3. A firm has a production function, q = k0.5 10.5, where q is daily output (in kilogram) and I and k are daily inputs of worker-hours and machine-hours respectively. In the short run, the firm's machine-hours is fixed at k = 40 hours. The rental rate for k is v= £3, and the wage rate for I is w = £12. In a perfectly competitive market, the output yields a price P = £12 per kilogram. Derive the firm's short-run total cost (STC) function. (a) (b) (c) Derive the firm's profit function. What is the necessary first-order condition for the firm to maximise profit when /> 0? Compute the profit-maximising amount of labour as a function of the wage. What is the effect of an increase in wage on the firm's optimal employment level?

3. A firm has a production function, q = k0.5 10.5, where q is daily output (in kilogram) and I and k are daily inputs of worker-hours and machine-hours respectively. In the short run, the firm's machine-hours is fixed at k = 40 hours. The rental rate for k is v= £3, and the wage rate for I is w = £12. In a perfectly competitive market, the output yields a price P = £12 per kilogram. Derive the firm's short-run total cost (STC) function. (a) (b) (c) Derive the firm's profit function. What is the necessary first-order condition for the firm to maximise profit when /> 0? Compute the profit-maximising amount of labour as a function of the wage. What is the effect of an increase in wage on the firm's optimal employment level?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter10: Cost Functions

Section: Chapter Questions

Problem 10.9P

See similar textbooks

Similar questions

Suppose the production function for high quality brandy is given by : Q = √KLWhere q is the output of brandy per week and L is labor hours per week ., in the short run K is fixed at 100, so the short run production function is Q = 10√La. If the capital rents for 10$ and the wage are 5$per hour ., write the short run total cost function. b. How much will the firm produce at a price of 20$ per bottles of brandy ?c. How many labor hours will be hired per week?
The production function for a product is given by q= 10K^(1/2)L^(1/2) where K is capital, and L is labor and q is output. a) Find the marginal product of labor and the marginal product of capital. b) Find the marginal rate of technical substitution between labor and capital. c) Denote the wage of labor by w and the rental of capital by r. What is the cost minimization condition for a firm? Show it diagrammatically. d) Now suppose w =30 and r = 120. What is the minimum cost of producing q=1000. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.) Part A, B, C
Consider a production function of two inputs, labor and capital given by Q=LK. Let w =5 and r = 1, where w is the price per unit of labor and r is the price per unit of capital. a) Suppose that the firm is required to produce Q units of output. Show how the cost-minimizing quantity of labor depends on the quantity Q? b) If the firm is operating in the short run, with K fixed at 10 units, calculate the short-run total cost of producing 100 units of output?
Suppose a short-run production function is described as Q=30L-0.05L^2 where L is the number of Labors used each hour. If each unit of output (Q) has a marginal revenue (price) of $5 and the marginal cost of labor is $40 per labor unit (L), how many units of labor (L) should be hired to maximize profit?
E1 Suppose that production for good X is characterized by the following production function, Q = K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $15 and the per-unit wage, w, is $25, then the average fixed cost of using 9 units of capital and 81 units of labor is $75. incalculable since there is insufficient information to determine the average fixed costs. $80. $5.
Consider a purely competitive firm that has two variable inputs L (labor hour) and K (machine) for production. The price of product is $p. The production function is Q (K; L) = 4L^1/4 K^1/4 . Assume that the hourly wage of workers is fixed at $w and the price per machine is $r Write out the optimal inputs quantities, L and K, as a function of parameters, p, w, and r
The production function for a product is given by q = 10 K1/2L1/2 where K is capital, and L is labor and q is output. a) Find the marginal product of labor and the marginal product of capital. b) Find the marginal rate of technical substitution between labor and capital. c) Denote the wage of labor by w and the rental of capital by What is the cost minimization condition for a firm? Show it diagrammatically. d) Now suppose w =30 and r = 120. What is the minimum cost of producing q=1000. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.) e) Now suppose that the firm is in the short run and cannot vary the amount of capital. That is, it must use the same amount of capital as in part d. However, the firm wants to produce 1200 units of output. How much labor should it use to minimize its cost and what is the minimum cost of producing q =1200?
A perfectly competitive rm produces output q with capital K and labor L according to the production function: q = f(K, L) = 4K 1 4L 1 4 The price of labor is w = 4, and the price of capital is r = 4. The rm has xed costs equal to 8. The current market price is 8. (a) Prove that this production function has decreasing returns to scale. (b) Find the optimal (cost-minimizing) ratio of capital to labor inputs K L for any level of output (use scale expansion path). (c) For any q, nd the cost minimizing inputs as functions of output: i.e., derive functions L(q) and K(q).
Sugar Enterprises manufactures high speed personal computers as its main product line. Its short run production function is given by Q = 3600K - 4.5K² where Q is the number of computers produced in a week and K pertains to the number of units of capital. c.) Compute the marginal product of the 200th unit of capital. d.) How many computers can be produced by the firm in a week if there are 200 units of capital? e.) Compute the marginal product of the 300th unit of labor. f.) How many computers can be produced by the firm in a week if there are 300 units of labor?