Expected Utility Questions
1. Exercise 4.2 from Lengwiler.
2. Consider the following ìportfolio choiceî problem. The investor has initial wealth w and utility v(x) = ln x: There is a safe asset that has a payo§ of 1 in both states. There is also a risky asset with a random payo§, it pays Rh with probability p and Rl < Rh with probability 1 p. Let A be the amount invested in the risky asset, so that w A is invested in the safe asset. Both assets cost 1, i.e. q = (1; 1) : Rh and Rl are such that there is no arbitrage.
(a) Find A as a function of w. Does the investor put more or less of his portfolio into the risky asset as his wealth increases?
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(b) Another investor has the utility function v(x) = e x. How does her investment in the risky asset change with wealth?
(c) Find the coe¢ cients of absolute risk aversion A(x) = v00 (x) for the two v0 (x)
investors. How do they depend on wealth? How does this account for the qualitative di§erence in the answers you obtain in parts a: and b:?
3. Consider the following lotteries on the outcomes {5,1,0}
p = (0:00; 1:00; 0:00); q = (0:10; 0:89; 0:01); r = (0:10; 0:00; 0:90); s = (0:00; 0:11; 0:89):
(a) Show that there are lotteries on the outcomes {5,1,0}, say x and y, and a number 2 [0; 1] such that
p = p+(1 )p; q= x+(1 )p r = x+(1 )y; s= p+(1 )y
(b) Show that an agent who satisÖes the expected utility hypothesis will rank these lotteries as follows:
p q () s r:
4. Consider the expected utility function of a risk-averse decision-maker:
s u(xs) s=1
For S = 2 and given , draw the indi§erence curves of the expected utility function in a state-contingent outcome diagram.
5. In a two-period economy, a consumer has Örst-period initial wealth w. The consumerís utility level is given by
u (c1; c2) = u (c1) + v (c2) ;
where u and v are concave functions and c1 and c2 denote consumption levels in the Örst and the second period, respectively. Denote by x the amount saved bytheconsumerintheÖrstperiod(sothatc1 =w xandc2 =x),andletx0 be the optimal value of x in this problem.
We now introduce uncertainty in this economy. If the consumer saves x in the Örst period, his wealth in the second period is given by x + y, where y is distributed according to . In what follows, E[] always denotes the expectation of y with respect to . Assume that the vNM utility function over realized wealth levels in the two periods (w1; w2) is u(w1) + v(w2). Hence, the consumer now solves
maxu(w x)+Ey [v(x+y)]: x
Denote the solution to this problem by x.
(a) Show that if the coe¢ cient of absolute risk aversion of v (w), A (w) ; is decreasing with wealth w, then v000 (w) > 0; that is P (w) > 0. Hence DARA implies prudent behaviour.
(b) Show that if v000 > 0, and E[y] = 0, then E[v0(x + y)] > v0(x) for all values of x.
(c) Show that if E[v0(x0 + y)] > v0(x0), then x > x0:
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