Lecture 3 in Contracts
Moral Hazard
Our final lecture analyzes optimal contracting in
situations when the principal writing the
contract has less information than the agent
who accepts or rejects it. In this scenarios the
principal is not only limited by a participation
constraint, but also by incentive compatibility
and truth telling constraints as well.
Read Chapters 17 and 18 of Strategic Play.
Moral hazard
Moral hazard arises when the unobserved choices of
one player affect but do not completely determine
the payoff received by another person.
Since the player’s choice is not observed, a contract
cannot direct him to make a particular choice.
However linking the player’s payments to the
consequences of his action, can help align his
incentives with those of the other players, even
though the consequences are only partly
attributable to or caused by the action itself.
Examples of moral hazard
Managers are paid to make decisions on behalf of
the shareholder interests they represent. If they
were paid a flat rate, why would they pursue the
objectives of shareholders?
Lawyers representing clients are more likely to win
if they are paid according to their record, and also
whether they win the case in question or not.
The extent of warranties against product defects is
limited by the uses the product is put, and how
much care is taken.
Insurance against accidents discourages care.
Settle up at the end
Sometimes the unobserved action can be
inferred exactly at some later point in time.
In this case a moral hazard problem does not
exist, providing the contract period is
sufficiently long.
For example construction companies can
sometimes be sued for structural defects that
are found after the project is completed.
In this case we might expect large companies
with deep pockets and collateral to have an
advantage over small companies that can
more nimbly evade punishment.
. . . but settle up as soon as possible
Credit entries are financial assets.
Since debtors have an incentive to evade their
liability (through bankruptcy, flight or death), such
assets typically have a low or negative, pro-cyclical
rate of return.
Therefore non-banking institutions typically shun
long term credit positions with others, unless there
is new information about past performance that
should be incorporated into the contract.
For example retirement plans might include stock
options if the manager’s current decisions can
affect the stock price at some future date.
Managerial compensation
Managerial compensation comes in the form of:
1. Cash and bonus
2. Stock and option grants
3. Abnormal return on stocks and options
held by the manager
4. Pension and retirement benefits
5. Compensation for termination
A moral hazard problem
To illustrate the nature of optimal contracting
under moral hazard, we consider a wealth
maximizing group of shareholders who contract
with a risk averse CEO to manage their firm.
The CEO has 3 choices He can:
1. work for another firm (j = 0).
2. accept employment with the shareholders’
firm, but pursue his own interests rather
than theirs (j = 1).
3. accept employment with the shareholders’
firm, and pursue their interests (j = 2).
Manager’s preferences
Suppose the manager gets a utility of
w 1/2
from following the directions of his employer and
a utility of
w 1/2
from adopting a preferred managerial lifestyle to
his job. If
 1
the manager benefits from say ignoring
shareholder interests and doing his own thing.
Signals about the managerial effort
Suppose x is the abnormal return on the firm’s
equity, f2(x) is the probability density function of x if
the manager works “diligently”, and f1(x) is the
density if the manager “shirks”.
We define the likelihood ratio of f1(x) and f2(x) as
g(x) = f1(x)/f2(x) .
If the shareholder observe the realization x* then it
is more likely that the manager shirked than worked
diligently if g(x*) > 1 and vice versa.
Thus g(x) partitions x into sets whose elements are
signals about the manager’s diligence.
An example
Suppose
f2(x) = 1/2 for –1  x  1
f1(x) = ¾ for –1  x  0 and f1(x) = ¼ for 0  x  1
Then the signal only takes two values since
g(x) = 3/2 for –1  x  0 and g(x) = 1/2 for 0  x  1
In this case the optimal compensation to the manager is a
two tiered contract in which he receives a base wage for
all x and a bonus as well if 0  x  1.
Alternatively suppose f2(x) is a triangular increasing
density, but f1(x) is uniform. Then g(x) is a linear
decreasing function and each value of x might justify a
different level of compensation.
A binary signal
Suppose shareholders observe a signal about whether
the manager is diligently working for them or not.
Denote the signal by the variable s, and suppose it only
takes two values. Either s=1 or s=2.
When the manager works diligently, the probability that
s = 1 is p1, and the probability s = 2 is p2 = 1 – p1.
When the manager shirks, the probability that s = 1 is
q1 and the probability that s = 2 is p2 = 1 – q1.
We suppose that s = 1 is more likely if the manager
shirks, and s = 2 is more likely if the manager works
diligently. Thus p1< q1 and p2 > q2.
Shareholders objectives
We assume the objective of shareholders is minimize the
expected payments to the manager subject to the
constraints that he:
1. Chooses to work for them (called the
participation constraint)
2. Decides to work diligently rather than shirk
(called the incentive compatibility constraint)
When s = 1 they pay the manager w 1 and when s = 2
they pay him w 2 .
Thus shareholders minimize:
p 1 w 1 p 2 w 2
subject to his participation and incentive compatibility.
Participation
Let w 1 , w 2
denote the compensation in each state.
Suppose the manager could take a position with
another firm paying w 0 .
It is straightforward to demonstrate that at the
optimal contract the participation constraint is
satisfied with equality.
Then the participation constraint may be expressed
as:
1/2
p1w1
1/2
p 2 w 2
1/2
w 0
Is the signal redundant?
Notice there is no conflict of interest between
shareholders and the manager if  1
Consider the unconstrained optimum 
.
w 1 , w 2 
The solution to this problem can be found by
minimizing Lagrangian
p 1 w 1 p 2 w 2 
1/2
p1w1
1/2
p 2 w 2
1/2
w 0
where λ is the Lagrange multiplier.
The first order conditions are
1/2
2p 1  p 1 w 
1
1/2
2p 2  p 2 w 
2
This implies full insurance
w 1 w 2 w 0
Incentive compatibility
When the incentive compatibility constraint is binding
we can express it as
1/2
1/2
q 1 w 1/2

q

w
w
2
1
2
0
Noting that:
1/2
q 1 w 1
1/2
q 2 w 2
1/2
1/2
p 1 w 1 w 1
1/2
p 1 w 1
1/2
p 2 w 2
1/2
1/2
p 2 w 2 w 2
1/2
1/2

q 1 p 1 
 w 1 w 2
incentive compatibility requires the expression to be
negative. Hence w 1 w 2 .
Thus we interpret w 1 as the base pay and w 2 w 1
as the bonus.
When is shirking inevitable?
If p 2  q 2 then:
1/2
p1w1
1/2
p 2 w 2
1/2
q 1 w 1
1/2
q 2 w 2
for all w 1 , w 2  and we cannot meet the incentive
compatibility constraint.
In this case incentives cannot be used to motivate the
manager. He will shirk regardless of the contract.
The optimal contract is then found by minimizing
compensation subject to the participation constraint.
The solution is to set w
1/2
2
w 1/2
0 or w   w 0 .
Optimal compensation
The only other case is that
1  
.
p2
q2
In this case both the participation and incentive
compatibility constraints are met with equality.
We can find the optimal contract by solving the two
equations in the two unknowns w 1 , w 2 to obtain:
w1 
q 2 p 2 /
p 1 q 2 q 1 p 2
2
w2 
q 1 p 1 /
2
p 2 q 1 p 
2 p1
w0
w0
Illustrating the optimal contract
Lecture Summary
Private information and outside options available to
agents working for principals are captured through the
truth telling, incentive compatibility and participation
constraints.
These constraints help determine the shape of the
contract but limit its value. The more attractive the
outside alternative to the agent, the better informed he
is about the project relative to the principal, the harder
it is to monitor the agent’s activities, then the lower the
value of the contract to the principal.
However ignoring these constraints is even more costly
to the principal, because the agent may reject the
contract, misinform the principal about the business
situation, or not pursue the firm’s interests.