Ralph’s Limited Liability
Ralph wishes to employ a manager who, like her, is risk neutral. So incentives are a
snap. Help her demonstrate this by completing the following exercise.
Ralph’s manager has two options; H or L. H will provide greater gross benefit to
Ralph than will L. If the manager chooses H, the probability of low “performance”
(x1) is .1 and the probability of high “performance” (x2) is .9. However, if he chooses
L the probability of x1 is .3, and the probability of x2 is .7. As you can see,
performance is an imperfect source of information about the manager’s input.
In addition to being risk neutral, the expected utility maximizing manager is not
indifferent between the two actions, captured by the utility function I – ci, where I is
the payment received for Ralph and ci is the “private cost” to the manager of action i,
i = H, L. Naturally, we want cH > cL; let cH = 40 and cL = 20. Finally, the manager is
willing to accept employment with Ralph if he receives an expected utility of U = 100
or greater.
1. Initially assume Ralph can observe the manager’s action, and so can may
payment depend on the action (so obvious notation to use for the payments is IL
and IH). Find the optimal contract that induces the manager to accept
employment with Ralph and prefer H to L, and make a note of the expected
compensation paid to the manager
2. Now assume Ralph cannot observe the manager’s action, so she may only use the
performance measure (so obvious notation to use for the payments is I1 and I2).
Find the optimal contract that induces the manager to accept employment with
Ralph and prefer H to L.
3. Now solve for an optimal contract as in 1 and 2 if U = 10. Compare the solutions
under each value of U. Notice which constraints (participation and incentive
compatibility) are tight.
4. Rework part 3, but instead assume the manager’s payments cannot be negative.
Again consider the cases where U is low and high. Notice which constraints are
tight now. Explain what you see.