Contract Theory: Midterm
學號：R01323008
姓名：游捷名
1.
In a standard Principal-Agent Problem, the main instrument for the principal (P) to provide
incentives to the agent (A) is, of course, the wage.
P wishes A to put more effort so that P can earn more profit. If effort is observable and verifiable, P
simply design a contract that gives A the minimum wage A would accept and requires A to devote
his effort at the level P wants. However, this is not the case in the real world because effort is
usually unobservable.
In such cases, the wage in the optimal contract should be contingent on the output level, (which, in
a standard Principal-Agent Problem, is assumed to be observable and verifiable.) The key link is
that the PDF of output level should depend on A’s effort level. Typically, if A devotes more effort,
we would expect the output level is more probable to be large; thus when the wage is contingent on
the output level, it provides incentives to A to give more effort.
An additional complication is that now we must take care of the Incentive Compatibility (IC)
constraint. Since now A could choose his effort level, he must choose the one which maximizes his
expected utility. IC constraint requires A to choose such an effort level.
A tradeoff is between the incentive and the willingness of A to work for P. If the incentive is too
low, A would work somewhere else rather than work for P. Another tradeoff is between the
incentive and the risk. If the incentive is strongly contingent on the output level, it means A bears
most of the risk and he is likely to devote more effort; on the other hand, if the incentive is
contingent on the output level only weakly, it means P bears most of the risk and A is likely to
devote less effort.
2.
w , if y  y H
Denote: w   H
and u L  wL , uH  wH
 wL , if y  y L
(1) If P wants to implement e  0 , the expected profit of P would be: 0.9 yL  wL   0.1 yH  wH ,
given that e  0 is optimal for A. Thus, P faces a cost minimization problem:

02
0
.
9
w

0
.
1
w

0
L
H

2
m i n 0.9wL  0.1wH , s.t. 
2
2 , and wH , wL  0
wL , wH
0
2
0.9 wL  0.1 wH   0.2 wL  0.8 wH 

2
2
which is equivalent to:
 0.9u L  0.1u H  0
, and uH , uL  0
m i n 0.9u L2  0.1u H2 , s.t. 
uL , uH
0.7u L  0.7u H  2
The solution would be uL  0 , uH  0 , so wL  0 , wH  0
The expected profit of P would be: 0.9 yL  wL   0.1 yH  wH   1.8  0.1yH
(2) If P wants to implement e  2 , the expected profit of P would be: 0.2 yL  wL   0.8 yH  wH  ,
given that e  2 is optimal for A. Thus, P faces a cost minimization problem:

22
0.2 wL  0.8 wH 
0

2
m i n 0.2wL  0.8wH , s.t. 
2
2 , and wH , wL  0
wL , wH
0.2 wL  0.8 wH  2  0.9 wL  0.1 wH  0

2
2
which is equivalent to:
 0.2u L  0.8u H  2
, and uH , uL  0
m i n 0.2u L2  0.8u H2 , s.t. 
uL , uH
 0.7u L  0.7u H  2
2
2
20
400
 2 

The solution would be uL  0 , u H 
, so wL  0 , wH  
 
0.7 7
49
 0.7 
The expected profit of P would be:
400 
300.4

0.2 y L  wL   0.8 y H  wH   0.4  0.8 y H 
  0.8 y H 
49 
49

(3) If P would want to implement e  2 , it must be
0.8 y H 
300.4
388.6
388.6
 1.8  0.1y H  0.7 y H 
 yH 
 11.3294
49
49
49  0.7
Thus we have yH  11.3294
3.
The paper provided a two-step method to solve the Principal-Agent Problem: In the first step, P
n
adjusts the wage wi W to minimize the expected cost (   i e wi ,) given the effort level, e .
i 1
In the second step, P then chooses the optimal e to maximize the expected profit
n
(   i e  yi  wi  .)
i 1
This can only be done when P is risk-neutral. If P is not risk-neutral but has some general form of
utility function U  , the expected profit of P would be
n
  e  U  y
i 1
i
i
 wi  , where the portion of
cost cannot be taken out alone thus the two-step method fails to apply.
4.
Consider an example in a team work, say, a basketball team. If a player shoots a lot of scores, it
probably means his teammates are playing well, too, (maybe they have good defense or assistance.)
Thus the payment of a player should positively depend on the performances of his teammates. More
formally, since the good performances of the other agents probably means that the agent devotes
more efforts, the wage should be positively contingent on performances of the others so that the
agent has more incentives to work harder (to “help” the others.)
Consider an example of a real estate salesman. If all the other salesmen are making good sales, it
probably means that the market is very hot. So if only this salesman has a bad sale, it probably
means that he does not work hard enough. On the other hand, if all the other salesmen are making
bad sales, it probably means that the market is very cold. So if only this salesman has a good sale, it
probably means that he works really hard. Thus the payment of a salesman should negatively
depend on the performances of other salesmen. More formally, since the good performances of the
other agents probably means that the agent devotes less effort, the wage should be negatively
contingent on performances of the others so that the agent has more incentives to work harder.
Mathematically, under the context of Holmstrom (1982), the principal’s problem is:



 ui si x g x; edx  vi ei   u i

m a x   E y x; e   si x  g x; e dx s.t.
e , si  x 
ei  arg m a x  ui si x g x; ei, ei dx  vi ei 
i



ei

 u s x g x; edx  ve   0
If we replace the IC constraint with its F.O.C:
i
i
ei
i
i
We would have the Lagrarangian:






L    E y x; e  si x  g x; e dx   i  ui si x g x; e dx  vi ei   u i    i  ui si x g ei x; e dx  vi ei 
i
i
i


The F.O.C for si x  would be:
g e x; e 
1
, x, i
 i   i i
ui si x 
g x; e 
Take derivative w.r.t. x , we have:


g e  x; e  

  g ei  x; e  g  x; e  
0  si  x   uisi  x i   i i
  ui si  x   i

g  x; e  
x



  g ei  x; e  g  x; e  
 1 
 0  si  x   uisi  x 
 ui si  x   i


x
 ui si  x  


  g ei  x; e  g  x; e  
 ui si  x   i

x


 si  x  
uisi  x 
ui si  x 




In the first example, when x (which represents the performances of the others) increases, it
probably means that the agent devotes more efforts. Suppose the term “probably” is in the sense
 g ei x; e  g x; e 
 0 , thus si x   0 ,
that the likelihood ratio increases in x , then we would have
x
(note that uisi x   0 , uisi x   0 , i  0 ,) suggesting wage should be positively contingent

on x .

In the second example, when x increases, it probably means that the agent devotes less efforts.
Suppose the term “probably” is in the sense that the likelihood ratio decreases in x , then we would
have

  0 , thus sx  0 , suggesting wage should be negatively contingent on
 g ei x; e  g x; e 
x
i
x.
5.
 yi  ei  e j  

yi ~ N ei  e j ,   F yi ; ei , e j   



where  is the CDF of a standard normal distribution.
 yi  ei  e j  
 y  ei  e j  
   i
, ei  ei
 






So f  yi ; ei , e j  satisfies the first-order stochastic dominance property.
The likelihood ratio:
f ei  yi ; ei , e j 
f  yi ; ei , e j 

yi  ei  e j 

2



 f ei  yi ; ei , e j  f  yi ; ei , e j 
1
yi
2
0
So f  yi ; ei , e j  satisfies the monotone likelihood ratio property.
It is obvious that an increase in ei would shift the whole distribution f  yi ; ei , e j  to the right,
suggesting that an increase in ei would make higher yi more probable. Thus a contract
contingent on yi would provide incentives for agent i to exert ei . The other agent’s effort level
e j may also shifts the distribution f  yi ; ei , e j  . If   0 , an increase in e j would make higher
yi more probable, too; if   0 , it would be the other way round; if   0 , e j has no effect on
f  yi ; ei , e j .
In an optimal team contract, we should look at the joint distribution of y1 and y 2 . Suppose the
joint PDF is given by:
  y1  e1  e2 2   y 2  e2  e1 2 
g  y1 , y 2 ; e1 , e2   f  y1 ; e1 , e2  f  y 2 ; e2 , e1  
exp 

2 2
2 2


1
We would have the likelihood ratio:
g ei
g

y  e  e   y  e
i
i
j
j

j
 ei 
2
Thus the wage of agent i should positively depend on yi because

.
 g ei g
yi
  0 . And

 g ei g
y j


, so the wage of agent i should positively depend on yi if   0 , negatively
2
depend on yi if   0 , and has no dependence if   0 .
If   0 , an increase in ei would make higher y j more probable. Thus a contract contingent on
y j would provide incentives for agent i to exert ei . And vise versa.
6.
Under the context of the paper, we have
f x; a 
G x  sx 
 a
U sx 
f x; a 
where Gw is the principal’s utility function, U w is the agent utility on wage, a is the effort,
and  and  are Lagrarange Multipliers.
In the example: Gw  w , U w  2 w , f x; a  
1 a 1  x  a
e    2 e
2
xa
x  a
a a 
a2


    2  s  x      2 
x
a
a 
1 a

e
a
x
1
sx 0.5
x
1 a
e , we have
a
x
where goes the equation (10) of the paper.
7.
When designing an individual contract, we have to evaluate the double integrals over the joint
distribution of zi and  (denote as H zi , ; e ) in the IC and IR constraints. In the original
assumption that zi and  are independent, the joint distribution may decompose into two
separate parts ( H zi ,; e  F zi ; e G; e ); but if zi and  are dependent, we are not able to
decompose the joint distribution, thus complicates the calculations.
A more serious problem arises when designing a tournament. In the original assumption, the j-th
n!
j 1
n j
order statistic of z is given by  jn z; e  
f z; e F z; e  1  F z; e  , which is
n  j ! j  1!
independent of  . However, under the new assumptions, the PDF of z conditional on  is
given by: f z; , e  
hz, ; e 
 hz,; e
, thus the j-th order statistic of z would be
z
 jn z; , e 
n!
j 1
n j
, which depends on  . The form
f z; , e F z; , e  1  F z; , e 
n  j ! j  1!
of H zi , ; e would thus affect the solution of an optimal tournament. In the original setting, the
optimal tournament does not depend on the form of the distribution of  , ensuring that an optimal
tournament must be no worse than optimal individual contracts when the distribution of  is very
dispersed. This statement, however, may not always be true under the new settings.
If the correlation between z and  is positive, it is like the common shock becomes more
dispersed, so the effect of the common shock would be more important. Thus we should expect that
an optimal individual contract would tend to be less efficient, while an optimal tournament would
tend to be more efficient. Consider a particular example such that z  x   , where x and  are
independent so Correlationz,   1 . Thus y  z    x  2 , it is like the model reduces to the
original setting but the common shock becomes twice as large.
On the other hand, if the correlation between z and  is negative, it is like the common shock
becomes less dispersed, so the effect of the common shock would be less important. Thus we
should expect that an optimal individual contract would tend to be more efficient, while an optimal
tournament would tend to be less efficient. Consider a particular example such that z  x   ,
where x and  are independent so Correlationz,   1 . Thus y  z   x , it is like there is
no common shock at all, because the effects of the common shock are cancelled out. In this
particular case, optimal individual contracts must be no worse than an optimal tournament, no
matter how dispersed the distribution of  is.