Question

Suppose a consumer i faces a two-period lifespan and must choose a consumption plan, ?0 and ?1, in order to maximize his total utility, which is time separable of the form: ?(?0, ?1 ) = ?(?0 ) + ??(?1 ), where 0 ≤ ? ≤ 1 is the consumer’s constant “time preference”, or discount factor, for period 2 consumption. The consumer receives income of ?0 at the beginning of period 1 and ?1 at the beginning of period 2. Finally, there is a capital market, so the present value of the consumer’s lifetime consumption must equal the present value of the consumer’s lifetime wealth, ?0. Thus: ?? = ?0 + ?1 (1 + ?) = ?0 + ?1 (1 + ?) .

b. If the consumer has log utility, ?(?) = ??(?), show the first order conditions for ℒ.

Answer 1

Here, FOCs are First Order conditions