Microeconomics 2nd Edition by Douglas Bernheim

Exercise 2

Repeat calculus Worked-Out Problem 21.3 (page 782) assuming instead that the salesperson's personal cost is C ( H ) = 1,000 + 2 H 2.
Worked-Out Problem 21.3
The Problem A salesperson works for a car dealership for 40 hours per week, but may choose not to work hard all of the time. The dealership's owner cannot observe the salesperson's effort, but can observe the number of cars sold. The salesperson's personal cost of working at the dealership is C ( H ) = 1,000 + H 2 , where H is the number of hours during which he works hard. The corresponding marginal cost of effort is MC = 2 H. Without any effort, the salesperson will, on average, generate a profit of $800. With each hour of high effort, he has a 4 percent chance of selling a car. Each car sale generates a profit of $1,000. What is the efficient number of hours of effort How much surplus does the relationship generate
Suppose the owner gives the salesperson $1,200 in base pay plus a bonus of $500 for each car he sells. How hard will the salesperson work Describe an incentive scheme that leads to the efficient effort level and allows the owner to keep all of the surplus.

The Solution We first derive the dealership's benefit function. Each hour of hard work has a four percent chance of generating $1,000 in profit, so it produces $40 in profit on average. The benefit function is therefore B ( H ) = 800 + 40 H and the marginal benefit is MB = 40.
The efficient number of hours of high effort equates the marginal benefit and marginal cost of effort (provided this effort level results in a nonnegative net benefit). Setting MB = MC, we have 40 = 2 H. The solution is H = 20, which generates a net benefit of [800 + (40)(20)] (1,000 + 20 2 ) = $200.
If the salesperson faces an incentive scheme with a base pay of $1,200 and a bonus of $500 for each car he sells, then each hour of effort yields, on average, $20 in extra compensation. The salesperson's expected earnings are E ( H ) = 1,200 + 20 H and his marginal earnings (marginal benefits) are ME = 20. His best choice equates this marginal benefit and marginal cost (provided his net benefit is positive). Setting ME = MC, we have 20 = 2 H, which implies H = 10. His net benefit is $[(1,200 + (20)(10)) (1,000 + 10 2 )] = $300.
For the salesperson to choose the efficient number of hours, he must receive all the benefits and absorb all the costs of his actions on the margin. Thus, his marginal benefit at 20 hours must be $40. Consider a linear incentive scheme with base pay of K and a bonus of $1,000 per car (the full profit). In that case, the salesperson's marginal benefit from an hour of effort will be $40 on average. To ensure that the owner keeps all of the surplus, we set K so that the salesperson's net benefit when he works hard for 20 hours is zero (his total earnings equals his total cost):
K + (40)(20) = 1,000 + 20 2
The solution is K = 600.