Question

Consider an infinitely lived agent who has one unit of a commodity and she consumes it over lifetime. The commodity is not perishable and she receives no dividend or interest on saving. Consumption of the commodity in period t is denoted Xt. Let the lifetime utility function be given by: u(x0, x1,...,) = Σ B^t (ln Xt).[Limit 1 to infinity]. Here, 0 < B < 1.

Compute the consumer’s optimal consumption level for each period.

Answer 1

Consumption Xt

$U=Σ B^t (ln Xt).$

Optimal consumption

$U=XB$