Economics of Corporations (ECON 3123)

Problem Set 1

This is due on August 15 by midnight.

Question 1

In this problem, we will consider a multitasking problem where the principal can only incentivizethe agent on one task.

There are two tasks (task 1 and task 2). The principal benets from the agent's eort inboth tasks: Up = x1 + x2 w where x1 = e1 and x2 = e2. The agent has payo functionUa = w 12(e21 + e22 + e1e2). Notice that the two tasks are substitutes in the agent's eort costfunction.

(Note that the agent may choose negative eort levels, potentially resulting in negative output.)The principal cannot reward the agent for total output; instead, he can only reward the agent

for his performance in the rst task. That is, the principal can oer the agent an incentive schemeof the form w = + x1.

The timing is as usual:

1. Principal oers agent an incentive scheme w = + x1.

2. Agent may accept or reject the oer. If he rejects, he recieves his outside option U0.

3. If agent accepts, then he chooses e1 and e2.

We'll go through the problem step-by-step.

a) For step 3, given the principal's oer (w = + x1), write down the agent's maximizationproblem, and calculate his payo-maximizing eort choices e1 and e2 as a fucntion of and.

b) For step 1, write down the principal's maximization problem, and calculate his payo-maximizing choice of incentive scheme ( and ).

c) What eort levels does this incentive scheme induce in the agent?

d) Calculate the ecient eort levels (i.e. the eort levels e1; e2 that maximize total payosUp + Ua).

e) Explain, in words, why your answers to (c) and (d) dier.

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f) Suppose that Vodaphone decides to pay its customer-service agents a bonus for quick res-olution of customer-service calls. What does the above model predict would happen? (Ingeneral, for open-ended questions in the problem sets, there is no correct answer; the key isinterpretation.)

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Question 2

In this question, we'll work through the model with noisy output, with the following twist: theprincipal is risk-averse, while the agent is risk-neutral.

The principal's payo function is Up = E[x w] 12V ar[x w] with x = e + where E[ =0]; V ar[] = 2. The agent's payo function is Ua = E[w] 12e2. The principal can oer the agentan incentive scheme of the form w = + x.

The timing is as usual:

1. Principal oers agent an incentive scheme w = + x.

2. Agent may accept or reject the oer. If he rejects, he recieves his outside option U0.

3. If agent accepts, then he chooses e.

We'll go through the problem step-by-step.

a) For step 3, given the principal's oer (w = + x), write down the agent's maximizationproblem, and calculate his payo-maximizing eort choice e as a function of and .

b) For step 1, write down the principal's maximization problem, and calculate his payo-maximizing choice of incentive scheme ( and ).

c) What eort levels does this incentive scheme induce in the agent?

d) Calculate the ecient eort levels (i.e. the eort level e that maximizes total payos Up+Ua).Verify that the ecient outcome is achieved regardless of the principal's risk aversion ().

e) Why doesn't the principal's risk aversion result in any ineciency (whereas, in the modelwhere the agent is risk-averse, the outcome is inecient)?

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