Problem Set 8
Economics 703
Spring 2016
Due: Friday April 1
1. Consider a principal who earns profits from two different tasks undertaken by the
agent. His profits from task 1, denoted π1 , depend randomly on the level of effort
undertaken by the agent in task 1. More specifically, if the agent puts high effort eH into
task 1, then
(
A with probability 1/2
π1 =
0 otherwise.
If the agent puts low effort (eL ) into task 1, then π1 = 0 with probability 1. The
principal’s profits from task 2, π2 , are a nonstochastic function of the agent’s effort on
task 2. More specifically, if the agent puts high effort into task 2, then π2 = B, while
if he puts in low effort, π2 = C where B > C. The agent can only put high effort into
at most one task. That is, the only possibilities are low effort on both tasks, eH on task
1 and eL on task 2, or eH on task 2 and eL on √
task 1. The agent’s payoff if he is paid
w and puts high effort into one of the √
tasks is w − g, while his payoff if he puts low
effort into both tasks and is paid w is w. The agent’s payoff if he does not work for
this principal is 0. The principal is risk neutral.
Finally, assume that A/2 > B − C > g 2 . In other words, the expected marginal value
of high effort on task 1 is greater than the marginal value of high effort on task 2 and
both are greater than the utility cost of high effort.
(a) What is the first–best contract?
(b) Now suppose that the principal observes both π1 and π2 but not the agent’s
effort choice. Find the cost–minimizing contracts for inducing each possible effort choice.
Under what conditions is the effort choice changed from the first–best?
2. A principal hires an agent to work for him. The agent can choose either high effort,
denoted eH , at a cost to him of c > 0 or low effort, denoted eL , at a cost of 0. Effort is
not observable by the principal. If the agent chooses high effort, the principal will receive
revenues of πH with probability p ∈ (0, 1) and πL otherwise. If the agent chooses low
1
effort, the principal receives πL with probability
1. The agent’s reservation utility is 0.
√
The agent’s utility if he receives wage w is w minus c if he chooses high effort. Assume
πH − π L >
c2
.
p2
(a) Find the optimal contract for the principal to offer.
(b) Now suppose that there is a probability q that the principal observes the agent’s
effort. More specifically, if the agent chooses high effort, then with probability q ∈ (0, 1),
the principal observes this. If the agent chooses low effort, the principal observes nothing.
Suppose also that the principal is not allowed to base the payment he makes to the worker
on his revenue level, but can only base it on whether or not he observes the worker
choosing high effort. Find the optimal contract for the principal to offer. Under what
condition is the principal better off in (b) than in (a)?
3. Consider the following three type version of the adverse selection model. The three
types are θL , θM , and θH where θL < θM < θH and the marginal cost of effort is strictly
decreasing in θ. Determine which individual rationality and incentive compatibility constraints are binding. Characterize the optimal contract, giving first–order conditions
where an explicit solution cannot be computed in general.
4. Suppose we modify the adverse selection model discussed in class by allowing the
principal to not contract with the agent. That is, suppose that instead of requiring the
principal to satisfy the individual rationality constraint for both types, he can satisfy it
for only one type and earn zero profits if it turns out to be the other type. Assume ū = 0
and v(0) = 0.
(a) Show that under these assumptions plus the ones we used in class, this does not
alter our characterization of optimal contracts.
(b) Change our assumptions so that there is a “fixed cost” for effort. More specifically,
now assume that g(0, θL ) > 0. Can this change our characterization of optimal contracts?
Why or why not? What happens if g(0, θL ) = 0 but v −1 (ū) > 0?
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