Precautionary saving

Question

Precautionary saving Assume a representative consumer who lives for 2 periods, t = 0, 1, and whose lifetime utility is given by

U = ln c₀ + E ln c₁,

where E is the expectation operator. The interest rate is normalized to zero. The current income y₀, earned at time t = 0, is known to the consumer, y₀ = 10. The future income, ye1, to be earned at period t = 1, is known to be a random variable:

ye₁ = 20 with probability 50 % (optimistic scenario), and ye₁ = 0 with probability 50 % (pessimistic scenario). 1) Compute the permanent income yp given

by yp ≡ 1 2 (y0 + Eye1) 2) Show that

c0 > yp. Explain the intuition.

Answer 1

This is a bit of a Tricky one. E(ln(C1)) less than ln(E(C1) the difference is a Variance term.so ln(C0) + ln(E(C1) is Greater Than ln(C0) +E(ln(C1).

If C0 and E(C1) were the same Then Utilty could be increased by increasing C0 and reducing E(C1) by the same amount. In the example if C+ = E(C2) and Y1 = E(Y2) = permanent income C)= E(C1) implies no saving. Utility could be increased By dissaving or C)>permanent income