3. 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 uo = u and u = du for a common function u and discount factor 8. Suppose the investor has labor income Ỹ at the end of the period, so she chooses Co to maximize u (Co) + 6E [u (Wo – Co) R + Y)] . Suppose the investor has convex marginal utility (u" > 0) and suppose that E Y = 0. Show that the optimal Co is smaller than if_ Y = 0. Note: This illustrates the concept of precautionary savings the risk imposed by Ỹ results in higher savings Wo - Co.
3. 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 uo = u and u = du for a common function u and discount factor 8. Suppose the investor has labor income Ỹ at the end of the period, so she chooses Co to maximize u (Co) + 6E [u (Wo – Co) R + Y)] . Suppose the investor has convex marginal utility (u" > 0) and suppose that E Y = 0. Show that the optimal Co is smaller than if_ Y = 0. Note: This illustrates the concept of precautionary savings the risk imposed by Ỹ results in higher savings Wo - Co.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:3. 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
uo = u and u1 = du for a common function u and discount factor d. Suppose the investor
has labor income Ỹ at the end of the period, so she chooses Co to maximize
u (Co) + SE u ((Wo – Co) R¡ + Ỹ
Suppose the investor has convex marginal utility (u" > 0) and suppose that E Y = 0.
Show that the optimal Co is smaller than if Y = 0. Note: This illustrates the concept of
precautionary savings the risk imposed by Y results in higher savings Wo – Co.
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