Consider an environment with one consumer, two firms, and two goods, a and b. (a) The consumer's utility from consuming xa R+ of good a and x R+ u(xa, xb) = and the consumer's wealth is 10. Solve the consumer's optimization problem for prices pa, Pb > 0: max (xa,xb)R xa + xb s.t. xa 0, xb 0, Paxa Pbxb 10. (b) Firm 1 produces good a and does not produce good b. Firm 1 can use capital, which has a fixed cost of 0 and constant marginal cost of 1, and labor, which also has a fixed cost of 0 and constant marginal cost of 1, to produce good a, and the quantity of good a that Firm 1 can produce from KER+ of capital and L = R+ of labor is qa(K1, L1) = KL. Solve the Firm 1 optimization problem for price pa > 0: max KL - K - L (K1, L1) ER2 s.t. K10, (c) Firm 2 produces good b and does not produce good a. Firm 2 uses just labor to produce good b, and the quantity of good b that Firm 2 can produce from L = R+ of labor is 3 96(42) = Solve the Firm 2 optimization problem for price p > 0: max L2R 3 PbL - L L2 s.t. L2 0. (d) Solve for the prices p* and pt at which, for both goods, demand and supply match, and give the corresponding good amounts.