Principal - Agent Games
Sometimes asymmetric information develops after a contract has been signed
In this case, signaling and screening do not help, since they are only effective at separating
types before signing a contract
The classic example – an owner of a firm hires a manager and wants the manager to
maximize the profits of the firm
There are two main sources of asymmetric information
(1) the owner cannot observe the manager’s effort
(2) the manager might obtain better information about firm’s opportunities than owner
Problems of this type are known as principal agent problems
- the principal hires the agent and wants the agent to take actions that maximize the
principal’s welfare,
- but the agent’s preferences might differ from the principal
- the agent might be more risk averse and dislike effort
Information problems
The two parties attempt to design a contract that deals with these problems
(1) Hidden action – the principal does not observe the agents actions
- ex: in the firm, the action is work effort
- in insurance, the action is safe driving
- this is known as the moral hazard problem
(2) Hidden information – the agent acquires better information
- ex: new investment opportunities
Suppose an owner wants to hire a manager for a one-time project
Suppose that the project’s profits are affected by the manager’s actions
- if the manager’s actions are observable, both to the owner and to a third party enforcer,
then the contract could be quite simple:
it would specify the manager’s actions and corresponding wage
The contract would be a forcing contract, where the manager is paid only if actions are
performed.
Hidden Action
Even if the P and A can observe actions,
this is not enough as they must be verifiable by a third party.
If the manager’s actions cannot be observed,
then the contract cannot specify particular actions, since they cannot be verified.
In this case, the manager’s compensation can only be based on observable variables
Profits may or may not be observable and verifiable, but we will assume that they are
This may be more appropriate for the top managers than anyone else
- as they affect the profits of the whole company and these profits might be public
- or good proxies like share price are public
For regular workers involved in team production, this only provides a rough guide.
Hidden Action Model
Let π denote the project’s profits -observable and verifiable.
Let e denote the manager’s effort – usually e is assumed as a one-dimensional real number,
- but it could also be a vector
If profits are a deterministic function of e and presumably a monotonic function,
- then e could be inferred perfectly just by observing profits.
π
π(e)
π1
e1
This case is then no different from the
case where effort can be observed.
More realistically,
Shocks can affect π
-we cannot perfectly infer e from observing π
e
Shocks
To add to the model, assume that    ,   and that there is a conditional density fn
f  e  such that f  e   0  e and     ,  
Given any e, we can observe any  in the interval  ,   (some  levels more likely)
Assume that the manager only has 2 possible effort choices e H , e L
We will assume that the distribution of  conditional on e H first order stochastically
dominates the distribution of  conditional on e L
F  e H   F  e L  at all    ,  
with the inequality being strict for some open set    ,  
So
High profits are more likely with eH
F  e H   F  e L 
F is the CDF (cumulative density function)
- so it gives the probability of getting profit levels less that or equal to  .
The inequality says that low profit levels are more likely with e L than with e H .
This implies that expected profits are higher when the manager chooses e H .
  f  e  d     f  e  d 
H
L
Manager’s expected utility fn
Assume that the manager maximizes expected utility
Given a utility function U ( w , e) defined over certain outcomes ( w , e).
Assume that:
U w  w , e   0 so that higher wages are preferred
U ww  w , e   0 at all  w , e  diminishing marginal utility of money (weak risk aversion)
U  w , e H   U  w , e L  at all w effort e H is more costly then e L
and e L is preferred holding w constant
U  w , e   V  w   g  e  additively separable in w and e this simplifies the analysis
- the first assumption implies that V   w   0, V   w   0 and that g  e H   g  e L 
Owner maximizes net revenue (π-w)
Assume that the owner receives the profits minus the wages.
- assume that the owner is a risk neutral expected payoff maximizer.
- owners can diversify, so they are less risk averse than the manager is
Consider a game where,
1- the owner moves first and chooses a contract to offer the manager
2- then the manager observes the contract and either accepts or rejects
3- if the manager accepts, then he chooses an effort level
4- then shocks are realized
5- payoffs are determined.
Initially, suppose that effort is observable, then the contract would specify
- manager’s effort level e  eL ,eH  and manager’s wage as a function of profits w(π)
- Assume that the manager has a reservation utility level U .
- So the owner must provide manager with an expected utility of at least U .
- or the manager will not accept the contract. .
When effort is observable
The optimal contract for the owner solves the following problem.
max
ee L ,e H , w  
   w    f  e  d 
the firm's expected profits net of wages
s .t .
 V  w    f  e  d   g  e   U
the manager's exected utility must be at least U
Since there are only 2 possible levels of e , can solve this max problem by:
. .
(1) set e  e L , solve for w   given e  e L
(2) set e  e H , solve for w   given e  e H
(3) choose e L or e H and the associated optimal w  
depending on which one results in a higher payoff for the owner
Transform optimization problem
To solve, we can first transform the optimization problem
    w    f  e  d     f  e  d    w   f  e  d 
Since, in (1) and (2) we hold e fixed and choose w   , we can ignore   f  e  d 
The
. . problem becomes min  w   f  e  d 
w  
s .t .
 V  w    f  e  d   g  e   U
We can solve this problem by choosing the optimal w for each possible  .
It is more intuitive if you think in terms of a finite # of possible profit levels ( 1 , ... n )
n
min  w i Prob  i e 
w 1 ,... w n
i 1
s .t .
n
V  w  Prob  e   g e   U
i 1
i
i
Lagrangian
 n

min  w i Prob  i e     V  w i  Prob  i e   g  e   U 
w 1 ,... w n ,
i 1
 i 1

n
To solve
And we would get a FOC for each w i
Prob  i e    V   w i  Prob  i e  =0
. . the case where    ,   , we would get a similar FOC
In


The wage w   at each level of    ,   must satisfy f  e    V   w    f  e  =0
Divide by f  e  and rearrange to obtain
1
  ( <0, when minimized)
V   w    a positive
constant
Risk Sharing result
From this condition, we know that if the manager is strictly risk averse V   w   0 
the optimal w   is a constant wage that does not depend on 
This is just a risk sharing result,
- since the contract specifies an effort level and there is no need to provide incentives,
- the optimal contract has the less averse party bearing all of the risk.
By paying the manager a constant wage,
- the owner insures the manager against fluctuations in income.
The owner will offer a fixed wage w e *, s. t. the manager receives his reservation utility level
V  we *   g e   U
or V  w e *   U  g  e 
Since g  e H   g  e L  , the manager's wage will be higher if the contract specifies e H .
If manager is risk neutral
Suppose the manager is risk neutral V  w 
1
then V   w   1 and
  always holds (   -1)

V  w   
There is no need for insurance because both the P and the A are risk neutral
So any wage funciton w   that results in any expected wage of
E V  w    E  w   g  e   U
 g  e   U  is optimal
Regardless, in equilbrium the expected wage satisfies w  V 1  g  e   U 
Given the expected wage, the P maximizes expected profits minus expeced wage payments
by choosing e
max   f  e  d   V 1  g  e   U 
ee L , e H 
When effort is not observable
When effort is observable it is easy to get the A to choose the desired effort
- just use a contract ( w , e ) where w  V 1  g  e   U 
- and e is chosen to maximize expected profits minus expected wage.
- the agent is insured against income risk
When effort cannot be observed, the contract cannot specify e.
- the only way to make the agent choose the desired e is to make the wage a fn of  .
- since  is random, the agent is no longer fully insured against income risk.
There is a conflict between providing incentives and having an efficient allocation of risk.
- If the agent is risk neutral, then there is no need to insure the agent at all
- in this case, the principal can end up with the same payoffs are before.
Risk neutrality  V  w   w ,
we established that in the observable effort case,
e would be chosen to solve
max
ee L ,e H 
  f  e  d   g  e   U
this is the principal's payoff
Optimal contract w/unobservable effort
We have the following result for the case with unobserved effort.
Prop: In the PA model with unobservable effort and a risk neutral A, an optimal contract
generates the same effort choice and expected utilities for the manager and the owner
as when e is observable.
Proof:: Note that the P can always do at least as well when e is observable as when it is not,
since the P could always ignore e in choosing a contract. So if we find a contract
when e is unobservable that generates the same expected payoffs as the optimal
contract when e is observable, then the contract must be optimal. This is the plan.
Consider w       , where  is a constant.
This contract is like selling the firm to A,
- because A pays  and in return gets whatever profits are generated.
Risk neutral agent
If A accepts w   , then A will choose e to solve
max  w   f  e  d   g  e     f  e  d     g  e 
Note that the e that maximizes this is the same e that maximizes
max   f  e  d   g  e   U (which is what was maxed when e was obs)
So the optimal e under w(  ) is the first best optimal e (fully observable case).
A accepts the contract as long as it pays at least his residual utility, U .
  f  e *  d     g  e *   U
Now choose  so that this constraint binds, then we get
*
   f  e *  d   U  g  e * 
this is what the P gets
when e is unobservable
this is the P's payoff when e is observable
So both the P and the A get the same payoff as before. QED.
Risk adverse agent
The intuition for the prior result is since A is risk neutral, there is no need for risk sharing.
- Efficient incentives can be provided by selling the A the firm.
- Then the A gets the full marginal return from e.
With a risk adverse agent, proceed by taking 2 steps
1) characterize the optimal contract for each effort level
2) choose the effort level to maximize the P’s payoff
In solving 1), the P’s problem reduces to the wage minimization problem discussed above;
the difference is now there is an added constraint because the contract cannot specify e.
The wage function w(π) must be chosen so that when the A chooses e to max
max  v  w    f  e  d  g  e 
e
- the optimal e is the effort level the P is trying to get A to choose.
So the Principal’s problem is
min  w   f  e  d 
w  
s .t . (1)
 v  w    f  e  d   g  e   U
participation constraint
and (2) e solves max  v  w    f  e  d   g  e 
e
incentive constraint
To implement e L , P optimally offers the A a fixed wage w e *  v 1  u  g  e L  
- where w e * is the same wage that P would pay if e L was observable
Since effort is costly and g  e H   g  e L  , if the wage is fixed then the A will choose e L
- contract results in the same payoffs to P and A as in the case when effort is observable
- so it is optimal.
To implement e H is more challenging
Implementing high effort
In order to induce the A to choose the more costly effort level, wage must depend on profit.
Rewrite constraint (2) as
 v  w    f  e  d   g  e    v  w    f  e  d   g  e 
H
A's expected payoff given
w   and the choice of e H
P's problem max   w   f  e  d  s .t . (1)
w  
H
L
A's expected payoff given
w   and the choice of e L
L
 v  w    f  e  d   g  e   U  0
H
H
(2)  v  w    f  e H  d   g  e H    v  w    f  e L  d   g  e L  0
As before, choose optimal w   for each 
 a FOC must be satisfied at every    ,   


 f   e H    v   w     f   e H    f   e H   f   e L  v   w     0
where   0
multiplier on
constraint (1)
,
 0
multiplier on
constraint (2)
rearranging, we get

f  e L  
1
    1 



f  e H  
v  w   

Proof that constraints bind
Claim: In any solution to P's problem when P implements e H , both   0 and  >0.
So, both constraint (1) and (2) are binding at the optimal wage.

f  e L  
1
Proof: Suppose  =0. Then
  1 
.

f  e H  
v   w   

Because F  e H  first order stochastically dominates F  e L  , there must exist
an open set of profit levels    ,   such that f  e L   f  e H  at all    .
This implies that
f  e L 
f  e H 
 1 at all    ,
which, along with the fact that   0,
implies that v   w     0 at all    .
But by assumption, A always strictly prefers more money to less, so v   w     0,
which leads to a contradiction. Therefore   0.
Optimal contract when P chooses eH
1
  & the optimal contract specifies a fixed wage
v   w   
- we know that A strictly prefers e L to e H when the wage is fixed,
- so this contract cannot implement e H . Therefore,   0. QED
Now suppose that =0. Then
More details on the optimal contract when P chooses e H .

f  e L  
1
- we know that
    1 


f

e
v   w   
 H  

- as a benchmark, let wˆ be the wage that sets
- optimal w   satisfies: w   > wˆ if
1


v  w   
f  e L 
f  e H 
1 ;
w   < wˆ if
f  e L 
f  e H 
1
Likelihood ratio
The optimal w   pays relatively more for levels of  that are relatively
more likely to occur when A chooses e H .
f  e L 
f  e H 
is a likelihood ratio as it corresponds to a ratio of probabilities in the discrete case
By paying A more for outcomes that are more likely to occur when e H is chosen,
- P ensures that A chooses e H .
This suggests P should pay more for higher  , but w   doesn't have to be increasing in  .
- if
f  e L 
f  e H 
is decreasing in  , then it is optimal for w   to be increasing in  .
- but otherwise the optimal w   would be non-monotonic in  .
Monotone likelihood ratio property
Often economists assume that f  e L  f  e H  is decreasing in  .
If f  e  has this property, then said to have the monotone likelihood ratio property.
- Note that the monotone LRP is not implied by first order stochastic dominance.
- So w( ) is not monotonic in general; even when it is it is not linear in general
f  e L 
f
f  e H 
w
F
1
F  e L 
F  e H 

0
1

Middle values of  ,    0 ,  1 
are relatively more likely with e H

0
1

F  eH  stochastically dominates F  eL 

0
1

w   pays more for    0 ,  1 
(values relatively more likely with e H )
Principal’s payments
Because the A is risk averse, and the P needs to make w   vary with  ,
- the expected wage payment is strictly higher than the fixed wage payment
made in the case where effort is observable.
- A must be compensated in utility terms for taking the extra risk.

 
Strict risk aversion  v     0  implies E v  w    e H  v E  w   e H 

Jensen's inequality




Since, E v  w    e H  U  g  e H  , we must have v E  w   e H   U  g  e H 


Since, when effort is observable, v w eH *  U  g  e H  , and since v   w   0,


We must have E w   e H  w eH *
Summing Up
The wage payment for implementing eL is exactly the same as when effort is observable.
The expected wage payment for implementing eH is strictly larger in the non-obs. effort case.
So, non-observable effort raises the cost of implementing eH and does not change the cost of
implementing eL . This could lead to an inefficiently low level of effort being implemented.
When eL is optimal under obs, it is still optimal under non-obs, and non-obs causes no losses.
If eH would be optimal under observability, then under non-observability either
1) could still be optimal to implement, but A must bear risk and the expected wage rises
2) the risk bearing costs may be high enough that P decides that it is better to implement eL
In either case, non-observability causes a welfare loss to P.
- A gets an expected utility equal to his reservation utility no matter what.
Note: a social planner can’t improve welfare unless the planner can observe effort when P can’t