## Quantitative Problems

 1. There are two types of workers with abilities , where . The value of output produced by a worker, y, depends on his ability, a, and the speed of the conveyer belt, s, according to . Worker utility, U, depends on ability and the pace of work according to .(a) Assume that ability is publicly observable, what wage offers would emerge in competitive equilibrium? Depict the optimal conveyer belt speed for both worker types.(b) Now suppose that workers know their abilities, but employers do not. Depict the signaling equilibrium that arises in this environment, assuming that firms choose the lowest speed that discourages lowability applicants.(c) Suppose that a sudden technological improvement raises the productivity of (only) low-ability workers from to , where k > 1. What happens to the equilibrium? 2. In Problem 1 suppose that and . Calculate the optimal choices made by each worker type under conditions of complete information (case(a)) and asymmetric information (case (b)).Hint: For a worker with ability a ∈ A, the marginal benefit of a faster speed is MB = a, and the marginal cost is MC = e/a. 3. The facts are as follows: there is a single worker and firm, employment lasts forever, the interest rate is r, workers can either work (e = 1) or shirk (e = 0) (and derive the goofing-off utility \$G in this event), output depends on effort as y (1) = y and y (0) = − k, the firm can see y but it is unverifiable in court (and so cannot be included in a contract), and finally the worker and the firm earn zero if they do separate.Suppose that at the beginning of each period the worker and firm negotiate a contract (w, b), where \$w is an explicit wage guarantee that the firm must honor, and \$b is a performance bonus that it can choose not to honor.Is it possible for the firm to propose a deal (w, b) for which (a) the worker sets e = 1 and (b) the firm actually pays the bonus if the worker exerts effort?Hint: Appendix C shows that the present value of a constant payment stream x—from t = 1, 2, . . . —is just x/r. The value of a stream from t = 0, 1, 2, . . . is, therefore, (x + x/r). Given this, calculate the values to the firm and worker when they behave themselves and when they do not. 4. In the fall of 2008, during the worst financial crisis to hit the United States in over 70 years, there was enormous public indignation that financial institutions honored their huge bonus payments to analysts. (Rightly or wrongly, the blame for the calamity was laid at their feet.) Use the relational contracting framework to explain what the likely consequences of default would have been. 5. Suppose there are 10 fair dice and two boxes labeled P and NP. Suppose that after a random throw, any die that scores 4 or greater is assigned to the box labeled P; otherwise, it is put in the box labeled NP. On the first throw, we get 1 six, 2 fives, and 1 four; the other dice score 3 or less. What is the average score among the dice placed in box P? What is the expected score of these dice on the next throw? What do these observations potentially have to do with promotions and the Peter principle? 6. What is the fundamental economic distinction between an employee and an independent contractor? 7. Some organizations (universities, for example) meet their printing needs (pamphlets, brochures, end-of-year reports) using their own dedicated printing center, others use Kinko's. What are the main factors that govern the choices between these two options?