Here are some answers to the questions I received about the final exam. The values for alpha and e are posted in the Course Materials section of the web site.
I'm going to be away from the computer for the next few hours; if you have a real emergency, please call me. Otherwise, good luck and I hope to see you tonight! (If you're done with the exam and plan on coming by, please do drop a note so I have a sense as to how many people will be here.) CRM
Note: for question 1.1, there are some different ways of calculating V(0), V(1), etc. Just document however you find these values. Your answers to the later questions will be graded on the basis of consistency with your answers with those questions. Therefore, even if your value of V(0) is wrong, if your answers to later questions are consistent with the values you found, they will be given full credit. Note in particular that in the hint, the value x/r for a stream of x received forever is true if you receive that x after one year. I should have specified that in the hint. We will accept answers that either specify the value as of the date of discovery, or the value as starting one year from the date of discovery. Please specify in your answer which it is.
Question 1: Prize vs. Patent
Question: You say that all firms who make the discovery have an equal probability of reaping the reward. Do you mean that they have a probability of getting lucky, and getting all the benefits from the patent, and the other firm (even though they got it at the same time) gets nothing? Or do you mean to say what it has been in the past where if 2 invent, they split the profits each getting 1/2, if 3 invent they split and each gets 1/3, and so on? I know you say that if 2 firms invent they each get z/2, but I wasn't sure if that was the expected value of their profits or not.
Answer: Your first analysis is correct. If two or more firms make the discovery at the same time, each has an equal chance of being selected the winner of the prize or being granted the patent. If it were the other way, then if two firms had a patent for it, Bertrand competition would reduce price to cost and the patent would be worthless!
Question: On the very first question, part 1 of question 1, I was wondering if we are to assume that for V(0) a firm will have earned the prize of Y since no patent is being offered.
Answer: For 1.1, disregard any welfare effects of a prize. These enter the question later.
Question: Is the demand function P=50,000-0.01Q a yearly demand function?
Answer: Yes.
Question: In calculating V(T), the net present value of discovery, do we
include the investment costs?
Answer: No; it's once the discovery has been made.
Question: For question 1 part 1, does the firm charge monopoly price while it has the patent and then unit cost?
Answer: Yes.
Question: The first period that the firm will get a cash flow from the production after its discovery today is 1 year from now, right? That is the first cash flow of the profits should be discounted by rho rather than have no discount. Is that correct?
Answer [AMENDED]: You can do this either way; please specify in your answer how you calculated it.
Qustion 2: The Principal-Agent Problem
Question: In question 2.2, when you say assume 0 costs, does that mean you're also not subtracting the chef's wage from their profits?
Answer: For question 2.2, yes; ignore the chef's wages. For question 2.3, you should factor those wages in.
Question: What happens if my square root of wage value is negative? Do I just keep going with the equation using its positive squared value or try to explain why that wage cannot exist?
Answer: For this question, it's a good idea to work in utilities first, not wages: that is, let v=sqrt(w) and solve the question using v's, then substitute in w = v^2 at the end. That ensures everything is positive. You can see the posted solution for the final problem set for an explanation of this technique.
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