We wish to compare the savings behavior of two individuals. We interpret time t = 0 as the present, and time t = 1 as the future, and study preferences over the consumption stream (c0, c1). The first individual, named Uthbert, values the stream (c0, c1) with the utility function U(c0, c1) = u(c0) + u(c1) where u(x) = 2√x, while the second individual, named Valbjørg (Val to her friends), values the stream (c0, c1) with the utility function V(c0, c1) = v(c0) + v(c1) where v(x) = 10x - x^(3/10). By assumption, all consumption levels are between 0 and 10, and both Uthbert and Valbjørg have W = 8 to be allocated between the two periods and face the constraint that c0 + c1 ≤ W. Throughout, X is a mean 0 random variable.
D.1. Give the first three derivatives of u(·) and v(·).
D.2. When there is no risk, both individuals prefer equal consumption in both periods. Explain why this is true, and be complete. [The best answers will use some of the properties of the derivatives that you found in the first part of this problem.]
D.3. Risk in either period lowers both individuals' utility, that is, both Uthbert and Valbjørg prefer (c0, c1) to either (c0 + X, c1) or to (c0, c1 + X) where X is a mean 0 random variable. Explain why this is true, and be complete.
D.4. Suppose now that Uthbert's consumption in the future is subject to an additive mean 0 shock. Specifically, time 1 actual consumption, ca1, is equal to planned consumption, c1, plus X where X is uniformly distributed on [-1, +1]. Show that c1 > c0 in the problem maxc0,c1 u(c0) + E[u(c1 + X)] s.t. c0 + c1 ≤ W.
D.5. Suppose now that Valbjørg's consumption in the future is subject to an additive mean 0 shock. Specifically, time 1 actual consumption, ca1, is equal to planned consumption, c1, plus X where X is uniformly distributed on [-1, +1]. Show that c1 < c0 in the problem maxc0,c1 v(c0) + E[v(c1 + X)] s.t. c0 + c1 ≤ W.