ECON0029 Problem Set 4
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ECON0029 Problem Set 4
This problem set is due on Sunday 11 November 2021
For all problems: You may use all results and observations that were discussed in the lecture. You should not replicate the derivations unless the problems explicitly asks for it!
1. (Lecture 3) Consider a model with 3 possible effort levels e1 , e2 , and e3 , and two possible
outcomes xH = 10 and xL = 0. The probabilities of the high outcome conditional on the
effort levels are pH(1) = 2/3, pH(2) = 1/2, and pH(3) = 1/3. The effort cost function is given
by v(e1 ) = 5/3, v(e2 ) = 8/5, and v(e3 ) = 4/3. Finally, U = 0, U (w) = ^w, and the
principal is risk-neutral.
What is the optimal contract if effort is not verifiable? (Hint: you cannot just use the formulas from the lecture because now there are three effort levels. To solve the problem, start with the following steps. (Some more are required to complete the analysis.)
(a) Identify the effort level that can be implemented with a constant wage.
(b) For the other effort levels, formulate the principal’s problem. The difference to the
lecture is in the IC constraints!
(c) Set up the Lagrangian. Show first that the IR and at least one IC constraint must be binding.
(d) Derive solutions from the binding IR and one binding IC constraint. Check if the other constraint is satisfied.)
2. (Lecture 4) Suppose a firm can achieve high or low (gross) profit, denoted by x1 < x2 . The CEO can exert effort e = 1 or “shirk” e = 0. The effort is not contractible. The profit is affected by the effort as well as general market conditions denoted by m, which can be “good” m = 1 or “bad” m = 0. The probability that market conditions are good is 1/2. The probability of achieving high profit as a function of effort and market conditions is given by
V 5(3)
Suppose that the compensation of the CEO can be contingent on both x and m. That is, both profits and market conditions are contractible. Assume that the CEO is risk-averse and the owner of the company is risk-neutral.
(a) Show that in order to implement high effort, it is optimal to make the CEO’s wage
dependent on both the profit and market conditions. (Do not solve for an optimal contract for this question!)
(b) What is the difference to the example (Ice cream in Hyde Park) in the lecture? How
is this reflected in the optimal wage schedule?
From now on, assume that U (x) = ^x , U = 1.4, v(0) = 0, v(1) = 0.15
(c) Consider the optimal contract under the restriction that the wage does not depend on market conditions. Verify that it is optimal to use w1 = 1 and w2 = 4.
(d) Now consider the problem of finding the optimal wage schedule if the wage can depend on market conditions. Let wxm denote the wage paid if the profit is equal to x and market conditions are equal to m.
i. State a system of 6 equations that determine the optimal wages.
ii. The solution is:
λ ≈ 3.1
µ ≈ 2.97521
w10 ≈ 0.912
w11 ≈ 1.388
w20 ≈ 4.601
w21 ≈ 3.233
Verify that this solution solves the 6 equations you specified in (i).
iii. Suppose that x2 is high enough so that it is optimal to implement e1 in part (c). Verify that the net profit of the firm is higher if it uses a wage that is contingent on market outcomes.
(e) Harder: Suppose now that market conditions are observed by the CEO and the firm
before she accepts the contract.
i. Check if the wage schedule from (d) still implements e1 . Think carefully about the constraints you need to check given that market conditions are observable for both parties before the contract is proposed by the principal!
ii. Argue verbally that it may be optimal to implement different effort levels de- pending on the market conditions. Under which market conditions would you expect the optimal effort to be higher in this case? (Hint: It is helpful to review the answer to problem 1 on problem set 2 for this question.)
2022-12-16