# Problem Set 2

Problem Set 2

1. A risk neutral principal (she) owns a Örm and hires a manager (him) to run the business. The principal cares about Örmís proÖts net of managerís compensation. The managerís utility is e r(w ma2), where w is managerís compensation, a is his level of e§ort, and r and m are positive constants. Man- agerís reservation wage equals w. Firmís proÖts are given by a + , where is a random shock which is normally distributed with mean 0 and variance 2. The principal cannot observe managerís e§ort and is restricted to linear wages.
(a) Write the principalís problem using managerís certainty equivalents
(b) Find whether managerís pay-performance sensitivity increases or de- creases with each of these variables (w, r, m, and 2) and explain the intuition.
2. A risk-neutral principal (she) hires an agent (him) to work on two projects with observable payo§s. The principal cares about the sum of the payo§s from the projects net of managerís compensation. The managerís utility is e 2(w g(a1;a2)), where w is managerís compensation, a1 is his level of e§ort on project 1, a2 is his level of e§ort on project 2, and g(a1;a2) is his monetary cost of e§ort. Managerís reservation wage equals 0. The payo§ from project i is given by ai + i, where i is a random shock which is normally distributed with mean 0 and variance 2i . The shocks 1 and 2 are independent. The principal cannot observe managerís e§ort on either project and is restricted to linear wages. Hereafter, assume that the Örst order approach is valid.
(a)Letg(a1;a2)=k1a21+k2a2,wherek1 andk2 aresomepositiveconstants. How would the sensitivity of managerís pay to the payo§ of the Örst project depend on the variance of the shock to the second project? Explain.
(b) Assume that measuring the payo§ from the second project is close to impossible, i.e., 2 ! 1. Let g(a1; a2) = a21 ka1a2 + a2, where k is a positive constant.
(i) How would the sensitivity of the managerís pay to the payo§ of the Örst project change if k increases? Explain.
(ii) How would the sensitivity of the managerís pay to the payo§ of the second project change if k increases? Explain.
(c) Assume that measuring the payo§ from the second project is close to impossible, i.e., 2 ! 1. Let g(a1; a2) = k (a1 + a2)2, where k is a positive constant.
(i) How would the sensitivity of the managerís pay to the payo§ of the Örst project change if k increases?
(ii) How would the sensitivity of the managerís pay to the payo§ of the second project change if k increases?
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3. Explain why an insurer may Önd it optimal to o§er contracts with coin- surance provisions.
4. When do you expect Örms to employ their own sales force rather than using outside agents or companies?
5. Consider a local monopolist on the market for wine. She can produce wine of quality q 2 (0; 1) at a cost q2 . Her utility from selling a bottle of wine
2
is the price of the bottle less the cost of producing it. Each customer wants to buy at most one bottle of wine. If he buys wine of quality q for a price p, his utility is given by q p, where > 0. If he does not buy any wine, his utility is 0. The wine seller knows that two types of customers enter her shop: customers with a sophisticated taste for wine for whom = 2 and customers with a frugal taste for wine for whom = 1, where 2 > 1.
(a) Assume that the principal can recognize a sophisticated from a frugal customer. Find the price and the quality of a bottle of wine (as functions of the preference parameters ) that the principal would o§er to:
(i) sophisticated customers
(ii) frugal customers
(b) Now assume that the principal cannot observe the type of the customer.
She believes, however, that a fraction 2 (0; 1) of her customers are frugal and the rest are sophisticated.
(i) How many types of bottles would she o§er?
(ii) Find the price and the quality of the wine in each bottle (as functions of and the preference parameters ). How do they compare with the results you obtained in (a). Comment.
(iii) Who receives an informational rent and how much is it? Does the informational rent increase, decrease, or stay the same if increases? Comment. (iv) Find whether the quality and the price of each bottle of wine [as obtained in b(ii)] increase, decrease, or stay the same if increases. Comment.

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