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The problem of a principal who wants to maximize profits by hiring workers with two different effort levels (el and eh) under full information and individual rationality constraints. The document derives the optimal wage contract for each effort level and shows that it consists of a fixed wage. The principal then selects the optimal effort level and offers a compensation schedule to guarantee the worker's choice.
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Suppose that effort is observable and verifiable, so that a contract can be conditioned on effort.
The principal’s problem is:
max w(π),e∈{eL ,eH }
Z^ π π
[π − w(π)] · f(π | e)dπ
s.t.
Z^ π π
[v(w(π)) · f(π | e)dπ] − g(e) ≥ u.
(Individual Rationality (IR))
(Only have IR constraint, since principal can impose an arbitrarily large penalty for taking the other effort level.)
Solution:
First, for eL and eH , find the contract that maximizes profits s.t. the IR constraint for that effort level. For effort level e :
max w(π|e)
Z^ π π
[π − w(π | e)] · f(π | e)dπ
Since mean profits are indepedent of w(π | e), this is the same as:
min w(π|e)
Z^ π π
w(π | e) · f(π | e)dπ
s.t. Zπ π
[v(w(π | e)) · f (π | e)dπ] − g(e) ≥ u.
(Individual Rationality (IR))
We have a “fixed wage contract” w∗(π | e) = w∗ e , where w e∗ is solves:
v(w∗ e ) − g(e) = u; ie., w∗ e = v−^1 [u + g(e)].
The risk-neutral principal absorbs all of the risk of the uncertain profits, and we have the first-best level of risk sharing.
We will have w e∗ (^) H > w∗ eL, since g(eH ) > g(eL).
Second, select the optimal effort level from the point of view of the principal, e∗, by solving:
max e∈{eL ,eH }
Z^ π π
π · f (π | e)dπ − w∗ e.
Once the principal determines e∗, it can offer the fol- lowing compensation schedule to the agent:
w e∗ for e = e∗
−∞ for e 6 = e∗.
Threrefore, the principal guarantees that the worker will choose the wage w∗ e for e = e∗.
