Question Question 1. A firm needs to hire an employee to

04 Jun Question Question 1. A firm needs to hire an employee to

Question

Question 1. A firm needs to hire an employee to complete a project. The

project may be successful or not depending on how hard the employee works.

Suppose there are two e?ort levels the employee can choose from, high e?ort

(e = eH) and low e?ort (e = eL), if she chooses eH the project is successful

with probability 0.8, while if she chooses eL the project is successful with probability

0.4. A successful project yields revenue of x = 2500 for the firm and an

unsuccessful project yields x = 0.

The employer cannot observe the employee’s e?ort choice, but can only observe

whether the project is successful or not. Therefore, the employer can only base

the wage w on the success of the project, i.e. w = w(x).

The firm (employer) is risk neutral, with profit equal to x ! w(x), and since the

success of the project is random, the expected profit of the firm is E[x ! w(x)].

The employee is risk averse with utility equal to pw(x) ! c(e), and again since

the success of the project is random, the expected utility of the employee is

E[

pw(x)]!c(e). Exerting high e?ort is costly, in particular, suppose c(eH) = 16

and c(eL) = 0.

(a) If the wage is w(x) = x, i.e. the employer pays the employee all the revenue

the project brings along, what e?ort level would the employee choose?

(b) Suppose the employee has no outside opportunity, so that her reservation

utility is 0. What is the optimal incentive contract the firm will provide?

(c) Suppose the employee has an outside o?er that guarantees her a reservation

utility of 20, what is the optimal incentive contract the firm will provide?

Question 2. Firm A is the sole producer of a sport drink. A’s marginal cost

equals average cost MC = AC = 30, and it faces market demand given by

inverse demand function P = 120 ! 0.5Q.

(a) Suppose A produces quantity q = 120 units at price p = 60. Is there any

dead weight loss at current price and quantity? If yes, how much is the DWL?

(b) What is the monopoly price? What is the monopoly DWL?

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Question 3. The inverse market demand for mineral water is P = 200 ! 10Q,

where Q is total market output and P is the market price. Two firms have

complete control of the supply of mineral water and both have zero costs.

(a) Find the Cournot quantity, price, and each firm’s profit.

(b) Denote the Cournot quantity for each firm by qa, and denote half of the

monopoly quantity by qb. Suppose that the two firms interact with each other

for infinite periods, and in each period they choose quantities simultaneously.

Consider the following collusive strategy, the same as discussed in class: produce

qb only if no one has cheated so far, and to produce qa forever if some

has cheated before. Assume each firm acts to maximize its sum of discounted

profits where the interest rate is r. For what values of r can such collusion be

sustained?

(c) Find the Bertrand price, quantity, and each firm’s profit.

(d) Denote the Bertrand price by pa, and denote the monopoly price by pb.

Suppose that the two firms interact with each other for infinite periods, and

in each period they set prices simultaneously. Consider the following collusive

strategy, the same as discussed in class: set price pb only if no one has cheated

so far, and to set price pa forever if some has cheated before. Assume each firm

acts to maximize its sum of discounted profits where the interest rate is r. For

what values of r can such collusion be sustained? Compare your answer to (b)

and explain.

Question 4. Two firms produce candies that are imperfect substitutes. This

is reflected in the demand curves of the two firms’ candies, D1(p1, p2) = 100 !

p1 + 0.5p2 and D2(p1, p2) = 100 ! p2 + 0.5p1. Suppose each firm has constant

marginal cost of 20.

(a) Interpret the demand curves, in particular, explain from the demand curves

why the candies produced by the two firms are imperfect substitutes.

(b) Suppose the two firms compete by making simultaneous price decisions.

Calculate the equilibrium price, quantity and profit for each firm.

(c) Suppose the two firms compete by making sequential price decisions, where

firm 1 is the leader. Calculate the equilibrium price, quantity and profit for

each firm. Compare the profits of the leader vs. the follower.