Figure 1: Smooth curve of through the six points x = 200, 100, 25, 100, 300 and 600.

Transcription

1 AMS 221 Statistical Decision Theory Homework 2 May 7, 2016 Cheng-Han Yu 1. Problem 1 PRS Proof. (i) u(100) = (0.5)u( 25) + (0.5)u(300) 0 = (0.5)u( 25) u( 25) = 1 (ii) u(300) = (0.5)u(600) + (0.5)u(100) 1 = (0.5)u(600) u(600) = 2 (iii) u(100) = (0.5)u( 100) + (0.5)u(600) 0 = (0.5)u( 100) + (0.5)2 u( 100) = 2 (iv) u( 100) = (0.5)u( 200) + (0.5)u(300) 2 = (0.5)u( 200) + (0.5)1 u( 200) = 5 Figure 1: Smooth curve of through the six points x = 200, 100, 25, 100, 300 and 600. (a) What is the CE of a lottery that gives a.5 chance at $300 and a.5 chance at $600? (0.5)u(300) + (0.5)u(600) = (0.5)2 = 1.5. So we need to find z such that u(z ) = 1.5. According to the smooth curve of the utility function, z 428. Note that CE < EMV (Expected monetary value), i.e, z < z = 450 because the ulitily function is concave. 1

2 (b) What is the CE of a lottery that gives a.75 chance at $400 and a.25 chance at -$200? First, u(400) 1.4, and so (0.75)u(400)+(0.25)u( 200) = (0.75)(1.4)+(0.25)( 5) = 0.2. The CE z such that u(z ) = 0.2 is z 70. Again, z < z = 250, and the difference between them is larger than that in (a) since the decision maker is more risk averse in this range. (c) You are a compound lottery with a canonical chance at the lottery l of part (b) as one prize and a complementary chance at no net gain. What would your chance of winning lottery l have to be before you would accept the offer? I am confused about this problem but the following is how I interpret this question. I assume the canonical chance is just 0.5 and winning lottery l means that I get the lottery and my amount of money becomes 400. Suppose the chance of winning lottery l is p. Suppose the current amount of money I have is x between 200 and 400. Otherwise, if x > 400, I will not accept the offer no matter how large p is. Also, I will definitely accept the offer no matter how small p is when x < 200. Assume that no net gain means the current amount of money does not change. To accept the offer, the current utility without the offer should be less than the utility before accepting the offer, i.e., u(x) < (0.5)[pu(400) + (1 p)u( 200)] + (0.5)u(x) u(x) < p(1.4) + (1 p)( 5) u(x) < p(6.4) 5 Hence p > u(x) + 5. The chance of winning the lottery have to be higher than 6.4 u(x)+5 depending how much money I have now. 6.4 (d) What is the insurance premium of a lottery that gives a 0.5 chance at $0 and a.5 chance at -$200? The EMV z = (0.5)($0) + (0.5)($ 200) = $100. The CE z is such that u(z ) = (0.5)u(0) + (0.5)u( 200) = (0.5)( 0.7) + (0.5)( 5) = And so the CE z $140. Hence the insurance premium is ( z z ) = ( $100 ( $140)) = $40. (e) Given that the CE of a lottery is $325, and the lottery gives a π chance at $500 and a (1 π) chance at $300, find π. u(ce) = u(325) = πu(500) + (1 π)u(300). Given that u(325) = 1.11, u(500) = 1.7 and u(300) = 1, we have and hence π = π(1.7) + (1 π), (f) What is the CE of a lottery that ofers a.375 chance at $500, a.125 chance at $600, and a.5 chance at $0? 2

3 u(z ) = (0.375)u(500) + (0.125)u(600) + (0.5)u(0). Given that u(500) = 1.7, u(600) = 2, and u(0) = 0.7, we have and hence z u(z ) = (0.375)(1.7) + (0.125)2 + (0.5)( 0.7) = (g) You are offered the lottery of part (f) $200; would you buy it? If you were an EMVer would your choiuce change? I would not buy it because I am just willing to pay to avoid the uncertainty caused by the lottery. If I were and risk-neutral person, since the EMV is (0.375)(500) + (0.125)(600) + (0.5)(0) = 262.5, which is higher than the price of lottery, I would buy it in this case. (h) Consider the lottery l = {(.2 : $0), (.5 : $150), (.3 : $600)}. For how much would you just be willing to sell this lottery if you owned it? CE can be seen as the selling price of a lottery. That is, we would just be willing to sell the lottery l, if we had it, for the amount CE. So u(z ) = (0.2)u(0) + (0.5)u(150) + (0.3)u(600) = (0.2)(0.7) + (0.5)(0.333) + (0.3)(2) = Hence z 280. I will be sell the lottery at price $280 if I owned it. (i) For how much would you just be willing to buy the lottery of part(h) if you did not own it? Again, the amount I am just willing to buy the lottery if I did not own it is the CE, which is $280 derived in (h). But we need to keep in mind that the amount adecision maker would just be willing to pay a certain lottery assuming he does not have it is in general different from the amount he would just be willing to sell this same lottery for assuming he does have it. 2. Problem 2 PRS I start with u($0) = 0 and u($10000) = 1. After using the method of section with total 21 points, I got my utility function shown in Figure 2. It is not being smoothed. 3. Problem 4 PRS u(don t drill) = u(0) = (0.14)u($100000) + (0.86)u( $50000). u(keep all) = (0.6)u( 50) + (0.2)u(100) + (0.1)u(200) + (0.07)u(500) + (0.03)u(1000) = ( )u($100000) + ( )u( $50000) u(sell 1/4) = (0.6)u( 37.5) + (0.2)u(75) + (0.1)u(150) + (0.07)u(375) + (0.03)u(750) = (0.2088)u($100000) + (0.7912)u( $50000) 3

4 Figure 2: Utility function of Problem 2 PRS. u(drill and sell 1/2) = (0.6)u( 25)+(0.2)u(50)+(0.1)u(100)+(0.07)u(250)+(0.03)u(500) = ( )u($100000) + ( )u( $50000) u(sell 3/4) = (0.6)u( 12.5) + (0.2)u(25) + (0.1)u(50) + (0.07)u(125) + (0.03)u(250) = ( )u($100000) + ( )u( $50000) Based on this result, the wildcatter should choose keep all because this option gives them the highest expected utility. 4. Problem 6 PRS I am facing two options: Option A: In addtional to your regular income you will receive a tax-free gift of z dollars per year for the rest of my life. Option B: A single toss of a fair coin will determine whether you get nothing or the fabulous privilege of an unlimited ability to wrtite checks in any amount you wish for the rest of your natural life. Based on my current economic situation, z = $200, 000 satisfies me although I know I will have one-half probability of getting checks in any amount I want. This kind of explain why my utility function is bounded from above. If I could get ckecks in any amount I d like and my utility function is unbounded, then the expected utility of this game should go to infinity. But here merely $10000 gives me the same satisfaction as the game. 5. Problem 4.4 PI 4

5 Let z be defined in R +. Bernoulli s utility function is Then Hence Bernoulli is risk-averse. Then we compute the absolute risk aversion: u(z) = c log(z) log(z 0 ). u (z) = c/z, u (z) = c/z 2 < 0 z > 0. λ(z) = u (z) u (z) = c/z2 c/z λ (z) = 1/z 2 > 0 z > 0, = 1/z, and so λ(z) is increasing in z and hence Bernoulli is not decreaingly risk-averse. 6. Problem 4.6 PI From the textbook corollary 1 of chapter 4, we know that the constantly risk averse utility function is one of following forms { az + b if λ(z) = 0 u(z) = ae λz + b if λ(z) = λ > 0 where a > 0 and b are constant. Suppose u(z) = az + b. The certainty equivalent z = z = $1000(1 p), which is not equal to 0.25, 0.6, 0.85 and 0.93 for p = 1/10, 1/3, 2/3, and 9/10. Now suppose u(z) = ae λz + b. Then If p = 1/10, ae λz + b = p( ae λ0 + b) + (1 p)( ae λ b) = pa + b (1 p)( ae λ1000 ) ae λ0.25 = ( 1/10)a (9/10)ae λ1000 This implies e 0.25 (9/10)e 1000 = 1/10, which cannot be true because e 0.25 (9/10)e 1000 = Hence the utility function cannot be the form of u(z) = ae λz + b. So the person s utility function is not consistent with constant risk aversion. 5