Consider the production function fl k 2lk 5k13 where l

Question

Q1. Consider the production function f(L, K) = 2LK + 5K^1/3, where L is the amount of labor and K is the amount of capital. Calculate the average product of labor and the marginal product of capital. Is the marginal product of capital diminishing?

Q2. Consider the production function f(L, K) = 3L² + 4K^1/3. Calculate the isoquant for production level q^- = 10 and express this in the form of K as a function of L.

Q3. Consider the production function f(L, K) = √3(K^1/2L^2/3). Find the level of production q^- for which the MRTS equals -16 when L = 1.

Q4. Consider the production function f(L, K) = (max{3L, K})^3/4 (min{√L, 5√K})^x, where x > 0. Is this production function homogeneous of any degree? If so, what value of x makes it exhibit constant returns to scale?

Q5. Suppose that the short-run cost function of a firm is c^SR(q) = 100 + 3q + q². Calculate the average cost, the average variable cost, and the marginal cost at production level q = 10.

Q6. Consider a firm operating in the short run and using the production function f(L, K) = min{3√L, K²}. Suppose that the wage rate is w = 2 and the rental rate is r = 1.

(a) What is the maximal level of output q_max that the firm can produce in the short run when the capital is fixed at level K^-?

(b) Suppose that capital is fixed at K^- = 5 in the short run. Calculate the function L(q, K^-) that gives the quantity of labor required to produce any level of output q that the firm can actually produce.

(c) Calculate the short-run cost function of the firm and draw it in a graph.

(d) Calculate the average cost, the average variable cost, and the marginal cost of the firm.

Q7. A firm operates in the long run using the production technology f(L, K) = 3√(LK) + L + 4K.

(a) Calculate the marginal product of labor and capital.

(b) For each level of output q ≥ 0, derive the level of inputs L^∗(w, r, q) and K^∗(w, r, q) that allow the firm to produce q at the lowest cost in the long run, given that the prices of labor and capital are w = 20 and r = 80.

(c) Calculate the long-run cost function of the firm.

(d) Suppose that the firm expects that, in three years, the price of its product will be p = 10 and the prices of labor and capital will continue to be w = 20 and r = 80. Would it make sense for the firm to invest over the next three years so as to grow and achieve a level of output q = 140?