Personnel EconomicsExercise 3Exercise 3: The Basic Agency Problem1. Stefan has just been hired as a consultant to advise...

Personnel Economics

Exercise 3

Exercise 3: The Basic Agency Problem
1. Stefan has just been hired as a consultant to advise the Campus Construction
company on how to cut its costs. The compensation scheme offered by CC Inc. is
such that Stephan’s pay depends on his output, i.e. on the total dollars saved by CC
after following Stephan’s advice. Stephan’s output, Q, (in dollars) depends on the
amount of effort he exerts, E, in a very simple way: Q = E. Stephan’s pay, Y, is a
linear function of his output, i.e. Y = a + bE . In other words, Stephan gets paid a just
for taking the assignment, and in addition is paid b dollars for every dollar he saves
CC Inc. Thus, b is his commission rate.
Stephan’s utility (U) depends on two things, income and effort. In particular, U = Y –
C(E), where C(E) is the disutility of effort. For the purposes of this question, assume
C(E) = E3/3.
a) Graph Stephan’s disutility of effort curve, C(E). Graph his marginal disutility of
effort curve, dC/dE. Does he exhibit increasing or decreasing marginal disutility of
effort?
b) Assuming Stephan is a utility maximizer, derive his utility-maximizing effort (and
output) level as a function of the parameters of his pay scheme, a and b. How much
effort will he choose if b =.36? If b = 1? Does Stephan’s choice depend on the level
of a?
c) Illustrate your answer to part (i) graphically. In one diagram, show Stephan’s total
levels of income (Y) and disutility from effort (C(E)). Directly below this, show his
optimal choice using marginal benefit and cost curves.
2. Now assume that CC Inc. is hiring Marie as a consultant to computerize its office
operations. Everything is the same in Marie’s situation as in Stephan’s except that
her disutility of effort is different. Specifically, C(E)=E2/10.
a) How much effort will Marie exert if her commission rate is 20%? 100%? Does effort
depend on a?
b)

Now suppose that Marie and CC are negotiating the terms of her contract (i.e. the
level of a and b) before she starts. Before they settle on who gets exactly what in the
end, they agree that, whatever contract they ultimately choose, it must be efficient, i.e.
it must maximize the sum of CC’s profits and Marie’s utility. If the contract is
efficient, what will be the level of b? Comment.

c) Suppose now that, to retain Marie, CC Inc. must offer her a utility level of at least 1
unit (because this is what she can get working for another firm). What must a be? If
Marie’s alternative utility is 5 units, will she work for CC Inc?

1

Personnel Economics

Exercise 3

3. This simple agency problem is designed almost entirely for a spreadsheet. Below you
will see a printout of a spreadsheet with mostly empty cells. Your goal is to fill in the
blank cells. You will be thinking of this firm as experimenting with different values
of the commission rate, b, ranging from zero to 1.5, as shown in the first column of
the table.
Assume:
Effort = E
Output = Q = dE (d is a “productivity parameter”; higher levels of d correspond to
higher productivity)
Disutility of Effort = C(E) = E2
Agent’s reward = Y = a + bQ
Agent’s Utility = Y – C(E)
In your “base” calculation, let d=20. (But allow the spreadsheet to input alternative
values for d).
a) Given the different values of b in column one, fill in the agent’s choices of effort, his
output, and his total commission income in the next three columns. (By commission
income we mean only the part of his income that depends on how much he produces,
i.e. bQ).
b) Suppose that, to get the agent to work at this firm at all, he must attain an overall
utility level of 20 (this is what he can get at another firm). For each row of the table
(i.e. for each different value of b under consideration), calculate the level of a the firm
must offer the agent to induce him to come to the firm. Hint: remember that the
agent’s total income must be at least 20 because he gets disutility from effort too.

Commission
Rate (b)
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

Agent’s
Effort

Output

Commission
Income

2

a

U

Worker’s
Income

Profit

Personnel Economics

Exercise 3

c) Fill in the remaining columns in the table (in the “utility” column, use the definition
of utility (Y – C(E)) to check that you set a in just the right way to keep the agent’s
utility at 20). Noting that the worker is indifferent between all the rows of the table
(all the possible levels of b), which level of b yields the highest profit to the firm?
Why?
d) Does a profit-maximizing firm select the compensation plan in which income of the
agent is the lowest? Does the agent always prefer those schemes that give him the
highest income? Why or why not?
4. This question is identical to Question 3 with one exception. We now introduce
uncertainty and risk aversion, and aim to show the effects of different levels of risk on
the optimal levels of a and b, and the corresponding profit maximizing levels of Q.
We now suppose that, when the agent exerts effort, he can no longer be sure of the
outcome. In particular, when he exerts effort E, he produces an output of dE + k/2
with probability one half, and an output of dE – k/2 with probability one half. Thus k
indexes the amount of output uncertainty. The firm (principal) is risk neutral and
cares only about the expected value of the profit it earns. We introduce risk aversion
for the agent, however, in the following simple way: utility equals expected income,
minus the disutility of effort, minus a “risk aversion” term that depends on the
difference between his income in the “good” state (when he produces dE + k/2) and
his income in the “bad” state (when he produces dE + k/2). If we call this income
difference ?, his utility is given by: U = [Y - r?2] - C(E). Think of r as an index of
how “risk averse” the agent is (if r = 0 he is risk neutral and the problem is identical
to number 3 above).
a) Assuming the agent is still compensated according to the output he produces, via the
reward schedule Y = a + bQ, derive an expression for his utility as a function of a, b,
d, E, r and k.
b) Please fill in the spreadsheet below in Excel. Allow for inputtable values for k and r,
but set k = 10 and r = 0.4 in your “ baseline” case. (Keep d = 20 and alternative
utility = 20 as in question 3).
c) What is the profit-maximizing commission rate now (in your “baseline” case)?
Compare this to question 3 and explain the difference.
d) How does the optimal commission rate change when output uncertainty (k) rises to 20
units (i.e. when there is now a difference of 20 units of output between the “good”
and “bad” states)? Under the optimal commission rate when k = 20, what is the
agent’s fixed income, a? Is it positive or negative? Why?
e) Now put k back to its original value of 10, and see what happens when the agent’s
aversion to risk parameter, r, rises from .4 to 1. What is the optimal commission rate
now? Explain.

3

Personnel Economics

Commission
Rate (b)

Agent’s
Effort

Exercise 3

Output

Commission
Income

Uncertainty
Cost
r(bk)2

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

4

a

U

Worker’s
Income

Profit