151

Game Theory

Definition: The players are the decision-makers whose behavior we are trying to predict.

Definition: The strategies are the potential choices that can be made by the players.

Definition: The payoffs are the outcomes, or consequences, of the strategies chosen.

Definition: A strategic form game is a tuple 1 1 1 1[ , ... , , ( ,..., ), ... , ( ,..., )]n n n nG S S u S S u S S , where

for each player 1,...,i n , iS is the set of strategies available to player i , and

1( , ..., )i nu S S describes player 'i s payoff as a function of the strategies chosen by all

players.

A strategic form game is finite if each players strategy set contains finitely many elements.

Example: The Prisoners Dilemma Game

Penalty B

Confess Not confess

A Confess $7m, $7m $1m, $10m

Not confess $10m, $1m $2m, $2m

{confess, not confess}

(confess, confess) (confess, confess) $7

(confess, not confess) (not confess, confess) $1

(not confess, confess) (confess, not confess) $10

(not confess, n

A B

A B

A B

A B

A

S S

u u m

u u m

u u m

u

ot confess) (not confess, not confess) $2 Bu m

Note: A 2-player strategic form games with finite strategy sets can always be represented in

matrix form.

Definition: is , for player i , is a strictly dominant strategy if ( , ) ( , ) ( , )i i i i i i i iu s s u s s s s .

Note: A strictly dominant strategy is a strategy a player will adopt no matter what strategy the

other player adopts.

Definition: A strictly dominant strategy equilibrium is the resulting equilibrium of all players

adopt their dominant strategy.

152

Note that in this game, the best outcomes for both 2 players together is when neither one of them

confesses. The worst outcomes for both 2 players together is when both of them confess.

The Prisoners Dilemma game predicts that both player will confess.

In this game,

i) the dominant strategy for A is confess.

ii) the dominant strategy for B is confess.

iii) the dominant strategy equilibrium is (confess, confess).

Cheating in the cartel

Profit B

Cheat Comply

A Cheat $10, $10 $25, $5

Comply $5, $25 $20, $20

Note that in this game, the best outcomes for both 2 players together is when neither one of them cheats.

The worst outcomes for both 2 players together is when both of them cheat.

The cheating in the cartel game predicts that both player will cheat.

In this game,

i) the dominant strategy for A is cheat.

ii) the dominant strategy for B is cheat.

iii) the dominant strategy equilibrium is (cheat, cheat).

Example: High output or low output (a game with no DS equilibrium)

Profit B

Low output High output

A Low output $22, $20 $9, $30

High output $20, $17 $18, $25

In this game,

i) there is no strictly dominant strategy for A.

ii) the strictly dominant strategy for B is high output.

iii) there is no strictly dominant strategy equilibrium.

Definition: Player 'i s strategy is , strictly dominates another of his strategies is , if

( , ) ( , ) i i i i i i iu s s u s s s . In this case, we also say that is is strictly dominated.

153

Solution by iterated elimination of strictly dominated strategies

Example:

2

L C R

T 3, 3 2, 6 3, 1

1 M 2, 4 1, 4 0, 4

B 1, 5 0, 2 6, 0

2

L C R

1 T 3, 3 2, 6 3, 1

B 1, 5 0, 2 6, 0

2

L C

1 T 3, 3 2, 6

B 1, 5 0, 7

2

L C

1 T 3, 3 2, 6

2

C

1 T 2, 6

Definition: Let 0 for each player i iS S i , and for 1N , let N

iS denotes those strategies of player i

surviving after the thN round of elimination. That is 1, if N Ni i i is S s S is not strictly

dominated in 1NS .

Definition: A strategy is for player i is iteratively strictly undominated in S (or survives iterative

elimination of strictly dominated strategies) if 1Ni is S N .

Example:

2

L C R

T 3, 3 2, 1 3, 1

1 M 2, 4 2, 4 0, 4

B 1, 5 0, 2 6, 5

both players do not have

dominate strategies.

For 1: M is strictly

dominated by T.

For 2: R is strictly

dominated by either L or C.

For 1: B is strictly

dominated by T.

solution

neither player has strictly

dominated strategy.

154

Definition: Player 'i s strategy is , weakly dominates another of his strategies is , if

( , ) ( , ) with at least 1 strict inequalityi i i i i i i iu s s u s s s S . In this case, we also

say that is is weakly dominated.

Definition: Let 0 for each player i iW S i , and for 1N , let N

iW denotes those strategies of player i

surviving after the thN round of elimination. That is 1, if N Ni i i is W s W is not strictly

dominated in 1NW .

Definition: A strategy is for player i is iteratively weakly undominated in S (or survives iterative

elimination of weakly dominated strategies) if 1Ni is W N .

Solution by iterated elimination of weakly dominated strategies

2

L C R

T 3, 3 2, 1 3, 1

1 M 2, 4 2, 4 0, 4

B 1, 5 0, 2 6, 5

2

L C R

1 T 3, 3 2, 1 3, 1

B 1, 5 0, 2 6, 5

2

L

1 T 3, 3

B 1, 5

2

L

1 T 3, 3

neither player has strictly

dominated strategy.

For 1: M is weakly

dominated by T.

For 2: L and R are weakly

dominated by L.

solution

155

Example

2

L C R

T 3, 3 2, 1 3, 6

1 M 4, 5 2, 7 0, 4

B 1, 5 0, 2 6, 5

Definition: A Nash equilibrium is a set of strategies (or choices) such that each players choice is the best one possible given the strategy chosen by the other players.

Formally, given a strategic form game 1 1 1 1[ , ... , , ( ,..., ), ... , ( ,..., )]n n n nG S S u S S u S S ,

( 1 , ..., ns s ) is a Nash equilibrium if for each player i , 1 ( , ..., ) ( , ) i n i i i i iu s s u s s s S .

Example

2

L C R

T 3, 3 2, 1 3, 6

1 M 4, 5 2, 7 0, 4

B 1, 5 0, 2 6, 5

(1, 2) 12 12

(T, L ) TR X

(T, C) TR X

(T, R) TR BR X

(M, L) MC X

(M, C) MC T,MC Nash equilibrium

(M, R) MC X

(B, L) BL,R ML X

(B, C) BL,R X

(B, R) BL,R BR Nash equilibrium

neither player has weakly

dominated strategy.

neither player has weakly

dominated strategy.

156

Example: (A game with no Nash equilibrium)

B

I II

A I $0, $0 $0, $1

II $1, $0 $1, $3

(A, B) AB AB

(I, I ) II III X

(I, II) II X

(II, I) IIII X

(II, II) IIII III X

157

Mixed Strategies Nash Equilibrium

Definition: Fix a strategic form game 1 1 1 1[ , ... , , ( ,..., ), ... , ( ,..., )]n n n nG S S u S S u S S . A mixed

strategy for player i is a probability distribution over iS .

iM : set of mixed strategies for player i

iS : set of pure strategies for player i

Example:

2

L R

1 T 3, 1 2, 4

B 2, 2 3, 1

The game has no pure-strategy Nash equilibrium.

Lets assume the players have mixed strategies, in which a player randomizes by choosing the probabilities for playing the possible pure strategies.

Let p be the probability for 1 to choose T ,

1 p be the probability for 1 to choose B .

Let q be the probability for 2 to choose L ,

1 q be the probability for 2 to choose R .

Expected payoff for 1: 1( ) (3) (1 ) (2) (1 )(2) (1 )(1 )(3) 2 3E pq p q p q p q pq p q

Expected payoff for 2: 2( ) (1) (1 ) (2) (1 )(4) (1 )(1 )(1) 4 3 1E pq p q p q p q pq p q

Problem for 1:

1max ( ) 2 3p

E pq p q

FOC: 1( ) 1

2 1 0 *2

dEq q

dp

Problem for 2:

2max ( ) 4 3 1q

E pq p q

FOC: 2( ) 1

4 1 0 *4

dEp p

dq

1

1 1 1 1( ) 2( )( ) ( ) ( ) 3 2.5

4 2 4 2E

2

1 1 1 1( ) 4( )( ) 3( ) ( ) 1 1.75

4 2 4 2E

158

Remark: There is a problematic feature of a mixed-strategy equilibrium: neither player has any

positive incentive to play the equilibrium strategy.

Player 1: 11 1

( ) 2( )( ) ( ) ( ) 3 2.5 (no matter what is!!!)2 2

E p p p

Player 2: 21 1

( ) 4( )( ) 3( ) ( ) 1 1.75 (no matter what is!!!)4 4

E q q q

Theorem: Every n-player game with finite pure-strategy sets has at least 1 pure or mixed strategy

Nash equilibrium.

159

Oligopoly

Properties:

1. a few sellers (firms)

2. many buyers (consumers)

3. homogeneous products

4. difficult to enter into the market

Cournot model with 2 identical firms

1Firm i's share (1 j's share)

2

: competitive outputCQ

P P

MC MC

Q Q 0

MR D MR D

Residual D for firm 2 Firm 1 Firm 2

1 1

2cQ

0

2 1

2cQ

1 1 1( )

2 2 4c c cQ Q Q

3 1 1 3( )

2 4 8c c cQ Q Q

1

4cQ

4 3

8cQ

1 3 5( )

2 8 16c c cQ Q Q

5 1 5 11( )

2 16 32c c cQ Q Q

5

16cQ

6 11

32cQ

1 11 21( )

2 32 64c c cQ Q Q

7

8 1

3cQ

1

3cQ

Nash equilibrium:

1 1 1 1 1 1Firm 1: ( ) Firm 2: ( )

2 3 3 2 3 3c c c c c cQ Q Q Q Q Q

Nash

Equilibrium

160

Cournot model of 3 identical firms

Firm 1 Firm 2 Firm 3

1 1

2cQ

0 0

2 1

2cQ

1 1 1( 0)

2 2 4c c cQ Q Q

0

3 1

2cQ

1

4cQ

1 1 1 1( )

2 2 4 8c c c cQ Q Q Q

4 1 1 1 5( )

2 4 8 16c c c cQ Q Q Q

1

4cQ

1

8cQ

5 5

16cQ

1 5 1 9( )

2 16 8 32c c c cQ Q Q Q

1

8cQ

6 5

16cQ

9

32cQ

1 5 9 13( )

2 16 32 64c c c cQ Q Q Q

7

8 1 Nash Equilibrium

4cQ

1 Nash Equilibrium

4cQ

1 Nash Equilibrium

4cQ

Nash Equilibrium

1 1 1 1Firm 1: ( )

2 4 4 4c c c cQ Q Q Q

1 1 1 1Firm 2: ( )

2 4 4 4c c c cQ Q Q Q

1 1 1 1Firm 3: ( )

2 4 4 4c c c cQ Q Q Q

Cournot model with n identical firms

Since the firms are identical, hence the shares are the same.

Let x be firm is share.

Firm is share: 1

( ... )2

cx Q x x

1 'n x s

1 ( 1) 1[ ( 1) ] [1 ]

2 2 2 2 2

* Nash Equilibrium1

c c

c

c

c

Q Qn nx Q n x x x

QQ

n

Total output1

c

nQ

n

As , total output cn Q Competitive market is the limiting case of oligopoly.

161

Example

Consider a duopoly market with a demand function : 100D P Q . Let the cost function of firm 1 be 2

1 1 110 4TC Q Q and the cost function of firm 2 be 2

2 2 210 2TC Q Q .

Firm 1: 2 2# #

1 1 2 1 1 1 1 1 1 2max [100 ( )] [10 4 ] 10 96 2Q Q Q Q Q Q Q QQ

FOC: #1 1 21

96 4 0d

Q QdQ

Reaction function: #

21

96*

4

QQ

Firm 2: 2 2# #

2 1 2 2 2 2 2 2 1 2max [100 ( )] [10 2 ] 10 99 3Q Q Q Q Q Q Q Q Q

FOC: #2 2 12

99 6 0d

Q QdQ

Reaction function: #

12

99*

6

QQ

Nash equilibrium: # #

1 1 2 2* , *Q Q Q Q

1 2

1 2

96 4 * * 0

99 * 6 * 0

Q Q

Q Q

1 2

477 300* 20.74, * 13.04

23 23Q Q

Collusion equilibrium

1 2

2 2

1 2 1 2 1 1 2 2,

2 2

1 2 1 2 1 2

max [100 ( )]( ) [10 4 ] [10 2 ]

20 96 99 2 3 2

Q QQ Q Q Q Q Q Q Q

Q Q Q Q Q Q

FOC:

1 2

1

1 2

2

96 4 2 0

99 2 6 0

Q QQ

Q QQ

1 2 1 2

189 102 291* , * * *

10 10 10Q Q Q Q Q

Stackelberg Equilibrium (Firm 1 is the leader)

211 1 1 1

99max [100 ] [10 4 ]

6

QQ Q Q Q

1 2

477 567* 21.68, * 12.89

22 44Q Q

162

The dominant Firm Model (price leadership)

1 dominant firm: behaves as if it is a monopoly price-setter a few small firms: behave as if they are competitive firms price-takers P

Ssmall

MCbig

P*

Dbig = Residual demand facing the dominant

firm

D

MRbig

Q

Qbig* Qsmall*

Example:

10 small firms: 22 2TC Q Q P S(one firm)

1 dominant firm: 210TC Q Q

Demand function: 100Q P S(10 firms)

small firm: 22 2 2 2TC Q Q MC Q

For a competitive firm, MC curve is the supply curve.

2: 2 2 1

2 2small

P PS P Q Q

Q

10 small firms: 10( 1) 5 102

PQ P

Residual demand for the big firm:110

(100 ) (5 10) 110 66

QQ P P P P

Problem facing the big firm: 2 2

2 2110 110 104 7max ( ) (10 ) 10 106 6 6 6 6Q

Q Q QQ Q Q Q Q Q Q

FOC:

52110

7 104 52 110 718 35970 * *3 6 7 6 6 42 21

big

d QQ Q P

dQ

359

359 317211 1 12 2 42 42

small

PQ

163

Kinked demand curve

Assumptions:

1. When a firm lowers its price, every competitor will follow (lower its price).

2. When a firm raises its price, no competitor will follow (raise its price).

P

MC

D

MR

Q

Q*

Prediction:

When MC changes slightly, * and * will not be affectedP Q .

164

Monopolistic Competition

Properties of a monopolistic competitive market

1. Many sellers (firms)

2. Many buyers (consumers)

3. Heterogeneous product

4. Free entry/exit

Short-run equilibrium ( 0 ) P MC

P* AC D

MR

Q Q*

Long run equilibrium ( 0 )

P

MC

AC

P*

D

MR

Q

Q*

165

A Model of Monopolistic Competition

Consumers

Consider a single (representative) consumer whose preferences exhibit the love-for variety property.

The utility function of the representative consumer is given by a constant-elasticity-of substitution

(CES) utility function:

1 2

1

( , ,..., )M

M i

i

U x x x x

where M is a large number.

This type of utility exhibits love for variety since the marginal utility of each brand at a zero

consumption level is infinite.

The indifference curves (for the 2 goods case) are convex to the origin, indicating the consumers will

usually buy more than 1 good.

The indifference curve (for the 2 goods case) will cut the horizontal axis and vertical axis, therefore it is

possible that the consumers do not purchase the entire variety of products.

The consumers income (I) is composed of the total wages paid by the producing firms plus the sum of profits (if any).

( ) : i ix the profit of the firm producing brand i

We also normalize the wage rate to $1.

Problem:

1 2

1

1 1

max ( , ,..., )

. . ( )

i

M

M ix

i

M M

i i i i

i i

U x x x x

s t p x I L x

Lagrangian: 1 1

[ ]M M

i i i

i i

L x I p x

FOC for each brand i :

10, 1,2,...,

2i

i i

Lp i N

x x

2 2

1

4i

i

xp

166

22 2 11 1 12

1 1 2 2

2 2

1 11 1 2 1 12 2

1 2

2 2 2 2

1 1 1 11 1 1 1 1 1

2 2

2

1 1 1211

1

1 1 1... ...

2 2 2

( ) ... ( )

... ( ... )

1( ) or

1( )

M M i

iM M

M

i i n

i M

M M

M

Mi i

i i

pp x p x x x

pp x p x p x

p pI p x p x p x p x I

p p

p p p pp x x x I x p I

p p p p

Ix p I x

Pp

P

2

1

1

( )i M

i

i i

Ix

pP

From the FOC, we have 1

2 i ip x

1

2

2 1 12

1

1

1 1 1

1 122

114 ( )( )

4

M

i ii M M

ii Mi ii i

ii

i i

p Ix

IIpp

p ppp

Ip

When M is large, 0ip

3

2 2 3

2 3

2 2

1 1( 2)

4 2

1( ) 2

12

4

ii

i i

i i i

i i i

i

xp

p p

x p p

p x p

p

Brand-producing firms

Each firm is produced by a single firm. All (potential) firms have identical technologies (identical cost

structure) .

The total cost of a firm producing ix units of brand i : if 0

( )0 if 0

i i

i i

i

F cx xTC x

x

Definition: The triplet , , , 1,..., }mc mc mc mci iM p x i M is called a Chamberlinian monopolistic-competition equilibrium if

1) Firms: Each firm behaves as a monopoly over its brand; that is, given the demand for brand i ,

2 2

1

4i

i

xp

, each firm i chooses mcix to solve the problem max ( ) ( )i

i i i i ix

p x x F cx

more firms iM x

167

2) Consumers: Each consumer takes his income and prices as given and solve the utility

optimization problem.

3) Free entry: Free entry of firms (brands) will result in each firm making zero profits;

( ) 0 1,2,...,mci ix i M .

4) Resource constraint: Labor demanded for production equals the total labor supply,

( ) total wage bill $1N

i

i

F cx L L

,

2mcip c

( )i ii

FAC x c

x

( )i iMC x c

( , )i iD p M ix

mc

i

Fx

c

The demand facing each (existing) brand-producing firm depends on the total number of brands in the

industry, M . When M , the demand facing each brand-producing firm shifts downward, reflecting the fact that consumers partially substitute higher consumption levels of each brand with a lower

consumption spread over a large number of brands.

Therefore, free entry increases the number of brands until the demand facing each firm becomes tangent

to the firm's average cost function. At this point, each existing brand-producing firm makes zero profit,

and entry stops.

The equilibrium condition in which demand becomes tangent to the average cost of each firm is known

as Chamberlin's tangency condition.

Note:

i) In equilibrium the price of each brand equals average cost.

ii) In equilibrium all brand-producing firms produce on the downward sloping part of the average

cost curve. Thus, firms do not minimize average cost under a monopolistic -competition market

structure.

168

Solving for a monopolistic-competition equilibrium

A firm's profit-maximization problem is MR MC .

1 1( ) (1 ) (1 ) ( )

2 2

ii i i i i

pMR x MC p p c MC x

Hence, the equilibrium price of each brand is given by 2 (twice the MC)mcip c .

The zero-profit condition 0 ( ) ( ) (2 )mc mc mc mc mc mci i i i i i iF

x p c x F c c x F cx F xc

The resource-constraint condition ( ) ( )2

mc

i

F LM F cx M F c L M

c F

Proposition

1) In a monopolistic competition equilibrium with strictly positive fixed and marginal cost, only a

finite number of brands will be produced. The equilibrium is given by

2 ; ; 2

mc mc mc

i i

F Lp c x M

c F

2) When the fixed cost is large, there will be a low variety of brands, but each brand will be

produced/consumed in a large quantity.

When the fixed cost is small. there will be a large variety of brands, and each will be

produced/consumed in a small quantity.

169

Monopsony

Properties of a monopsony:

1. many sellers (wokers)

2. 1 buyer (firm)

P MCL S 1

2

8

P*=6

D

2 3 Q

Example:

Lau's company is the only company hiring economists in Lau's Island. Suppose its demand function is

D: 10P Q . Suppose the supply function for economists is S: 4P Q

Equilibrium

: 4S P Q Total cost of hiring2(4 ) 4Q Q Q Q

Marginal cost of labor 4 2Q

maximizing condition: Marginal cost of labor

4 2 10 3 6 * 2

*(wage*) 6

D

Q Q Q Q

P

A minimum wage of $7 will eliminate the DWL.

P MCL S

P'=$7

D

3 Q

$2 1$1

2DWL

If minimum wage $7 , # no DWLCQ Q

170

Example: Rent-seeking behavior

In many situations economic agents are willing to spend resources in pursuit of some economic gains.

For example: lobbying for tax breaks or trade protection.

2-player game

Consider 2 players who split an economic rent of $R . Each player spends money in pursuit of the rent.

Let 1 2 and x x represent the 2 players respective expenditures.

Assume the share of 1: 111 2

xs

x x

Assume the share of 2: 221 2

xs

x x

Problem of 1:

1

11 1 1 1

1 2

max ( )x

xs R x R x

x x

FOC: 1 1 2 1 22 2

1 1 2 1 2

( )[ ] 1 0 1 0

( ) ( )

x x x x RR

x x x x x

2

2 1 2

1 2 2

1 2 2

( )

* reaction function (nonlinear)

x R x x

x x x R

x x R x

Similarly, 2 1 1* reaction function (nonlinear)x x R x

Nash equilibrium: 1 1 2 2* and *x x x x

1 2 2 1 1 1 1

1 1 1

1 1 1

1 1 1

1 1

2

1 1

1 2

( ) ( )

( )

( )

2

4

* Similarly, *4 4

x x R x x R x R x R x

x R x R x R

x R x R x R

x R x x

x R x

x R x

R Rx x

4R

4

R

171

Production with Teams

The inability to monitor a worker's effort also generates an inefficiency when the output of the firm

depends on the effort levels of all workers assigned to work on a certain project(which we call the joint

effort of a team). This type of externality is called the free-rider effect, in which a worker, knowing

that all other workers in a team are putting a lot of effort into the project, will have an incentive not to

work hard, given that the group as a whole is rewarded on the value of the project, that is, when the

individual workers are not rewarded according to their individual effort levels.

Consider a lab developing the future product whose value is V.

In the lab there are N scientists (workers) who work on the project.

: the effort put in by scientist , 1,2,..,ie i i N

The value of the jointly developed product depends on the effort levels of all the N scientists and is

given by 1

(A)N

i

i

V e

The equation (A) can be viewed as a production function where the inputs are the efforts put out by the

scientists.

: the compensation given to scientist after the project is completed.iw i

We assume that the value of the product is distributed to the workers so 1

n

i

i

w V

.

All scientists have identical preferences: , 1,2,..., (B)i i iU w e i N

Optimal effort levels

Assume each scientist can observe the efforts of his other colleagues, and we suppose that they collude

to maximize their utility levels.

We now wish to calculate what the optimal symmetric allocations of effort and output shares (wages)

are, and therefore, we set and for every 1,2,...,i iV

e e w w i NN

.

If we substitute into (B), the representative effort level *e that maximizes a representative worker's utility.

max V N e

w e e e e eN N

172

FOC:( ) 1 1 1

1 0 2 1 *2 42

d w ee e e

de e

Note:

If the workers can (theoretically) collude, observe each other's effort, and adjust their efforts to

maximize their utility, each should put out 1

*4

e level of effort, and the resulting total value would be

1*

4 2

NV N

173

The equal-division economic mechanism

Suppose that the manager of this firm rewards the scientists according to their equal share of the total

value of output.

Let us suppose that the manager sets iV

wN

.

Assume that each scientist takes the effort levels of his colleagues as given and chooses his effort level

to maximize his utility (B).

Problem:

max j ij i

i i

e eU e

N

FOC:2

1 1 11 0 2 1 * (C)

2 42

nii i i

i i

UN e e e e e

e N NN e

Proposition:

Under the equal-division rule:

1. If the team consists of a single worker, the worker will provide the optimal level of effort. That

is, if 1

1, then *4

nN e e .

2. If the team consists of more than 1 worker, each worker would devote less than the optimal level

of efforts. That is, if 1

1, then *4

nN e e .

3. The larger the team is, the lower will be the effort put out by each worker (each would have a

greater incentive not to work hard). That is, as N increases, ne decreases.

Note:

1. This proposition shows that offering the workers equal shares of the value of the output is

insufficient to induce them to devote the optimal level of effort to their work. So, why not offer them a

higher share of the output? Although it may be possible to induce them to work harder, if all workers

are offered a higher share of the output (i.e. iV

wN

), the total wage bill will exceed the value of output.

2. The above equilibrium is called a Nash equilibrium.

174

Effect of the size of the workforce on the total output as well as the worker's welfare level

1. 2

1 1( ) ( ) (D)

4 2

n nC A V N e NN

2

1 1 1 1* *

4 4 2 2 2

n n N NV V N e N e N NN

2. 2 2

2 3 3

11 1 12( ) ( )

4 2 4

1 1 10 if 1

2 2 2

nn

i

i

VC B U e

N N N N N

U NN

N N N N

Proposition:

1. An increase in the number of workers on the team will increase the difference between the

optimal output level and the Nash equilibrium output level.

2. An increase in the number of workers will reduce the welfare levels of each worker.

Note:

1. Part 2 of the proposition shows that the free-rider effect intensifies when the number of workers

increases, causing a further deviation from the optimal output level.

2. The optimal output level ( *2

NV ) increases with the team size, but the equilibrium level

(1

2

nV ) does not with the size of the team.

An economic mechanism that works

Following Holmstrom (1982)1, we now discuss a (rather tough) incentive mechanism that would induce

all the N workers to put forth the optimal effort level.

Suppose the team sets the following rule: If the team as a group achieves the optimal output level *V ,

then each team member receives *V

N. If the teams output is different from *V , then all team members

receive 0.

1

* if *

0 otherwise

N

i

ii

Ve V

Nw

1 Holmstrom, B. (1982): "Moral Hazard in Teams", Bell Journal of Economics, 13, pp. 324-340.

175

This mechanism makes each team member responsible for the entire output level of the team. Under the

equal-division rule, the marginal effect of each team member is lower than the marginal social value. In

this mechanism, however, the marginal value of each worker effort is the entire enterprise.

Does this work?

1. This allocation mechanism may suffer from a problem known as time inconsistency. That is,

this mechanism can work only if after each time the output is produced, the manager fires and

replaces all the workers.

2. If workers continue to work on a new project, it seems unlikely that workers would agree to let

the manager confiscate all the output just because somebody has intentionally or unintentionally

deviated from the optimal effort level.

3. Even if some deviation has occurred, it looks as if the workers would be able to negotiate with

manager or among themselves a re-division of the output, given that some output has already

been produced. Since workers anticipate that the manager will re-negotiate the contract, the

workers may not take this contract too seriously.

176

General Equilibrium II

Example: Suppose there are 2 price-taking consumers, consumer A and consumer B. Consumer A

has an initial endowment of ( , ) (1,0)X Y and a utility function of 1( , ) a aU X Y X Y . Consumer B has

an initial endowment of ( , ) (0,1)X Y and a utility function of 1( , ) b bU X Y X Y .

Calculate the competitive equilibrium price.

Consumer A:

,max ln (1 ) ln

s.t. (1) (0)

X Y

X Y X Y X

a X a Y

P X P Y P P P

Cobb-Douglas Utility function

(1 )

1and (1 ) (1 )

(1 )

X X A

XY X A

Y

aP X M aP X a

a a

a PP Y M a P Y a

a a P

Excess demand:

(1) 1 0

(1 ) (0) (1 ) 0

A

X

A X XY

Y Y

E a a

P PE a a

P P

Consumer B:

,max ln (1 ) ln

s.t. (0) (1)

X Y

X Y X Y Y

b X b Y

P X P Y P P P

Cobb-Douglas Utility function

(1 )

1and (1 ) 1

(1 )

YX Y B

X

Y Y B

PbP X M bP X b

b b P

bP Y M b P Y b

b b

Excess demand:

(0) 0

1 (1) 0

B Y YX

X X

B

Y

P PE b b

P P

E b b

177

Market:

( 1) ( )

(1 ) ( )

A B YX X X

X

A B XY Y Y

Y

PE E E a b

P

PE E E a b

P

At equilibrium: 0X YE E

10 1 0 (1)

0 (1 ) 0 (2)1

Y YX

X X

X XY

Y Y

P P aE a b

P P b

P P bE a b

P P a

Note that (1) and (2) are exactly the same, hence it is impossible to find XP and YP separately.

178

Example: Suppose there are 2 price-taking consumers, consumer A and consumer B. Consumer A

has an initial endowment of ( , ) (2,2)X Y and a utility function of ( , ) min( , )U X Y X Y . Consumer B

has an initial endowment of ( , ) (1,2)X Y and a utility function of 1/ 2 1/ 2( , )U X Y X Y .

Calculate the competitive equilibrium price.

A: A( , ) min( , ) (2,2) 2 2A X YU X Y X Y M P P

2 2( , ) min( , ) * * =2 X Y

X Y

P PU X Y X Y X Y X Y

P P

The individual will buy his endowment!!

(2) 2 0

(2) 2 0

A

X

A

Y

E

E

B: 1 1

2 2( , ) (1,2) 2B B X YU X Y X Y M P P

1

21 12 *1 1 2 2 2 2

2 2

1

212 * 11 1 2 2 2 2

2 2

X Y YX

X X X

X Y XY

Y Y Y

P P PMP X M M X

P P P

P P PMP Y M M Y

P P P

1 1( ) 12 2

( 1) 2 12 2

B Y YX

X X

B X XY

Y Y

P PE

P P

P PE

P P

Market:

1 1(0) ( )

2 2

(0) ( 1) 12 2

A B Y YX X X

X X

A B X XY Y Y

Y Y

P PE E E

P P

P PE E E

P P

At equilibrium:

1 10

2 2

1 0 1 22 2

Y YX

X X

X X XY

Y Y Y

P PE

P P

P P PE

P P P

179

Walras Law: 1

0n

j j

j

P E

Proof:

From the budget constraint of individual i , we have 1

0 1,...,n

i

j j

j

P E i m

1 1 1 1 1

( ) 0 ( )m n n m n

i i

j j j j j j

i j j i j

P E P E P E

Note: i) By the Walras Law, 0 0X YE E

ii) If 1 markets are in equilibrium the remaining market is also in equilibriumn .

iii) There is no way to solve out the n individual prices, we can only solve out 1n relative prices.

Pareto optimality

Example: Consider an economy composed of 2 1N individuals, N own one right shoe and 1N own one left shoe. Shoes are indivisible. Every one has the same utility function which is

min[ , ]U R L , where and R L are the quantities of right and left shoes consumed respectively. Find

the set of Pareto Optimal allocations.

Solution: A Pareto Optimal allocation is an allocation where all the shoes are in pairs. (It does not

matter who own the shoes. It is possible that a single individual own all N pairs.)

There will be an individual which owns the remaining left shoe.

180

Pareto Optimality in a Consumption Economy

,max ( , )

. . ( , )

A A

A A AX Y

B B B B

A B

A B

U X Y

s t U X Y U

X X X

Y Y Y

Method 1

,max ( , )

. . ( , )

A A

A A AX Y

B A A B

U X Y

s t U X X Y Y U

( , ) [ ( , )]A A A B B A AL U X Y U U X X Y Y

FOC:

0

0

A B

A A B

A B

A A B

U UL

X X X

U UL

Y Y Y

A A

A A

B B

B B

U U

X Y

U U

X Y

B A

B A

B A

B A

U U

Y Y

U U

X X

A B

XY XYMRS MRS

Method 2

( , ) [ ( , )] [ ] [ ]A A A B B B B A B A BL U X Y U U X Y X X X Y Y Y

FOC:

0

0

A

A A

A

A A

UL

X X

UL

Y Y

A

A

A

A

U

X

U

Y

0

0

B

B B

B

B B

UL

X X

UL

Y Y

B

B

B

B

U

X

U

Y

A B

A B

A B

A B

U U

X X

U U

Y Y

B A

B A

B A

B A

U U

Y Y

U U

X X

A B

XY XYMRS MRS

181

Pareto Optimality in a Production Economy

,max ( , )

. . ( , )

X X

X X XK L

Y Y Y

X Y

X Y

F K L

s t F K Y Y

K K K

L L L

or

,max ( , )

. . ( , )

X X

X X XK L

Y X X

F K L

s t F K K L L Y

( , ) [ ( , )]X X X Y X XF K L Y F K K L L

FOC:

0

0

X Y

X X Y

X Y

X X B

F F

K K K

F F

L L L

X X Y X

X X Y X

Y Y Y X

Y Y Y X

F F F F

K L L L

F F F F

K L K K

X Y

KL KLMRTS MRTS

182

Pareto Optimality in General

Consider an economy with m consumers, n producers, r primary factors and s produced goods.

Assume each consumer consumes all produced goods, and each producer user all primary factors and

produces all goods.

The consumers utility functions are 1* * 1 1* *( , ... , , , ... , ) 1,...,s r ri i i i i i i iU U X X Y Y Y Y i m

The firms implicit production functions are 1 1( ,..., , ,..., ) 0 1,...,s rh h h h hF X X Y Y h n

At equilibrium,

a) the aggregate amount of primary factors supplied by the consumers is equal to the aggregate

amounts used by producers: *

1 1

1,...,m n

j j

i h

i h

Y Y j r

b) the aggregate consumption levels of produced goods is equal to their aggregate output levels:

*

1 1

1,...,m n

j j

i h

i h

X X k s

quantity of

produced goods kX consumed

by consumer i

j

iY : endowment of the jth

primary factor of

consumer i *j

iY : quantity of primary input j that

consumer i supplied to the producers *j j

i iY Y : quantity of primary input j that

consumer i consumes

quantity of

produced goods kX produced by

firm h

quantity of input k that firm h uses

183

To attain Pareto Optimality, consumer i is to solve the following problem:

1* * 1* *1 1 1 1

1* * 1 1* *

1 1 1 1 1 1 1,..., , ,...,

1* * 1 1* *

1

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

. . ( , ... , , , ... , ) 2,...,

( ,...,

s r

s r r

X X Y Y

s r r

i i i i i i i i

h h h

U X X Y Y Y Y

s t U X X Y Y Y Y U i m

F X X

1, ,..., ) 0 1,...,s rh hY Y h n

*

1 1

1,...,m n

j j

i h

i h

Y Y j r

*

1 1

1,...,m n

k k

i h

i h

X X k s

1* * 1 1* * 1* * 1 1* *

1 1 1 1 1 1 1

2

1 1 * *

1 1 1 1 1

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

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

ms r r s r r

i i i i i i i i i

i

r m n ns r j j k k

h h h h h h j i h k h i

j i h h i

L U X X Y Y Y Y U U X X Y Y Y Y

F X X Y Y Y Y X X

1 1

n s m

h k

FOC:

1

* *

1 1

0 1,...,kk kUL

k sX X

* *0 2,..., ; 1,...,ii kk k

i i

ULi m k s

X X

0 1,..., ; 1,...,hh kk kh h

FLh n k s

X X

1

* *

1 1 1

0 1,...,( )

jj j j

ULj s

Y Y Y

* *0 2,..., ; 1,...,

( )

ii jj j j

i i i

ULi m j s

Y Y Y

0 1,..., ; 1,...,hh jj jh h

FLh n j s

Y X

The constraints also have to be satisfied.

From the FOCs, we have the following conditions for Pareto Optimality:

a) 1 1... ...m njk jk jk jkMRS MRS MRTS MRTS for each pair of produced goods

1 1

**

1 1

1 1

* *

1 1

... ... , 1,...,

m n

j jj jj m n

m nkk kk k

m n

U FU F

X XX Xj k s

U U F F

X XX X

184

b) 1 1... ...m njk jk jk jkMRS MRS MRTS MRTS for each pair of primary goods

1 1

* ** *

1 1 1

1 1 1

* * * *

1 1 1

( )( )... ... , 1,...,

( ) ( )

m n

j j jj j jj m m n

nkk k k k k k

m m n

U FU F

Y Y YY Y Yj k r

U U F F

Y Y Y Y Y Y

c) between factors and commodities (consumers)

between factors and commodities (producers)

MRS

MRTS

1 1

* ** *

1 1 1

1 1

* *

1 1

( )( )... ... 1,..., ; 1,...,

m n

j j jj j jj m m n

m nkk kk k

m n

U FU F

Y Y YY Y Yj r k s

U U F F

X XX X

185

External Effects in Consumption

Interdependent utility function

Assume that the utility level of one consumer depends upon the consumption of another. Extreme

altruism may increase the satisfaction of the ith

consumer if the consumption level of the jth

consumer is

raised. The desire to keep up with the Joneses may have the opposite effect.

,max ( , , , )

. . ( , , , )

A A

A A A B BX Y

B A A B B B

A B

A B

U X Y X Y

s t U X Y X Y U

X X X

Y Y Y

or

,max ( , , , )

. . ( , , , )

A A

A A A A AX Y

B A A A A B

U X Y X X Y Y

s t U X Y X X Y Y U

( , , , ) [ ( , , , )]A A A A A B B A A A AU X Y X X Y Y U U X Y X X Y Y

FOC:

0

0

A A B B

A A B A B

A A B B

A A B A B

U U U U

X X X X X

U U U U

Y Y Y Y Y

A A A A A A A A

A B A B A B A B

B B B B B B B B

A B A B B A B A

U U U U U U U U

X X Y Y Y Y X X

U U U U U U U U

X X Y Y Y Y X X

The Pareto optimal allocations are no longer points where the indifference curves are tangent to

each other!!

186

Public Goods

2 consumers: and A B who produce and consume a public good X and a private good Y , using a given amount of labor

,max ( , )

. . ( , )

A

A A AX Y

B B B

U U X Y

s t U U X Y

( )

( )

X

A B Y

X X L

Y Y Y L

production function

X YL L L

1 2 3 4( , ) [ ( , )] [ ( )] [ ( )] [ ]A A B B B X A B Y X YU X Y U U X Y X X L Y Y Y L L L L

FOC:

1 2 0A BU U

X X X

(1)

3 0A

A A

U

Y Y

(2)

1 3 0B

B B

U

Y Y

(3)

2 4 0X X

X

L L

(4)

3 4 0Y Y

Y

L L

(5)

(4) and (5) 24 2 3

3

Y

X Y

X

Y

LX YMRT

XL L

L

(6)

(2) and (3) 31

A

A

B B

B B

U

Y

U U

Y Y

(7)

2 1(1)

A

A B A A B

B

B

U

U U U Y U

UX X X X

Y

marginal rate of transformation

187

(2), (6) and (8) 2

3

A

A A B

BA B

B

A A B

A A B

U

U Y U

UX X U UY X X MRT

U U U

Y Y Y

A B

YX YXMRS MRS MRT

188

Theory of Second Best

Consider a simplified system with 1 consumer, 1 implicit production function, n commodities, and a fixed supply of 1 primary factor not desired by the consumer.

The necessary conditions for Pareto optimality are obtained by maximizing the consumers utility subject to the production function.

0

1 1( ,..., ) ( ,..., , )n nL U x x F x x x

FOC:

0 1,...,i ii

LU F i n

x

or i i

j j

U FMRS MRTS

U F

Assume that institutional conditions prevent the attainment of one of the FOC.

Lets say, 1 1 0 where U kF k (**)

The conditions for second-best welfare optimum are obtained by maximizing utility subject to the

aggregate production function and (**).

0

1 1 1 1( ,..., ) ( ,..., , ) ( )n nL U x x F x x x U kF

FOC:

1 1

0

1

1 1

( ) 0 1,...,

( ,..., , ) 0

0

i i i i

i

n

LU F U kF i n

x

LF x x x

LU F

1 1

1 1

( ) , 1,...,

( )

i i i i

j j j j

U F U kFi j n

U F U kF

In general, nothing is known a prior about the signs of the cross partial derivatives. Therefore,

the usual conditions for Pareto optimality no longer hold.

The theory of second best has been used to questions the desirability of partial-equilibrium

policies that might be used to attain the Pareto conditions on a piecemeal basis for markets

considered in isolation.