Adv. Micro Theory, ECON

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1 Av. Micro Theory, ECON Assignment 4 Ansers, Fall 00 Due: Wenesay October 3 th by 5pm Directions: Anser each question as completely as possible. You may ork in a group consisting of up to 3 members for each group please turn in only set of ansers an make sure all group member names are on that set of ansers. All group members ill receive the same grae.. Consier a risk neutral iniviual. Sho that this iniviual s Arro-Pratt measure of absolute risk aversion, R a (), is equal to zero. (Hint: What type of vn-m utility function must the iniviual have in orer to be risk neutral?) Using the hint, e kno that a risk neutral iniviual must have a linear vn-m utility function. Since this is the case, let: u () + x be of the general form for the risk neutral iniviual s utility function, ith > 0 (so that it is increasing). We then have: So that: u 0 (x) u 00 (x) 0 R a () R a () u 00 (x) u 0 (x) 0 0. An iniviual has ealth W. Her von Neumann-Morgenstern utility function over non-negative levels of ealth is u() ; here 0 < <. The iniviual is o ere the folloing bet. If she pays x, ith probability / she receives nothing an ith probability / she receives x( + s), here s >. Ho much ill she bet (as a function of s)? The iniviual maximizes expecte utility. If she ins she receives ( x) + x ( + s) + xs. If she loses she receives x because she pays x. Her expecte utility is then: u (g) ( + xs) + ( x) She chooses x, the amount to pay, to maximize utility so e have: max x ( + xs) + ( x)

2 Assume that 0 < x < so that: u x ( + xs) s + ( x) ( ) 0 ( + xs) s + ( x) ( ) 0 ( + xs) s ( x) ( x) ( + xs) s ( x) s ( + xs) s x + xs s + xss x x x + xss s + s x s s + s Note that x > 0 since > 0, +s > 0, an s is s > 0, but s < since 0 < <. As! 0 e have s! s e have!, an so s > 0. The only one that may be i cult to see < since s >. As!! 0 <. Also note that x < since > 0 an s +s <. 3. (Harer) Consier an investor ho has initial ealth an has to ecie ho to invest it. There is a riskless asset ith rate of return r. The risky asset has return x i ith probability i ; i ; : : : ; n. Denote by the fraction of ealth that the investor puts into the risky asset, so that is the fraction he invests in the riskless asset. a Write on the investor s optimization problem. The investor s problem is max [0;] i i u(( )r + x i ) b Sho that if the investor has constant relative risk aversion (CARA) (note: shoul have been CRRA), then the fraction of ealth investe in the risky asset, oes not change ith (that is, ( ) 0, here enotes the solution to the investor s problem). Note that you may assume an interior solution. The rst orer conition is: i u 0 (i ) (x i r) 0 i i u 0 (i ) (x i r) 0 i

3 here i ( )r + x i. We can then n, hich is: i iu 0 (i ) (x i r)) (P n i iu 0 (i ) (x i r)) If an investor has constant relative risk aversion, then: here R r is some constant. Then: i iu 0 ( i ) (x i r)) i iu 0 ( i ) (x i r)) i iu 0 ( i ) (x i r)) i iu 0 ( i ) (x i r)) R r u00 () u 0 () u 00 () R r u 0 () i u 00 (i ) (x i r) (( ) r + x i ) i i u 00 (i ) (x i i i u 0 (i ) (x i i R r r) i r) R r () i u 0 (i ) (x i r) But this must equal zero since P n i iu 0 ( i ) (x i r) 0 by the rst orer conition. Thus, i Consier the quaratic vn-m utility function u () a + b + c. a What restrictions, if any, must be place on paramters a, b, an c for this function to isplay risk aversion? First, a simply shifts utility up or on so e o not nee to place any restriction on it other than it is some nite real number. Next, e kno that e nee: u 0 () > 0 u 00 () < 0 for risk aversion. We have: u 0 () b + c u 00 () c Since e nee c < 0, e nee c < 0. Since e nee b + c > 0, an c < 0, e nee b > jcj, here I have note the absolute value of c just to make certain everyone is clear that c is a positive number. b Over hat omain of ealth can a quaratic vn-m utility function be e ne? Using our restriction on b above, the quaratic utility function can be e ne for 0 jcj. Again, I am making sure that everyone is clear about the sign of c. Once > b jcj, our quaratic utility function is no longer increasing. Consier the function: u () + 45 b 3

4 u Once > 45 c Given the gamble or > :5 the function begins ecreasing. g sho that CE < E (g) an that P > 0. ( + h) ; ( h) We kno that: u (g) u ( + h) + u ( h) Using u () a + b + c, e have: u (g) a + b ( + h) + c ( + h) + a + b ( h) + c ( h) u (g) a + b + bh + c + ch + ch + a + b bh + c ch + ch u (g) a + b + c + ch By e nition, the certainty equivalent is the certain amount of money hich gives the same utility as the gamble, so e have: No, e nee to n E (g): u (CE) u (g) u (CE) a + b + c + ch E (g) ( + h) + ( h) E (g) Thus, the utility of the expecte value of the gamble is: u (E (g)) u () u (E (g)) a + b + c 4

5 No e can compare u (E (g)) ith u (CE). We ant: u (E (g)) > u (CE) If this is the case, then e can say that E (g) > CE since u () is increasing. Comparing e have: a + b + c > a + b + c + ch 0 > ch No, this looks like a violation, but recall that c < 0 an h > 0, so ch < 0. Since E (g) have P > 0. CE > 0, e Sho that this function, satisfying the restrictions in part a, cannot represent preferences that isplay ecreasing absolute risk aversion. For the quaratic function satisfying the restrictions in part a e cannot have ecreasing absolute risk aversion because e oul nee to have that the Arro-Pratt risk aversion parameter ecreases hen ealth increases. We can n that the Arro-Pratt risk aversion coe cient for this quaratic is: No, i erentiate ith respect to : R a () R a () u 00 () u 0 () c b + c So as increases, so must R a (). R a () c (b + c) R a () c ( ) (b + c) c R a () (c) (b + c) 5