Practice problem set 3 solutions – David Figlio
(1) You are manager of a firm that is considering making a certain investment. You don’t know whether the investment will be successful or not, but you do have a good sense as to what your profits will be if the investment is successful or unsuccessful. If the investment is successful, you expect to earn profits of $12 million. If the investment is unsuccessful, you expect to earn a profit of -$3 million (yes, that’s negative.)
Based on your intuition, you expect that your investment has a 25 percent probability of success. But you also know that you can buy additional research that would provide you more clarity about the likely success of your investment. If your research is “favorable”, you would expect a 40 percent chance of success. But if your research is “unfavorable”, you would expect a 10 percent chance of success. The research firm that is offering to sell you this information tells you that the report will be “favorable” 50 percent of the time.
(a) Is it worth it for you to make the investment if you don’t buy the additional research?
Solution: Find the expected value of the investment: EV=(0.25)(12)+(0.75)(-3)=0.75 million. This is positive, so as long as you are risk neutral (which we assume in this class) you will make the investment.
(b) Suppose that the price of the research is $0.5 million. Would you pay the money for it? Why or why not?
Solution: You need to solve this with backward induction. To do this, you compare the expected value of the investment without the research (0.75 million, solved in (a)) to the expected value of the investment with the research. If the difference exceeds 0.5 million, you’ll buy the research.
How do you find the expected value of the investment with the research? You look ahead and think about what you would do if you learned the research was “favorable” and what you would do if you learned the research was “unfavorable.” Your expected value of the investment with the research is the probability the research will be favorable (0.5) times the your expected profit from the investment if the research is favorable (solved below) plus the probability the research will be unfavorable (0.5) times your expected profit from the investment if the research is unfavorable.
The expected value of the investment if the research is favorable is (0.4)(12)+(0.6)(-3)=3 million. Since this is positive, you would definitely want to carry out the investment if your research was favorable, and your expected profit given favorable research is 3 million. The expected value of the investment if the research is unfavorable is (0.1)(12)+(0.9)(-3)= -1.5 million; since this is negative you wouldn’t carry out the investment when presented with negative information; the expected profit given unfavorable research is zero, as you’d walk away from the investment.
So your expected value of the investment with the research is (0.5)(3)+(0.5)(0)=1.5 million. The difference between this and the expected value of the investment without the research is 1.5-0.75=0.75 million. It’s worth it to buy the research.
(2) You are interested in hiring an assistant. You believe that there are four types of assistants out there, and you “know” their type after interviewing them. After each interview you decide whether to hire the assistant or keep searching. It costs you $1,000 to keep searching.
The four types of assistants, their values to you, and your perceived probabilities that an interview candidate will be each of the types, are as follows:
Type / Value to you / Perceived probabilityA / 20,000 / 0.2
B / 16,000 / 0.3
C / 14,000 / 0.3
D / 12,000 / 0.2
(a) Think about what would happen if you discovered that your interview candidate was each of the four types. Which types of assistants would you hire, rather than continuing to search? Explain.
Solution: Compare the expected value of the next interview candidate to the “realized” value of the current interview candidate. This expected value is (0.2)(20,000)+(0.3)(16,000)+(0.3)(14,000)+(0.2)(12,000)=15,400. Subtract 1,000 search costs from this figure and the expected value of the next candidate is 14,400. You’ll stick with types A and B and keep searching if you encounter a Type C or D.
(b) Suppose that you discover that your interview candidate is “Type D”. Suppose further that this changes your priors about the probability that the next candidate is a given type. Specifically, suppose that this changes your perceived probability that the next candidate is Type A with probability 0.1, Type B with probability 0.2, Type C with probability 0.3, and Type D with probability 0.4. How does this change your answer to (a)?
Solution: Now the expected value is (0.1)(20,000)+(0.2)(16,000)+(0.3)(14,000)+(0.4)(12,000)=14,200. Subtract 1,000 search costs from this figure and the expected value of the next candidate is 13,200. Now you’ll still stick with types A, B or C and keep searching if you encounter a Type D. (Note that the search costs would have to get to 2,200 before you’d settle for a Type D candidate.)