Personnel EconomicsExercise 3Exercise 3: The Basic Agency Problem1. Stefan has just been hired as a consultant to advise the Campus Constructioncompany on how to cut its costs. The compensation scheme offered by CC Inc. issuch that Stephan’s pay depends on his output, i.e. on the total dollars saved by CCafter following Stephan’s advice. Stephan’s output, Q, (in dollars) depends on theamount of effort he exerts, E, in a very simple way: Q = E. Stephan’s pay, Y, is alinear function of his output, i.e. Y = a + bE . In other words, Stephan gets paid a justfor taking the assignment, and in addition is paid b dollars for every dollar he savesCC Inc. Thus, b is his commission rate.Stephan’s utility (U) depends on two things, income and effort. In particular, U = Y –C(E), where C(E) is the disutility of effort. For the purposes of this question, assumeC(E) = E3/3.a) Graph Stephan’s disutility of effort curve, C(E). Graph his marginal disutility ofeffort curve, dC/dE. Does he exhibit increasing or decreasing marginal disutility ofeffort?b) Assuming Stephan is a utility maximizer, derive his utility-maximizing effort (andoutput) level as a function of the parameters of his pay scheme, a and b. How mucheffort will he choose if b =.36? If b = 1? Does Stephan’s choice depend on the levelof a?c) Illustrate your answer to part (i) graphically. In one diagram, show Stephan’s totallevels of income (Y) and disutility from effort (C(E)). Directly below this, show hisoptimal choice using marginal benefit and cost curves.2. Now assume that CC Inc. is hiring Marie as a consultant to computerize its officeoperations. Everything is the same in Marie’s situation as in Stephan’s except thather disutility of effort is different. Specifically, C(E)=E2/10.a) How much effort will Marie exert if her commission rate is 20%? 100%? Does effortdepend on a?b)Now suppose that Marie and CC are negotiating the terms of her contract (i.e. thelevel of a and b) before she starts. Before they settle on who gets exactly what in theend, they agree that, whatever contract they ultimately choose, it must be efficient, i.e.it must maximize the sum of CC’s profits and Marie’s utility. If the contract isefficient, what will be the level of b? Comment.c) Suppose now that, to retain Marie, CC Inc. must offer her a utility level of at least 1unit (because this is what she can get working for another firm). What must a be? IfMarie’s alternative utility is 5 units, will she work for CC Inc?1Personnel EconomicsExercise 33. This simple agency problem is designed almost entirely for a spreadsheet. Below youwill see a printout of a spreadsheet with mostly empty cells. Your goal is to fill in theblank cells. You will be thinking of this firm as experimenting with different valuesof the commission rate, b, ranging from zero to 1.5, as shown in the first column ofthe table.Assume:Effort = EOutput = Q = dE (d is a “productivity parameter”; higher levels of d correspond tohigher productivity)Disutility of Effort = C(E) = E2Agent’s reward = Y = a + bQAgent’s Utility = Y – C(E)In your “base” calculation, let d=20. (But allow the spreadsheet to input alternativevalues for d).a) Given the different values of b in column one, fill in the agent’s choices of effort, hisoutput, and his total commission income in the next three columns. (By commissionincome we mean only the part of his income that depends on how much he produces,i.e. bQ).b) Suppose that, to get the agent to work at this firm at all, he must attain an overallutility level of 20 (this is what he can get at another firm). For each row of the table(i.e. for each different value of b under consideration), calculate the level of a the firmmust offer the agent to induce him to come to the firm. Hint: remember that theagent’s total income must be at least 20 because he gets disutility from effort too.CommissionRate (b)0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.5Agent’sEffortOutputCommissionIncome2aUWorker’sIncomeProfitPersonnel EconomicsExercise 3c) Fill in the remaining columns in the table (in the “utility” column, use the definitionof utility (Y – C(E)) to check that you set a in just the right way to keep the agent’sutility at 20). Noting that the worker is indifferent between all the rows of the table(all the possible levels of b), which level of b yields the highest profit to the firm?Why?d) Does a profit-maximizing firm select the compensation plan in which income of theagent is the lowest? Does the agent always prefer those schemes that give him thehighest income? Why or why not?4. This question is identical to Question 3 with one exception. We now introduceuncertainty and risk aversion, and aim to show the effects of different levels of risk onthe optimal levels of a and b, and the corresponding profit maximizing levels of Q.We now suppose that, when the agent exerts effort, he can no longer be sure of theoutcome. In particular, when he exerts effort E, he produces an output of dE + k/2with probability one half, and an output of dE – k/2 with probability one half. Thus kindexes the amount of output uncertainty. The firm (principal) is risk neutral andcares only about the expected value of the profit it earns. We introduce risk aversionfor the agent, however, in the following simple way: utility equals expected income,minus the disutility of effort, minus a “risk aversion” term that depends on thedifference between his income in the “good” state (when he produces dE + k/2) andhis income in the “bad” state (when he produces dE + k/2). If we call this incomedifference ?, his utility is given by: U = [Y – r?2] – C(E). Think of r as an index ofhow “risk averse” the agent is (if r = 0 he is risk neutral and the problem is identicalto number 3 above).a) Assuming the agent is still compensated according to the output he produces, via thereward schedule Y = a + bQ, derive an expression for his utility as a function of a, b,d, E, r and k.b) Please fill in the spreadsheet below in Excel. Allow for inputtable values for k and r,but set k = 10 and r = 0.4 in your “ baseline” case. (Keep d = 20 and alternativeutility = 20 as in question 3).c) What is the profit-maximizing commission rate now (in your “baseline” case)?Compare this to question 3 and explain the difference.d) How does the optimal commission rate change when output uncertainty (k) rises to 20units (i.e. when there is now a difference of 20 units of output between the “good”and “bad” states)? Under the optimal commission rate when k = 20, what is theagent’s fixed income, a? Is it positive or negative? Why?e) Now put k back to its original value of 10, and see what happens when the agent’saversion to risk parameter, r, rises from .4 to 1. What is the optimal commission ratenow? Explain.3Personnel EconomicsCommissionRate (b)Agent’sEffortExercise 3OutputCommissionIncomeUncertaintyCostr(bk)20.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.54aUWorker’sIncomeProfit

# Exercise 3 – The Basic Agency Problem

by writings | Apr 6, 2019 | Uncategorized

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