1. A manufacturing firm's production function is Q=KL+K+L. For this production function, MPL=K+1 and MPK=L+1. Suppose that the price r of capital services is equal to 1, and let w denote the price of labor services. If the firm is required to produce 5 units of output, for what values of w would a cost-minimizing firm use
a) only labor?
b) only capital?
c) both labor and capital?
2. Q=L1/2+K1/2+M1/2. For this production function, the marginal products of labor, capital, and materials are MPL =1/(2 L1/2), MPK =1/(2/K1/2), and MPM= 1/(2M1/2). Suppose that the input prices of labor, capital, and materials are w=1, r=1, and m=1, respectively.
a) Given that the firm wants to produce Q units of out- put, what is the solution to the firm's long-run cost- minimization problem??
b) Given that the firm wants to produce Q units of output, what is the solution to the firm's short-run cost- minimization problem when K = 4? Will the firm want to use positive quantities of labor and materials for all levels of Q??
(c) Given that the firm wants to produce 12 units of output, what is the solution to the firm's short-run cost- minimization problem when K= 4 and L =9? Will the firm want to use a positive quantity of materials for all levels of Q?
3. A hat manufacturing firm has the following production function with capital and labor being the inputs: Q= min(4L, 7K )-that is, it has a fixed-proportions production function. If w is the cost of a unit of labor and r is the cost of a unit of capital, derive the firm's long-run total cost curve and average cost curve in terms of the input prices and Q.