2. Cost Functions: A firm operates with a production function, y = 4√lk and input pricesPe = 6 and Pk = 7. In the short run, capital is a fixed input, k = 36.(a) Find the firm's cost function, c(y), the minimum cost to produce a given amount ofoutput.(b) In the long run, both labor and capital are variable inputs. Suppose thatwhat will the firm do in the long run? What if the price is py:=0.50?¹=Py 20;

A mathematical function that depicts the relationship between the input factors of production and…

Answered: Cost Functions: A firm operates with a…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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(a) Find the short-run cost function, SRTC(q) and marginal cost function, MC(q), assuming that k = 27, and letting w and v denote the price of labor and capital. (b) Find the long-run cost function and the long-run marginal cost function (L and k are both chosen optimally). (c) Give a formal definition of increasing, decreasing, and constant returns to sale, and determine if the production function above exhibits increasing, decreasing or constant returns to scale. (d) Derive the firm’s contingent labor demand function
Find the production which has the following cost function (q is output, v= cost of capital, w=labour cost): C = qv^1/3w^2/3
b) Suppose a business faces a production function which is of the Cobb-Douglas form: Q(L,K) = AL“ Kß where Q represents total product, L represents labour units, K represents capital units, A is some fixed constant representing technology and a and B are fixed parameters. Show that if a + B> 1 there is increasing returns to scale. Also show that the output elasticities with respect to labour and capital are constants and equal to a and B, respectively.
If the production function is f(x1,x2) = x1,x2.a. Show whether this production function exhibits increasing, decreasing or constant returns to scale.b. Show whether the average cost is increasing, decreasing or constant.
Suppose a firm has the following production function Q = f(K,L) = (1/2)L(2/3)K(1/3) and a total cost function TC = wL + rK The rental cost of capital is 2 euros per hour and the labor wage is 4 euros per hour. Suppose also that in the short-run, the firm uses 8 machines in the production process. (a) Show that the firm’s short-run average cost (SRAC) is SRACK=8 = (16/Q)+ 4√? (b) Draw the SRAC curve(c) By solving the first order conditions that the global minimum of production is 4(d) Show that the firm’s long-run average cost (LRAC) is 12. Hint: Start by obtaining the firm’s long-run expansion path(e) Draw both the SRAC and the LRAC and explain why the SRAC is always equal or above the LRAC.
Suppose a firm’s cost function is given by: C= 50 + 4q2 a) Find the equations for AFC, AVC and MC. b) If w=$4, what’s the short run production function for this firm?
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A firm estimates its production function as Q = -2.4 + 20*L -.5*L2 + 30*K -1.00*K2 + 15*F -.3*F2. Total cost is TC = $2*L + $2.5*K + $1.75*F. The firm wishes to produce 400 units of output. Use Excel to find the cost minimizing amounts of L, K and F. How much F (Fuel) should the firm use?
Given the following production function: q = 10KL. Assume that w = 25, r = 75 and C = 1200. (a) Mathematically find the minimum cost combination of capital and labour to produce a given level of output.b) Does the production function in part (a) show increasing returns to scale, decreasing returns to scale, or constant returns to scale? Explain. (c) Using isoquants and isocosts, graphically illustrate the effect of an increase in the wage rate, assuming the firm is producing at the same level of output.
Which of the following production function would exhibit increasing returns to scale? f(K, L) = (K+ L)1/2 f(K, L) = K + L f(K, L) = K1/4+ [1/4 Of(K, L) = KIL None of the above
Consider the following short run cost function: TC = 4Q³ - 24Q² + 16Q + 100 Derive the expressions for marginal cost and average total cost. What can we infer about the range of output over which production shows increasing returns to labor and diminishing returns to labor?
If a firm has the production function q = f(L, K) = L + 2K, then its technology exhibits: constant returns to scale increasing returns to scale None of the above. decreasing returns to scale increasing marginal product

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