Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. Suppose the investor has time-additive utility with u0 = u and u1 = δu for a common function u and discountfactor δ. Supposethe investor has labor income y˜ atthe end ofthe period, so she chooses c0 to maximize u(c0) +δE[u((w0 −c0)Rf + ˜y)].

Suppose the investor has convex marginal utility (u > 0) and suppose that E[˜y] = 0. Show that the optimal c0 is smaller than ify˜ = 0. Note: This illustrates the concept of precautionary savings—the risk imposed by y˜ results in higher savings w0 −c0.