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Solutions ps 5 part 1 - Economics of Organizations...

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Economics of Organizations 2010-2011 Prof.: Pablo Ruiz Verdú Problem Set 5 - Solutions Part 1 1) Patricia installs computer networks and hires Alberto to provide technical support by phone to Patricia’s clients. Patricia’s future revenues ( z ) depend on how satisfied clients are with the technical support they get. Therefore, z=e+x , where e is the effort that Alberto puts into solving the clients’ problems and x is a random variable that represents random factors that also affect revenues. We will assume that E(x)=0 and Var(x)=s 2 . The psychological cost (expressed in euros) for Alberto of exerting effort e is c(e)=qe 2 , where q>0 is a constant. If Alberto does not work for Patricia, he can get a certainty equivalent of B . Suppose that Alberto’s certainty equivalent to an uncertain pay W is E(W) . a) Is Alberto risk averse? No, Alberto is risk neutral, since his certainty equivalent of uncertain pay W is equal to its expected value. b) Patricia cannot observe how much effort Alberto puts at work, so she decides to give him incentives to exert effort by tying his pay to future revenues. In particular, Alberto’s pay is equal to I(z)=F+pz . Given this pay contract, how much effort will Alberto exert as a function of F and p ? Discuss your results. Alberto’s certainty equivalent if he exerts effort e is: E(I)-c(e)=E(F+pz)- qe 2 = E(F+pe+px) - qe 2 = F+pe+ E(px) - qe 2 = F+pe - qe 2 He will choose the level of effort that maximizes his certainty equivalent. The first order condition of the maximization problem yields: p – 2qe* = 0 e* = p/(2q) = 0.5p/q Therefore, Alberto’s effort as a function of F and p is e(F,p)=0.5p/q. As we would expect, Alberto’s effort does not depend on F, is increasing in p (the marginal benefit of effort for Alberto) and decreasing in q. c) Patricia’s profits are z – I(z) . Suppose that Patricia is risk neutral. What are Patricia’s expected profits (as a function of F and p ) if she pays Alberto according to the formula I(z)=F+pz and Alberto accepts the contract? ( Hint: you need to use the result you obtained in b) to solve this part. ) If Alberto accepts the contract, he exerts effort e(F,p). Therefore: Expected pay = E(I)= F + pe(F,p) = F + 0.5p 2 /q = F + 0.5p 2 /q Expected sales = E(z) = e(F,p) = 0.5p/q Expected profits = E(z) – E(I) = 0.5p/q – F – 0.5p 2 /q d) Suppose that Patricia earns 0 if she does not hire Alberto. What contract/s (i.e., what values of F and p ) maximize/s Patricia’s expected profits? ( Hint 1: you need to make sure that Alberto will accept the contract. Hint 2: Patricia will not pay Alberto more than what is strictly needed to have him accept the contract. ) How does the optimal contract depend on s 2 , B, and q? To make sure that Alberto accepts the contract; his certainty equivalent has to be at least B:
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E(I)-c(e) B F + pe(F,p) – c(e(F,p)) = F + 0.5p 2 /q – q (0.5p/q) 2 = F + 0.25p 2 /q B Patricia will not pay Alberto more than strictly needed. Therefore, for any p, Patricia will set F in such a way that: F + 0.25p 2 /q = B F = B – 0.25p 2 /q Therefore, for each p Patricia’s expected profits if Alberto accepts the contract will be: 0.5p/q – F – 0.5p 2 /q = 0.5p/q – (B – 0.25p
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