# A principal is hiring an agent to do a job. The agent produces output, Q, by expending effort

November 24, 2016
• A principal is hiring an agent to do a job. The agent produces output, Q, by expending effort, E, via the production function Q = dE. The agent’s utility is given by: U = Y –E3 /3, where Y is the agent’s income. The principal pays the agent according to the reward schedule Y = a + bQ.

Note: dU / dE = bd – E2

A) (Do this before you set up the spreadsheet). Derive an expression for the agent’s utility-maximizing effort level as a function of b and d. (Hint: the excel function for the square root of x is sqrt(x)).

B) Fill in a spreadsheet with the following rows and columns. (Think of the b’s as alternative hypothetical values of the commission rate the principal is thinking of adopting). Allow for inputable values of a and d.

Piece Rate (b) Agent’s Effort Output Commission Income Agent’s Utility Profit Utility +Profit

————————————————————————————————————————————————————-

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

C) Suppose that a is fixed at zero (the principal neither “charges” the agent for the job nor pays him for just showing up), and that d = 30. What is the profit maximizing commission rate (to the nearest 10%)? What is the level of profits and of the agent’s utility?

D) What is the “economically-efficient” commission rate (the one that maximizes the sum of profits plus worker utility). Why does this differ from your answer in part c? How much profit does the firm earn if it chooses the economically efficient commission rate?

E) Now assume the agent is much more productive than in parts c and d: Raise d from 30 to 90 (a productivity increase of 300%). How do the profit-maximizing and efficient commission rates change? Explain.

F) Put d back to its “baseline” level of 30, used in part c. Using your spreadsheet, come up with a combination of a and b that make both the principal and the agent Personnel Economics Midterm 2015: Part I 4 better off than they were in part c (i.e. when the firm maximizes profits subject to a = 0). What are the levels of profits and utility after your Pareto-improving (“win-win”) proposal? Explain why your proposal works.

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