BUS-G304 Managerial Economics
Problem Set #5 Solutions
Part 1 (10 points)
1. Suppose the marginal cost of writing a contract of length L is MC(L) = 10 + 2L. Find the optimal
contract length when the marginal benefit of writing a contract is:
a. MB(L) = 100
When the MB = 100, the optimal contract length is 45.
b. MB(L) = 150
When the MB = 150, the optimal contract length is 70.
c. What happens to the optimal contract length when the marginal benefit of writing a
Increases in the marginal benefit to writing a contract increase the contract length.
Decreases in the marginal benefit to writing a contract decrease the contract length.
2. Suppose a firm’s production is tied to effort by Q = 2e + x, where Q is quantity produced, e is
effort, and x is a random variable with mean 0 and variance 4. Also assume that the market
price is 10. Finally, suppose this firm hires one worker at a wage of a + b*Q. The worker’s cost
of effort is C(e) = e and her coefficient of risk aversion is 2. The worker’s outside option has
a. If effort is observable, what piece rate (b) should the firm offer?
The firm should only offer a fixed wage in order to avoid exposing the worker to risk, so b =
b. If effort is observable, what wage should the firm offer?
The wage just needs to make the worker willing to accept the contract, so we need U = a +
2 2 2 2
2b*e – e – (1/2)rbs = 0, and if b = 0, then we just need a = e . To determine what this
number will be, we need to know the firm’s optimal choice of effort for the contract. This
will be the effort level that maximizes average profits = 10*2e – a – b*2e = 20e – e . This is
maximized where e = 10, so the wage should be 100.
c. If effort is unobservable, solve for:
i. The optimal wage
P P P 2
For a given level of effort, e , the optimal wage solves U = a + 2b*e – (e ) –
2 2 2 P 2
(1/2)rbs = 0. Plugging in for b, r, and s , we have a = 3(e ) . So, a = 12.ii. The…