Problem Set
You are allowed to work in groups, but each student must write up his or her work independently; please note on your answer sheet with whom you worked. Be sure to provide the reasoning behind your answer in order to receive full credit.
Incentivizing Effort
You are a division manager for the logging firm of Karrde Incorporated. You are planning to offer a contract to Jade Enterprises for constructing logging roads in the Myrkr Forest. It is imperative that the roads are wide enough; if the roads are wide enough, Karrde's profit will be 120, while it will only be 60 if the roads are too narrow. (Karrde's profits are 0 if it does not hire Jade.) You have the opportunity to make a take-it-or-leave-it offer to Jade, but you can not set the payment below 0 for any outcome.
Whether or not the roads are wide enough depends crucially on Jade's effort: If Jade exerts low effort, then the probability of success is only one third; if Jade exerts medium effort, the probability of success is two thirds; and if Jade exerts high effort, success is guaranteed.
Jade Enterprises is a small concern. If Jade does not take the job, its utility is 2. If Jade takes the job, and exerts low effort, its utility is , where is the payment to Jade. If Jade takes the job and exerts medium effort, its utility is given by - 1. Finally, if Jade takes the job and exerts high effort its utility is given by - 3.
a) Suppose that you can observe the effort that Jade exerts, and so can write a contract where payment is contingent on effort. If you wish to incentivize low effort, what is the most profitable contract you can offer? If you wish to incentivize medium effort, what is the most profitable contract you can offer? If you wish to incentivize high effort, what is the most profitable contract you can offer? Given your answers to the previous questions, which is the most profitable contract for Karrde to offer Jade?
b) Now suppose (for the rest of this problem) you can not observe effort. If you wish to write the most profitable contract that incentivizes low effort, what contract will you write? Does it differ from the contract for low effort in part a)?
c) If you wish to write the most profitable contract that incentivizes medium effort, what contract will you write? How does it differ from the contract in part a) for medium effort?
d) If you wish to write the most profitable contract that incentivizes high effort, what contract will you write? How does it differ from the contract in part a) for high effort?
e) Consider your answers to the previous parts of this question. What contract will you offer to Jade-that is, what contract maximizes your profits? Does this contract incentivize the same level of effort as the contract you offered in part a)? Why or why not?
Signalling Through Prices
Lex Chemicals has developed a new plastic usable for insulation in new homes. Production costs for the new plastic are = 6 per unit. Demand for the insulating material is given by = 16 - , where is the price set by Lex. Hence, per-period profits are given by( - ) . Lex Chemicals will have 5 periods (including this one) of being the only producer of this particular type of insulation before the patent runs out and lower cost competitors enter and take the market. (For simplicity, assume no discounting of future periods.)
Unfortunately for Lex, a competitor, Edge Plastics, has developed a competing product and is considering entering the market immediately after this period (and would then remain in the market for the remaining four periods); if Edge Plastics does not enter after the first period, then Edge Plastics will never enter the market (and so Lex will have a monopoly for the final four periods). The plastic developed by Edge Plastics has a production cost of 7. There is a fixed cost of 2 for Edge Plastics to enter the market.
Edge, however, is uncertain of the cost structure of Lex Chemicals. Edge believes there is a 50% chance that Lex has a marginal cost of production of 8, and a 50% chance that Lex has a marginal cost of production of 6.
a) What are your per-period profits as a monopolist? What if you had the higher marginal cost of 8?
b) Will you attempt to signal to Edge that you are, in fact, the low cost producer? If so, what price will you set in the first period?