Principal - Agent Games

1
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
• the owner cannot observe the managers effort
• the manager might obtain better information about
  firms 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
• principals welfare,
• - but the agents preferences might differ from
  the principal
• - the agent might be more risk averse and
  dislike effort

.
2
Information problems

• The two parties attempt to design a contract that
  deals with these problems
• 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 projects profits are affected
  by the managers actions
• - if the managers actions are observable, both
  to the owner and to a third party enforcer, then
  the contract could be quite simple
• it would specify the managers actions and
  corresponding wage
• The contract would be a forcing contract, where
  the manager is paid only if actions are
  performed.

.
3
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 managers actions cannot be
observed, then the contract cannot specify
particular actions, since they cannot be
verified. In this case, the managers
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.

.
4
Hidden Action Model
Let p denote the projects profits -observable
and verifiable. Let e denote the managers
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.
This case is then no different from
the case where effort can be
observed. More realistically,
Shocks can affect p
-we cannot perfectly infer e from observing p
.
5
Shocks
.
6
High profits are more likely with eH
.
7
Managers expected utility fn

.
8
Owner maximizes net revenue (p-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
- managers effort level and managers wage
as a function of profits w(p) - Assume that the
manager has a reservation utility level . -
So the owner must provide manager with an
expected utility of at least . - or the
manager will not accept the contract. .

.
9
When effort is observable
.
. .
10
Transform optimization problem
.
. .
11
Lagrangian
.
. .
12
Risk Sharing result

.
13
If manager is risk neutral

.
14
When effort is not observable

.
15
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.

.
16
Risk neutral agent

.
17
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
Ps payoff In solving 1), the Ps 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(p) must
be chosen so that when the A chooses e to max
- the optimal e is the effort level the P is
trying to get A to choose.
.
18
So the Principals problem is
.
19
Implementing high effort

.
20
Proof that constraints bind

.
21
Optimal contract when P chooses eH

.
22
Likelihood ratio

.
23
Monotone likelihood ratio property
.
24
Principals payments
.
25
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 cant improve welfare unless the
planner can observe effort when P cant

.