San Francisco State University ECON 851 Summer Problem Set 1

Transcription

1 San Francisco State University Michael Bar ECON 85 Summer 05 Problem Set. Suose that the wea reference relation % on L is transitive. Prove that the strict reference relation is transitive. Let A; B; C be any three lotteries in the lottery sace L, such that A B and B C. We need to rove that this imlies A C. By de nition of, we have A B ) [A % B but not B % A] B C ) [B % C but not C % B] Thus, by transitivity of % we have A % C. It is left to rove that "not C % A". Suose, by contradiction, that C % A. Then, from the above we have C % A % B and by transitivity of % we have C % B. But this is a contradiction to B C.. Suose that the wea reference relation % on L is transitive. Prove that the indifference relation is transitive. Let A; B; C be any three lotteries in the lottery sace L, such that A B and B C. We need to rove that this imlies A C. By de nition of, we have A B ) [A % B and B % A] B C ) [B % C and C % B] Thus, by transitivity of % we have A % C and C % A (because C % B % A), which is the de nition of A C. 3. Find the certainty equivalents and ris remia of the lottery, $000 w.. = L = $500 w.. = for the following vnm utility functions: (a) u (x) = x. First, the exected value of the lottery is The exected utility from the lottery is E (x) = = 750 E [u (x)] = = 6:99

2 By de nition of certainty equivalent, By de nition of ris remium, CE = 6:99 (b) u (x) = ln (x). The exected utility from the lottery is ) CE = 6:99 = 78:55 RP = E (x) CE = :55 = :45 E [u (x)] = ln (000) + ln (500) = 6:56 By de nition of ris remium, ln (CE) = 6:56 CE = ex (6:56) = 707: RP = E (x) CE = : = 4:88 Remar. We showed in class that individual with u (x) = ln (x) is more ris averse than individual with u (x) = x, because u (x) is a concave transformation of u (x). In articular, u (x) = (u (x)), where (u) = ln (u) is a concave function. In this question we see that, consistent with our revious ndings, the more ris averse individual is willing to ay less for this lottery (707: v.s. 78:55), and is willing to ay higher ris remium in order to avoid laying the lottery. 4. Consider the gamble in St. Petersburg Paradox aradox examle, with ayo s,, = ; ; :::, and the associated robabilities = ; ; :::. Suose references over lotteries can be reresented with exected utility form, with vnm functional u (x). Calculate the exected utility of the gamble, and the its certainty equivalent for the following vnm utility functions: (a) u (x) = ln (x). The exected utility from the gamble is: E [u (x)] = X u (x ) = " X = ln () X ln X = ln () ( ) X # = ln ()

3 The second term in the bracets is X = = = = To comute the rst term in the bracets, we use the rule P a = a= ( a) : X = = ( =) = ln (CE) = ln () ) CE = (b) u (x) = x. The exected utility from the gamble is: X X E [u (x)] = u (x ) = X = " = X # =! 3 X 4 5 " # = = = :707 CE = :707 CE = = :94 (c) Exlain why individual with vnm utility function u (x) = ln (x) is willing to ay less for the gamble than individual with vnm utility function u (x) = x. We showed in class that individual with u (x) = ln (x) is more ris averse than individual with u (x) = x, because u (x) is a concave transformation of u (x). In articular, u (x) = (u (x)), where (u) = ln (u) is a concave function. In this question we see that, consistent with our revious ndings, the more ris averse individual is willing to ay less for this lottery ( v.s. :94 ). 5. Consider two lotteries: X U [0; ] and Y = 3 :475 w.. 0:8 0:0 w.. 0:

4 This means that X has a continuous uniform distribution on the interval [0; ]. It hels to recall that if X U [a; b], then the df, mean and variance are: for a x b f (x) = b a 0 otherwise E (x) = b + a V ar (X) = (b a) As an exercise, you should derive the above mean and variance. (a) Calculate the mean and variance of both lotteries. Lottery X: E (X) = b + a = + 0 = V ar (X) = ( 0) = 3 Lottery Y : E (Y ) = 0:8 : : 0:0 = V ar (Y ) = 0:8 (:475 ) + 0: (0:0 ) = 0:4503 Notice that both lotteries have the same mean, but Y has lower variance than X. (b) Which lottery is referred by an individual whose vnm utility function is u (x) = ln (x)? The exected utility from X: E [u (x)] = Z b a u (x) f (x) dx = {z } b a Z 0 ln (x) dx = [x ln (x) x] 0 = [x ln (x) x] 0 = [ ln () 0] = 0: Notice that to evaluate x ln (x) at x = 0 we must comute the limit: ln (x) lim x ln (x) = lim x!0 x!0 x The second ste used L Hôital s rule. The exected utility from Y : x = lim x!0 x = lim ( x) = 0 x!0 E [u (y)] = 0:8 ln (:475) + 0: ln (0:0) = 0:744 Thus, the utility from X is greater than the utility from Y, and the individual refers X over Y. 4

5 (c) Based on your answers to (a) and (b), is it true that any ris averse individual, when comaring two lotteries with the same mean, would always refer the one with the lower variance? In this examle we saw a case where a ris-averse individual actually referred a lottery with higher variance, to the one with lower variance and the same mean. This exercise illustrates that individuals with EUT references care not only about mean and variance of a lottery, but other things as well. If the investor had Mean- Variance references that are variance averse, she would refer lottery Y over X, because the lotteries have the same mean, but Y has lower variance. However, individuals with EUT references, care about the actual realizations of the ayo s, and not only about the variance of ayo s. In this examle, lottery Y has a low ayo of 0:0 which occurs with robability 0., and it has very negative imact on logarithmic utility as ln (x)! when x! 0. Thus, variance of a lottery does not reresent the entire risiness of the lottery, unless references are reresented by mean-variance utility. This examle therefore illustrates a case where EUT and MVT do not agree. As another examle where EUT and MVT do not agree, consider the following lotteries: 8 < w.. 0:5 X = 3 w.. 0:5 : 7 w.. 0:5 8 ><, Y = >: w.. 0:5 3 w.. 0:5 4 w.. 0:5 6 w.. 0:5 Once again, both lotteries have the same mean, but Y has lower variance. You should be able to show that nevertheless, an individual with vnm utility function u (x) = ln (x) would choose X over Y, desite being ris averse and X having greater variance. E (X) = 0:5 + 0: :5 7 = 3:5 V ar (X) = 0:5 ( 3:5) + 0:5 (3 3:5) + 0:5 (7 3:5) = 4:5 E (Y ) = 0:5 + 0: : :5 6 = 3:5 V ar (Y ) = 0:5 ( 3:5) + 0:5 (3 3:5) + 0:5 (4 3:5) + 0:5 (6 3:5) = 3:5 Exected utility: E [u (x)] = 0:5 ln () + 0:5 ln (3) + 0:5 ln (7) = 0:5 ln (4) + 0:5 ln (3) + 0:5 ln (7) = 0:5 ln (84) E [u (y)] = 0:5 ln () + 0:5 ln (3) + 0:5 ln (4) + 0:5 ln (6) = 0:5 ln (7) 5