Question 1) A consumer must decide how much of a good to consumein period 0 and period 1, co and c1 respectively. Goods not consumedin period 0 become capital, k1, and are used to produce more goodsin period 1. There is an initial amount of capital, k0, that producesgoods in period 0. The production function is f(kt) = kαt Each periodthe consumer gets a utility that depends on consumption in that periodgiven by u(ct) = log ctfor t = 0, 1. The consumer's objective isto maximize the discounted expected utility. Solve the maximizationproblem faced by the consumer.(1) maxco,c1log co + β log c1s.t. co + k1 ≤ kαoc1 ≤ kα1co ≥ 0, c1 ≥ 0
Question 2) A price taking firm has a production function given byf(K, L) = K.25L.25. The price of output is normalized to 1, and theprices of L and K are w and r. Find the profit maximizing amountof K and L and verify that the second-order conditions or sufficient conditions are satisfied.