1. Lisa Listens supplies services for which the demand function is p(Q) =1 -1/2 Q. The technology exhibits constant returns to scale. Labor is the only variable input and the firm employs L workers. An individual worker’s productivity cannot be costlessly observed but it is known that the expected output of a shirking worker is 1 unit per hour (Q = L) whereas that of a hard working employee is q (q > 1) units per hour (Q = qL). The workers and the firm are risk neutral.
a. Draw the `marginal revenue product of labor’ curve for the case of shirking employees and for the case of hard working employees. Do these curves intersect? Give an interpretation of your findings.
b. At wage w = 1, how much labor is hired if Lisa Listens knows that its workers will shirk?
c. Suppose Lisa Listens decides to induce its workers to work hard by paying an efficiency wage and monitoring the workers. As a consequence the cost per worker equals we + M(p) where W is the efficiency wage and M(p) is the monitoring cost per worker incurred to ensure a probability p of catching a shirking worker. A worker suffers a disutility c from working. If she works hard, this disutility is c= 0.75 whereas if she shirks it is c=0. Lisa Listens pays its workers w` unless they are caught shirking in which case they get nothing. A worker who is caught cheating cannot get a job elsewhere. Workers have a choice between a job with Lisa Listens or an outside low effort job (giving disutility c=0) paying w=1.
i. Given a detection probability p, what is the minimum wage we Lisa Listens can pay and still attract workers if the workers anticipate that they will have to work hard?
ii. How large should rv` be to induce the workers to work hard?
d. Show that the solution to Lisa Listens’ minimum cost implementation problem is p =1/7 and w` = 21/4 for monitoring cost M(p) =147 p/4. How much labor is hired in this scenario for q=12?
2. Many charities offer differential membership fees. Depending on whether the member wishes to receive the charity’s magazine or whether only membership is desired, a different fee is charged. It is usually not possible for non-members to purchase a subscription to the magazine only. Assume that among potential members, willingness to pay for membership ve is uniformly distributed on [O,E] and willingness to pay for the magazine vQ is uniformly (and independently of v,) distributed on [O,A].
a. OUT, a charity for overworked university teachers, charges a `package’ price p for membership and a subscription to its magazine Termtime Blues. Indicate on the [O,EJx[O,A] rectangle which potential members are interested in this package. Determine the revenue maximizing price p and OUT’s optimal revenue per potential member. Assume that 2A/3
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