Problemset 52018 - Problem set 2018 PDF

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Economics for Business Studies. Problem Set 5. (1) A competitive firm has production function: y = 2x1 + 3x2 . Factors prices are [1, 3], respectively. What is the minimum total cost for producing y = 100? (2) For each of the following functions, determine whether the function is a cost function. If it is a cost function, find the marginal cost function and the derived demand for each input. Explain whether returns to scale are increasing, constant, or decreasing. Note that wi is the price per unit of factor i, while y is output. √ • (w1a + w2a) y for a = 1 • (w1 )α(w2 )β y γ where α, β, γ are any positive numbers. (3) Prove rigorously that profit maximization implies cost minimization (4) For each of the following production functions, calculate the cost function and the corresponding conditional demand functions. Find and plot marginal cost. Explain whether returns to scale are constant, decreasing or increasing. 0.3 • y = x0.3 1 + x2 • xα1 x2β , where α, β > 0 (5) A firm has a production function: f (x1 , x2 ) = x1αxβ2 , with α, β > 0 and α + β < 1. Let wi be the price of input i. • Compute the firm’s conditional factor demand functions and the cost function. • Assume that in the short run, x2 is fixed and equal to x¯2 . Compute: The short-run conditional demand function for input 1, the short-run total cost function, the total, average and marginal variable and fixed cost functions. (6) Let C(y) be the total cost function of a firm. If C(y) = 144 + 16y 2 . Determine the minimum average cost (7) The total cost function of a competitive firm is c(y) = 2 + (y 2 /3). At what market price the firm is producing 30 units of y ? (8) Suppose that a firm has a cost function c(y) = ay α, with α > 1 . For this cost function: • Find the profit-maximizing output level as a function of the price of output, p. • Assume that there are N firms in the industry with the same technology and derive the industry supply function. • Assume that α = 2 and that the industry demand function is given by D(p) = b − dp, with b and d > 0, and find the competitive equilibrium of the industry. • What is the competitive equilibrium with free entry? 1...