1. Mr Stock has a wealth $100 to invest. He can buy shares of an umbrella factory which for one dollar investment pays $10 if the year is rainy and $3 if the year is dry. He can buy shares of an ice cream factory which for one dollar investment pays $2 if the year is rainy and $9 if the year is dry. The probability that the year is rainy is 0.5. His vNM utility function is u(x) = x1/2 where x is his realized final wealth.
(a) If Mr Stock invest a fraction λ of his wealth in the Umbrella company and 1-λ in the Ice-Cream company what is
1. his expected return 2.his expected utility
(b)what value of λ maximizes 1.his expected return 2.his expected utility
2. Suppose that July is risk averse vNM utility maximizer. I would like to offer following lotteries: A = (.25, $8; .75, $0) or B = $2. Which one do you expect her to choose?
3. A suspect is brought into a court room and it is announced that he committed a crime but judge does not know whether he is guilty or not. Judge has to listen a witness and give his final decision-send the suspect to jail or let him free.
If suspect is guilty and judge sends him to jail then judge's payoff will be +1. If suspect is guilty but judge let him free then judge will receive payoff of -2. However, if suspect is not guilty but judge sends him to jail then his payoff is -2. If suspect is not guilty and judge lets the suspect free, his payoff is 0 (all payoffs are Judge's payoff).
Assume that it is known by the judge that suspect may have been committed the crime with probability 1/3. Moreover it is also known that the witness may lie with probability 1/4. Assume that judge is risk-neutral vNM utility maximizer and according to witness' testimony suspect committed the crime. What should judge do? Show all your work!
4. Assume that July is graduated from a university and planning to apply to a job. She may pursue a career in one of the following sectors; she can work in a financial firm or in a real estate firm. She knows that in both sectors there are large number of job opportunities. If she choose to work in a financial firm she may get wage distributed according to a p.d.f (probability distribution function) f1 on interval [w1, w1] (assume that w1 > 0). If she choose real estate sector, she may earn wages which is distributed with a p.d.f. f2 over the interval [w2, w2] (assume that w2 > 0). Assume that July is risk neutral and expected utility maximizer. Before deciding which firm to work, she needs to choose a sector.
(i) Write down July's problem.
Suppose now that f1 and f2 are uniform over [w1, w1] = [2, 4] and [w2, w2] = [1, 5] respectively.
(ii) Which sector does July choose?
(Note: I believe you know what uniform distribution is. In this question im- portant point is that the set of alternatives or "states" of the world or prizes is not finite. For that reason you cannot use the vNM utility functional form given in the lecture notes. So, what is the form of vNM utility function in this case? I want you to think about it first. But remember in calculus we use another mathematical operation for infinite sum. You may take these wages as present value of her lifetime income, i.e., she earns wage once and for all.)
5.(We will solve this in class) Suppose July has to choose one of the following lotteries A = (.25, $3000; .75, $0) or B = (.2, $4000; .8, $0) and she chooses B over A. Now, suppose that she has to choose one of the following lotteries C = (1, $3000; 0, $0) or D = (.8, $4000; .2, $0) and she chooses C over D. Show that July cannot be vNM utility maximizer.
6.(We will solve this in class) Suppose that I toss an unfair coin. Head comes up with probability 1/4 and tail comes with probability 3/4. Then, I ask you to choose Head or Tail. If your guess is correct, I pay you $1 and if it is wrong, then you pay me
$1. Assume that you are a vNM utility maximizer and risk neutral.
(i) Do you choose head or tail? Now, after tossing the coin and before asking your choice, I show the coin to July (so, she knows whether coin came up head or tail). I let July to tell you what she saw and she always tells the truth.
(ii) What would you pick in this case, head or tail?
Now, assume that July has a memory problem (that is, she may say Head although she sees tail). The probability that what July says is correct is 0.8 (so, if coin comes up head, she will say head with probability 0.8 and tail with probability 0.2). Suppose that July tells you that she saw head.
(iii) After this information, what do you choose?
7.(We will solve this in class) Consider the following game. I will toss an unfair coin (head comes with probability 1/4 and tails comes with probability 3/4). If head comes, I give you $100 and game ends, if tail comes I'll toss the coin second time. In the second toss, if head comes, I pay you $100 and game ends. If tail comes up, I will toss the coin third times. The game continues like this as long as tail comes up. Suppose that I ask you $120 in advance to play this game with me. Do you accept my offer. If yes why? if not why not?
Now suppose that July (who is vNM utility maximizer) wants to play this game with me. What price should I charge so that she plays this game. (Note two things; make your analysis with which your and July's utility functions are linear on wealth i.e., u(x) = x. Do you think that your analysis will change if we assume risk averse or risk lowing utility function. Moreover, assume that I do not have any money constraint.)
8. Now, consider a more interesting variant of the previous game: Again, I will toss an unfair coin (head comes with probability 1/4 and tails comes with probability 3/4). For the first toss, if head comes, I give you $100 and game ends, if tail comes I'll toss the coin second time. In the second toss, if head comes, I pay you $50 and game ends. If tail comes up, I will toss the coin third times. In the third toss, if head comes, I pay you $50 and game ends. If tail comes up, I will toss the coin fourth times. The game continues like this as long as tail comes up. Note that the game ends only if head comes up at some point and you receive the prize (either $100 or $50) when game ends. The only difference between this question and the previous one is that after the first toss, the prize is $50. So if the game finishes in the first round you will get $100. If head comes up at some later period then you will get $50.
Suppose that I ask you $60 in advance to play this game with me. Do you accept my offer. If yes why? if not why not?
Now suppose that July (who is vNM utility maximizer) wants to play this game with me. What maximum price can I charge so that she plays this game?
(Note two things; make your analysis with which your and July's utility functions are linear on wealth i.e., u(x) = x. Do you think that your analysis will change if we assume risk averse or risk lowing utility function. Moreover, assume that I do not have any money constraint.)