Question

1. Consider the cost function, C = 2q^2(wr)^1/2. Suppose the firm with this cost function is perfectly competitive in its output market and faces an output price, p. A. Find the marginal cost function for output. B. Find the average cost function. Show your work. C. Is this a long-run or short-run cost function? Explain. D. Derive the cost elasticity with respect to output. Show your formula and all calculations. What does this value of cost elasticity tell you about the degree of scale economies? Explain. E. Derive the supply function for output. Show all your logic and work. 2. Consider the cost function, C(q) = 0.5 q^2 − 10q + 200. A. Is it a short or long run cost function. Explain. B. Calculate the short-run average total cost function. C. Find the output level at which average total cost is minimized. Show your logic and work. D. Calculate the output supply function for a competitive firm whose market determined price of output is given by p. Show your logic and work. E. What is the shut-down condition? Explain its logic. 3. Consider the production function, q = L^1/4E^1/4K^1/4, where L is labor, E, energy, and K, capital. Their prices are respectively w, u, and r. The price of output is given by p. Output and input prices are competitively determined so that they are parametric (constants). Capital, K, is a fixed input. The short-run variable cost function is given by CV(q, w, u, K) = 2q2(wu/k)1/2. A. Use the short-run variable cost function to find the profit-maximizing output supply function. B. Find the maximum profit function. C. Compute the derivative of the maximum profit function with respect to output price, p. Does it look familiar? D. Compute the derivative of the maximum profit function with respect to input price, w, and multiply it by − 1: − ∂π/∂w where π denotes the maximum profit function. This derivative gives the profit-maximizing demand for labor. Together, the derivatives, − ∂π/∂w, − ∂π/∂u, and ∂π/∂p, constitute Hotelling’s lemma, the analog to Shephard’s lemma for the cost and expenditure functions.

Answer 1