Question: (a) A firm uses K and L units of two inputs to produce √(KL) units of a product, where K > 0, L > 0. The input factor costs are r and w per unit, respectively. The firm wants to minimize the costs of producing at least Q units. Formulate the nonlinear programming problem that emerges. Reformulate it as a maximization problem, then write down the Kuhn-Tucker conditions for the optimum. Solve these conditions to determine K∗ and L∗ as functions of (r, w, Q).
(b) Define the minimum cost function as c∗(r, w, Q) = rK∗ +wL∗. Verify that ∂c∗/∂r = K∗ and ∂c∗/∂w = L∗, and give these results economic interpretations.