Reference no: EM133577036

Consider a consumer who consumes a single good in two periods. The income in period 0 is I0 > 0, and the income in period 1 is I1 > 0. Let x0 ≥ 0 be the consumption at period 0, and x1 ≥ 0 be the consumption at period 1. The utility function 1 is u = ln x0 β ln x1, where 0 < β < 1, and the consumer can use a risk-free total return ¯r ≥ 1 in period 0 to save s ≥ 0. (a) Use the Kuhn-Tucker condition to find the optimal consumption-savings plan (x∗ 0, x∗ 1, s∗), expressed as the function of I0, I1, β, and ¯r. Under what conditions will saving s∗ be strictly positive? (b) Is the optimal consumption-savings plan (x∗ 0, x∗ 1, s∗) homogeneous of degree one in income (I0, I1)