)=0. Sketch AVC, short-run
AC and MC curves on the same graph.
Relationships between the SR and LR cost curves
TC LR Q   TC SR Q, K  for any Q and K ,
TC LR Q   TC SR Q , K , where K - the level of capital optimal for Q .
Implication: AC LR Q   AC SR Q, K  for any Q and K ,
AC LR Q   AC SR Q , K  , where K is the level of capital optimal for Q and


SMC Q , K  LMC (Q ) .
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Cost minimization with two plants
Suppose firm has two plants that produce the same output with different technologies. The
resulting cost functions are given by TC 1 q  and TC 2 q  .
Question: what is the firm’s cost function?
To find out the firm’s cost function we have to allocate any given output Q in a cost
minimizing way between the two plants:
TC
firm
Q   min TC 1 q1   TC 2 q2 
s.t. q1  q 2  Q, q1  0, q 2  0
As q2  Q  q1 , then TC
firm
.
Q  qmin
TC 1 q1   TC 2 Q  q1  .
0, Q 
1
If both plants are used in production (i.e. we deal with interior solution), then the FOC implies
TC1 q1   TC 2 Q  q1 , i.e. MC1 q1   MC 2 q2  . It means that in case when both plants are
used, output is allocated in such a way that marginal costs of production are equalized. If for
MC of production at one plant is always less than MC of production of the other plant, then
only the plant with lowest marginal cost would be used.
5.4 Profit maximization in case of perfect competition
Profit maximization problem max  pQ  TC w, r, Q  .
Q 0
Rules of profit maximization:
1. produce only if Q   0 or p  TC / Q  TC 0/ Q .
2. if Q  0 , then produce at a point, where   p  MC Q   0 (First order condition
for interior solution);
3. produce at a point, where   0 (Second order condition);
Implications for the long-run:
1. If TC LR 0  0 . then produce if p  AC LR Q   0 , i.e. p  AC LR Q  .
2. if Q  0 , then produce at a point, where p  MC LR Q 
3. produce at a point, where MC LR Q  Q  0 , i.e. at non-diminishing part of long-run
MC.
Implications for the short run:
1. produce only if p  AC SR Q    AFC , i.e. p  AC SR Q   AFC  AVC Q  as
TC LR 0  FC .
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2. if Q  0 , then produce at a point, where p  MC SR Q 
3. produce at a point, where MC SR Q  Q  0 , i.e. at non-diminishing part of short-run
MC.
Example
Suppose that AC and MC are U-shaped and TC(0)=0. Let us sketch the long run supply curve.
NOTE: As AC is U-shaped, then p  AC LR Q  for all p  min AC LR Q  . As a result we get
the following LR supply curve:
Q
LR
LR

0, p  m in AC Q 
 p  

LR
LR
LR

Q : MC Q   p, MC Q   0, p  m in AC Q 


$
Firm’s LR
supply
0
Consider an example with U-shaped short-run AC, AVC and MC. As AVC is U-shaped, then
p  AVC Q  for all p  min AVC Q  . As a result we get the following SR supply curve:
0, p  m in AVC Q 
Q SR  p   

Q : MC SR Q   p, MC SR Q   0, p  m in AVC Q 


5.5 Sample exercise with solution
Consider a perfectly competitive industry that produces good X. All firms in this industry
have identical technologies with cost function с q  , where с' 0  0 , с' q  0 , с q   0 for
q  0 . Unfortunately a fraction  of the output produced by each firm is defective and cannot
be sold. Moreover firm experiences some utilization cost for unsold output and the
corresponding cost function is given by l z  , where z is the volume of utilized output,
l 0  0 , l' z   0 and l z   0 for z  0 . Both production and utilization cost are zero if
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output is zero. Suppose that improvement in management brings a reduction of  while cost
function stays the same. What is the impact on individual supply of each firm? Provide both
graphical and analytical solution (Note: change in  is not necessarily small).
Solution
Profit maximization problem 1    pq  cq   l q   max
q 0
Function is strictly concave (second derivative is negative due to the assumptions), thus FOC
is both necessary and sufficient.
1  p  c q  l q  0 and 1  p  c q  l q  0 if q  0 .
Note that q  0 for any p  0 as otherwise 1  p  c 0  l 0  1  p  0 which
violates FOC.
Algebraic analysis
Let us prove that reduction in  for given price results in an increase in quantity supplied by
any firm. The LHS that represent marginal revenues goes up and so should do the RHS. If
q  0 then marginal production cost would fall or stay the same (as MC is increasing) and
the second term (marginal utilization cost) definitely falls as: q   0 and MUC is
increasing then l q   0 and with smaller  we have l q  0 . Thus RHS goes
down while LHS up and we get a contradiction. It means that q  0 .
Graphical analysis with comments
Supply is given by nondiminishing part of total MC per efficient unit (i.e. sold unit) that lie
above AC. Here as cost function is convex and goes from the origin than at any point
MC>AC and MC is increasing. Thus total MC per efficient unit represents supply curve. With
reduced  under the same output we utilize less and due to increasing marginal utilization
cost assumption we get lower value of MUC. Thus total MC falls at every q. Moreover, as we
sell more, the level TMC per efficient unit goes down which strengthens the effect of
reduction of MUC.
As TMC per efficient unit shift downward, it means that firm is willing to produce the same
output under lower price (i.e. supply curve shifts down or to the right).
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p
q
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6. PERFECT COMPETITION
6.1 Perfect competition
Fundamental assumptions of perfect competition
 Buyers and sellers are price takers (each agent chooses its actions under the assumption
that it cannot influence the prices of the goods)
 Entry into the market is free (new suppliers can enter the market without any
restrictions on the process of entry)
We will look at equilibrium in some particular market ignoring all other markets, that is we
will perform partial equilibrium analysis. This analysis is valid under quite restrictive
assumptions: absence of the feedback effects and negligible income effects.
To find equilibrium market price and quantity we need information about market demand and
market supply curves.
Market demand could be obtained by summing up (horizontally) the individual’s demand
curves.
p
p
p
20
20
Market demand
q1(p)+ q2(p)
10
10
q1(p)
q2(p)
q1
10
Q
q2
10
20
30
Market supply is different in the SR and in the LR as in the SR new firms cannot enter the
market as they cannot obtain the needed fixed inputs. In the LR new firms can enter and
existing firms can exit.
The short run
In the SR the number of firms in the industry is fixed and we get SR industry supply by
N
summing up the firms SR supply curves: Q S  p    qiS  p  .
i 1
As each firm produce at non-diminishing part of its MC the resulting SR industry supply is
upward sloping. The result does not depend on whether suppliers are homogeneous (have the
same cost functions) or heterogeneous (have different cost functions).
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p
p
p
SR Industry supply
Q
q2
q1
Market is in equilibrium if:
(1) buyers are choosing their optimal purchase levels, given the prevailing market price;
(2) sellers are choosing their optimal output levels, given the prevailing market price;
(3) suppliers are willing to produce as much as buyers wish to purchase.
SR equilibrium corresponds to the intersection of market demand with SR industry supply.
Equilibrium price p  is a solution of the equation Q D p    Q S p   .
Industry supply
Market demand
The long run
In the LR new suppliers can enter the market and old suppliers can exit.
To find the market quantity supplied at a given price, we need to find both the quantities
supplied by each firm in the market and the number of suppliers who choose to be in the
market at that price.


LR equilibrium is given by p , Q  , N  such that:
D
N
p   Q p , where Q p    q p ;

S

S

S
i


Q

p q  C q  0  i but it becomes negative for N   1


i
 
i 1

i
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(a) Homogenous firms
We start with homogenous firms case, i.e. the case, where all firms have identical cost
functions.
Consider the constant cost industry - industry, where individual firm’s cost function remains
unchanged as industry output expands.
Note that as all firms have the same technology, nobody is willing to produce at price that is
below the minimum of AC. Thus quantity supplies is zero at any price below p   min AC .
Now, let us take any price above min AC . At this price p a  AC q a  and firm gets positive
profit. New firms will be attracted to the industry by the prospect of earning positive
economic profit (due to assumption of constant cost industry the expansion of industry output
has no effect on cost curves of individual firm). Thus industry supply is unlimited at this
price. The same argument applies for any other price that exceeds p  . If p   min AC , then
each firm produces q  and gets zero economic profit. Thus firm is indifferent between being
in and out of the market, there is no incentive to enter or exit the market.
p
p
Constant cost industry
LR Supply
Q
q
Firm
Market
Conclusion: in case of constant-cost industry LR industry supply is zero for p  min AC and
the curve is horizontal at p   min AC .
As a result the quantity produced by the industry is determined by the market demand and the
number of firms in the industry N  is obtained as a ratio of industry and individual output:
 
p   min AC , Q   Q D p  , q   argmin AC q  , N  Q  / q  .
So far we assumed that an increase in the industry output has no effect on individual firm cost
function. It may not be the case. We can observe both external diseconomies (an increase in
industry output brings an increase in LRAC) or external economies (an increase in industry
output results in a reduction in LRAC). There are two reasons for the presence of external
economies/diseconomies:
(1) pecuniary external economy/diseconomy is the result of interaction between industry
output and firm’s cost function through the changes in the market prices of inputs;
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(2) technological external economy/diseconomy is the result of interaction between
industry output and firm’s cost function through the physical possibilities of
production (i.e. the production function).
Examples: pecuniary diseconomy may result from increased competition for specific factor;
technological economy may come from innovations that represent a by-product of an
increased industry output.
Increasing cost industry - an industry in which external diseconomies take place, i.e. LRAC
rise with the industry output level.
The LR industry supply will be upward sloping as min AC LR rises with expansion in
industry output. Note: although min AC LR rise with increase in Q , the minimum efficient
scale may stay constant, fall or rise.
Decreasing cost industry - an industry in which external economies take place, i.e. LRAC fall
as industry output rises.
The LR industry supply will be downward sloping as min AC LR falls with expansion in
industry output. Note: although min AC LR falls with increase in Q , the minimum efficient
scale may stay constant, fall or rise.
p
p
LR Supply
LR Supply
Q
Increasing cost industry
Q
Decreasing cost industry
(b) Heterogeneous firms
Let us assume that industry output has no effect on the firms cost functions, i.e. we deal with
constant cost industry.
Suppose that there are two types of producers of some good. Producers of type 1 have lower
costs but the number of type 1 producers is limited (for example they use specific resource
with restricted access). Assume for simplicity that there is only one firm of type 1. Producers
of type 2 have higher costs of production but any firm can become a type 2 producer.
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p
p
p
LR Industry supply
Q
q2
q1
Type 1 firm
Type 2 firm
Market
If price is below the minimum of LRAC of the first firm (firm with the lowest cost of
production), then nobody is willing to produce at all. As a result industry supply is zero for
any p  p1  min AC1LR . If price is below the min AC 2LR but above min AC 1LR other firms
would be willing to produce had they access to the technology used by firm 1, but the access
is restricted so that firm1 will be the only producer and the industry supply corresponds to the
part of MC of firm 1 (if there are more firms of type 1, we should sum up their supply curves
for any p  p1 , p2 . As price reach the level of min AC 2LR , then any type 2 firm is able to


produce but its profit would be 0. Thus it is indifferent between coming in and staying out. As
a result industry supply is horizontal at p2  min AC 2LR .
Conclusion: industry supply can be upward sloping even in case of constant cost industry if
firms have different cost functions.
6.2 Equilibrium and efficiency
We want to know not only how a competitive market works but also whether the resulting
equilibrium allocation is ‘desirable’ for the society.
We will use the concept of total surplus (TS) to measure the social welfare. TS is a difference
between social benefit (measured by total willingness to pay) and social cost:
Q

~ ~
TS Q   TB Q   TC Q    P D Q dQ  C Q  .
0
Total willingness to pay (or gross consumers’ surplus) gives total benefit from consumption
of given quantity of a good. Graphically it can be represented as the area below the market
demand function. For example if Q  Q0 , then total benefit is represented by grey area:
 
CS gross Q 0  A  B .
Total cost can be represented as the area below the market supply curve for given number of
firms in the industry (integral from MC). If Q  Q0 , the cost of production are given by
dashed area (B). As a result TS=(A+B)-B=A.
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From TS maximization we get the socially efficient output: MBQ    MC Q  . This
condition is known as condition of allocative efficiency. In addition to this condition there are
two more requirements. When we calculate TS we should be sure that TC Q  represents the
minimum cost of production for given output Q [i.e. total output should be allocated
efficiently between the firms] and TB Q  represents the maximum benefit from consumption
[i.e. Q should be allocated efficiently between the consumers].
p
Supply
A
Demand
B
Q
Another approach to TS
Note that in the absence of government intervention TS can also be calculated as a sum of net
consumers’ surplus (CS) and producers’ surplus (PS):


Q
Q

~ ~
~ ~
TS Q   CS Q   PS Q     P D Q dQ  QP Q   QP Q   C Q    P D Q dQ  C Q  .


0
0

PS is the revenue the agent receives in excess of what he would require to produce a given
quantity. If additional unit of output is produced then costs increase by the value of MC, thus
the difference between the market price and MC corresponds to a net gain of producer from
this additional unit. By summing up over all units produced, we get the PS. Note that in the
absence of fixed cost PS is equal to profit.
If we implement this approach, then consumers will purchase Q0 units of output at price p 0
(note that at this price producers are willing to supply more than Q0 so that p 0 does not
correspond to equilibrium). The resulting CS is the area below the demand function above the
market price (grey) and together with PS we get exactly the same value of TS as before.
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p
Supply
CS
PS
Demand
Q
The key question is whether TS is maximized at competitive equilibrium. If some other
allocation had a higher TS, then output would be called inefficient because it would be
possible to make society better off.
Claim: competitive equilibrium is efficient.
To prove this claim let us look at output level, Qa , which is less than competitive. Then by
moving from Qa to Q  , total benefit increase by (C+D), while costs increase only by D and
TS goes up by C, which implies that initial allocation was inefficient.
Similarly if we look at output level that exceeds equilibrium one, Qb  Q  , then we can
increase TS by moving from Qb to Q  as total benefit falls by F, while costs go up by (E+F).
As a result TS is increased by E.
p
Supply
C
E
D
F
Demand
Q
The same result we can derive algebraically. Profit-maximization implies p s  MCQ  , while
utility-maximization
MBQ   MCQ  .
requires
MBQ   pd .
Since
in
equilibrium
p s  pd
then
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Implication: as competitive equilibrium output is efficient, then any government interventions
that result in deviation from competitive output would reduce total surplus, i.e. the
corresponding output would be inefficient. The corresponding reduction in TS is called
deadweight losses.
Question: Prove algebraically that perfectly competitive outcome is efficient
Application 1: per unit tax analysis
Suppose that per unit tax with tax rate of t on sales of good X is introduced. Let demand
and supply curves be linear. Let us denote by p the price paid by consumers and suppose that
tax is paid by producers, then the price received by producers is p  t . As a result the FOC in
profit maximization problem looks like p  t  MC q or p  MC q  t . Thus this tax can
be treated as an increase in MC. As this increase is the same at every level of output, the MC
shifts up parallel, which implies that industry supply curve shifts up by the value of tax.
As a result consumers’ price goes up and quantity produced falls.
p
t
A
B
E
C D
G
F
H
Demand
Q
Initially
With tax
Change
CS
A+B+C+D
A
-(B+C+D)
PS
E+F+G+H
B+E=H
-(E+F+G)
Gov.Rev.= t  Qt
0
B+C+E+F
B+C+E+F
TS=CS+PS+GR
A+B+C+D+ E+F+G+H
A+B+C+ E+F+H
-(D+G)
Conclusion: this policy results in DWL  D  G
Alternative approach to calculation of DWL: Total benefit is reduced by (D+G+I), while cost
goes down only by I and as a result TS  ( D  G  I )  I  ( D  G)
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p
t
DWL
D
G
I
Demand
Q
Note that both producers and consumers are worse off due to the tax. Distribution of the tax
burden between consumers and producers depends on relative slopes of demand and supply
curves. Note that relationship b/w slopes implies the corresponding relationships of price
elasticities.
In equilibrium under per-unit tax: QS  pS t   QD  pD t  , where pS t   pD t   t . Consider
a tax rate change dt assuming that initially t  0 : QS  dpS  QD  dpD . Then we obtain the
relative tax burden
Q D
dp S

and proceed to elasticity (multiply and divide by before-tax
dp D
Q S
price and quantity):
D
Q D p / Q  p
dpS
Q  p
Q  p


 S , where  Dp  D D and  Sp  S S .
dp D
Q S
p/Q p
Q
Q
Conclusion: greater share of tax burden corresponds to market side with lower (in terms of
absolute value) price elasticity.
p
Loss in CS
t
t
Loss in CS
Loss in PS
Demand
Loss in PS
Demand
Q
Q
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Application 2: price ceiling
Under effective price ceiling there is a gap between quantity demanded and quantity supplied.
The quantity sold is determined by the short side of the market (in this case given by the
quantity supplied) but different rationing mechanisms could be used. We will consider two
extreme cases: in the first buyers with highest valuation obtain the good and in the second
case the good produced is sold to the buyers with lowest valuation.
A
B
A
C
A
E
D
C
B
D
F
B
E
D
D
Q
Q
Case 1
Case 2
Initially
Case 1 (High-valuation)
Case 2 (low-valuation)
With price ceiling
Change
With price ceiling
Change
CS
A+B
A+C
CB
F
F(A+B)
PS
C+D+E
E
(D+C)
E
(D+C)
TS
A+B+C+D+ E
A+C+E
(B+D)
F+E
F(A+B+C+D)
Conclusion: the change in CS and the resulting DWL crucially depends on the rationing
mechanism used. If the good is sold to the high-valuation consumers (case 1) then the value
of DWL is minimized and equal to A+D. This mechanism is called the efficient rationing as it
allocates the good efficiently between the consumers. In case of inefficient allocation of the
good the values of social loss may increase significantly and the upper bound of DWL is
demonstrated by case 2, where DWL=A+B+C+DF.
Application 3: price support program
Suppose that the government wants to help the producers of the competitively supplied
product. Government sets the support price above the equilibrium one and buys up any excess
supply at the support price.
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A
B
E
C
A
F
D
H
G
E
I
D
Q
Initially
With price support
Change
CS
A+B+C
A
(B+C)
PS
E+F
E+F+B+C+D
B+C+D
Government
Surplus
0
(C+D+H+F+G+I)
(C+D+H+F+G+I)
TS=CS+PS+GS
A+B+C+E+F
A+B+C+ E+F+H
(C+H+F+G+I)
This program increases PS but requires huge government spending and finally results in
tremendous value of DWL=C+H+F+G+I. This is due to the wrong incentive created by the
policy: with high price producers increase output while demand is reduced. Government
spends a lot of resources for price support and most part of the resources is wasteful since
government cannot resale the good without dropping the price.
Alternative policy in the form of lump sum subsidy equal to PS will generate that same
increase in producers’ welfare without DWL.
6.3 Sample exercise with solution
Consider perfectly competitive constant cost industry with identical firms. Suppose that, a
per-unit sales subsidy is replaced by a lump sum subsidy that every active firm gets. The
lump-sum subsidy leave the equilibrium price the same as it was under the per-unit subsidy
scheme. Compare the total government expenditures under per unit and lump-sum subsidy
schemes: (i) graphically assuming U-shaped AC and (ii) analytically (i.e. for any type of AC
consistent with the conditions of the problem).
Solution
Graphical analysis
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p
p
LS subsidy
LR Supply
per unit subsidy
Q
q
Firm
Market
Analytical approach
 
p LR  AC q   s  AC q LS  
AC q LS   
S
q LS 
2
S
,
q LS
 
AC  q   0  AC q LS  
S
q LS 2
,
which
implies

 0 , i.e. AC is not minimized at q  q LS but AC is minimized at q  q .
Thus AC q    AC q LS  and
 
S
 s  AC q LS   AC q   0 .
q LS
Cost of subsidy comparison: S  n LS  S
Q LS
Q
 S s  sQS .
q LS
q LS
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7. GENERAL EQUILIBRIUM AND WELFARE ECONOMICS
7.1 General equilibrium in exchange economy
Consider an economy, where there are K consumers k  1, 2,, K  and N consumption
goods:
i  1, 2,, N  .
consumption
Assume that every consumer k has an initial endowment of

e k  e1k , e2k ,, e Nk
goods

that
can
be
traded
at
market
prices
p   p1 , p2 ,, pN  . We assume that there is no production so that this is an exchange
economy.


Denoting the consumption bundle of agent k by x k  x1k , x2k ,, x Nk we get the following
budget constraint p1 x1k  p2 x2k    p N x Nk  p1 e1k  p2e2k    p N e Nk .


Prices p   p1 , p2 ,, pN  and allocation x 1 , x 2 ,, x K constitute general equilibrium in
exchange economy if:


constraint under given p   p1 , p2 ,, pN  ,
K


for every consumer k bundle x k  x1k , x2k ,, x Nk maximizes uk  subject to his budget
all markets clear:
K
 x  e
k 1
k
i
k 1
k
i
for every i  1, 2,, N .
Example. General equilibrium in 2×2 economy with Cobb-Douglas preferences.
Consider an exchange economy with two goods ( x and y ) and two consumers ( A and B )
with identical preferences u( x, y  xy .
We start with derivation of individual demands for agent A:
m ax x A y A
px x A  py y A  p x e xA  py e yA
x A  0, y A  0
We can notice that there are no corner solutions for positive income. The explanation is
simple: under zero quantity of some good the resulting utility is zero, while it could be
positive if, for example, we spend half of the income on each commodity. Thus zero
consumption of any of the good doesn’t maximize utility.
In interior solution MRS xy 
y A px
, which implies equal spending on each good

x A py
py y A  px x A . Plugging into the budget constraint and rearranging we get the demand
functions:
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px e xA  py e yA
2 px
 0.5e xA  0.5
py
px
e yA and y A  0.5
px A
e x  0.5eyA .
py
In the similar way we derive demand of agent B from his utility maximization problem:
max x B y B
px x B  py y B  px e xB  py e yB
x B  0, y B  0
Due to the symmetry we get: x B  0.5e xB  0.5
py
px
e yB and yB  0.5
px B
e x  0.5e yB .
py
To get equilibrium prices we should find total demand for every good and equate with total
supply given by the total endowment.
For x - market we get the following equation:
x A  x B  0.5e xA  0.5
py
px
e yA  0.5e xB  0.5
Solving this equation we get the following price ratio:
py
px
py
px

e yB  e xA  e xB .///
exA  exB
. It could be easily
eyA  eyB
verified that this price ratio will also equilibrate the second market
y A  yB  0.5e yA  0.5
px A
p
e x  0.5e yB  0.5 x e xB  eyA  e yB .
py
py
py
e xA  e xB

Plugging the equilibrium price ratio
into demands we get the equilibrium
p x e yA  e yB
allocation
A
B

1  A e xA  e xB A 
1  e y  e y A
x A  ex  A
e x  e yA 
ey , y A 
A
B
B


2 
2  e x  e x
ey  ey


and
A
B
1  B
B ex  ex
x B  ex  ey A
2 
e y  e yB

 e A  e yB B

B 
 , yB  1  y
.
e

e
x
y


2  e xA  e xB


We can note that equilibrium prices are not unique. If price vector p   p1 , p2 ,, p N  and


allocation x  x 1 , x 2 ,, x K constitute a general equilibrium in some economy then price
vector p and allocation x is also a general equilibrium in this economy. It means that prices
are relative and we can normalize one price to 1.
From the previous analysis we can derive another important result. It is enough to equilibrate
all markets but one and the last market will be automatically in equilibrium. This is an
implication of the Walras law that we discuss below.
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Walras law
For any price vector p   p1 , p2 ,, p N  the total value of excess demands summed up over
N
all markets is always zero:
 p  ED
i 1
i
i
 0 , where EDi stays for excess demand for good i.
K
K
k 1
k 1
In case of exchange economy: EDi   x ik ( p)  eik
Let us prove Walras law for the case of exchange economy. Since demand of agent k is
derived from utility maximization problem it should satisfy the consumer’s k budget
constraint. Moreover, due to non-satiation all income is exhausted:
p1 x1k  p2 x 2k    p N x Nk  p1e1k  p2 e2k    p N e Nk .
Summing up over all consumers and re-arranging we get:
 M k M k
 M k M k
M k M k
p1   x1   e1   p2   x 2   e 2     p N   x N   e N   0 .
k 1
k 1
k 1
 k 1

 k 1

 k 1

Edgeworth box
We can illustrate the equilibrium in exchange economy graphically. We are interested in
feasible allocations, that is, those allocations, where consumption of each commodity equals
to the amount available in the economy.
As total supply of each commodity is fixed then all feasible allocations in exchange economy
can be illustrated graphically by Edgeworth box.
OA
We can also illustrate the consumers’ preferences by indifference curves.
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OB
Increase in
utility of A
Increase in
utility of В
OA
To find consumer’s choice we illustrate the budget set and then indicate the bundle that gives
the highest utility. The budget line should go through initial endowment. At the figure below
we illustrate a general equilibrium.
OB
endowment
Budget set B
equilibrium
Budget set A
OA
Existence and uniqueness of equilirium
It is important to establish the existence if GE before we proceed to the discussion of its
properties. The existence theorem was proved by Kenneth Arrow, Gerard Debreu and Lionel
McKenzie in 1951. The proof is based on a fixed point theorem applied to mapping of a set of
prices to itself (it requires restrictions on preferences & endowments).
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Equilibrium is not necessarily unique. The following example demonstrates that we might
have multiple equilibrium allocations and price vectors. Consider an exchange economy with
two goods ( x and y ) and two consumers ( A and B ). Assume that these consumers have
identical preferences and treat the two goods as perfect substitutes: ux, y  minx, y .
Suppose that there are 3 units of each good in the economy. Agent A has 3 units of good x
and one unit of good y and the remaining units belong to agent B.
OB
D
C
endowment
OA
In this example any allocation from CD segment is an equilibrium and the corresponding
prices are given by the slope of the budget line that connects this allocation with the
endowment point. Thus for every allocation we have a unique (normalized) price vector but
we have continuum of equilibrium allocations and prices: x A  1, 2 , y A  x A ,
x B  yB  3  x A and px / py  y A  1/x A  2 .
7.2 Pareto optimum in exchange economy
An allocation of resources is Pareto efficient (Pareto optimal) if it is not possible to reallocate
resources to make one person better off without making someone else worse off.
Pareto improvement is a reallocation of resources that makes at least one person better off
without making anyone else worse off.
Pareto optimal allocations (PO) in exchange economy with 2 goods (x and y) and 2 agents (A
and B) could be derived from the utility maximization problem of one consumer2 under fixed
utility of the other and feasibility constraints:
2
This is the case with well-behaved preferences that are represented by continuous utility function that increases
in each good.
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u A (x A , y A )
m ax
x A  0, y A  0, x B  0, y B  0
u B (x B , y B )  u
x A  x B  e xA  e xB
y A  y B  e yA  e yB
To solve the problem we write down the Lagrangean:
ℒ  u A ( x A , y A )  u B ( x B , y B )  u   x exA  exB  x A  x B   y eyA  eyB  y A  y B  .
FOCs for interior solution:
u A
 x  0 ,
ℒ/x =
x A
A
u A
ℒ/y = A   y  0 ,
y
A
u B
ℒ/x =  B   x  0 ,
x
B
u B
ℒ/y =  B   y  0 .
y
B
A
From the first two conditions we get MRS xy 
B
MRS xy

x
and from the next two conditions we get
y
x
. Thus in case of interior solution, in addition to feasibility conditions
y
x A  x B  exA  exB , y A  y B  eyA  eyB , the following requirement should be satisfied:
A
B
MRS xy
 MRS xy
.
This condition could be also derived graphically. Consider allocation C. It is not efficient
since both persons would be better off by moving to any point that lies above IC A and below
IC B .
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Pareto
improvement over C
OB
Preferred by A
Preferred by В
OA
A
B
Efficiency is achieved if the two IC are tangent, that is, MRS XY
. If it is not the
 MRS XY
A
B
case, and in some interior allocation, for example, MRS XY
, then a Pareto
     MRS XY
improvement is possible. As agent A values good X more relative to agent B, we should
reallocate good X in favour of agent A. Suppose we take small unit of good X from B and
give it to A. Then A is willing to sacrifice up to  units of good Y. Let us take only
(  ) / 2 units of good Y from agent A. As a result he would be better off as (  ) / 2  .
If we give (  ) / 2 units of good Y to agent B, he would be better off as well since he
would be as well off by getting  units of good Y instead of one unit of X but gets more:
(  ) / 2   . Thus by reallocation of resources we were able to improve the position of
both agents, which means that initial allocation was not Pareto efficient.
The set of all interior PO allocations in exchange economy is given by the locus of tangency
of agents’ indifference curves and is called contract curve in consumption.
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OB
Contract curve
OA
Tangency condition can be violated in case of corner solution. For example, a feasible
A
B
allocation with MRS xy  MRS xy can be optimal if y A  0 and an allocation with
A
B
MRS xy
 MRS xy
can be optimal if x A  0 as it is illustrated below for the case of perfect
substitutes.
7.3 Welfare theorems for exchange economy
The First Fundamental Theorem of Welfare Economics
If there is a market for every commodity, optimizing behaviour of consumers and firms under
perfect competition leads to a PO allocation.
This theorem is also known as `Invisible hand' theorem since the market price mechanism
ensuring that each agent is acting in pure self-interest results in a Pareto optimal outcome.
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Informal proof for 22 economy.
If an equilibrium is not PO then there exists some other allocation in the Edgeworth box that
makes at least one agent better off.
Since equilibrium is the best point in the budget set, to make the agent better off we must
choose a bundle outside of his budget set.
To make the other agent at least as well off we should either move along the budget line or
outside of his budget set.
There are no feasible allocations that satisfy the last two requirements.
Implications of the FFWT
If we have complete system of perfectly competitive markets then there is no case for
`efficiency-enhancing' government intervention.
Reasons for the market failure that explains the government:

Incomplete market system due to asymmetry of information or externalities/public
goods

Absence of perfect competition.
In such cases, government intervention is desirable (to reduce or eliminate the market
inefficiency).
Second Fundamental Theorem of Welfare Economics
Usually in the economy we have many different PO allocations. Can any arbitrary PO
allocation be attained by some set of competitive prices, assuming there is a suitable
assignment of initial endowments? The answer to this question is provided by the second
welfare theorem.
If all agents have convex preferences, any efficient allocation can be obtained as an
equilibrium under suitable redistribution of incomes.
Proof for differentiable case.
~
p
A ~A ~A
x ,y
Choose prices ~x  MRS xy
p


A
px ~
xA ~
py ~
yA  ~
px exA  ~
py eyA ,
and transfers: T  ~
y
T ~
px ~
xB  ~
py ~
yB  ~
px exB  ~
py eyB . Due to the given definitions, the financial balance is
A
B
p ~
xA ~
x B  e A  eB  ~
p ~
yA ~
yB eA eA  0 .
satisfied: T  T  ~
B
x

x
x

y

x

y




A
B ~B ~B
x A,~
y A  MRS xy
x , y , therefore
Since in PO indifference curves are tangent MRS xy ~
~
p
B ~B ~B
MRS xy
x , y  ~x , which together with T B  ~
px ~
xB  ~
py ~
yB  ~
px exB  ~
py eyB under the
p


y
convexity of preferences guarantees that ~
x B ,~
y B  maximizes utility of consumer B under
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~

~px , ~py  . Similarly, as ~px  MRS xyA ~x A , ~y A
p

 and
TA ~
px ~
xA ~
py ~
yA  ~
px exA  ~
py eyA then
y

~
x A,~
y A maximizes utility of consumer A under ~
px , ~
py  .
~ A ~B
Both markets are in equilibrium as ~
xA ~
x B  e x and y  y  e y due to the feasibility of
PO allocation.
Importance of convexity and other problems
The following example demonstrates that the theorem might fail if convexity assumption is
violated. At the following graph point C stays for PO allocation but it cannot be decentralized
(cannot be achieved as GE) since the only price vector at which agent B will choose C is
given by the tangency line but agent B will choose D rather than C at these prices.
x
B
yA
OB
D
C
OA
xA
yB
Problems with implementation:


lump-sum taxes are difficult to impose in practice;
people might not fully trust the government to make use of the proceeds in the right
manner.
7.4 Production economy
Consider a simplest version of production economy with one consumer. This economy is also
called a Robinson Crusoe Economy. Assume that this representative agent consumes two
goods and his preferences are represented by utility function u( x, y) .
Production opportunities are represented by production possibilities frontier (PPF) that is also
called transformation curve: T ( x, y)  0 . Any point on PPF shows the maximum amount of
one good attainable under any given output of the other good for given technologies and
factors’ availability.
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By definition all points on PPF are technically efficient. The absolute value of PPF slope
shows the marginal rate of transformation (MRT).
MRT - the minimum quantity of one good that must be sacrificed to produce an additional
(small) unit of the other good: MRT xy  
dy
T / x

. MRT xy shows the rate at which
dx PPF T / y
the economy can transform one output into another by shifting its resources, i.e. the
opportunity cost of additional (differentially small) unit of x in terms of forgone y .
y
PPF
x
Pareto optimum in Robinson Crusoe economy
To find PO allocation we should maximize utility of Robinson subject to the PPF:
max u( x , y)
x  0, y  0
T ( x, y)  0 .
To solve this problem we setup a Lagrangean: ℒ  u( x, y)  0  T ( x, y) .
FOCs for interior solution:
u
T

 0,
x
x
u
T
ℒ/y =    0 .
y
y
ℒ/x =
By re-arranging we get
u / x T / x

, that is PO is given by the point on PPF, where
u / y T / y
indifference curve is tangent to the PPF: MRS xy  MRTxy and T ( x, y)  0 .
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y
PPF
PO
x
GE in Robinson Crusoe economy
To find equilibrium prices in Robinson Crusoe Economy we should derive supply of the
goods from profit maximization problem and equilibrate it with demand that comes from
utility maximization.
Note that along PPF total costs are fixed (all the resources are used). Thus profit
maximization could be equivalently restated as revenue maximization under given costs:
max px x  py y
x, y
s.t. T ( x, y)  0 .
Setup a Lagrangean: ℒ  px x  py y  0  T ( x, y) .
FOCs for interior solution: ℒ/x = px  
Rearranging we get
T
 0 and
x
ℒ/y = py  
T
0.
y
px T / x

. Thus the profit maximizing bundle is given by a tangency
py T / y
point of PPF and isoprofit line given by px x  py y  const .
y
PPF
Production
x
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Now, we proceed to the derivation of demand. Robinson as a consumer tries to maximize his
utility subject to the budget constraint:
max u( x , y)
x, y
p x x  py y  M
Since Robinson owns all the resources and the firm his income M is equal to the sum of the
value of initial endowment and profit, that is px x   py y , where ( y  , x  ) is the firms’
production plan. Thus the budget constraint of Robinson coincides with the iso-profit line that
goes through the profit-maximizing production plan. The point of tangency of this budget line
p
with indifference curve indicates the utility-maximizing consumption bundle: MRS xy  x
py


and px x  py y  px x  py y .
If we look at the figure below, then we can see that the given prices do not equilibrate the
markets: there is excess demand at y market and excess supply at x market, that is the
relative price of x is too high.
y
Consumption
excess
supply
excess
demand of y
PPF
Production
x
Economy is in equilibrium at a lower relative price of x , at which optimal production and
consumption decisions coincide.
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y
PPF
x
Note that in equilibrium Robinson attains the highest indifference curve on his PPF, which
implies that GE is PO.
7.5 Pareto efficiency in economy with production
Any interior Pareto efficient allocation has to satisfy the following efficiency conditions:
 efficiency in consumption,
 efficiency in production,
 efficiency in output mix.
Consider an economy with two consumption goods x, y  , two factors of production ( L , K )
in fixed supply and two individuals ( A, B ).
Efficiency in consumption
An allocation of commodities is efficient in consumption if given the total amount of each
commodity available for consumption, the only way to make one person better off is to make
another person worse off.
This condition was extensively discussed in the section on exchange economy. The key
A
B
conclusion is that efficiency in consumption requires equality of MRS: MRS xy  MRS xy .
Efficiency in production
An allocation of inputs is efficient in production if, given the total amount of each input
available for production, the only way to increase the output of one commodity is to decrease
the output of another commodity.
Note that this condition is necessary for Pareto efficiency. Indeed, if this condition is violated,
then by reallocating inputs we can produce more of one good under the same quantity of the
other. Let us distribute this additional quantity produced equally between all consumers and
each one would be better off, which means that initial allocation was not Pareto efficient.
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Let us denote the fixed supply of capital and labour by K and L . Then all feasible
allocations could be represented by production Edgeworth box.
Contract curve
in production
Following the same logic as in the analysis of efficiency in consumption we can show that
efficiency in production takes place if the two isoquants are tangent, i.e.
x
y
. The locus of all points that are efficient in production is called
MRTS LK
 MRTS LK
production contract curve.
Efficiency in production and PPF
Once the economy produces efficiently, the only way to produce more of x is to give up
some y . Thus we can represent efficient production in terms of produced outputs. Production
possibility frontier (PPF) is derived from the production contract curve. Any point on PPF
shows the maximum amount of one good attainable under any given output of the other good
for given technologies and factors’ availability.
qy
0
Production
possibility
frontier
qx
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The absolute value of its slope MRT xy can be calculated as a ratio of marginal products of
the same factor in production of different outputs: MRT xy 
MPLy
MPKy
MRT

or
. Due
xy
MPLx
MPKx
MPKy MPLy

to efficiency in production
.
MPKx MPLx
This result is obtained immediately for one factor economy. Suppose production requires only
 x  Q L 
x
x

dy
one input - labour, then y  Q y L y  and MRT xy  
dx

 L x  L y  L

PPC
MPLy dLy
MPLx dL x

PPC
Alternatively MRT xy could be calculated as a ratio of marginal costs: MRT xy 
MPLy
.
MPLx
MC x
.
MC y
Efficiency in output mix (allocation efficiency)
For allocation efficiency the rate at which producers can convert y into x has to be equal to
the rate at which consumers are willing to sacrifice y for x : MRTxy  MRS xy . Suppose this
is not the case and MRS xy      MRTxy , then we should reallocate resources in favor of
good x production as consumers value additional unit of good x more than it costs. Indeed,
if we produce additional unit of x , then we have to reduce production of good y by  . Let
us change the bundle of consumer A leaving bundle of agent B unchanged. Then A will
increase his consumption of x by one unit and reduce consumption of y by  ; he was
willing to sacrifice up to  units of good Y for additional unit of x but the actual reduction
in consumption of y was less, which means that he is better off, while welfare of B is
unaffected. Thus, the initial allocation was not PO.
qY
PPF
Pareto efficient
allocation
0
qX
General equilibrium with production
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A set of prices constitutes general equilibrium if
(1) every firm maximizes profits given its technology,
(2) every consumer maximizes utility subject to his/her budget constraint,
(3) demand equals supply for each commodity.
The First Fundamental Theorem of Welfare Economics
If there is a market for every commodity, optimizing behaviour of consumers and firms under
perfect competition leads to a Pareto optimal allocation of resources.
Let us show that interior GE is efficient in consumption, production and output mix.
Equilibrium allocation is feasible as demand for each commodity equals supply.
In equilibrium, the MRS between the two goods equals the price ratio. As both consumers
face the same prices we get equality of MRS in equilibrium. As each consumer maximizes his
B
A
 px / py and MRS xy
 px / py , which
utility subject to his budget constraint, then MRS xy
A
B
implies MRS xy  MRS xy . It means that the equilibrium allocation is efficient in
consumption.
Each firm maximizes profit taking prices as given. Cost minimization is a necessary condition
of profit maximization. Cost minimization implies that MRTS between capital and labour in
x
production of each good equals the ratio of factors’ prices: MRTS LK
 w / r and
x
y
y
, i.e. efficiency in production takes
MRTS LK
 MRTS LK
 w / r , which implies MRTS LK
place.
Then, from the consumer’s utility maximization problem MRS xy  px / py . The profit
maximization by firm x implies p x  MC x and profit maximization by firm y implies
py  MC y .
MRT xy 
As
a
result,
efficiency
in
product
mix
takes
place:
MC x p x

 MRS xy .
MC y p y
Thus, the price system allows Pareto efficiency to be achieved in a decentralized setting.
Nobody directs agent to choose particular combinations of inputs or consumption goods.
Efficiency arises automatically as the outcome of a process in which each consumer and each
producer observes prices and privately makes the decisions that maximize his or her wellbeing. This is the outcome of the ’invisible hand’ of the market.
The Second Fundamental Theorem of Welfare Economics
Provided that all indifference curves and isoquants are convex to the origin, for each Pareto
efficient allocation of resources there is a set of prices that can attain that allocation as a
general competitive equilibrium.
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~ ~
If (~
x, ~
y, K , L ) is Pareto efficient allocation, then it can be achieved as Walrasian equilibrium
~ and ~
with consumption good prices ~
r.
p ,~
p and input prices w
x
y
~
p
A
B
 MRS xy
1) Choose ~x  MRS xy
. Then the FOC for UMP of each consumer is
p
y
px ~
xA ~
py ~
y A and I B  ~
px ~
xB  ~
py ~
yB.
satisfied. Required incomes: I A  ~
~~
r ~
py MPKy . Then FOC for firm y are satisfied.
py MPLy and ~
2) Choose w
3)
A
xy
Efficiency in output mix implies MRS
Thus
MRTS
~
~
px MPLx  ~
py MPLy  w
x
LK
 MRTS
y
LK
and
due
~
p
p
MPLy ~
 ~x or MRT xy 
 ~x .
x
py
py
MPL
 MRT xy
to
efficiency
in
production
we
have
MPLy MPKy
px MPKx  ~
py MPKy  ~
r . Thus we get

or
. It means that ~
x
x
MPL
MPK
FOC for profit maximization by firm y as well.
7.6 Sample exercise with solution
Consider an economy with two agents A and B. The diagram below shows the indifference
curves for the two agents in an Edgeworth box and the endowment point (E). In this case, the
result of the Second Welfare Theorem does not hold. Is this true or false? Explain your
answer.
A
B x2
OB
x1
E
ICA
ICB
OA
G
x1A
x2B
True.
SFWT
G is PO (preferred bundles for A lie above ICA and preferred bundles for B lie below ICB and
these two sets do not have common points).
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xB
yA
OB
E
ICA
F
ICB
G
O
A
x1A
x2B
Consumer B will max his utility in G iff BL is tangent to his IC at G, i.e. under BL given by
GF. It means that if G can be decentralized then it can be done only under the prices given by
this tangency line.
At the same time under GF consumer B maximizes his utility at F, which results in excess
demand for good x (and excess supply of y).
Thus it is impossible to find the prices that will decentralize allocation G.
The failure is due to nonconvexity of A’s preferences. Preferences are convex if for any u
the set of weakly preferred bundles B(u )  x : ux   u is convex, i.e. for any x  B(u ) and
x  B(u ) we have x  (1  )x  B(u)    0, 1 .
x2
H
(H+G)/2
Consider B(u )  x : ux   u - blue IC and above
G
x1
H  B(u) and G  B(u ) but  H  G  / 2  B(u )
NOTE: convexity is a sufficient (not necessary) condition, that is, even in the absence of
convexity some PO might be decentralized.
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8. MONOPOLY
8.1 Pure monopoly
Fundamental assumptions of pure monopoly
 Single firm is a price maker (affects the price by changing its output level)
 Buyers are price takers
 Price discrimination is impossible (each consumer pays the same price)
 Entry is blocked
Profit maximization problem of a monopolist
Let P Q  represent inverse market demand function, then monopolists’ problem can be
written as maxTR Q   TC Q  , where TR stays for total revenue: TRQ   P Q Q .
Q 0
 
The necessary condition for interior solution is given by MR(Q  )  TR(Q)
Q
and the SOC: MR(Q  )  MC Q   .
 MC Q 
Q Q 
MRQ  - marginal revenue, i.e. revenue that the firm gets from additional unit sold:
MR(Q) 
TR(Q) P (Q)  Q 

 P(Q)  P (Q)Q .
Q
Q
Relationship between MR and inverse demand function:
 if Q  0 , then MR0  p0  p0  0  p0
 as demand is diminishing, then p(Q)  0 and MR(Q)  P(Q)  P (Q)Q  P(Q) , i.e.
MR curve lies below the inverse market demand curve.
Special case of linear demand function pQ   A  bQ . In this case MR curve is also linear
and two times steeper than inverse market demand MRQ  A  bQ  bQ  A  2bQ .
$
MRQ 
Q
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TR(Q) P (Q)  Q 

 P (Q)  P (Q)Q 
Q
Q

 P (Q)Q 
1 
P (Q)  1 
  P (Q)  1  d
  (Q) 
P (Q) 

p


,
where  dp denotes price elasticity of demand.
As profit maximization implies MR(Q  )  MC (Q  ) , we can write down the following
formula for profit maximizing monopolist price:

1 
P (Q  )  MC (Q  ) 1  d   .
  (Q ) 
p


Note:
MC  0 , which implies that monopolist will produce at a point, where

1 
MC (Q  )  MR(Q  )  P (Q  )  1  d    0 . As price is positive, this inequality requires
  (Q ) 
p


monopolist to produce where 1  d 1   0 , i.e. d 1   1 . Solving this inequality we get
 p (Q )
 p (Q )
 dp (Q  )  1 . As price elasticity is negative this is equivalent to  dp (Q  )  1 .
Conclusion 1: monopolist produces only at (price) elastic part of market demand.
Conclusion 2: monopolist’s price is a markup over the marginal cost:

d

1 
1
P (Q  )  MC (Q  ) 1  d    MC (Q  ) as  p (Q )  0 , which implies that 1  d   1 .
  (Q ) 
 p (Q )
p


Equilibrium in case of monopoly:
$
MC
p
profit
B
A
Q
P(Q)
Q eff
Q
MR
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Inefficiency of monopoly
Note: monopolist’s output does not coincide with the competitive one as pQ    MC Q  ,
while at competitive output pQ comp   MC Q comp .
Monopolist solution results in DWL due to underproduction relative to the efficient level
(consumers are prevented from buying units of the good that they value more than it costs to
produce them).
By moving from Q  to Q eff  Q comp gross CS increases by area A+B, while costs rise only
by A, so TS increases by B. It means that equilibrium is inefficient: DWL=TS=B.
Besides DWL, we may observe additional loss that comes from rent-seeking activity. As
monopoly gets abnormal profit, each producer is willing to become a monopolist. As a result
several firms compete for this monopoly rent and this rivalry involves resource costs that
must be added to the value of DWL.
8.2 Sources of monopoly and regulatory responses:
 government franchise monopoly
 resource-based monopoly
 patent monopoly
 technological or natural monopoly
 monopoly by good management
A franchise monopoly arises when a government grants the exclusive right to do business in a
specified market to some firm.
Resource-based monopoly power comes from the exclusive ownership of a natural resource
essential in a particular production process.
Regulator can force the monopolist to sell off some portion of essential resource, which
makes competition a feasible market alternative to monopoly.
Patent monopoly arises from the government action to enable inventors and authors to gain
the exclusive right to their respective discoveries and writings by means of patent or
copyright.
Patents serve a potentially useful economic purpose by stimulating the invention and
developments of new products. Eliminating patenting rights does not necessarily increase
efficiency because the economic benefit to consumers of patented products might outweigh
the economic cost of monopolistically exploited innovations. Regulation requires the optimal
length of patent.
Natural monopoly is usually defined as an industry where economies of scale make it cheaper
to produce when there is one firm rather than several. (Example: public utilities)
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Monopoly by good management occurs when the monopolist can deter entry and finds it
profitable to do so. In fact this type of monopoly results from oligopolistic strategic
interactions between the established monopoly and a potential entrant.
In this case the monopolist will not produce the output that corresponds to equality of MR and
MC. The firm would maximize the profit taking into account additional constraint that comes
from the possibility of entry that results in oligopolistic competition. To deter entry firm may
decide to produce output that exceeds the standard monopoly output.
Regulatory responses to monopoly
1) Average cost pricing
Objective: to limit monopoly to a fair, or normal, rate of return
Problem: even though average cost pricing (if successful) does eliminate monopoly profit, it
does not induce the monopolist to produce the efficient level of output. Both overproduction
(a) and underproduction (b) relative to efficient output may take place.
$
$
MC
MC
P(Q)
P(Q)
AC
AC
Q
Q
(a)
(b)
Moreover, the average cost pricing does not provide incentive to minimize the cost of
production as under this policy monopoly covers the cost but does not get any profit.
Finally, knowing that the regulator is going to use AC information to set price, a rational firm
would misreport its AC curve in order to get positive profit. As regulation is based on private
information, monopoly will use this information to its own advantage and report AC that
induces the regulator to set monopoly price.
2) Rate of return regulation
Rate of return regulation is aimed at limiting the rate of return a regulated monopoly can earn
on its invested capital. Suppose that monopolist produces output with 2 factors of production,
capital owned by the firm and labour hired by the firm. The monopolist’s total return on its
capital is given by the difference between total revenue and labour costs: TRQ   wLQ  .
Rate of return regulation imposes the following constraint on the firm’s behaviour:
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TR( K , L)  wL  K ,
where  is the allowed rate of return on capital Regulatory agency will choose  that is not
less than r as otherwise the monopolist’s profit would be negative and it will be forced out of
business:
K , L   TR( K , L)  wL  rK    r K  0 if   r .
If   r , the monopolist will chose the combination of inputs that is not cost minimizing: it
has an incentive to use too much capital and too little labour as with increase in the quantity
of capital used the monopolist can get higher profit.
3) Marginal cost pricing combined with lump sum subsidy
Regulator may set efficient price and if AC is above this level (it might be the case with
natural monopoly due to diminishing AC) the policy could be combined with the lump-sum
subsidy that allows to cover the gap.
$
MC
$
P(Q)
MC
P(Q)
AC
AC
Lump sum subsidy
Q
(a)
Q
(b)
Implementation problem is similar to the one discussed with respect to AC pricing: knowing
that the regulator is going to use private information on MC to set price, a rational firm would
misreport its MC and induce regulator to set monopoly price.
4) Per unit subsidy
As unregulated monopoly under-produces relative to efficient output, government can create
an incentive for output expansion by offering per unit subsidy.
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$
MC
P(Q)
MC-s
Q
MR
But the problem with private information is not eliminated. The efficient subsidy rate is given
by s eff  MC Q eff   MRQ eff  and Q eff is a solution of the equation pQ eff   MC Q eff  .
Thus we need information on private marginal cost to find efficient subsidy rate, which
creates an incentive for misreporting.
5) Two-tier pricing scheme
Produce Q 0 and incur per unit costs AC 0 . Sell in two blocks: Q M at per unit price p M and
Q 0  Q M at per unit price p0 such that profit from Q M balances the losses.
p
D
MR
pM
AC 0
AC
MC
p0
QM
Q0
Q
6) Efficient regulatory mechanism based on market demand only (optional)
The scheme proposed by Loeb and Magat (1979)3 does not require private information to
attain efficient output. The idea is to give monopoly a subsidy, equal to the value of CS. In
this case, monopolist’s profit will be given by the sum of profit from sales (PS) and subsidy
(CS). Thus, profit maximization would be equivalent to the maximization of TS. By
definition, the output that brings maximum TS is efficient. Thus monopolist will choose
3
Loeb M. and W.A.Magat (1979) A Decentralized Method for Utility Regulation. The Journal of Law and
Economics 22 (2), 399-404.
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efficient level of output. To implement this policy regulator needs information on demand
curve (which is not related to monopolist), so we eliminate the problem of misreporting.
8.3 Monopolistic price discrimination
Price discrimination - selling different units at two or more different prices for the reasons
not associated with differences in costs.
Necessary conditions for profitable price discrimination:
 the firm must be a price maker;
 the firm must be able to identify which consumer is which (i.e. identify willingness to
pay);
 the firm must be able to prevent consumers from engaging in arbitrage (arbitrage is the
process whereby customers whom the firm charges low prices make purchases that
they then resell to customers who would otherwise have to pay high prices);
 the transaction costs (the costs of meeting the second and third requirements) must be
less than the benefits.
Perfect or first degree price discrimination - the practice of selling each unit of output at a
price just equal to the buyer’s maximum willingness to pay for that unit.
Under first degree price discrimination monopolist leave consumers with zero CS as each unit
is sold at a price equal to maximum willingness to pay. Thus monopolist gets TS and, trying
to maximize it, he will end up with efficient level of output.
Example. Suppose that the good is discrete, MC=const and there are two consumers with
different demand functions.
$
Profit from sales to
agent A
$
Profit from sales
to agent B
MC
MC
5
3
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Monopolist will sell 2 units to agent A at prices 14 and 8, which bring profit of 22-10=12 and
3 units to agent B at prices 12, 9 and 6, that brings profit of 27-18=9. Thus total profit is 21.
He is not willing to produce the sixth unit, as additional cost of 5 can’t be covered since the
maximum willingness to pay for additional unit is only 4.
In case of continuous demand the same outcome could be achieved via two-part tariffs
scheme, or bundling, that will be considered at the end of the chapter.
Market separation (third-degree) price discrimination
Assumptions: seller can observe the consumer’s willingness to pay and can charge different
prices (based on differences in willingness to pay) but he is able to discriminate only between
the groups of consumers; within the group each unit is sold at the same price.
Third-degree price discrimination - the practice of identifying separate groups of buyers of a
good and charging different per unit prices to these groups.
Suppose there are two groups of consumers with inverse demand functions p1 q  and p2 q  ,
correspondingly. Note that if we know the price charged, then we can find the corresponding
quantity sold to this group and vice versa. That is why we can maximize monopolist’s profit
either with respect to prices or with respect to quantities. Let us follow the second approach,
then the monopolist solves the following problem
max TR1 q1   TR2 q2   TC q1  q2 , where TRi qi   qi pi  and i  1,2 .
q1 0,q2 0
 MR1 (q1 )  MC (q1  q2 )
The FOCs for interior solution: 
.



 MR2 (q 2 )  MC (q1  q2 )
Conclusion 1. If both groups are served then marginal revenues have to be equal:
MR1 (q1 )  MC (q1  q2 )  MR2 (q2 ) .
Conclusion 2. The segment with more elastic demand will be charged a lower price.
 q  p 

1
MRi qi   pi  qi  pi  pi 1  i i   pi 1 
pi 

 i
elasticity of group i demand.
Proof.

As MR1 (q1 )  MR2 (q2 ) , then p1 1 


1
price elastic 1   2 , then 1    1 
1


1 
  pi 1 
, where  i -price

 i 



1 
1 


p
1

. Thus if group 1 demand is more
2
 

1 

2


1
, which implies p1  p2 .

2
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Graphical presentation of third-degree price discrimination
$
$
$
p2(q)
MR1 = MR2
MC
p1(q)
MR1
group 1
Q
q2
q1
MR2
group 2
Market
Welfare effects of third degree price discrimination
The output produced under market segmentation is inefficient as at any segment
pi qi   MRi qi   MC .
Let us compare TS under pure monopoly and under market segmentation.
The total output produced under market segmentation may rise, fall or stay the same as under
pure monopoly.
But we should take into account that under market segmentation, the output is inefficiently
allocated among consumers.
Conclusion 1. If price discrimination does not bring an increase in output, then
TS segm  TS monopoly.
Conclusion 2. If price discrimination results in an increase in output, the change in TS is
ambiguous: it may rise, fall or stay the same.
Special case of linear demand functions and constant MC.
Let us consider a special case of third-degree price discrimination, when demand curves are
linear and marginal costs are constant.
Claim 1. If all markets are served under uniform price then discrimination lowers (total)
welfare in comparison with pure monopoly.
Proof is based on direct calculation of total quantities produced in both cases. Let demand
function at segment i ( i  1,, N ) be given by qi  p  ai  bi p and MC  c , where
c  ai / bi . To find output under pure monopoly, we need to find aggregate demand, derive
the corresponding marginal revenue function and then equate MR with MC.
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We look only at the segment of market demand where sales are positive at all markets, thus
i ai  Q
d
,
inverse
market
demand
is
given
by
. As inverse
Q   ai  p bi
pQ  
b
i
i
 i
i
demand function is linear, MR would be also linear and two times steeper
i ai  2Q
. Monopolist’s output could be found from the equation
MRQ  
 bi
i

 a  2Q

b
i
MR Q
М
i
М
 c . Thus we get Q
М

a
i
i
 c  bi
i
.
2
i
i
Now let us find the sales under market segmentation. As we know, at each segment marginal
revenue has to be equal marginal cost. Inverse demand function at segment i is given by
pi q  ai  q/ bi . Thus marginal revenue at segment i is also linear and two times steeper
MRi q  ai  2q  / bi . Profit maximizing sales corresponds to equality of group i marginal
revenue
q
discr
i
and
monopolists’
marginal
 ai  cbi / 2 . Thus Q   q
discr
i
i

 
 a  c b

MRi qidiscr  ai  2qidiscr / bi  c
cost:
  ai  cbi / 2 
i
i
i
i
2
i
or
 QM .
As in both cases output is the same, TS discr  TS M since under market segmentation the
output is inefficiently allocated among consumers.
Graphical proof for the case with 2 groups
p
p
p
p2(q)
MR1=MR2
p1(q)
Market
demand
MR
MR2
MC
q1
Q
q2
Claim 2. If one group of consumers is not served under uniform price but is served under
discrimination, then TS discr  TS M .
As Q M  q2M , then MRtotal Q M   MR2 q2M   c .
At
the
same
time
 
MR2 q2d  c ,
which
implies
that
q2M  q2d
but
Q M  q2M  q2d  q2d  q1d  Q d as by assumption q1d  0 .
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Market separation
Pure monopoly Comments
q1d
>
q1M
q 2d
=
q 2M
Qd
>
QM
Pd
<
PM
due to diminishing demand
CS1d
>
CS1M
As q1d >0 and q1M =0
CS 2d
=
CS 2M
As q2d  q2M >
PS d
>
PS M
Monopolist under segmentation could
charge the same price as pure monopolist
but chooses different pricing policy
TS d
>
TS M
p
q1d >0 and q1M =0
p
p
p2(q)
MR1=MR2
MC
MR
p1(q)
MR1
MR2
q1
q2
Market
demand
Q
Multi-part pricing: two-part tariff
In case of two-part tariff each consumer pays a fixed fee ( F ) for consuming any amount of
the good plus a price per unit ( p ) for each unit consumed.
If there is only one consumer, then for each per unit price p the maximum fixed fee equals to
his CS. Thus firm is able to appropriate TS. As a result, firm finds optimal to set linear price
p   MC q   and F  CS q  . As a result firm’s output is efficient.
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P(q)
F
MC
q
In case of several homogenous consumers the optimal tariff is similar: the per unit price
should be set at the level of marginal cost while the fixed fee should be equal to the
individual’s CS. We can derive it algebraically. With N identical consumers market demand
is given by N  q d ( p)  Q p . Then the profit-maximization problem of the monopolist could
be written as:
maxN  F  pQ( p)  C (Q( p))
p,F

s.t . F  CS ind  p    q d ~
p d~
p.
p
It could be shown that the optimal contract will satisfy this constraint as equality, which
results in the following equivalent setup


 ~ ~

d ~
~


max  Nq  p dp  pQ( p)  C (Q( p))  max  Q p dp  pQ( p)  C (Q( p)) .


p 
p 
p

p

From the FOC  Q( p)  Q( p)  pQ( p)  C Q Q( p)  0 we get the per unit price p  CQ and
F  CS ind  p .
Non-linear pricing in case of heterogeneous consumers
Assume that monopolist can offer only one two-part tariff (alternative case with the menu of
tariffs is considered later).
In case of heterogeneous customers and unique two-part tariff, the monopolist may still find it
optimal to set linear part of the tariff equal to MC.
For example, this is the case if monopolist faces two (equal) groups of customers with large
difference in the marginal willingness to pay. Then it is more profitable to deal with the high
valuation group only (see figure below).
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p
p2(q)
F
p1(q)
MC
q1
q2
Another possibility corresponds to the situation, when the difference in marginal willingness
to pay is not too big, but the number of customers with higher marginal willingness to pay is
much bigger than the number of buyers with lower willingness to pay.
Now, let us assume that we deal with two groups of equal size and the difference in
willingness to pay is not too high. Let us illustrate that monopolist will benefit from charging
linear price above marginal cost.
$
P2(q)
P1(q)
A
C
B
E
D
F
MC
q
First of all we can show that it is more profitable to deal with both groups of consumers rather
than only with the high valuation one. The optimal tariff in case of selling to the highest
valuation (i.e. second) group only is to set linear price equal to MC (as a result the profit from
sales is zero) and the fee equal to the corresponding value of consumer surplus: p0  MC
and F0  A  B  C  D  F  E . Resulting profit equals to the value of fee:  0  F0 .
If the monopolist will decrease fee up to the value of CS of group 1 ( F   A  B  D) under
the same linear price, its profit will increase:   2F   2( A  B  D)   0 as
A  B  D  C  E  F . So it will sell to both groups.
Now, we can illustrate that it is not optimal to set price equal to MC. If monopolist increases
~
~p
linear
price
to
and
sets
then
its
profit
equals
F  A,
~
  2 A  B  B  D  E    ( E  D)   . So, it is profitable to raise linear price above
MC.
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Optimal 2-part tariff in case of heterogeneous consumers
There are two options that must be considered.
Option 1: sell to high-valuation group by charging p  MC and setting fee equal to the CS of
the high-valuation group F  CS2  p  MC  .
Option 2: sell to both groups by charging per-unit price above MC and setting fee equal to the
CS of the lowest valuation group:
m ax2 F  p  q1  p   q2  p   C q1  p   q2  p 
p, F
F  m inCS1  p , CS2  p 
Finally, we should compare the resulting profits and choose the option that generates higher
profit.
We can note that in any case the total output produced under this scheme is below the socially
efficient one.
Second-degree price discrimination
In reality no firm can observe every individual buyer’s willingness to pay for its product. In
this case a monopolist can use the consumer’s own actions as a basis of discrimination.
Second-degree price discrimination - is the practice in which the same price schedule is
offered to all buyers but they sort themselves through self-selection.
Example with bundling. Suppose we have two groups with linear demands but the second
group has higher willingness to pay (see graph below). For simplicity let’s assume that we
have one agent in each group and MC=0. Each consumer knows his/her willingness to pay but
the monopolist cannot identify the type of each customer.
Had the types been observable, then as we know from the previous discussion, the
monopolist would sell under linear price equal to MC (zero in this case) and charge fees equal
to values of individual’s CS: Fee1  A and Fee 2  A  B  С . In this case monopolist’s
profit is 2 A  B  С . In fact it can achieve the same profit by bundling: if it offers to the first
group the bundle with q1 units at price of A and to the second group bundle q 2 units at price
of A  B  С , the outcome would be the same.
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$
P2(q)
P1(q)
A
B
C
q
But as the types are unobservable, the monopolist is unable to implement this scheme. Now it
has to offer the same menu of bundles to all customers. If it offers the two bundles found
above, the high valuation agent (agent 2) would definitely take the bundle designed for group
1 as it gives him net CS of B , that is, full information bundles are not incentive compatible.
We say that a contract is incentive compatible if it gets the agent to make intended choice. As
a result, the monopolist’s profit would go down to 2 A .
In fact, the monopolist can earn more by changing the contract designed for consumer 2 in
such a way that consumer would find it optimal to choose it (that is, by proceeding to
incentive compatible contracts). Incentive compatible contract should provide agent 2 the net
CS not less than B . This can be done by reducing the price for the large package with q 2 units
up to A  C . In this case each consumer self-selects by choosing the contract designed for
his/her type and monopolist’s profit would be 2 A  C . Is it the maximum possible profit? No,
we can change the package designed for group 1 to make it less attractive for the second
group.
In order to do this we reduce the quantity included in this package and the price:
q

1

 q1 , A  A . Now, if agent 2 takes this contract, his CS will be equal to B  B . Thus
this contract is less attractive that before. This gives a possibility to increase the price of the
large package by B : q2 , A  C  B . The resulting profit is 2 A  C  B  A . Thus,


the monopolists’ profit was increased by B  A  0 .
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$
P2(q)
P1(q)
B
B
A
A
C
q
Now, we can generalize the result of this exercise. At the optimum, the marginal profit should
be zero. When q1  0 then A  P1 q1  and B  P2 q1   P1 q1  . Thus, the optimal
~   P q
~   P q
~ .
quantity in the small package should solve the following equation: P q
2
1
1
1
1
1
Now, let us consider a more general case, when the two groups may differ in size. Denote the
share of high-valuation group by  , then the share of low-valuation group is 1   and the
total number of customers – by N. The optimal contracts could be derived from the following
profit maximization problem:
max N  1   T1  T2 
T1 ,q1 ,T2 ,q2
s.t.
CS1 q1  T1  0
(PC1)
CS2 q2  T2  0
(PC2)
CS1 q1   T1  CS1 q2   T2
(IC1)
CS2 q2   T2  CS2 q1   T1
(IC2)
The first two constraints guarantee that agents will purchase the good since the CS from
purchasing is non-negative. In addition to participation constraints we need incentive
compatibility constraints (IC) that guarantee that agents cannot gain from purchasing the
‘wrong’ bundle, that is, the bundle that was designed for the consumer of other type.
Our previous analysis suggests that (PC1) and (IC2) are binding. From (PC1) we get
Plug
it
into
(IC2)
to
obtain
T1  CS1 q1  .
T2  CS2 q2   CS2 q1   T1 
 CS2 q2   CS2 q1   CS1 q1  . Finally we plug both T1 and T2 into objective function and
rearrange:
max CS1 q1   CS 2 q2   CS 2 q1 
q1 ,q2
The FOCs are given by:
CS1 q1   CS 2 q1   0 and CS 2 q2   0 .
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As a result we get the following equations for q1 and q 2 : P1 q1   P2 q1  and P2 q2   0 .
If we compare these bundles with the first-best ones (i.e. the contracts obtained under full
information) we can see that the quantity for high-valuation consumers is the same, while the
quantity designed for the low-valuation is reduced below the socially efficient level.
8.4 Sample problem with solution
A price discriminating monopolist allocates its output between domestic market and foreign
market. The monopolist’s marginal cost schedule is rising and marginal revenue curves are
declining on each market and current domestic price is below the foreign price. Suppose that
the government decided to tax export by introducing a per-unit export duty equal to t .
Assume that sales at both markets are positive both before and after the introduction of export
duty.
(a) Analyze graphically the impact of this policy on domestic sales assuming that all curves
are linear (you are expected to provide some comments to the graphs).
(b) Derive the analytical solution for general (non-linear) case for the impact of the policy on
domestic sales. [Differentiability of MR and MC is not assumed].
Solution
$
Foreign
$
Home
$
MRF = MRd
PF(q)
MC
Pd(q)
qF
MRF-t
MRF
qd
MRF –t = MRd
Q
MRd
Step 1. We equalize the values of MR across the two markets (monopolist will sell at both
only if MR is the same).
Step 2. Find the optimal production by intersecting equalized MR with MC.
Per unit tax reduces MR from export. As a result it affects the locus of points where MRs are
equalized and this new (blue) line intersects increasing MC at lower value of MC. Domestic
MR curve is unaffected and we need lower value of MR which happens at increased domestic
sales.
(b) Domestic sales will go up.
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 MR F (q F )  t  MC (q F  q d )
Proof. In equilibrium with export duty we have 
.
 MRd (q d )  MC (q F  q d )
Note that initially t  0 and then it increases by t  0 .
Let us prove that q d  0 . Suppose that this is not the case and q d  0 , then MRd  0 as
MR is diminishing. In equilibrium MC  MRd  0 . (**)
As MC is increasing, then Q  qd  q F  0 . Thus q F  Q  qd  0 . MR F is
diminishing then MRF  t  0 . This implies MC  MRF  t  0 which contradicts to
(**). Contradiction proves that q F  0 .
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9. OLIGOPOLY
We start our analysis of oligopolistic markets by looking at firms that compete with
homogeneous products.
9.1 Cournot model with N firms
Assumptions:
 firms compete by setting quantities,
 firms make their choice simultaneously,
 firms produce homogeneous product,
 entry into the market is completely blocked.
We are going to look at the simplest example of Cournot model with N firms and assume that
both firms have identical CRS technologies (cost function TCq   cq ) and market demand is
linear pQ  A  Q , where A  c .
To find Nash equilibrium we need to derive the best response function for each firm. But as
the firms have identical technologies, the corresponding best response functions would be
symmetric.
The best response function of firm i could be derived as a solution of its profit maximisation
problem, where rivals’ outputs are treated as exogenous parameters:



m ax  A  qi   q j qi  cqi  .

qi  0 
j i



Note that this function is strictly concave in q i , thus FOC is both necessary and sufficient.
FOC:
A  2qi   q j  c  0 .
j i
Note that we can rewrite it as qi  A  Q  c . As the RHS is the same for any i then qi  q 
for all i  1,, N and we get q  A  Nq   c , which gives q 
Q 
N A  c
1 N
 i  ( P  c )q  
and
price
equals
P   A  Q 
A  Nc
.
1 N
Ac
. Industry’s output is
1 N
Profit
of
firm
i
is
 A  c 2 .
1  N 2
Note that perfect competition can be treated as a limiting case of Cournot oligopoly when N
A  Nc A / N  c
goes to infinity: P   A  Q  

 c .
1 N
1 / N  1 N 
The case of N  2 could be illustrated graphically as an intersection of the two best response
functions.
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Cartel
NE
Both firms get
q1
higher profit
Isoprofit lines of firm 1. Equation  A  q1  q2 q1  cq1  1 or q 2   A  c  q1  

straight line
1
q1

hyperbola
q2
A-c
A-c-q1
A  c  q1   / q1
q1
  / q1
Note, that profit of firm 1 increases as we move down along its reaction curve (i.e. when
competitor’s output falls and as a result market price goes up).
Point of maximum of iso-profit curve always lies on the reaction function. The reason is
simple: for any output of the competitor (firm 2) the profit maximising output of firm 1 is
given by the corresponding point on reaction function and any other output results in lower
profit and so should lie on the higher iso-profit line.
As we can see both firms can get higher profits by producing less (shaded area). Thus we
have a problem similar to prisoners’ dilemma: if firms could collude, they would benefit from
the collusion.
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Collusion
If the total output is chosen collectively by the two firms (rather than individually), i.e. firms
create a cartel and choose the output by maximising joint profit, then
max A  Q Q  cQ .
Q 0
The first order condition requires the equality of market marginal revenue to marginal cost,
which implies MRQ   A  2Q  c and Q cartel   A  c / 2   A  c / 3  Q Cournot .
Note that as both firms have identical CRS technologies the total output can be produced at
any plant. Suppose the cartel’s profit is split equally and q1Cartel  q2Cartel   A  c  / 4 . This
point is illustrated on the above diagram. Note that cartel solution is characterized by a
tangency of the firms’ isoprofit lines.
Comparison of Cournot oligopoly, perfect competition and monopoly
Cournot Duopoly: Q Cournot 
A  Nc
N A  c
, p Cournot  A  Q  
.
1 N
1 N
Monopoly (collusion): Q M   A  c / 2 , p M   A  c / 2 .
Perfect competition: Q comp   A  c  , p comp  c .
QM 
 A  c   Q Cournot  N  A  c    A  c   Q comp
2
1 N
and p M  p Cournot  p comp .
As a result TS M  TS Cournot  TS comp since all the market structures are efficient in production
(marginal costs are equalized) and the only reason for efficiency loss comes from the
violation of allocative efficiency in case of monopoly and oligopoly.
9.2 The Stackelberg model
Assumptions:
 firms compete by setting quantities,
 firms make their choice sequentially,
 firms produce homogeneous product,
 entry into the market is completely blocked.
In the Stackelberg model one firm, the leader, sets its output first and the other firm (or firms)
reacts. This is an example of sequential game and we will look for the perfect Nash
equilibrium that requires using backward induction.
The leader uses its first mover advantage by forcing the rival to produce the quantity that,
combined with the leader’s output, results in the most profitable outcome for the leader.
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To find the equilibrium, we need to derive the best response function of the follower. Let firm
1 be the leader and firm 2 – the follower. If we use the same assumptions about demand
function and technologies as in Cournot model ( P Q   A  Q, AC i  MC i  c  A ), then
the reaction function of the follower is exactly the same is the one in Cournot model:
A  c q1
q 2 q1  
 .
2
2
Now let us look at the leader’s problem. He is trying to maximize his profit subject to best
response function of the follower:
m ax A  q1  q 2 q1  cq1 
q1
s.t . q 2 
.
A  c q1

2
2
By plugging the expression for q 2 into objective function we get


  A  c q1 

A  c q1 
max  A  q1 
 q1  cq1   max 
 q1  cq1  .
q1
q1
2
2
2


 2

Ac
Ac
. Plugging
 q1  c , which gives q1St 
2
2
into the best response function of firm 2 we obtain the output of the follower
A  c q 2St
.
q 2St 

2
2
The first order condition takes the form
Comparison with Cournot
We can see that the leader expands his output in comparison with Cournot case
Ac Ac
q1St 

 q1Cournot , which induces the follower to produce less than in
2
3
A  c q 2St A  c A  c
simultaneous game q 2St 



 q 2Cournot . Total output increases
2
2
4
3
3( A  c ) 2( A  c )
and
price
goes
down
Q St  q1St  q2St 

 Q Cournot
4
3
A  3c A  2c
P St 

 P Cournot .
4
3
Profit of the leader goes up  leader 
A  c
2
goes down  follower 
16
 A  c 2
8
A  c

 A  c 2
9
 1Cournot and profit of the follower
2

9
.
  Cournot
2
Question. Explain intuitively, why the leader can increase the profit.
We can illustrate equilibrium graphically as a point of tangency of isoprofit line of firm 1
(leader) with reaction function of firm 2 (follower).
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q2
Cornout
Stackelberg
q1
9.3 Price-setting oligopolists: Bertrand model with homogenous good
Assumptions:
 firms compete by setting prices,
 firms make their choice simultaneously,
 firms produce homogeneous product,
 entry into the market is completely blocked,
 each firm has CRS technology, i.e. MC=AC=c.
We start with a classical duopoly model with symmetric firms, that is, the case where
MC1  MC 2  c . As a result the best response functions will be symmetric.
0 if pi  p j

Residual demand of firm i = Q pi  / 2 if pi  p j .

Q pi  if pi  p j
Let us consider the best response of firm i to any price chosen by firm j :

If firm j charges price less or equal to MC , than the best response is to charge
price, equal to MC (or higher). If firm i responds by lowering the price it will
keep all the market and incur losses as the price is below its marginal costs.

Monopoly
If firm j charges price above MC but less (or equal) to P
, then it is
optimal to undercut this price by  4 and serve all the market.
4
Here we implicitly assume that prices are measured in discrete units and  is the smallest unit like one cent.
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If firm j charges price that exceeds the monopolist one, then firm i responds by
setting the monopolist price. This policy attracts all the customers and results in the
maximum possible (monopolistic) profit.
These best response functions are illustrated below.
p2
MC
NE
45
MC
p1
Nash equilibrium: ~
p1  ~
p2  MC . As a result each firm serves one half of the market
~
~
~
q  q  Qp/ 2 and profit of each firm equals zero.
1
2
Conclusion: price competition is more severe than quantity competition. The reason is that
with identical products the firm that charges lower price takes all the market (not just part of
the market).
Bertrand paradox: if MС1  MC 2  с then p1  p2  с and firms do not get any profit.
Solution:

firms may have capacity constraints

firms interact repeatedly and collusive outcome can be sustained in the infinitely
repeated game if firms are patient enough.
9.4 Price leadership or Dominant firm model (optional)
Assumptions:
 firms make their choice sequentially,
 leader sets the price and competitive fringe responds by choosing quantity,
 firms produce homogeneous product,
 entry into the market is completely blocked.
Instead of setting the quantity, the leader may set the price. To find the optimal price the
leader has to take into account the response of other firms (competitive fringe) that take this
price as given and choose the quantity produced. Output produced by competitive firms
determines the residual demand that could be served by the leader.
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As usually we look for the perfect equilibrium by solving this model backward. The problem

of competitive fringe: max pq f  TC
qf
f
q  . The first order condition is
f
p  MC f q f  .
Thus, the non-diminishing part of MC f gives the supply of competitive fringe S  p  .
Now, we can find the residual demand as the difference between market demand and supply
of competitive fringe: Q res  p  maxQ p  S  p,0.
Dominant firm will choose price by maximising its profit subject to the residual demand, that
is, it acts like monopolist on residual demand. Problem of dominant firm:



max pQ res  p  TC dom Q res  p
p
or it could be equivalently rewritten as


max QP res Q   TC dom Q  .
Q
The first order condition requires equality of MR for the residual demand and dominant firm’s
marginal cost.
Let us illustrate equilibrium graphically for the case of linear demand and marginal cost
curves.
P
P(Q)
MCfringe
Pres(Q)
Qfringe
MCdom
Qdom
Q
res
MR
9.5 Repeated interactions
Bertrand paradox: in a symmetric Bertrand game each firm gets zero profit, while had both
firms agreed to charge the monopolist price, each would get half of the monopolist profit.
This problem is similar to prisoners’ dilemma.
In fact, firms set their prices each period, that is, they interact repeatedly. With repeated
decision making firms can base their decisions at a given time on the actions that have been
taken in the past, that is, firms can react to their rivals’ behaviour. This might help to achieve
the tacit collusion as, if one firm cheats on the agreement, another firm may be able to punish
it in the future. Whether this kind of strategy will be viable depends on whether the game is
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going to be played a fixed (known) number of times or indefinite (or unknown) number of
times.
Finitely repeated game
Let us suppose that both players know that the same (for example Bertrand) game will be
played N times ( N is finite). What will be the outcome?
As the game is dynamic, we look for the perfect Nash equilibrium using backward induction.
Consider the last round. As this round is the final one and everybody knows it, then there is
no incentive for cooperation and every player will choose the static game Nash equilibrium
strategies by charging price equal to MC. Now consider what will happen on round N  1 . As
at the last round there will be no cooperation there is no incentive to cooperate at this round as
well.
If one cooperates by charging the monopolist price the rival will find it optimal to cheat by
charging lower price and getting all the market. Each player has the incentive to deviate and
as a result the only equilibrium of this subgame is given by a static Nash equilibrium, where
each firm charges price equal to MC. The same logic proves that there would be no
cooperation at each round and the only perfect equilibrium corresponds to prices equal to
marginal cost.
The result is not surprising as players cooperate only if there is a punishment for cheating.
With finitely repeated game at the last round cheating cannot be punished and this creates
incentive for deviation at each round.
Infinitely repeated game
If the game is infinitely repeated, then the last round does not exist and as a result deviation at
any point of time could be punished in the future. For punishment strategies to be effective,
they must be both severe and credible.
The punishment is severe enough to deter cheating by a firm if the cost of cheating (i.e. loss
from punishment) outweighs the benefit. The cost is associated with the loss in profits due to
the punishment. As profit could be reduced in a number of periods, we should look for the
total reduction in profit, taking into account that profits of different periods have to be
summed up with discount factor (discount factor reflects the current value of future profit).
Thus, cost of punishment is the present value of the stream of forgone profits that results
when cheating is detected and punishment is implemented. Forgone profit in each period is
calculated as a difference between the profit that is earned under collusion and profit that is
earned if punishment takes place. For example, if cheating takes place at period t and
punishment starts from t  1, then the cost of punishment is given by

collusion
  punish
Loss from punishment  
where  
 
t 1
 t 2 
  
collusion
  punish
t 1
 1  ,
1
and r stays for the market interest rate.
1 r
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The benefit of cheating is the present value of the stream of additional profits that are enjoyed
while cheating goes undetected. Each period benefit is given by a difference between the
profit earned while cheating and the profit under collusion. In our example cheating is
detected in the next period, so the benefit from cheating is derived only in one period:


dev
collusion
t .
Benefit from cheating    
Thus the punishment is severe enough to deter cheating if



dev  collusion t  dev   punish

Solving this inequality, we get  

t 1
or dev  collusion  collusion   punish
.
1
1

 

dev  collusion
. Note that this condition has to be satisfied
dev   punish
for each player.
The punishment is credible if it is in the non-cheating firms’ self interest to implement the
with pumishment
no pumishment
punishment when cheating is detected, that is,  nondeviating firm   nondeviating firm .
Infinitely repeated Bertrand duopoly (with symmetric MC)
Each firm uses the following grim-trigger strategy: it charges monopolist price in period t if
there were no deviations in previous periods and switches to one-period Nash equilibrium (i.e.
sets price equal to MC) otherwise:
m

 p , t  0 or
pi  t   

c otherwise
pi t  1  p j t  1  pm
The strategy of this sort is called Nash-reversion strategy.
Let us check whether these strategies are severe and credible. Assume that deviation is
revealed in the next period.
Optimal deviation corresponds to price which is a bit lower than monopolist and as a result
deviating firm will serve all the market and gets almost the monopolist profit  dev   mon .
Under tacit collusion each firm gets half of the monopolist profit collusion   mon / 2 . Thus, the


benefit from deviation is  dev   collusion   mon   mon / 2   mon / 2 .
Had the punishment been implemented, each firm gets zero profit and as a result cost of
 mon
cheating equals  mon / 2  0    2   

/2.
1 



1

 1/2.
mon / 2 which is equivalent to  
1r
1
This inequality is satisfied for any interest rate below 100%: r  1 .
No deviation condition: mon / 2 
Check credibility: if punishment is implemented, then profit of each firm becomes zero. If
non-deviating firm does not punish the cheater and continues to set monopolist price, then
nobody will purchase from this firm and its profit will be zero. Thus punishment does not
decrease the profit of the punisher, which implies that it is credible.
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Conclusion: under r  1 Nash reversion strategies constitutes perfect Nash equilibrium in
repeated Bertrand game.
Question. In the framework of infinitely repeated Bertrand model analyse how the likelihood
of collusion is affected by the number of firms in the industry.
9.6 Bertrand model with differentiated goods
So far we looked at the strategic interactions between firms that produce identical products.
Now we are going to look at product differentiation.
Differentiated products are goods that satisfy a particular need but differ in their individual
specifications.
Assumptions:
 firms compete by setting prices,
 firms make their choice simultaneously,
 firms produce differentiated goods,
 entry into the market is restricted by positive product development costs.
Suppose that firms 1 and 2 are duopolists that compete by setting prices simultaneously. They
have identical cost functions with MC=AC=c. Firms face the following demand functions
Q1  A  p1  p 2 and Q2  A  p2  p1 , where A  c . Thus, demand for the good produced
by firm i depends negatively on its own price and positively on the price of its close
substitute.
As the two firms act simultaneously, we look for the Nash equilibrium that can be represented
as an intersection of the two best response functions. Let us find the best response of firm i
given that firm j charges p j by solving the following profit maximisation problem:
max( pi  c)( A  pi  p j ) .
pi 0
The first order condition gives: A  pi  p j  ( pi  c)  0 or pi  0.5 A  c   0.5 p j . Thus,
the best response functions are upward sloping.
Since demand functions are symmetric and cost functions are identical, the resulting prices
would be equal: p1  p2  A  c  c .
The corresponding outputs and profits are given by: Qi  A and 1   2  A2  0 .
As we can see, symmetric price competition may result in positive profits if firms produce
differentiated products.
Note: under sufficiently high product development costs (F), firms earn positive profits even
in the long run.
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NE
Sequential pricing with differentiated goods
Suppose now that firms interact sequentially. Let firm 1 move first and choose p1 , then firm 2
observes p1 and chooses p2 .
We solve this game using backward induction. Best response function of firm 2:
p2 
A  c p1
 .
2
2
Firm 1 will maximize its profit taking into account the best response of the second firm:
max( p1  c)( A  p1  p2 )  max( p1  c)1.5 A  0.5c  0.5 p1 
p1
p1
s.t. p2  0.5 A  0.5c  0.5 p1
FOC: 1.5 A  0.5c  0.5 p1  0.5 p1  c   0 .
Thus p1  c  1.5 A and p2  0.5 A  0.5c  0.5c  0.75 A  c  1.25 A . Plugging into demand
functions we get the following equilibrium outputs: Q1  A  c 1.5 A  c  1.25 A  0.75 A ,
Q2  A  c 1.25 A  c  1.5 A  1.25 A
 2  1.25 A  1.25 A 
and
profits
9
1  1.5 A  0.75 A  A2
8
and
25 2
A .
16
We can see that profit of each firm was increased in comparison with the simultaneous game
and the firm that moves second has gained more than the first-moving firm:
25 2 9 2
2 
A  A  1 .
16
8
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To illustrate the sequential equilibrium graphically we need isoprofit lines. Isoprofit curve for
firm 1 is given by ( p1  c)( A  p1  p2 )   , which can be restated as
p2 

 ( p1  A). When profit increases the isoprofit line shifts upward.
p1  c
p2
p1
c
Equilibrium corresponds to the point of tangency of isoprofit line of the leader (firm 1) and
reaction function of the follower (firm 2).
Simultaneous
Sequential
9.7 Sample exercise with solution
Consider a repeated Bertrand model with two risk-neutral firms that have identical
technologies. Suppose that at each period there is a probability   0, 1 that the firms keep
competing next period and with probability 1   market can seize to exist in any future
period. Firms discount the future, so that, for each firm, a payoff of $1 received t periods from
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today is worth $  t today, where 0    1 .How an increase in the probability  affects the
likelihood of tacit collusion? Your answer must specify the strategies that the firms should
follow to sustain tacit collusion.
Solution
Punishment strategies (i=1,2):
m
m

 p , t  0 or pi t  1  p j t  1  p
pi  t   

c otherwise
Optimal deviation: pm   brings profit   m and the other (non-deviating) firm gets zero
profit as nobody is willing to purchase at higher price.
Credibility: non-deviating firm gets zero profit in both cases with punishment (as it sells at
p=AC) and without punishment (in this case it has zero sales). Thus punishment is credible as
doesn’t reduce its expected profit.
No deviation condition:

m
i 0
2

 i 
m / 2
1  

  m   0   , 1  2  2 ,   1/2  .
i
i 1
If  goes up then the minimum level of discount factor required to sustain collusion will be
lower so that the interval expands. Thus it would be more likely that the existing discount rate
belong to this interval.
Intuition is straightforward: deviating firm gain some profit today but is punished by the lost
future expected profits. An increase in  increases the expected value of these future profits
and make punishment more severe, as a result firm has lower incentive for deviation.
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10. FACTOR MARKETS
5
10.1 Demand for factors
We get individual firm’s demand by solving its profit maximization problem
max TRQK , L   TC K , L 
K 0, L 0
FOC for interior solution:
TR QK , L  TC K , L 
TR QK , L  TC K , L 


and
,

L

L

K
K

 

 
MRPL
MFC L
MRPK
MFC K
That is, the marginal revenue product of a factor (MRP) should be equal to the marginal
factor cost (MFC).
Marginal revenue product of factor i : MRPi 
TR Qx  TR Qx  Qx 

 MR  MPi .
x i
Q
x i
If output market is perfectly competitive, then MR  p and MRPi  pMPi .
If factor i market is competitive, then MFC i 
TC x 
 wi .
x i
Thus, if all (input and output) markets are perfectly competitive, then factor demands can be
found from the solution of the system pMPi  wi .
Consider a model with two factors of production, capital and labour. Assuming diminishing
marginal product of each factor and fixing the amount of capital used, the value of marginal
product of labour gives the inverse labour demand curve (see the figure below).
0
5
In the new syllabus there is no separate topic on factor markets: demand is studied in the theory of firm, supply
is studied in consumer theory and equilibrium is studied in different topics on market structure.
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Firms’ demand curve for a factor and industry demand curve.
To get the industry demand curve for a factor (in case of perfectly competitive industry) we
should take into account the product-price effect.
To simplify analysis let us assume that labour is the only factor of production. As individual
firm’s demand for an input is given by its MRP curve, then we should sum up the individual
demands by taking the horizontal sum of MRP curves. But the resulting curve does not
correspond to industry demand. Suppose initially industry economy was in the equilibrium
with output price p0 , input price w0 and market labour demand L0 . If wage rate falls, then
each firm will demand more labour and we move to L  . But as employment goes up, the
quantity produced increases, which brings a fall in output price from p0 to p1 . Thus,
employment rises only till L1 . This analysis suggests that the industry supply curve is steeper
(less price elastic) than the horizontal sum of the firms’ demand curves.
Industry demand
horizontal sum of
horizontal sum of
0
Conclusion 1. The lower the elasticity of demand for the product, the lower the elasticity of
demand for a factor.
Supply
(more
price elastic)
0
0
Explanation. With the more price elastic demand for the final product (flatter output demand
curve), the price fall would be smaller and the resulting downward shift of the MRP sum
would be less, which results in larger employment and flatter (more price elastic industry
factor demand).
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Firms’ demand curve for a factor in the SR and in the LR
In the short run capital is fixed and MRPL K 0  gives the firm’s demand for labour. But in
the LR capital adjusts and as a result MRPL curve shifts.
Two inputs are said to be complementary (anticomplementary) if the increased use of one
input raises (reduces) the marginal product of the other. If one input has no effect on the
marginal product of the other input, then factors are said to be independent.
Let the firm’s long run quantities demanded be given by L0 , K 0  and the initial wage rate is
w  w0 . Suppose wage rate falls till w1  w0 . In the SR firm will demand more labour,
which can be illustrated by movement to the right along the SR demand.
If factors are not independent, then MPK would be affected and capital will adjust in the LR,
which, in its turn, would affect the position of MRPL curve. Here, two cases should be
considered separately.
(a) K and L are complementary.
As labour increases in the SR, the marginal product of capital increases and as a result
MRPK  r . Thus firm will decide to use more capital, that is, K 1  K 0 . As a result marginal
product of labour goes up and brings an upward shift of
MRPL
curve (since
MRPL K 1   MRPL K 0  ). As marginal productivity of labour goes up firm finds it
profitable to use more labour, thus in the LR labour increases more than in the SR
(b) K and L are anticomplementary.
LR demand
0
LR demand
0
(a) complementary
factors
(b) anticomplementary
factors
As labour increases in the SR, then marginal product of capital falls and as a result
MRPK  r . Thus firm will use less capital, that is K1  K 0 . As a result marginal product of
labour goes up and brings an upward shift of MRPL curve (since MRPL K1   MRPL K 0  ).
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As marginal productivity of labour goes up, the firm finds it profitable to use more labour,
thus in the LR labour increases more than in the SR
Summary: demand curve is flatter (more price elastic) in the LR than in the SR.
Conclusion 2. Given either complementarity or anticomplementarity between inputs, the
demand curve for an input is flatter (more price elastic) in the LR than in the SR (when some
factors are fixed).
Conclusion 3. Factor demand curves always have negative slope (i.e. there is no such thing as
a “Giffen factor”.
Analytical proof. Let firm use two factors of production L, K , with corresponding prices w
and r . Let us fix the output price p and consider two different vectors of factors prices
(w 0 , r ) .and (w, r ) Denote by ( L0 , K 0 , Q 0 ) and ( L, K , Q) the solutions of profit
maximisation problems under (w 0 , r ) and (w, r ) , correspondingly. If ( L, K ) brings
maximum profit under given prices, then any other combination of factors cannot result in
higher profit and similarly for ( L0 , K 0 ) :
TR(Q )  wL   rK   TR(Q 0 )  wL0  rK 0
TR(Q 0 )  w 0 L0  rK 0  TR(Q )  w 0 L   rK 
.
Adding up these inequalities we get:
 wL   w 0 L0  wL0  w 0 L  or 0  w( L  L0 )  w0 ( L  L0 )  (w  w0 )(L  L0 ) .
Last inequality means, that if w  w 0 under constant price of other input, then demand for
labour either fall or stay the same: L  L0 .
The difference with consumer theory: when price of a factor goes up, the initial combination
of factors is still affordable for the producer as he does not face financial constraint.
Demand for a factor and supply of other factors.
To simplify the analysis let us assume that product-price effect is negligible.
Let the industry be in the long-run equilibrium initially with
w0 , r0 
and factors’
employment of L0 and K 0 . Suppose that wage rate falls till w1  w0 . In the SR industry
labour employment goes up from L0 to LSR
1 .
If factors are not independent, then MPK would be affected and capital will adjust in the LR,
which, in its turn, would affect the position of the industry labour demand curve. Here, two
cases should be considered separately.
(a) K and L are complementary.
As labour increases in the SR, the marginal product of capital increases and as a result
MRPK  r . Thus, the demand for capital goes up, which results in an increase in capital
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employment: K 1  K 0 . As a result marginal product of labour goes up and brings an upward
shift of MRPL curve for each firm (since MRPL K 1   MRPL K 0  ). As marginal
productivity of labour goes up each firm will use more labour and industry demand curve
shifts to the right. Finally, the labour increases from L0 to LLR
1 .
complementary factors case
(more
price elastic)
0
0
Labour market
Capital market
(b) K and L are anticomplementary:
This case brings the same result in terms of labour but K1  K 0 (FOR SELF-STUDY).
Let us analyze the role of elasticity of capital supply curve. With flatter (more price elastic)
supply of other factor its employment will change more in response to the increased demand,
which results in greater shift in the SR labour demand curve. As a result, LR industry demand
for labour would be flatter (more price elastic).
Conclusion 4. An industry demand for a factor is less elastic the less elastic is the supply of
other factors.
10.2 The supply of factors and competitive equilibrium
Individual supply curves could be derived from consumer’s utility maximisation problem (it
was studied in the theory of consumption).
Although individual labour supply could be backward bending, the market supply curve is
likely to be upward sloping. Explanation: even if some workers that are currently in the
industry may prefer to work fewer hours as the wage rises, new workers will be attracted
(from other industries or from leisure).
Elasticity of supply of a factor to a particular use depends:
 on the degree of specificity of the factor to this use;
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 on the length of time allowed for the factor to be reallocated to or away from that use.
Factor market equilibrium
Economic
rent
Transfer
earnings
L
Input earnings can be divided into two components: transfer earnings and economic rent.
Transfer earnings represent the amount that any unit of a factor must earn in order to prevent
from being transferred to another use.
Economic rent is any excess over transfer earnings that a unit actually earns.
Factor payments which are economic rent in the SR and transfer earnings in the LR are called
quasi-rents.
10.3 Monopsony and monopoly in factor markets
Monopoly on the supply side of the input market.
By combining into a trade union, workers (with no market power individually) may
collectively be restricting the supply of labour, raising their wages.
Examples of trade union objectives:
i. the rent maximization,
ii. the total wage bill maximization.
Consider labour market with linear demand curve for labor LD  A  bw and linear labour
supply curve: LS  cw , where A  c . Suppose that the labor market is controlled by
monopolistic labour union that maximizes economic rent.
Economic rent is any excess over transfer earnings that a unit actually earns
max
L 0
L
~ ~
w D ( L)  L   w S ( L )dL ,
0
where w S ( L)  L / c and w D ( L)  (a  L) / b .
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The first order condition for interior solution gives:
or
MRL  MC L
w D ( L)  Lw D / L  w S ( L) . Plugging expressions for w S () and w D () , we get
ac
ab  2ac  ac
a(b  c )
and w  w D ( L) 
.
(a  2L) / b  L / c . Finally we get: L 

b  2c
b(b  2c )
b(b  2c )
L
Now, suppose instead of rent trade union maximizes total wage bill:
max
L0
w D ( L)  L
Total wage bill maximization results in choosing employment level, where marginal revenue
equals 0: MRL  ( A  2L) / b  0 or L  A / 2 and w  0.5 A / b .
Note: employment is greater in case of wage bill maximization model:
Lwage bill 
A
A

 Lrent .
2 2b/c
The problem with wage bill maximization model: it might happen that the wage rate does not
cover the opportunity unit labour cost. This is the case if 0.5 A / b  Lw.bill / c  0.5 A / c , i.e.
b c.
Monopsony on the demand side of the input market.
Monopsony is a market situation in which there are many sellers but only one buyer that
exercises the market power.
Consider monopsonistic labour market.
Monopsonist
will
MFC
with
p  MPL ( L )  MFC ( L ) , where MFC  ( L  w ( L))L  w ( L)  Lw( L) .
0
choose
0
employment
that
S
equates
MRP:
S
If labour supply curve is upward sloping them MFC curve will lie above the inverse labour
supply at any L  0 and MFC 0  w S (0) . If labour supply is linear, then MFC is also linear
and two times steeper.
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L
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11. ASYMMETRIC INFORMATION
11.1 Types of asymmetric information problems
Asymmetric Information
HIDDEN INFORMATION
One side knows some
characteristics of itself that
the other does not
HIDDEN ACTION
One side can take an action that
affects the other side
but is not observable
ADVERSE SELECTION
uninformed side ends up
trading only with the wrong
types, as the good types are
driven out of the market
MORAL HAZARD
a party to a contract engages in
post-contractual opportunistic
behaviour (takes the wrong
action)
11.2 Adverse selection and the market for lemons
George A. Akerlof (1970) `The market for `lemons': quality uncertainty and the market
mechanism', Quarterly Journal of Economics 84, 488-500
Consider a market for second-hand cars. There are some high-quality cars (rarely breaks
down) and some low-quality cars. Assume that there are many more buyers (M) than sellers
(N) in the market: M >> N. Each buyer is willing to purchase, at most, one car. Half of the
sellers own high-quality cars and another half – low quality ones. A high quality car is worth
$1000 to the seller, while a bad quality one is worth only $300 to the seller. A buyer is willing
to pay $1300 for a high quality car, and $400 for a low quality one. All agents are riskneutral. Assume that markets are perfectly competitive. Sellers get the entire surplus from
trade (as N<<M)
Valuation ($)
Seller
Buyer
H type
1000
1300
L type
300
400
First of all, let us derive the efficient allocation of cars. Efficient allocation should maximize
total surplus: denoting by q H the number of high quality cars transferred to buyers and by q L
- the number of low quality cars transferred from the sellers to buyers, we get
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



1300q H  1000 0.5 N  q H  400q L  300 0.5 N  q L )  300q H  100q L  650 N 
max
0 q H  q L  M
0 q H  N / 2
0 q L  N / 2
Solving this problem we get: n H  N / 2, n L  N / 2 , that is, all cars should be transferred
from sellers to buyers.
Symmetric Information Case
Under symmetric information we have to separate the markets: one for low-quality cars and
another one – for high-quality cars. The supply of each type is zero if the price is below the
sellers’ valuation and supply equals N/2 for the prices above the sellers’ valuation. If the price
is equal to the sellers’ valuation, then each seller is indifferent and the market supply is any
number from 0 up to N/2.
In a similar way, the demand at every market is zero if the market price is below the buyers’
valuation and demand equals M for the prices below the sellers valuation. If the price equals
to the buyers’ valuation then every buyer is indifferent and total quantity demanded is any
number between 0 and M.
$
$
SH
1300
SL
DH
1000
DL
400
300
N/2
L market
M
QL
N/2
M
QH
H market
As a result in equilibrium all cars are sold. High-quality cars are sold at the price equal to
$1300, while low quality cars are sold at $400 per car. As we have a complete system of
perfectly competitive markets it is not surprising that the resulting allocation is PO.
Asymmetric Information Case
Now suppose that each seller knows the quality of his car but buyers are not able to
distinguish a high-quality car from a low-quality one. Then, instead of two markets, we have
only one.
We get the market supply by summing up horizontally the supply curves for high- and lowquality cars:
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 N , p  1000
N / 2, N , p  1000

Q S  p    N / 2, 300  p  1000
0, N / 2, p  300

0, p  300
We assume that buyers rationally expect only low-quality cars to be sold under price less than
$1000. As each buyer is willing to pay up to $400 for low-quality car, then the quantity
demanded is M if price is below $400. If the price equals $400 then every buyer is indifferent
and total quantity demanded is any number between 0 and M.
At price between $400 and $1000, the quantity demanded is zero as buyers expect only lowquality cars and they are not willing to buy these cars at price above $400.
Buyers expect average quality at price of $1000 and higher. For the average quality each
buyer is willing to pay $13001/2 + $4001/2 = $850. But at price of $850 only low-quality
cars appear at the market. Thus, the quantity demanded stays zero at price above $1000:
0, p  400

Q D  p   0, M , p  400 .
 M , p  400

S
$
1000
400
D
300
N/2
Q
M
The intersection of demand and supply appears at p=400, where only low-quality cars are
sold. We observe an adverse selection as high-quality sellers leave the market. This allocation
is inefficient as the potential gain from high-quality cars is lost: DWL=(1300 –
1000)N/2=150N.
Solution of the problem: separation through voluntary refunds
One of the solutions of the adverse selection problem at the used cars market is given by the
voluntary refund policy. Suppose that sellers can promise a refund of R if the car breaks
down. Assume that low-quality cars break down with probability 0.8, while high-quality cars
break down with probability 0.1.
Can this policy lead to a separating equilibrium?
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In separating equilibrium, high-quality cars are sold at price of $1300 with refund while lowquality ones are sold at price of $400 without refund. In equilibrium the two types of
constraints should be satisfies. First of all, sales should be profitable for both types, that is,
participation constraint (PC) should take place for every seller.
PC for high quality 1300 – 0.1R  1000 is satisfied if R  3000, while PC for low-quality is
satisfied automatically as 400 > 300.
In addition to PC, we need incentive compatibility constraint (IC): it should be profitable to
sell with R for H-type and without R for L-type:
1300 – 0.1R  400  R  9000,
400  1300 – 0.8R  R  9000/8=1125.
Finally, we get a separating equilibrium if 1125  R  3000.
Forced refunds policy
Now, assume that the government forces each seller to offer a full refund if the car breaks
down. As the policy is mandatory it cannot result in a separating equilibrium where both highquality and low-quality cares are sold. Either all sellers stay at the market and offer full refund
or some sellers leave the market.
If both types stay at the market, then they sell at the same price as consumers cannot identify
the car type. As the probability of breakdown is higher for a low-quality car seller, then the
expected profit of this seller is lower and we should analyze his participation constraint. Sales
are profitable for L-type sellers if P – 0.8P  300, that is, PC is satisfied for P300/0.2=1500
but nobody is willing to purchase a car at price above $1300. It means that only the sellers
with high-quality cars stay at the market and sell at price equal to $1300. For these sellers PC
is satisfied as 1300 – 0.11300=1170 > 1000.
The resulting outcome is inefficient as potential gain from low-quality cars is not realized:
DWL=(400 – 300)N/2 = 50N. Thus, obligatory refund destroys the separating role of this
policy.
11.3 Adverse selection at insurance market
Assumptions:
 price taking,
 two groups of customers: high-risk with probability of loss  H and low-risk with
probability of loss  L  L   H  , the share of high risk   0, 1 ,

customers have the same initial wealth W and have the same potential loss L ,
 customers are risk averse, preferences are represented by EUFs with the same
elementary utility functions but different probabilities of loss,
 insurance companies are risk-neutral,
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 there are no operation costs.
Equilibrium under symmetric information
As there are two different types of risk, identified by each agent, then there would be two
different insurance markets: market for high risk insurance and market for low risk one.
Consider insurance market of type t  H , L.
Demand side analysis.
If insurance is fair ( rt  t ), then each agent wants to purchase full insurance as agents are
risk averse.
If insurance is favourable ( rt  t ), then agents want to over-insure (if it is possible) or
purchase full insurance (if overinsurance is not allowed).
Finally, if insurance is unfavourable ( rt  t ), then less than full insurance would be
demanded.
Supply side analysis.
As insurance companies are risk neutral, then each company would maximize expected profit.
If insurance is fair ( rt  t ), then insurance company gets zero profit per dollar of insurance
sold and is willing to sell any quantity.
If insurance is favourable ( rt  t ), then insurance company gets negative expected profit and
is not willing to supply insurance at all.
Finally, if insurance is unfavourable ( rt  t ), then each unit of insurance sold brings positive
profit and the solution of profit maximization problem does not exist (supply is unlimited).
Thus the case of unfavourable insurance can never be observed in equilibrium.
Favourable price does not equilibrate the market as well, since supply is zero while demand is
positive.
It means that equilibrium price has to be fair, i.e. each agent would be insured completely but
prices are different.
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L-type
H-type
Equilibrium under asymmetric information
Suppose that each customer knows his probability of loss, but insurance companies do not.
As insurance companies are not allowed to differentiate between the two types of customers,
there would be one market with uniform price.
Note that price which is equal to the average probability of loss will never equilibrate the
market. Indeed, suppose r   AV   H  1    L . This price is in between  L and  H . As
r   H , then insurance is favourable for high-risk agents and they will purchase full
insurance (assuming that over-insurance is not allowed), i.e. x H  L . Low-risk agents find
this price too high and will purchase less than full insurance, i.e. x L  L . As the quantity
demanded by good customers (low risk) is less that quantity demanded by high-risk, the
resulting expected profit would be negative:








    AV   H L  1     AV   L x L    AV   H  1     AV   L L 

 
AV

  H  1    L L  0
Thus, the insurance company will never sell insurance under this price. As a result, the price
of insurance goes up and two types of equilibrium may take place.
Note that, in general, equilibrium price should bring zero expected profit: negative expected
profit results in zero supply, while positive expected profit leads to unlimited supply.
Equilibrium of type 1 corresponds to r   H . If low risk agents find this price too high and
prefer to stay at initial endowment, then only high risk stay at the market and purchase full
insurance at fair price.
Thus, only high risk agents may stay at the market, i.e., we deal with adverse selection
problem.
Equilibrium of type 2 corresponds to  AV  r   H . At this price, high risk agents purchase
full insurance (as price is favourable) while low risk find the price unfavourable and purchase
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only partial insurance. Moreover, the losses of insurance company from high-risk agents are
compensated by profits from low-risk so that total expected profit is zero.
cNL
cNL
Low-risk
W
Lowrisk
High-risk
High-risk
cL
cL
W-L
Whatever is the equilibrium, the risk is allocated inefficiently.
As firms are risk neutral while agents are risk averse, at any efficient allocation insurance
companies should bear all the risk, while agents have to be insured completely. But in
equilibrium only high-risk agents are insured completely, while low risk take at least part of
the risk.
11.4 Private and Government Response to Adverse Selection Problem
Private response

Screening: uninformed party moves first and offers a menu of contracts
Example: Monopolist offers menu of bundles to consumers that differ in WTP (see
second degree price discrimination)

Signaling: informed party moves first and signals its type

Student gets education to signal his talent to potential employer

Seller may offer a guarantee to signal high quality

On-line seller may invest in professionally-looking site to signal customers that the
business is not a fly-by-night operation
 Group insurance plans (employers offer health insurance as part of benefits. Insurance
company deals with a group of people and so it faces average risk since agents can’t
refuse from purchasing insurance)
 Targeted insurance rates (insurance premium is based on some observable characteristic
that is correlated with unobservable one)
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Government response:
 compulsory health insurance, compulsory public pension programs, OSAGO (as
program is compulsory, insurance company deals with average risk)
 information policies:
 prohibiting false advertising,
 setting quality standards,
 mandatory disclosure of certain facts (used car dealers are supposed to disclose
known defects)
11.5 Spence model of job market signaling
Assume that there are two groups of workers that differ in productivity MPL L  MPL H .
The share of high-productive group is  . There is no possibility for partial employment:
either worker gets a contract that guarantees annual wage w for n years or he stays
unemployed and gets 0 reservation utility.
The utility of worker of type t t  L, H  increases in his income   n  w but decreases in
education ( y  years of education): ut , y    ct  y , c L  c H and costs of education are
higher for low-productive worker. As a result indifference curves of a low-productive and a
high productive agent are upward sloping and indifference curves of L-type agent are steeper
(this property is known in economic literature as single-crossing property).
$
ICL
ICH
y
Firms compete by offering wages (Bertrand-type price competition) and as a result in
equilibrium under symmetric information each worker is paid in accordance with
wH  MVPL H and wL  MVPL L
Assume that each worker knows his/her productivity but firms cannot differentiate highproductive workers from the low-productive ones. To solve the problem workers acquire
education to signal their productivity.
Game timing:

workers choose education,

firms observe education and offer wage schedule,

workers decide whether to take the offer or not.
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Separating equilibrium
In separating equilibrium, each type chooses a different action, that is, in our case highproductive workers signal their productivity by acquiring education while low-productive
workers do not get education.
There is a threshold number y  of years of education such that if y  y  , then the worker is
considered to be high-productive and otherwise (if y  y  ) worker is considered to be a lowproductive one. Price competition between firms results in the following compensation
offered:
 L , y  y 
, where  L  wL  n and  H  wH  n ,
y   
 H , y  y 
Separating equilibrium exists if costs of signalling are less than benefits for high-productive
workers but higher for low-productive ones. Benefit from signalling is equal to the wage
premium B   H   L  wH  wL   n . Thus, the following conditions should take place:
B   H   L  c L y  and B   H   L  c H y  . Solving this system we get:
H  L
  L
 y  H
.
cL
cH
So far, we ignored participation constraints. These constraints are trivially satisfied due to the
assumption of zero reservation utility: for a low-productive worker PC requires L  0 and
for a high-productive worker we need  H  c H y   0 , which implies y    H / c H . This
condition is satisfied as y  
H  L H

.
cH
cH
Let us analyze the efficiency of separating equilibrium. It could be noticed that we have many
equilibria that differ in the level of signal of high-productive workers. In any equilibrium, the
expected profit of each firm is zero and utility of low-productive worker equals to  L but
utility of high-productive agent is different and it is the highest in the equilibrium with the
  L
lowest level of signal, that is, the equilibrium with y   H
is a Pareto superior one.
cL
But even in this equilibrium, the utility of high-productive workers is below the one observed
under symmetric information due to unproductive signaling costs that constitutes DWL.
$
H
(y)
L
y
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Pooling Equilibrium (both types acquire education)
Now we proceed to the analysis of pooling equilibrium. In pooling equilibrium all types
choose the same action, that is, in our game both types get education and as a result
compensation is based on average productivity:    H wH  1   H wL   n .
There is a threshold number y  of years of education such that if y  y  , then worker is
considered to have average productivity and otherwise (if y  y  ) worker is considered to be
a low-productive. Price competition between firms results in the following compensation
offered:
 L , y  y 
y   
.
, y  y 
Pooling equilibrium exists if costs of signalling are less than benefit for both types. Benefit
from signalling is equal to wage premium: B     L   H wH  wL   n . Thus, the
following conditions should take place: B     L  c L y and B     L  c H y . Deviation
is more profitable for low-productive than for high-productive worker, that is why only Ltype constraint is binding:
  L
0  y 
.
cL
$
H
(y)
L
y
Now we move to the analysis of participation constraints:   с L y   0 and   c H y  0 .
Since
y 
  L
cL
then
both
constraints
are
satisfied
automatically:
y      L  / c L   / c L   / c H .
11.6 Screening
Screening deals with uninformed party’s attempt to sort the informed parties by offering
special pricing schemes.
Screening requires some self-selection device: the uninformed party offers a set of options,
and the choice made by the informed party reveals his/her hidden characteristic.
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Example: low insurance premium in case of deductibles and high insurance rates for full
insurance. Deductibles: if damage is below certain limit, than agent pays for the bill himself,
if damage is above certain limit, then only damage in excess of this limit is compensated.
Low risk agents prefer insurance with deductibles with low insurance premium, while highrisk purchase expensive full insurance.
Another example of screening policy is given by the second degree price discrimination.
Below we consider a modified model of price discrimination, where the monopolist offers
quality-price instead of quantity-price contracts.
Assume that the cost of producing quality q is cq   q 2 . The seller sells a unit of quality q
at price t and gets the profit t , q   t  q 2 . His iso-profit lines t  q 2   are illustrated
below.
t
q
Each buyer purchases one unit of the good at most. Reservation utility of each buyer equals 0.
Buyers utility (consumer’s surplus) is increasing in quality and decreasing in price:
u, t , q  q  t , where  reflects quality preferences. There are two groups of consumers
that differ in quality preference parameter:  H   L  0 . Indifference curve of the type i
buyer is given by:  i q  t  u , which can be restated as t   i q  u . Thus, indifference
curves are straight lines with the slope dt / dq   i . It implies that the group with higher
quality preferences has steeper indifference curves.
t
uH
uL
q
Optimal Contracts under Full Information
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Under full information, different contract are offered to different types. Denote by qi , t i  a
i
contract offered to consumer of type . This contract should satisfy participation constraint
i
could be found from the following profit
 i qi  t i  0 . Optimal contract for type

max t i  qi2
maximization problem:
t i , qi

s.t.  i qi  t i  0 .
Participation constraint must be binding as otherwise we can increase the price and the
resulting profit will go up. Thus we can express the price from the constraint and plug it into
the objective function: max qi  qi2  .
qi
t
q
FOC:  i  2qi
Optimal contract for type i : qi   i / 2 and t i   i2 / 2 .
Optimal Contracts under Asymmetric Information
Now assume that monopolist cannot observe the preference parameter of the customers and
~ ~
~ ~
offers a menu of contracts to each customer: q H , t H and q L , t L .




Optimal menu of contracts could be derived from the following profit maximization problem
that takes into account both participation constraints (PC) and incentive compatibility
constraints (IC):
max t L  q L2   1   t H  q H2 
t i ,qi
s.t.
 L qL  t L  0
(PCL)
 H qH  t H  0
(PCH)
 L qL  t L   L qH  t H
(ICL)
 H qH  t H   H qL  t L
(ICH)
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PCL and ICH will be binding (this point is discussed below). From PCL we can find
t L   L qL . Plugging it into ICH we obtain
t H   H qH  qL   t L   H qH  qL    L qL .
Substituting prices into the objective function we can rewrite the profit maximization problem
as
max
qL ,qH
FOCs:



  L q L  q L2  1     H q H  q L    L q L  q H2

1   H  2q~H   0 and  L  2q~L   1    H   L   0 .

1 
~
~
Solving this system we get q H   H / 2 and q L  L 
 H   L    L . Plugging back
2
2
2
into PCL and ICH we obtain the prices:
~
~
~
~
~
~
t L   L q L and t H   H q H  q L   t L .
Note that the quality level for the consumer with high quality preferences stays the same as
~
under symmetric information q H  q H while the quality for the consumer with low
~   L  1         L  q  to make the contract
preference parameter was reduced q
L
H
L
L
2
2
2
unattractive to H-type. This is because full information contracts are not incentive compatible.
qi   i / 2 , t i   i2 / 2
Indeed, if both full information contracts
are offered under the
asymmetry of information then L-type will take the one that was designed for this type
q L , t L  since




u L q L , t L  0  u L q H , t H   L
 H  2H


  H   L  H
2
2
2
,
but H-type takes q L , t L  as


H
uH q , t

H
  0  u q
H

L
,t

L

 L  2L

 H

  H   L  L ,
2
2
2
that is, ICH is violated in this case.
Incentive Compatible Contracts
If we fix q L , t L  then ICH should lie along indifference curve that goes through q L , t L  or
below.
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t
q
Contracts that lie below this indifference curve are not profit-maximizing for seller as he can
increase t L .
Conclusion: ICH is binding.
Optimal Contracts
Optimal contracts calculated above could be illustrated graphically.
t
q
~
~
Optimal contracts are given by q H  q H   H / 2 , and we find tH from incentive
~ ~
~ ~
compatibily
constraint
of
H-type
which
implies
H q
tH   H q
tL ,
H
L
~ ~
~ q
~   H   L
t H  t L   H q
H
L
2


.

~
H - type gets surplus equal to t H  t H , which is called information rent.
~ we should solve profit-max problem with two binding constraints: PC and IC .
To find q
L
H
L
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11.7 Asymmetric information: hidden actions
In situations of hidden action one side of a transaction can take an action that affects the other
side but is not observable
Hidden actions may lead to moral hazard problem.
Moral hazard problem - a party to a contract engages in post-contractual opportunistic
behaviour, i.e. takes the wrong action.
Examples: worker may choose to shirk on a job; an insurance policy holder may fail to take
enough care to prevent an accident.
The principal-agent model gives another example of moral hazard problem.
Agent is hired by principle to perform certain tasks. Moral hazard problem arises when
principle and agent have different goals and actions of agent can’t be monitored by the
principle. As a result, the agent may take actions that are not in the interests of the principle.
Suppose that a principal hires an agent to carry out a particular project that might be
successful and bring  H or unsuccessful and bring  L , where L  H . Once hired, the agent
chooses an effort level: either high eH or low eL , e L  e H . Effort affects the outcome of the

 

project: higher effort leads to a higher probability of success P  H eH  P  H eL  0 ,
 
where P  e is the probability of obtaining gross profit  under effort level e . The
following numerical example illustrates this assumption:
H=100
L=20
eH
2/3
1/3
eL
1/4
3/4
We assume that principal is risk-neutral and maximizes net expected profit. Agent maximizes
expected utility, his vw, e   uw  ge  , where w is wage and ge  stays for costs of
efforts. We assume that utility is increasing in wage and marginal utility is non-increasing:
uw  0 , uw  0 . We will start with the case of a risk-averse agent u  0 and then
proceed to the case of a risk neutral one u  0 . We also assume that costs are increasing in
effort level ge   ge L  . The reservation utility is denoted by u 0 .
The principal pays the agent a profit-contingent wage schedule: w H if profit is  H and w L if
profit equals  L .
Full Information Case: Observable Efforts
Under full information case, the contract specifies the level of efforts and the corresponding
profit-contingent compensation wH , wL , e  .
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Let us derive the optimal contract for a risk-averse agent. We will proceed in two steps. First
we will find the wage schedule that implements the effort level e i , where i  L, H . Then we
will compare the resulting net profits and choose the effort level that generates higher profit.
To implement effort level e i , the principal should maximize his net profit subject to
participation constraint:
max P  H ei  H  wHi   1  P  H ei  L  wLi 
wHi , wLi
s.t. P  H ei uwHi   1  P  H ei uwLi   gei   u0
(PC)
Note that PC is binding (explain!).
Let us form a Lagrangean:
ℒ = PHi  H  wHi   1  PHi  L  wLi    i PHi uwHi   1  PHi uwLi   gei   u0  .
FOC:
ℒ/ wHi   PHi   i PHi uwHi   0 and
ℒ/ wLi  1  PHi    i 1  PHi uwLi   0
From this system we get uwHi   1 /  i and uwLi   1 /  i , which implies that marginal
utilities are equal in the two states: uwHi   uwLi  . Since u is diminishing this implies
that wHi  wLi , that is, the principal should offer full insurance to a risk-averse agent.
This result could be explained intuitively. If wage schedule is risky wHi  wLi , then a riskneutral principal can increase his expected profit by replacing this risky payment with a fixed
wage. If the fixed wage gives the same expected value, then a risk-averse agent will be better
off. Thus, the principal can replace a risky payment with a fixed wage with lower expected
value and make risk-averse agent as well off. At the same time, the expected profit will go up
due to cost economy.
We can conclude that under risky wage schedule there is inefficient risk sharing. Putting risk
on risk-averse agent is costly for the principal as risk should be compensated by higher
expected wage.
Thus, we should finish our calculation and find out the value of this fixed wage
wHH  wLH  w and wHL  wLL  w . To do this, we plug fixed wage into participation
constraint:
 
 
 
uw   1  P uw   ge   uw   ge   u
PHH u w  1  PHH u w  ge H   u w  ge H   u0 and
PHL


HL

L
L
0
.
Thus we get the following equations for wages under high and low efforts:
uw   ge H   u0 and uw   ge L   u0 .
Note that as ge H   ge L  then uw   ge H   u0  ge L   u0  uw  . Since utility is
increasing in wage it implies that higher efforts require higher compensation: w  w .
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Thus, we cannot predict unambiguously whether high effort will be more profitable as, on the
one hand, it brings higher gross profit:
 H P  H e H    L P  L e H    H P  H e L    L P  L e L 
but, on the other hand, it requires higher costs w  w .
Principal should compare:
 e H    H P  H e H    L P  L e H   w
and
 e L    H P  H e L    L P  L e L   w
and implement the effort that gives a higher net profit.
Inducing e H is profitable if  e H   e L   0 , that is


.

 H   L P  H e H   P  H e L   w
 w


 
exp ected gain from
exp ected
effort
cos t
Unobservable Efforts
Under unobservable efforts we cannot offer a contract contingent on effort level as monitoring
is impossible. Thus, the contract includes only wage scheme wH , wL  . This wage scheme
should be designed in such a way that makes desirable level of effort incentive compatible.
We will derive optimal contract in three steps. We start with a derivation of the contract that
implements low effort e L (requires PC but IC is satisfied automatically), then proceed to the
analysis of the contract that implements high effort e H (requires both PC and IC) and finally
compare the expected net profits to choose the profit-maximizing contract.
Contract that implements low effort
Let us start with the contract that implements low effort level. The principal maximizes
expected profit-maximization subject to participation and incentive compatibility (IC)
constraints:
max P  H e L  H  wH   1  P  H e L  L  wL 
wH , w L
s.t. P  H e L uwH   1  P  H e L uwL   ge L   u0
P  H e L uwH   1  P  H e L uwL   ge L   P  H e H uwH   1  P  H e H uwL   ge H 
We can verify that the contract that was optimal under full information satisfies IC constraint.
 
  uw   ge
If wH  wL  w then uwH   uwL   u w . Plug into IC and get
 
u w  ge L

H
 since ge L   ge H  .
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Conclusion: the principal does not need to provide incentives not to work hard. As a result
offer the same contract as under observable efforts: fixed wage wH  wL  w , where
 
u w  ge L   u0 .
Contract that implements high effort
Now we proceed to the analysis of the contract that implements high level of effort. The
principal maximizes expected profit-maximization subject to participation and incentive
compatibility (IC) constraints:
max P  H e H  H  wH   1  P  H e H  L  wL 
wH , w L
s.t. P  H e H uwH   1  P  H e H uwL   ge H   u0
P  H e H uwH   1  P  H e H uwL   ge H   P  H e L uwH   1  P  H e L uwL   ge L 
Now fixed wage doesn’t satisfy the incentive compatibility constraint. If wH  wL  w then
 
uwH   u wL   u w and (ICH) is violated: uw  ge H   uw  ge L  .
Conclusion: Putting some risk on the agent is unavoidable in providing the incentive for high
effort.
Thus we should find two different wages wH  wL and we have two binding constraints.
The optimal contract is derived from the system given by participation and incentive
compatibility constraints.
Optimal contract under unobservable efforts
We demonstrated that the full insurance contract implements low effort level:
wH  wL  w , where uw   ge L   u0 . The resulting expected profit is
 UNOBS e L    OBS e L    H P  H e L    L P  L e L   w .
To implement high level of effort, we need a state dependent payment scheme wH  wL ,
where
wH , wL 
is a solution of the system given by participation and incentive
compatibility constraints. The resulting expected compensation exceeds the one that

corresponds to the full information case: w  wH P  H e H   wL P  L e H   w since
 
uw  U w  ge H   u0  u w . Putting risk on the agent is good for incentives but
bad for risk-sharing, and bad risk-sharing is costly for the principal:
 UNOBS e H    H P  H e H    L P  L e H   w   OBS e H  .
To find the optimal effort level, the principal should compare  UNOBS e L  and
 UNOBS e H  and choose the one that generates higher expected profit.
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Case of risk-neutral agent
In case of a risk-averse agent there is a trade-off between incentives and risk sharing when
high level of effort is implemented. There is no such trade-off if the agent is risk-neutral, that
is, has a utility function vw, e   w  ge  , as in this case putting risk on the agent is not
costly and the principal can generate incentives for free.
Participation constraint takes the form: P  H ei wHi  1  P  H ei wLi  gei   u0 . It
implies that expected compensation must be equal to gei   u0 . Then the optimal effort level
under observable efforts could be derived from the following problem:
max P  H ei  H  wHi   1  P  H ei  L  wLi  
ei eH ,eL 
max P  H ei  H  1  P  H ei  L  gei   u0 .
ei eH ,eL 
Since constant u 0 doesn’t affect the solution, the problem could be restated as
max P  H ei  H  1  P  H ei  L  gei 
ei eH ,eL 
First best contract can be implemented even under unobservable efforts since the principal can
put all the risk on the agent and ask for a fixed fee F (agent becomes a residual claimant).
Agent as a residual claimant will choose e i by maximizing the expected utility:
max P  H ei  H  1  P  H ei  L  gei   F . As constant F
ei eH ,eL 
doesn’t affect the
solution this problem is equivalent to the one obtained under the full information case.
11.7 Sample exercise with solution
Consider a foreign firm that is willing to supply a unique product to country A. Suppose that
the firm’s cost function is C(Q)=cQ. The demand for its product is given by Q(p)=A-p.
Government of country A is going to propose a contract to this firm that will specify the
quantity of the good (Q) that this firm must sell at country A at a constant per-unit price
chosen by the firm and the fixed sum (T) that it should pay to (or receive from) the
government of country A. Government is looking for the contract that maximizes domestic
welfare (W) equal to the sum of consumers surplus and government budget surplus.
(a) Setup government optimization problem and derive the optimal tariffs for the two cases
(1) when c=A/2 and (2) c=A/4.
(b) Suppose that the potential entrant might be a high cost firm with c=A/2 or a low cost firm
with c=A/4 with equal probabilities. Government cannot identify whether the firm is highcost or low-cost but the firm learns its type before it chooses the contract. Assume that the
government is risk neutral.
(i) Suppose that government simultaneously offers two contracts identified in (a). Show using
your one graph that these contracts are not incentive compatible. Illustrate the resulting
expected welfare [do not make any numerical calculations].
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(ii) Find graphically the payment for L-type firm that will make the contracts found in (a)
incentive compatible and will provide the maximum possible expected welfare. What is the
resulting increase in expected welfare?
(iii) Reproduce your graph from (ii) and change the contract for the high-cost firm by
reducing the output a bit (by q) but keep the output for low-cost firm unchanged. Illustrate
the corresponding changes in fixed payments. What is the resulting change in expected
welfare.
(iv) Starting from the contracts represented in (iii) consider a differentially small reduction in
the output of high-cost firm. Identify (using your graph from (iii)) the resulting change in
expected welfare.
(v) Produce a new graph with demand and marginal cost curves. Based on the analysis
performed in (iv) show graphically the two contracts that maximize the expected welfare
under given cost uncertainty. Explain carefully.
Solution
(a) If the contract is accepted then the firm will charge the maximum price that consumers are
willing to pay for given Q, i.e. p=A-Q .
Thus PC is given by: (A-Q-c)Q-T0.
Under linear demand CS(Q)=Q2/2.
Government problem:
Q2 / 2  T  max
Q ,T
 A  Q  cQ  T  0
PC is binding (proof)
Q2 / 2   A  Q  c  Q  max , Solution: Q  A  c , T  0
Q 0
(1) QH=A-A/4=3A/4; TH=0
(2) QL=A-A/2=A/2; TL=0
(b) As the government is risk neutral it maximizes expected welfare.
(i) High cost firm gets negative profit (loss equal B+2F) if it takes the contract designed for
low cost firm and it gets 0 profit from the contract designed for high cost firm, so it will take
the right contract.
Low cost firm gets positive profit (B) if it takes the contract designed for the high cost firm
and it gets zero profit from the contract designed for the low cost firm, thus it takes the wrong
contract.
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$
A
D
A/2
B
F
A/4
Q
QH
A
QL
Expected wealth=D as CS=D in any case (firm of any type takes H-type contract) and T=0
(ii) TL=-B (government should increase the profit of L-type firm by providing lump sum
subsidy equal to B to make low cost firm indifferent b/w the two contracts)
CS in case of L-type firm increases by B+F but government pays B as a subsidy, thus
EW=F/2 since we have L-type firm with probability ½
(iii) Contracts:
QH-q; TH=G
Now if L-cost firm takes this contract its profit = B-B
Thus government should reduce the subsidy by B
QL; TL=-(B-B)
$
Gain
Loss
A
A/2
G
D
B
F
A/4
QH-q QH
QL
A
Q
EW=-D+B >0
(iv) EW=Gain –Loss
(v) Contracts:
QH= Q* should result in zero marginal expected welfare: Gain=Loss; TH=N
QL; =A-A/4=3A/4; TL=-M
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$
A
=
N
A/2
M
A/4
q*
QL
A
Q
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12. EXTERNALITIES AND PUBLIC GOODS
An externality refers to the effect when one economic agent’s action directly confers benefit
or imposes a cost on some other agent without that consequence being reflected directly in
market prices and exchange transactions.
Externalities can be positive or negative.
A positive externality occurs when one economic agent creates a benefit for another (without
compensation).
A negative externality occurs when one economic agent imposes a cost on another (without
compensation).
Externalities can be divided into 4 categories according to the source-recipient principle:
 Consumption-consumption (smoker sitting next to a table of non-smokers in a
restaurant),
 Consumption-production (healthy lifestyle increases productivity),
 Production-consumption (factory pollutes the air in the area where people live),
 Production-production (factory pollutes the water used by another firm).
12.1 Simple Model of Consumption Externalities
Consider
an
economy
with
two
agents
( i  1, 2 )
with
quasilinear
utility:
ui x, mi   i x   mi , where m i - money and x is a costless action taken by agent 1 that
affects utility of agent 2. Assume that utility of agent 1 increases in x but marginal utility is
diminishing: 1 x   0, 1x   0 . Depending on the sign of the marginal utility of agent 2 we
deal with positive 2 x   0 or negative 2 x   0 external effect. Marginal utility is still
assumed to be diminishing: 2 x   0 . Further we consider the case of negative external
effect, and the case of positive external effect is described in the Subject Guide.
Equilibrium and Efficiency
Externalities can adversely affect economic efficiency as complete market system assumption
or the assumptions of the FFWT is not satisfied, i.e., externalities results in the missing
market problem.
In unregulated economy agent 1 decides on the desirable level x by maximizing his utility:
max 1 x   m1 
x
Since the objective function is strictly concave, the FOC is both necessary and sufficient.
Thus, the equilibrium level of x  is given by the following equation:
 
1 x   0 .
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To derive Pareto optimum, we should maximize the utility of one agent subject to a fixed
level of utility of the other under given resource constraint:
max
m1 0,m2 0, x
  x   m 
1
1
s.t.  2 x   m2  u
m1  m2  em
We can express m 2 from the first constraint and plug it into the second one to find m1 :
m1  em  m2  em  u   2 x    2 x   const . Now we substitute it into the objective
function: max 1 x    2 x   const . As we assumed diminishing marginal utilities for both
x
agents, this function is strictly concave and the optimum level x  could be derived from the
FOC:
 
 
1 x   2 x   0 .
Now we could compare equilibrium level x  with efficient one x  . Since 2 x   0 then
 
 
 
1 x   2 x   0  1 x  . As 1 is diminishing, this inequality implies that x o  x  , that
is, in equilibrium x is overproduced since agent 1 doesn’t take into account the negative
external effect imposed on agent 2.
This result could be illustrated graphically. As x is costless then PMC  0 and agent 1
 
equates PMB  1 x  with zero MC when he chooses x  . Equilibrium: 1 x   0
$
PO
B
A
Equilibrium
x
 
 
To find PO we should solve the following equation: 1 x   2 x   0 that could be
equivalently restated as
 
 
1 x    2 x  .
 


MSB
MSC
The LHS gives the marginal social benefit that coincides with the private social benefit while
the RHS gives the marginal social costs that reflects disutility of agent 2 and exceeds zero
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private marginal cost of agent 1. Thus, as we can see from the graph, the resulting PO level of
activity is below the equilibrium one.
The blue triangle illustrates DWL resulting from negative externality. If

x
decreases from

x to x , then social benefit goes up by area A, while social cost rises by (A+B). As cost
increases more than benefit, total surplus is reduced by area of B, corresponding to
deadweight loss.
10.2 Private responses to externalities
Possible responses to externality problem could be separated into the government and private
response.
Some examples of private response:

Internalization via firms’ merger

Social conventions

Bargaining (Coase theorem).
Internalization
An externality can be internalized by combining the involved parties, i.e. putting the problem
in the hands of a single decision maker.
If firm A pollutes the water while firm B suffers from pollution, then the combined firm (if
merger takes place) will take into account the negative impact of water pollution as it would
maximize the total profit.
Unfortunately this method doesn’t work in case of consumption externalities.
Social conventions
Individuals cannot merge in order to internalize the externalities. Instead social conventions
are used to force people to take into account the externalities they generate.
Some moral norms induce individuals to internalize the externalities that they might create.
Property Rights Solution (Coase Theorem)
Suppose that property rights are assigned. Then parties may bargain and if bargaining is
costless they will reach the mutually beneficial solution. This is the essence of the Coase
theorem6:
6
Coase, R.H. (1960) The problem of social cost, Journal of Law and Economics 3, 1-44
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Under perfect competition (i.e. if there are no transaction costs), if property rights are welldefined, bargaining would lead to the efficient outcome irrespective of who the property
rights are allocated to (to the party generating the externality or to the party affected).
Let us demonstrate how this theorem works for our example of negative externality in
consumption.
We start with the case of permissive law, when property rights are given to agent 1 that
generates this external effect. It means that in case of disagreement x  x  .
Agent 1 asks agent 2 for a payment T to reduce x to some level below x  . Then optimal
contract is derived from the following optimization problem:
max 1 x   T 
x ,T
s.t. .  2 x   T   2 x  
You should be able to demonstrate that participation constraint is binding. Then we can find
T from this constraint: T   2 x    2 x   and plug it into the objective function:

 
max 1 x    2 x    2 x 
x
As objective function is strictly concave FOC gives the optimal level of x :
1 x   2 x   0 . We can note that this equation coincides with the one for PO allocation.
 
 
Now, let us move to the case of restrictive law, when property rights are given to agent 2 that
suffers from the external effect. In this case disagreement results in x  0 .
Agent 2 asks for a payment T to increase x to some level above 0 and the optimal contract
is derived from the following problem:
max  2 x   T 
x ,T
s.t. . 1 x   T  1 0 .
Similarly, it can be shown that the constraint is binding and T  1 x   1 0 . Substituting
into the objective function we get
max  2 x   1 x   1 0.
x
 
 
The solution should solve the following equation: 1 x   2 x   0 , which coincides with
the one derived for permissive law case. Thus in both cases we have x  x  .
Problems with implementation:

preferences – private information

even with just two parties, one or both could comprise of many individuals (many
fishermen may suffer from water pollution)  they should delegate decision to one
leader
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
a free rider problem may arise if committing to a negotiation is costly in terms of time
and effort

difficult to start negotiations if many parties are involved
Importance of the Coase ideas:

create the field of law and economics (contract law)

have influenced the creation of pollution rights markets
12.3 Government Regulation
Some examples of government response:



Direct regulation (for example, via emission quota in case of negative external effect)
Corrective (Pigouvian) taxes
Cap-and-trade policy
Direct regulation: quota
To correct negative externalities, the government can impose some maximum limit on
pollution that corresponds to the efficient level ( x  x  ) or require installing special
equipment that reduces pollution.
To correct positive externality, the government can introduce the minimum requirement on
the level of output.
Problems. To make direct regulation efficient, government needs private information on costs
of production, which is problematic.
Corrective tax
Inefficiency can be eliminated if per unit tax (in case of negative externality) or subsidy (in
case of negative externality) is introduced. This tax/subsidy is called Pigouvian tax (Pigou
1932).
This per unit tax/subsidy should be equal to the value of MEC (or MEB) evaluated at the
efficient level of output.
With a per unit tax this activity becomes costly for agent 1. Thus, he will take into account
these costs deciding on the level of activity: max 1 x   m1  t  . From FOC we get
 
 
x
1 x   t . If t  1 x  then x   x  .
The figure below illustrates how the Pigouvian tax works. It shifts the net private marginal
benefit curve downward exactly by the value of tax and, as a result, the equilibrium level of
x falls.
This solution still requires private information about preferences to find the right value of the
tax rate.
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$
x
Cap-and-Trade Policy
Government sets an overall quota (a cap) for pollution and then allows trade in pollution
rights. Firms with a lower MC of abatement would then be net sellers in the emissions trading
market, firms with higher MAC would be net buyers.
This policy achieves the same reduction of pollution with lower costs that uniform quota.
Examples:



EU Emission trading system is the largest greenhouse gas emission trading system in
the world
US: SO2 emissions trading system
Since 1995, trade in an SO2 allowance reduced emissions by half, saving $1 billion per
year compared with a conventional approach.
12.4 Efficient Provision of Public Good
Public good represents a special type of positive externality in consumption.
A good is called public good if it is non-rival in consumption, i.e. consumption by any one
person does not reduce the amount available for others. Examples: national defence, TV
broadcasting, roads.
Public goods may be non-excludable (if it is very costly to exclude non-payers from
consumption of the good) or excludable. Non-excludable good is called pure public good.
Examples: national defense, roads, basic public health, basic education, protection against
natural catastrophes.
Consider a model with N consumers and two goods: X – public good and m -private good.
Assume that all agents have quasilinear utility functions: ui  X   mi with positive but
diminishing marginal utility of public good ui  X   0 , ui X   0 .
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Assume that there is no initial endowment of the public good and each individual has some
i
endowment of the private good e m so that the total endowment of the private good is
N
em   emi . Suppose that public good could be produced from the numerior private good
i 1
with CRS technology that results in linear cost function: C  X   cX .
To find the efficient amount of the public good produced we should maximize utility of one
of the agents under fixed utilities of others and resource constraint:
maxu1  X   m1 
s.t.
ui  X   mi  ui , i  2, , N
N
m
i 1
 em  cX .
i
We can express m i from utility constraint of each agent mi  ui  ui  X  for i  2, , N
and plug into the resource constraint: m1 
N
m
i 2
N
i
 m1   ui  ui  X   em  cX . Then we
i 2
N
N
i 2
i 2
find m1 from this equation: m1  em  cX   ui  ui  X    ui  X   cX  const and
N




max
u
X

ui  X   cX  const . As constant
substitute into the objective function:

 1
i 2


does not affect the solution we proceed to the following optimization problem:
N

max  ui  X   cX  .
 i 1

The objective function is strictly concave as we assumed diminishing marginal utilities and so
we can restrict our attention to the FOC that takes the form:
 u X   c .
N
i 1

i
This condition suggests that at the PO the total willingness to pay (the sum of MRS over all
consumers of the public good) should be equal to the marginal cost. This condition is known
as Samuelson equation7.
12.5 Private Provision of Public Good
Consider a voluntary contribution mechanism of private provision of public good, where each
individual finances some fraction of the total amount of the public good consumed so that
7
Samuelson P.A. (1954) The Theory of Public Expenditure. Review of Economics and Statistics 36, 386–389.
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X  x i  x i , where x i - contribution of agent i to the public good provision and
x i   x j - total contribution of other agents.
j i
Assume that both markets are perfectly competitive markets and denote prices by p x  p
and pm  1.
Suppose that all agents simultaneously and independently decide on x i . Then the utility
maximization problem of agent i is given by:
max ui x i  x i   mi 
xi ,mi 0
s.t. px i  mi  emi .
Due to non-satiation, the budget constraint will be satisfied as equality and we can find
consumption of the private good mi  emi  px i and plug it into the objective function:
max ui x i  x i   px i  .
xi 0
Taking into account the possibility of corner solution, we get the following FOC:
 
ui X   p  0 and ui X    p  0 if x i  0 .
Suppose that the public good is produced in equilibrium X   0 . Then there exists at least

one agent that contributes to the production of this public good x j  0 . For this agent, we
 

have interior solution and so FOC takes the form of equality uj X  p . Since marginal
utility
of
public
good
is
positive
 u X   u X    u X   p .
N
i 1


i
j

i j
N
implies p  c . Then
i
 
 
ui X   0
for
every
consumer
i
then
Under perfect competition, the profit maximization
N
 
 ui X   p  c   ui X  . Since ui is diminishing for every i
i 1
i 1
then X   X  , that is, the public good is underproduced in comparison to the socially
efficient level.
Example
Consider an example with N consumers that have quasilinear utility functions ui  X   mi
and u1 X   u2 X     uN X  . Then socially efficient amount of the public good is given
 u X   c .
N
by:
i 1

i
Let us show that in the equilibrium only agent N that has the highest valuation of the good
will contribute to the public good production. Suppose that this is not the case and there exists

an agent j  N that makes some positive contribution x j  0 . Then FOC should be satisfied
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 
 
 
as equality uj X   p . Due to our assumption uN X   uj X   p , which implies that
FOC is violated for agent N. This contradiction proves the claim.
Thus, in the equilibrium x1  x 2    x N 1  0 and x N  X  . The equilibrium quantity is
 
derived from the FOC for agent N: uN X   p  c .
Below we illustrate PO and equilibrium levels of production for the case with 3 consumers.
MC
X
12.6 The Commons Problem
Finally we move to a special case of mutual external effect that deals with common property
resources that represent rival but not excludable goods.
The resulting negative external effect results in over-exploitation problem known as ‘tragedy
of commons’
Examples: water, fishery, pastures
Evidence: Stavins, R.N. (2011) The problem of the commons: Still unsettled after 100 years.
AER 101, 1-108


New England lobster fishery (1966): efficient number of traps - about 450,000 but
actual number is nearly 1 mln of traps;
Two lobster fisheries in eastern Canada (1979): losses due to unrestricted entry
amounted to 25% of market value of harvests, fishery effort exceeds the efficient level
by 350%
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Model of Resource Extraction
Consider a simple model of resource extraction with N users that simultaneously and
N
independently choose efforts. Production function is given by Q  L , where L   Li and
i 1
L i stays for the effort of user i . Assume that the price of final good is normalized to 1:
pQ  1. Cost of extraction (cost of efforts) for user i is given by C i Li   cLi .
The socially optimal level of efforts could be found from the maximization of total net
benefit:


max L  cL .
L 0
FOC takes the following form 0.5 / L  c and we get the socially optimal total level of

efforts L  1 / 2c  .
2
Now we proceed to the analysis of equilibrium. In the equilibrium each user decides on his
effort level taking the effort levels of other users Li   L j as given:
j i
 Li
max 
Li
 Li  L  i

Li  Li  cLi   max
Li

The FOC for user i is given by
 
1 / L  0.5Li / L
3/ 2


Li
 cLi  .

 Li  Li

L  0.5Li / L
 c  0 , which can be restated as
L
 
 c . Summing up over all users we get N / L  0.5L / L

. Solving these equation we find the equilibrium level of total efforts: L 
3/ 2
 Nc
1
1  1 /2 N 2 .
2
c
1
1  1 /2 N 2
2
c
1
 L  N  1  2  L we can
4c

Let us compare it with the socially optimal one. We can note that L 

increases when N goes up. Since L  N  
1
1  1 /2 N 2
2
c
conclude that the equilibrium level of efforts exceeds the socially efficient level.
Solutions to commons problem
The first solution deals with the assignment of property rights, that is, privatization. If
property rights are assigned to one person, than the equilibrium will result in socially efficient
1
level of efforts L  N  1  2  L . Unfortunately, for many resources (for example,
4c
fishing) it is impossible to assign the rights just to one person.
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Another solution deals with the creation of the market for extraction rights via the
introduction of individual transferable quotas (ITQ). In this case the government sets the
overall quota and issues individual quotas that can be traded. Due to trade, quotas will be used
by the users with the lowest cost of exploitation. Stavins (2011) mentioned that ITQ were
successfully used to regulate fisheries in 17 countries (Australia, Canada, New Zealand, USA,
and others). But this solution is only a partial remedy since monitoring is costly and not all
countries stick to the quotas.
Another approach deals with taxation. Suppose that a tax t is introduced per unit of effort
which increases tax-inclusive marginal cost to c  t . As a result, the equilibrium level of

efforts becomes L t  
1
c  t 
2
1  1 /2 N 2
. We should choose the tax rate in such a way

that the resulting level of efforts is Pareto optimal: L t  
Solving this equation we get t  c1 1 / N  .
1
c  t 
2
1  1 /2 N 2  L 
1
.
4c 2
This solution is not perfect as well, since the calculation of optimal tax rate requires private
information that the policymaker might not have.
Now we proceed from the government to private response. Tragedy of commons problem is
similar to Prisoners Dilemma as all users are better off if extraction is reduced. It means that
there is a scope for informal cooperation in equilibrium if the game is repeated. As we know
Folk theorem suggests that cooperative outcome can be sustained as SPNE for high .
Assume that every user applies the following trigger strategy. He starts by playing
Li  L / N (cooperate at t=0) and in period t > 1 plays L / N if L / N was played by every
user in the past and plays L / N otherwise.


, where is  user’s profit from
  DEV  
1 
1 
cooperation and   is user’s profit in NE (one-shot game), where     . Thus, informal
cooperation is sustained if    DEV 1      . Solving this equation with respect to
There is no incentive to deviate if
discount factor we get  
 DEV   
.
 DEV  
12.7 Sample exercise with solution
Consider two neighboring regions of the same economy that decide simultaneously and
independently on the own level of environment protection expenditures s i , i  1, 2 . The
resulting net benefit of region i is given by ki s1  s2  s i . Assume that 0  k1  k2 .
(a) Find the equilibrium.
(b) Demonstrate that the equilibrium outcome found in (a) is inefficient. Explain the reason
for inefficiency.
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(c) Is it possible to eliminate the loss from inefficiency via taxes/subsidies? Find the required
taxes/subsidies or prove that it is impossible.
Solution
(a) NE is given by the intersection of best response functions. Here we should take into
account the possibility of corner solution as region might find zero contribution to the
provision of the public good profitable under high contribution of the other region.
maxki s1  s2  s i . [The function is strictly concave, so we can restrict our analysis to the
si 0
FOC only]
FOC
ki
2 s1  s 2
 1  0 and
ki
2 s1  s 2
 1  0 if s i  0 .
Starting from this point equilibrium could be derived graphically OR algebraically.
Graphical analysis: BR curve of region i:
2
2

ki  / 4  s j , s j  ki  / 4
.
si  
2
0, s j  ki  / 4


We can find graphically (graph is skipped) that the only intersection is the one where s1  0
and s2  k2 2 / 4 .
Alternatively equilibrium could be obtained algebraically.
Let us prove that region 1 will never make a contribution.
k
k
k
Suppose that s1  0 and s2  0 then s1  s2  1 and 1  s1  s2  2 which implies
2
2
2
k1  k2 but it contradicts to the assumption k1  k2 .
We can also note that s1  s2  0 as net benefit at s1  s2  0 goes to infinity.
Thus in equilibrium s1  0 and s2  k2 2 / 4 .
(b) The equilibrium allocation found in (a) results in SMB that exceeds SMC:
SMB  MB1  MB2 
k1  k 2
2 s1  s 2

k1  k 2
k2
 1  MC .
This happens because each agent bases his/her decision on private MB that doesn’t take into
account the positive external effect (i.e. additional benefit enjoyed by the other region) and
equates it with MC. As a result, the social marginal benefit exceeds the private one and thus
exceeds the private marginal cost that results in insufficient of the public good.
(c) Is it possible to eliminate the loss from inefficiency via taxes/subsidies? Find the required
taxes/subsidies or prove that it is impossible.
k  k2
2
Efficiency requires SMB  MB1  MB2  1
 MC  1 , i.e. s1  s2  k1  k2  / 4 .
2 s1  s 2
Thus any allocation that results in total S  k1  k2  / 4 is PO.
To eliminate the efficiency loss the government may subsidies the environmental spending.
Moreover we need region-specific subsidy as the two regions create different external effects.
2
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Let  i stay for the subsidy rate for region i . Then
At PO level of protection
ki
2 s1  s 2

ki
k1  k2
ki
2 s1  s 2
 1   i  and  1   i if s i  0 .
. Thus we can take  i  1 
ki
k1  k2

kj
k1  k2
to
guarantee efficiency of equilibrium allocation.
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