Microeconomic Theory
12th Edition
ISBN: 9781337517942
Author: NICHOLSON
Publisher: Cengage
expand_more
expand_more
format_list_bulleted
Related questions
Question
thumb_up100%
You are assisting a small manufacturing firm in determining the optimal level of labor input (L) that maximizes profit. The analysis is based on the following production function:
Q = 10L – 0.5L2
Where:
- Q represents the output (units produced),
- L represents the variable input (labor hours).
Additional Information:
- Each unit of output is sold for $10.
- The firm can hire labor at a cost of $20 per hour.
Please derive the following results:
The Marginal Revenue Product
The Marginal Factor Cost
The Optimal Labor Input
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by stepSolved in 2 steps
Knowledge Booster
Similar questions
- Suppose that the production function for Hannah and Sam's home remodeling business is Q = F(L,K) Q = 10L0.1K0.4.Assume the wage rate is $8,000 per week and the cost of renting a unit of capital is $1,000 per week.a. What is the least-cost input combination for remodeling 400 square feet each week? Instructions: Round your answers to 2 decimal places. units of labor and units of capital. b. What is the total cost? Instructions: Round your answer to 2 decimal places. $ .revised jrl 08-11-2011arrow_forwardCraig and Javad run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's longrun production function is Q = 4KL, where Q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. The weekly cost function for the paper plant is C = 20K + 4L, where C is the total weekly cost. (a) What ratio of capital to labor minimizes Craig and Javad's total costs? (b) How much capital and labor will Craig and Javad 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?arrow_forwardSuppose a brewery uses a Cobb-Douglas production function for his production. He studies the production process and finds the following. An additional machine-hour of fermentation capacity would increase output by 600 bottles per day (i. e. MPK = 600). An additional man-hour of labor would increase output by 1200 bottles per day (i. e. MP₁ = 1200). The price of a man-hour of labor is $40 per hour. The price of a machine-hour of fermentation capacity is $8 per hour. 1. Is the brewery currently minimizing its cost of production? Check using the minimization condition. 2. It turns out, the brewery is not optimally choosing the factors of production. To lower its production cost, which factor of production should the brewery increase and which factor should he decrease? 3. Suppose that the price of a machine-hour of fermentation capacity rises to $20 per hour. How does this change the answer from part 1?arrow_forward
- 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.arrow_forwardMiguel 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= 4K¾L where q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. (K For this production, MP, L and MP = 3 K K The weekly cost function for the paper plant is C= 10K +2L where C is the total weekly cost What ratio of capital to labor minimizes Miguel and Jake's total costs? (Hint: Find the Marginal Rate of Technical Substitution (MRTS) for capital and labor.)arrow_forwardA firm produces plastic bins using labor (measured in man-hours) and capital (measured in machine-hours), according to the production function Q = f(L,K) = LK, where Q is the number of plastic bins produced. Suppose that the cost of labor is $20 per worker-hour and the cost of capital is $10 per machine-hour. What is the cost minimizing input combination if the firm wants to produce 28,800 plastic bins? Hint: The marginal products are MP, = K and MPg = L.arrow_forward
- Consider the table below that describes the production function for a good (Q) in terms of inputs labor (L) and capital (K) in the short run.arrow_forwardA firm can manufacture a product according to the production function Q = F(K, L) = K0.5L0.5. (a) What is the average product of labor, APL, when the level of capital is fixed at 36 units and the firm uses 16 units of labor? (b) What is the marginal product of labor, MPL, when capital is fixed at 36 units? (c) Suppose capital is fixed at 36 units. If the firm can sell its output at a price of $100 per unit of output and can hire labor at $40 per unit of labor, how many units of labor should the firm hire in order to maximize profits?arrow_forwardThe production function for a product is given by q = 100KL. Its marginal product functions are MPL = 100K and MPK = 100L. If the price of capital, r, is $120 per day and the price of labor, w, is $30 per day, what is the minimum cost of producing 20,000 units of output? The minimum cost of producing 20,000 units of output is $. (Round all intermediate results to 3 decimal places and enter your response rounded to two decimal places.)arrow_forward
- I haven't been able to answer this correctly, and neither has anyone I've shown, can you help me with this?arrow_forwardSuppose that a firm’s production function is Q =10 K^(3/4)L^(1/4). The cost of a unit of labor is $1 and the cost of a unit of capital is $3. The manager of this firm is interested in finding the following information: The manager now wants to know the K/L ratio to produce anyoutput level at the minimum total cost. Represent graphically the expansion path in the long run.arrow_forwardDerive an expression for the firm’s total cost function given that w= 4 and r= 6arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
