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 = δ u for a common function u and discount factor δ. Suppose the investor has a random endowment ỹ at the end of the period, so he chooses c0 to maximize:

u(c0) + δE[u((w0 - c0)Rf +  y̆ )]

Suppose the investor has convex marginal utility(u''' > 0) and suppose that E[ỹ] = 0. Show that the optimal c0 is smaller than if ỹ = 0

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