Assignment
Consider the principal-agent problem studied in class. Assume that the output levels are given by H = 24, L =6, Bernoulli function for the agent is square root of money, u(w) = (w)1/2, the probability of (H)igh output under high effort (e =1), is equal 5/6, while the same probability under low effort (e =0) is equal to 2/6, the reservation utility for the agent is 2, the cost of high effort (e =1) for the agent is 3/2.
(i) Write down the participation constraints under high and low effort.
(ii) Consider the observable effort case. Calculate the wage levels that implement (e=1), and (e=0), at the lowest cost for the principal. For each case, choose a wage schedule that (a) has the same expected cost for the principal, and then (b) gives the same expected utility to the worker. Show that the wage schedules you found in (a) and (b) either do not satisfy the participation constraint or do satisfy it, but has a higher expected cost for the principal (explain which one is the case for (a) and for (b)). Given the wages you found for low effort and high effort, implementing which level of effort is more profitable for the principal? Calculate!
From now on, consider the case in which the effort is unobservable.
(iii) Calculate the expected utility of the agent when he is being paid the profit maximizing wage for (e=1) you found in part (ii), when agent works hard (e=1) and when he shirks (e=0). Can principal pay that wage and implement high effort?
(iv) Based on your answer in (iii) write down the necessary incentive constraints.
(v) Now, clearly explain why we need only one incentive and one participation constraint to hold if the principal wants to implement high effort, e=1?
(vi) What if the principal wants to implement low effort, how many constraints do we need?
(vii) After identifying the necessary constrains, draw them on a graph where x-axis denotes the utility of agent from the wage that will be paid when output is low, and y-axis denotes the utility of agent from the wage that will be paid when output is high. Draw a set of indifference curves for the agent, what is the slope of these curves, his utility levels increase in which direction? Identify the set of (utility of) wage schedules that satisfy each of the necessary constraints, identify the set of wage schedules that satisfy both necessary constraints.
(viii) Identify the principal's indifference curves.
(ix) Identify the least (expected) costly contract that implements the high effort.
(x) Calculate the lowest possible cost of implementing e=1.
(xi) If the probability of High output under low effort was smaller, would it be possible that the contract with lowest (expected) cost that implements e=1, pays no wage when the output level is zero? If your answer is Yes, give an example with numbers. If your answer is No, clearly explain why?
(xii) Calculate the cost of implementing low effort e=0, when the effort is still not observable.
(xiii) Calculate and compare the expected profit of the principal when he implements e=0, and when he implements e=1. Based on your calculation, is the principal going to implement e=1, or e=0.
(xiv) (Hard) Is it possible to change H in the above story so that the principal would choose to implement an effort level other than you found in part (xiii)? If Yes, suggest such a change. If No, clearly explain why.
(xv) (Hard) Is it possible to change one of the parameters of the above model, so that in the wage schedule that implements e=1 with the lowest (expected) cost to principal, pays a wage of zero in case of Low output? If Yes, present one example. If No, clearly explain why.
Format your assignment according to the following formatting requirements:
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2. The response also include a cover page containing the title of the assignment, the student's name, the course title, and the date. The cover page is not included in the required page length.
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