A firm with Perfect Competition in the Labour Market and Perfect Competition in the Output Market. (Long Run) You are the manager of a business that faces perfect competition in the output market and perfect competition in the labour market. The production function of the business is given by: Q = 2K0.25L 0.50. The price of the product is $100. The wage rate is $5 and the price of capital is $10. a) Find an equation that describes the relationship between the optimal level of labour and capital. b) Draw a diagram of the solutions.c) Explain your approach in words

A firm with Perfect Competition in the Labour Market and Perfect Competition in the Output Market. (Long Run) You are the manager of a business that faces perfect competition in the output market and perfect competition in the labour market. The production function of the business is given by: Q = 2K0.25L 0.50. The price of the product is $100. The wage rate is $5 and the price of capital is $10. a) Find an equation that describes the relationship between the optimal level of labour and capital. b) Draw a diagram of the solutions.c) Explain your approach in words

Exploring Economics

8th Edition

ISBN:9781544336329

Author:Robert L. Sexton

Publisher:Robert L. Sexton

Chapter11: The Firm: Production And Costs

Section: Chapter Questions

Problem 15P

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New taskFor a producer under complete competition, the following production function applies. Q = F (K, L) = KL where Q is the size of production, K is the size of the capital investment and L is the amount of labor. Assume that the price of capital, r, is 5 and that the salary, w, is 10. a) Assume initially that the capital is locked at 4 units. How much labor will the producer use to produce 50 units of the final product? What will be the total cost of producing these 50 units? b) For the production function above applies The marginal product for work MPL = K The marginal product for capital MPK = L Enter the cost-minimizing combination of capital and labor in the production of 50 units. Also enter the size of the costs. c) Write down the equation for the isocost line for the manufacturer
3. a. b. C. d. Don Corneo's firm has a Cobb-Douglas production function given as: Y = AK 0.5 N 0.5 Show that the production function exhibits Diminishing returns with respect to Capital, K. Show that the production function can exhibit Constant returns to scale Estimate the output per worker, N. Calculate the elasticity of substitution, o.
Question 1. Suppose a small firm employs Capital (K) and labour (L) at its production of output. The production function of the firm is: q = -56³ + 10L²K + 100LK0.5. Assume that K-25 in the short run. a. What is its short-run production function? b. What are the firm's marginal product of labour and average product of labour in the short run? c. In the short run, when marginal output is maximized, what is the value of L? d. Calculate the short-run elasticity of output with respect to labour when L=10 and K-25 (Hint: the firm's elasticity of output with respect to labour in the short run is a function of marginal product of labour and average product of labour)
Economics The company A produces cars using three inputs, labour (L), capital (K)and talent (X). ACME's production function is Y(X, L, K) - х0.5LO.25к0.25 This problem only deals within the short run, where ACME's amount of capital is fixed at K = 9. Denote by w the labour wage, by r the price of capital and by q the price of talent. a) Find A's conditional input demand for talent and labour. b) Find A's cost function. What is its shutdown price (as a function of w, rand q)? c) Find A's short run supply function. Draw it together with any other curves. -х0.5L0.25 к0.25 %3D
Exercise 11 Question Help Suppose that a firm's production function is: 112- q = 10L0.5K0.5 98- Its marginal product functions are: 84- MP, = 5KO.5/L0.5 - 0.5g/L 70- and MPK = 5L0.5/K.5 - 0.5g/K. 56- The cost of a unit of labor, w, is $50 and the cost of a unit of capital, r, is $50. Isoquants for output of 140 and 280 are illustrated in the figure to the right. 42- 28- Initially, the firm is producing 140 units of output and has determined that the cost-minimizing quantities of labor and capital are 14 and 14, respectively. 14- -q= 280 q= 140 98 Suppose now that the firm wants to increase output to 280 units. If capital is fixed 0- in the short run, how much labor will the firm require? 70 84 L, Units of labor 14 28 42 56 112 The firm will now require units of labor. (Enter your response rounded to two decimal places.) K, Units of capital
Q87 Consider a firm that uses only labour and capital as inputs. At the present use of labour and capital, the MP of labour is four times the MP of capital and the price of labour is twice the price of capital. To minimise its costs for a given level of output, the firm should... a. Decrease both capital and labour. b. Increase both labour and capital. c. Decrease capital and increase labour. d. Stay at its present factor mix. e. Decrease labour and increase capital.
Suppose that labor is the only input used by a perfectly competitive firm. The firm’s production function is as follows: Number of Workers: 0, 1, 2, 3, 4, 5, 6, 7 Units of Output: 0, 7, 13, 19, 25, 28, 29, 29 a. Calculate the marginal product for each additional worker. b. Each unit of output sells for $10. Calculate the value of the marginal product of each worker. c. Compute the marginal profit if the wage is $100 a day. 2. Your enterprising uncle opens a sandwich shop that employs 7 people. The employees are paid $12 per hour, and a sandwich sells for $6. If your uncle is maximizing his profit, what is the value of the marginal product of the last worker he hired? What is that worker’s marginal product?
2. Suppose a firm faces the input prices of labor and capital of w = $15 and r = $2, respectively. The firm's production function is q = 4K°.5L°5. The firm currently has 40o units of capital, the amount of which cannot be easily changed. The firm has flexibility in adjusting the amount of labor used in the production process. Find the short-run average cost of producing 400 units of output. Round up your answer to the nearest cent.
4. Suppose that Leah's company is characterised by the following Cobb- Douglas production function z = f(x, y) = 50x¹/2²y¹/2 where is the number of units of labour, y is the number of units of capital and z is the units of production. Each unit of labour costs $25 in wages and each unit of capital is $5. Leah's company is also subject to a limit of $75,000 for the total cost of labour and capital. Find the number of units of labour and the number of units of capital that will allow Leah's company to maximise production and use this information to calculate the z for which production is maximised.
4. Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is Q=4K 0.75 0.25 "where Q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. The rental rate of capital is R = $10 per unit and wage rate is W $15. a. Write the constrained optimization problem for Miguel and Jake. b. How much capital and labor will Miguel and Jake need to rent and hire in order to produce 1,000 reams of paper each week? c. How much will hiring these inputs cost them (Total cost)? d. Graph the isocost line, isoquant curve, and the optimal capital and labor including intercepts.