Problem Set 7
Economics 703
Spring 2016
Due: Friday March 25
1. Determine which of the equilibria you identified on questions 3 and 4 of Problem Set
6 survive the Cho–Kreps intuitive criterion.
2. Consider the moral hazard problem with three possible effort levels E = {e1 , e2 , e3 }
and two possible profit levels, πH = 10 and πL = 0. Let the probability of πH given e be:



2/3, if e = e1
f (πH | e) =  1/2, if e = e2

1/3, if e = e3
The agent’s cost function for effort is



5/3, if e = e1
g(e) =  8/5, if e = e2

4/3, if e = e3
The agent’s utility of wealth function is v(w) =
√
w. His reservation utility ū is zero.
(a) What is the optimal contract when effort is observable?
(b) Show that if effort is not observable, then it is not possible to induce effort level
e2 . (Notational suggestion: Let v1 = v(w(πH )) and v2 = v(w(πL )).) For what values of
g(e2 ) can the principal induce effort level e2 ?
(c) What is the optimal contract when effort is not observable?
3. Suppose both the principal and agent are risk neutral. The agent has two possible
levels of effort, eH and eL . There is a probability that the project the agent is hired to
carry out succeeds or fails where this probability depends on the agent’s effort. More
specifically, if the agent chooses effort eH , the project succeeds with probability 1. If he
chooses eL , the project succeeds with probability p ∈ (0, 1) and fails otherwise. If the
project succeeds, then profits are πS . If it fails, profits are πF where πS > πF ≥ 0.
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The agent’s payoff is w − c if he chooses eH and is paid w, while his payoff is w if he
chooses eL and is paid w. The principal’s payoff is π − w if profits are π and he pays the
agent w. The agent’s outside option is 0.
(a) Find an optimal contract for the principal.
(b) Now assume there is a limited liability constraint. More specifically, the principal
must always pay the agent at least 0 regardless of whether the project succeeds or fails.
Now find an optimal contract for the principal.
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