ECON 3123: PROBLEM SET 1

PROFESSOR RICHARD HOLDEN

This is due at the start of lecture in Week 4. Late problem sets receive zero credit.

(1) Consider a setting with a risk-averse worker and a risk-neutral firm. The

worker puts in some effort level a which the firm does not observe. The

worker’s cost of effort is C(a) = ca2. Output is given by y = ka + ε, where

ε is a random variable with E(ε) = 0 and V ar(ε) > 0. Firms make workers

a take-it-or-leave-it wage offer of the form w = s + by. The firm maximizes

its expected net profit E(y − w) and the worker maximizes her certainty

equivalent given by CE = E(w) − r2V ar(w) − ca2, where r is a measure of

risk-aversion. The worker’s outside option is 0.

(a) Compute E(y), E(w), and V ar(w).

(b) Set up the maximization problem for the firm–i.e. write down the ob-

jective function, the choice variables and the constraints.

(c) Compute the efficient level of effort. Hint: add the firm’s expected profit

and the worker’s CE and maximize w.r.t. effort.

(d) Compute the firm’s optimal choice of s, b and a when c = 1, r = 1, k =

1, V ar(w) = 1.

(e) Compute the firm’s optimal choice of s, b and a when c = 1, r = 1, k =

1, V ar(w) = 2.

(f) Are the effort levels from the two above computations efficient. If not,

explain briefly why not.

1

2 PROFESSOR RICHARD HOLDEN

(2) Consider a setting where there are distortions with a risk-averse worker and

a risk-neutral firm. The firm’s (non contractible) benefit function is given

by y = f1a1 + f2a2 + εy, where aj is the worker’s action/effort on task j, for

j = 1, 2. There is another performance measure (which can be contracted on)

given by p = g1a1 + g2a2 + εp. Assume the firm offers a linear contract of

the form w = s + bp. Assume that f1, f2, g1, g2 are non-negative and that

E(εy) = E(εp) = 0. The worker’s cost of effort isa212

+a222

and her outside

option provides utility of u0. The firm maximizes expected profits and the

worker maximizes her certainty equivalent of E(w) − r2V ar(w) − a21

2+

a222.

(a) Set up the firm’s constrained optimization problem.

(b) Find the optimal level of incentives b. Hint: you may want to make use

of the formula f1g1 + f2g2 =√f 21 + f 2

2

√g21 + g22cosθ.

(c) Use the model above (using the parameters of the problem) to discuss

the following. There is no correct answer, the key is interpretation.

(i) A loan officer is typically paid based on the volume of loans that

she originates (not the performance of those loans), whereas a con-

struction manager is only paid on the successful completion of a

project. What reason would the above model offer for these differ-

ent incentive contracts?

(ii) Consider a hierarchy with a CEO on top, the division manager be-

low her and a plant manager below the division manager. Various

performance measures can be used for a plant manager of a pub-

licly traded firm. The performance measures are the stock price,

firm wide accounting profits, divisional profits, plant level profits,

and plant level costs. What is the main tradeoff shareholders face

in designing an incentive scheme for the plant manager?