Exercise 2 (16)Miranda owns a silversmith store.Running the store yields sales S=e+u,where e denotework effort and u~U(-3,3)is a uniformly distributed random variable on the interval[-3,3].She is risk neutral and values sales S by v(S)=S.Working in the store entailscost of effort given by c(e)=ae2,where a >0.a)Calculate the mean and the variance of u.[3 points]b)Derive the Pareto-efficient (first-best)level of effort by maximizing the expectedsurplus from the store business W E[Se]-c(e).How is the first-best surplusaffected by the parameter a?Explain the intuition.[3 points]Miranda would like to scale her business,which introduces some additional administrativetasks.She considers hiring Bob to help her with production.He offers Bob a wage contractwith a fixed wage w and a bonus b if sales exceed a sales threshold S.Suppose that Bobis risk neutral.c)Suppose that Bob has accepted the contract to work for Miranda.Derive her utility-maximizing effort level e for a given bonus contract (w,b,S).Verify that it is in-creasing in b and decreasing in a.Explain the intuition.[3 points]d)In the remainder,set a 1.Derive the optimal contract (w”,6″,S”)and verify thatit implements the Pareto-efficient (first-best)levels of effort and sales.For simplicityassume that Bob’s outside option is zero.Calculate Miranda’s expected profit andtotal welfare.Explain your results.[3 points]After a while,Miranda decides that she would like to focus even more to managerialactivities and considers hiring James to further help her scale production.Suppose thatJames’expected utility from employment under a bonus contract (w,b,S)is representedby w+bP[S S”le]-ae2-0.5r62Var(u)with r >0.e)Suppose that James has accepted the contract.What is his utility-maximizing efforte?How does it depend on r and Var(u)?How does it compare to that of Bob?[2points]f)Use the participation constraints to derive Miranda’s expected wage costs w+pE[Sle]for Bob and James,respectively.For given effort e,who is more expensive?Whatdo you think is the difference between the two workers?[2 points]
