# Suppose we are given a profit function Q = 12L^.5K^.5. The price of labor is $6 per unit and the...

## Question:

Suppose we are given a profit function {eq}Q = 12L^.5K^.5 {/eq}. The price of labor is $6 per unit and the price of capital (K) is $7 per unit.

The firm is interested in the optimal mix of inputs to minimize the cost of producing any level of output Q.

In the optimal mix, the ratio of labor to capital is _____.

## Profit Maximization:

In standard economic theories, firms are assumed to have the sole objective profit-maximization. The problem of cost-minimization and profit-maximization are called the dual problems.

## Answer and Explanation: 1

Become a Study.com member to unlock this answer! Create your account

View this answerAt the optimal input mix, the ratio of marginal product is equal to the ratio of input prices, i.e.,

- {eq}\dfrac{MPL}{MPK} = \dfrac{12 * 0.5...

See full answer below.

#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

from

Chapter 24 / Lesson 6Learn the profit maximization definition, its importance, and explore the profit maximization theory. See how to calculate profit maximization with examples.

#### Related to this Question

- Suppose we are given a profit function Q = 12L0.5K0.5. The price of labor (L) is $6 per unit and the price of capital (K) is $6 per unit. The firm is interested in the optimal mix of inputs to minimize the cost of producing any level of output Q. In the o
- Suppose a firm's production function is given by Q = 2L + K. Also, the price of Labor, w = 10, and the price of Capital, r = 4. If the firm minimizes the cost of production, how much will it cost the
- Assume a firm has a production function Q = 2 S L 5 K 5 and the price of labor is $3 and the price of capital is $12. a) What is the minimum cost of producing 1,250 units of output? b) Now show t
- Assume a firm has a production function Q = 25 L 5 K 5 and the price of labor is $3 and the price of capital is $12. a) What is the minimum cost of producing 1,250 units of output? b) Now show tha
- A firm's production function is given by Q = 2L - L^2 + K. The price of labor is w > 0 and the price of capital is r > 0. Assuming the firm uses both labor and capital. Suppose w = 1 and r = 1. At what output level is K = 3 a cost-minimizing choice of cap
- Suppose the production function for a competitive firm is Q = K^.75L^.25. The firm sells its output at a price of $32 and can hire labor at a wage rate of $2. Capital is fixed at 1 unit. a. What is the profit-maximizing quantity of labor? b. If the price
- A firm's production function is given by Q = 2L - L^2 + K. The price of labor is w > 0 and the price of capital is r > 0. Assuming the firm uses both labor and capital. Suppose w = 1 and r = 1. At what output level is L = 0.5 a cost-minimizing choice of l
- Suppose that a firm's production function is q = 10L1/2K1/2. The cost of a unit of labor is $20 and the cost of a unit of capital is $80. a. Derive the long-run total cost curve function TC(q). b. The firm is currently producing 100 units of output. Find
- Suppose a firm's production function is Q = 4K0.5 L0.5. Its level of capital is fixed at 4 units, the price of labor is PL = $16 per unit, and the price of capital is PK = $20 per unit. The firms aver
- Consider a profit-maximising, price-taking firm that only uses capital as input at a cost of v per unit. Its production function is given by f(k) = 11 - \frac{1}{1 + k}. What is its cost function (whe
- A firm is producing output Q using a mix of capital K and labor L. The production function is given by . A unit of capital costs $3 and a unit of labor costs $9. The firm wants to minimize the total c
- Suppose a firm produces output using the technology Q=K1/3 L2/3 Find a. The long run cost function b. The short run cost function if capital is stuck at 10 units. c. The profit maximizing level of out
- Suppose a firm's production function is given by the equation Q = 12L^.5K^.5 . This firm operates in the short run where capital (K) is fixed at a quantity of 16. If the price per unit of the good is $1.9 and labor costs $10 per unit. Then the profit-maxi
- A firm's product function is Q = 5L^{0.5}K^{0.5}. Labor costs $40 per unit and capital costs $10 per unit. K = 16 in the short run. Determine the level of output that minimizes SAC for the firm. A. 40 B. 80 C. 113 D. 135
- Suppose a firms production function is Q = 0.2K0.5 L0.5. Its level of capital is fixed at 25 units, the price of labor is PL = $8 per unit, and the price of capital is PK = $4 per unit. The firms aver
- Suppose a firms production function is Q = 2K0.5 L0.5. Its level of capital is fixed at 9 units, the price of labor is PL = $12 per unit, and the price of capital is PK = $10 per unit. The firms avera
- Suppose in the short run a firm's production function is given by Q = L^(1/2) x K^(1/2), and that K is fixed at K = 9. If the price of Labor, w = $12 per unit of Labor, what is the firm's Marginal Cost of production when the firm is producing 48 units of
- 1. Suppose that a firm's production function is q = 10L^{1/2}K^{1/2}. The cost of a unit of labor is $20 and the cost of a unit of capital is $80.a. If the firm wishes to produce 100 units of output,
- Suppose that a firm's production function is q = 5x^{0.5} in the short run, where there are fixed costs of $1,000, and x is the variable input whose cost is$1250 per unit. The total cost of producing a level of output q is C(q) = 1,000 + \frac{1250q^2}{25
- Suppose a firm has a production function given by Q = L1/2K1/2. The firm can purchase labor, L, at a price w = 8, and capital, K, at a price of r = 2. a) What is the firm's total cost function, TC(Q)? b) What is the firm's marginal cost of production?
- Suppose in the short run a firm's production function is given by Q = L1/2K1/2, and K is fixed at K = 36. If the price of labor, w = $12 per unit of labor, what is the firm's marginal cost of production when the firm is producing 48 units of output?
- Suppose a firm with a production function given by Q = K0.25L0.75 spends $6000. The firm pays a wage of $30 per units and pays a rental rate of capital of $10 per unit. The maximum output that can be
- Suppose that a firm's production function is Q = min{K, L}. Currently, the wage is w = 8 and the cost of capital is r = 8. (a) What is the minimum cost method of producing Q = 40 units of
- A firm's production function is given by Q = 2L - L^2 + K. The price of labor is w > 0 and the price of capital is r > 0. Assuming the firm uses both labor and capital, derive the long-run total cost function.
- Suppose a price-taking firm has a cost function given by C (q) = q + 1 / 2 q^2. If the price of output is 2, what is the profit-maximizing choice of q? (a) 0. (b) 1. (c) 2. (d) None of the above.
- Consider a firm whose production function is Q = 0.4K^0.5L^0.5. Its level of capital is fixed at 100 units, the price of labor is PL = $4 per unit, and the price of capital is PK = $2 per unit. Given this information, the firm s total cost function is A)
- A firm produces according to the following production function: Q = K0.25L0.75. The price of K is $4 per unit, and the price of L is $6 per unit. a. What is the optimal capital/labor ratio? b. Derive the amount of capital and labor required to produce 400
- Suppose that capital costs $10 per unit and labour costs $5 per unit. For a profit-maximizing firm operating at its optimal factor mix, if the marginal product of capital is 50, the marginal product of labour must be A) 10 B) 20 C) 25 D) 50 E) 100
- Consider a firm with production function f(L,K)=3L1/3K2/3. Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the variable cost of producing q units is what?
- Suppose that a production function of a firm is given by Q= min{2L,K}, where Q denotes output, K capital, and L labor. Currently the wage is w=$10, and the rental rate of capital is r=$15. a. What is the cost and method of producing Q=20 units of capital
- What is the cost-minimizing level of capital that the firm must use to produce a target level of output, Q = 1600? A firm's production process uses labor, L, and capital, K, and materials, M, to produce an output, Q according to the function Q= KLM, where
- Suppose that capital costs $10 per unit and labour costs $4 per unit. If the marginal product of capital is 50 and the marginal product of labour is 50, then in the long run the firm should in order to minimize its costs of producing its output. A) emplo
- Consider a firm with the production function f(L,K)=L^{1/5}K^{4/5}. Assume that the price of capital r=3 and the price of labor w=2. If L^* and K^* are the amounts used by the firm to produce q units of output when both L and K are variable, then what is
- Suppose a firm has a production function given by Q = K^{0.5}L^{0.5}. The firm pays a wage of $64 per unit and pays a rental rate of the capital of $5 per unit. K = 256 in the short run. The maximum level of profit this firm could make in the short run if
- A firm produces output with the production function F(K,L) = K^{1/2}L^{1/2}. The price of capital is p_K = 10 and the price of labor is p_L = 40. Find the cost-minimizing input bundle for producing Q = 50 in the long run.
- Suppose in the short run a firm's production function is given by Q = L^{1/2}K^{1/2}, and that K is fixed at K = 36. If the price of Labor, w = $12 per unit of Labor, what is the firm's marginal cost of production when the firm is producing 48 units of ou
- Consider a firm with production function f(L,K)=3L+8K. Assume that capital is fixed at K=12. Assume also that the price of capital r=10 and the price of labor w=3. Then, the average cost of producing q units is what?
- Output is produced according to Q = 4LK, where L is the quantity of labor input and K is the quantity of capital input. The price of K is $10 and the price of L is $5. Determine the cost minimizing co
- A firm has a total cost function of: TC(Q) = 100 + 100Q + Q^3/100, where 'Q' is the firm's output level. A) Find the function that gives the average cost of production. B) What output level minimizes
- Suppose the cost function of a firm is given by C(Q) = 250 + 2Q2. If the firm sells output in a perfectly competitive market at a price of P = $8, what level of output should the firm produce to maximize profits or minimize losses in the short run? What w
- Consider a firm with production function f(L,K)=3L1/3K2/3. Assume that capital is fixed at K=1. Assume also that the price of capital r=5 and the price of labor w=3. Then, the average fixed cost of producing q units is what?
- Suppose a production function is q = K^(1/2)L^(1/3) and in the short run capital (K) is fixed at 100. If the wage is $10 and the rental rate on capital is $20, the short run marginal cost is _____.
- Suppose a firm's production function is given by Q = LK^2. Suppose the firm is producing 16 units of output by using 1 units of Labor and 4 units of Capital. What is the slope of the isoquant at this
- The production function for a product is given by q = K1/2L1/4 where K is capital, L is labor and q is output. a. Find the marginal products of labor and capital. b. Is the marginal product of labor increasing or decreasing with labor? Is the marginal p
- Suppose the production function is q = 12L0.25K0.75. Determine the long-run capital-to-labor ratio \frac{K}{L} if the cost of a unit of capital ''(r)'' is three times the cost of a unit of labor ''(w)''.
- Suppose the production function is q = 12 L^{0.25} K^{0.75}. Determine the long-run capital-to-labor ratio (K/L) if the cost a unit of capital (r) is three times the cost of a unit of labor (w).
- Given a firm demand function, P 24 0.5 Q and the average cost function, AC Q2 8Q 36 3 Q. Calculate the level of output Q which: a maximizes total revenue b maximizes profits
- Suppose a firm with a production function given by Q = 30 K^{0.5}L^{0.5} produces 1,500 units of output. The firm pays a wage of $40 per unit and pays a rental rate of capital of $640 per unit. How
- Suppose the firm has fixed costs of $10 and the output price is $8. Using the marginal approach, the firm will maximize profits by producing {Blank} units of output. At the profit-maximizing output level, the firm's total revenue will be ${Blank}, its tot
- A firm's product function is Q = 5L^{0.5}K^{0.5}. Labor costs $40 per unit and capital costs $10 per unit. K = 16 in the short run. Determine the firm's short-run cost function. \text{A.}\; C = 160 + 2Q^2\\ \text{B.}\; C = 640 + Q^2/20\\ \text{C.}\; C = 1
- Suppose a firms production function is given by Q = L^{1 / 2} K^{1 / 2}. The marginal product of labor and the marginal product of capital are given by: MP_L= {K^{1 / 2 / {2 L^{1/2 and MP_k = {L^{1/2 / {2K^{1/2. What is the firms marginal cost?
- Suppose a firm can use either capital (K) or labor (L) in a production process. The firms production function is given by Q = 5L + 15K. The price of capital is $20 per unit and the price of labor is $8 per unit. a. What is the firm's total cost function?
- 2. Given a firm s demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 8Q + 36 + 3/Q, calculate the level of output Q which a) maximizes total revenue b) maximizes profits
- Given a firm's demand function, P = 24 - 0.5Q and the average cost function, AC = Q2 8Q + 36 + 3/Q, calculate the level of output Q which a) maximizes total revenue b) maximizes profits
- Suppose that a firm's production function is q = 10L^{0.5}K^{0.5} . The cost of a unit of labor is $10/hour and the cost of capital is $40/hour, and the firm is currently producing q=1000 units
- Suppose that the production function of a firm is given by the equation Q = 2K1/2L1/2, where Q represents units of output, K units of capital, and L units of labor. What is the marginal product of labor and the marginal product of capital at K = 40 and L
- a) Suppose the marginal produce of labor is: MPL = 10, the marginal product of capital is: MPK = 20, while the cost of labor is: W = $5 and the cost of capital is: r = $30. Is the firm minimizing cost? If not, what should the firm do to produce the sa
- Suppose that a firm has a production function given by q = 10L0.5K0.6. The firm has 10 units of capital in the short run. Which of the following will describe the marginal product of labor (MPL) for this production function? a. increasing marginal returns
- Suppose that a firm has the following production function: Q(K, L) = 2LK^{1/2} a. If the price of labour is $2/unit and the price of capital is $4/unit, what is the optimal ratio of capital to labou
- Suppose a firm with a production function given by Q = K^{0.25}L^{0.75} produces 1,500 units of output. The firm pays a wage of $50 per unit and pays a rental rate of the capital of $50 per unit. To produce 1,500 units of output, the firm should use: a. 1
- The long-run production function for a firm's product is given by q = f(K; L) = 5 K L. The price of capital is $10 and the price of labor is $15. a. Suppose the firm wishes to produce an output of 500. List 5 combinations of capital and labor that the fi
- Consider the production function F(l,K)=3l^{.25}K^{.75} a) Find the cost-minimizing bundle and the long-run total cost if w = 64 and v = 1 and total output = q = 36. b) Change the price of capital to
- The production function for a firm is given by q = L^{.75} K^{.25} where q denotes output; Land K labor and capital inputs. (a) Determine marginal product of labor. Show whether or not the above production function exhibits diminishing marginal produ
- Assume that a firm's production function Q = K1/2L1/2. Assume that the firm currently employs 200 units of capital and 100 units of labor. Determine the Average Product of Capital, Average Product of Labor, Marginal Product of Capital, and Marginal Produ
- A firm's product function is Q = 5L^{0.5}K^{0.5}. Labor costs $40 per unit and capital costs $10 per unit. K = 16 in the short run. Derive the long-run cost function for the firm. A. C = 4q B. C = 17q C. C = 8q D. C = 10q
- Suppose that a competitive firm's marginal cost of producing output q is given by MC(q) = 3 + 2q. Assume that the market price of the firm s product is $9. \\ a) What level of output will the firm produce to maximize profit? b) What is the firm's produc
- Consider a profit-maximizing firm that uses labor, L, as an input to produce its output, Q, according to the production function Q = L^1/2. Labor is paid an hourly wage w. The firm's total revenue is
- Given a firm has a demand function of: P = 45 - 4Q and average cost function of: AC = 1.5Q + 25 + 26/Q a. Find the level of output (Q) which: i. Maximizes total revenue (TR) ii. Makes marginal cost
- Suppose a monopolist has a production function given by Q = L1/2K1/2. Therefore, MPL = , and MPK = The monopolist can purchase labor, L at a price w = 16, and capital, K at a price of r = 9. The dema
- Consider the following production function: Q = 10KL. If w = 25, r = 75, and c = 1200, find the minimum cost combination of capital and labor to produce a given level of output.
- If output is produced according to Q = 5Lk (L is the quantity of labor and k is the quantity of capital), the price of K is $12, and the price of L is $6, then the cost minimizing combination of K and L capable of producing 4,000 units of output is A. L
- Suppose a firm's production function is given by Q = L1/2*K1/2. The Marginal Product of Labor and the Marginal Product of Capital are given by: MPL = (K^1/2)/(2L^1/2), and MPK =(L^1/2)/(2K^1/2). a) S
- Suppose that a firm had a production function given by q = L0.25K0.75. The wage rate (w) is $5 and the rental rate (r) is $10. Calculate the amount of labor the firm would hire when it produces 400 units of output in a cost-minimizing way. (Round to the n
- A firm's product function is Q = 5L^{0.5}K^{0.5}. Labor costs $40 per unit and capital costs $10 per unit. K = 16 in the short run. Suppose the production function of a firm is Q = 5 + 2K + L. Which of the following statements is correct? A. The firm's LA
- A firm has a production function, q = L^{0.6}K^{0.4}. It wants to minimize cost for a given production q. The wage rate and rental rate on capital are w and r, respectively. a. Write the Lagrangian ex
- Suppose a firm with a production function given by Q = 30K^0.5L^0.5 produces 3,000 units of output. The firm pays a wage of $40 per unit and pays a rental rate of capital of $640 per unit. How many units of labor and capital should the firm employ to mini
- Suppose a firm with a production function given by Q = 30 K^{0.5}L^{0.5} produces 1,500 units of output. The firm pays a wage of $40 per unit and pays a rental rate of capital of $640 per unit. How many units of labor and capital should the firm employ to
- A firm's production function is Q = 8L^(1/2) and this firm sells each unit of its product at a price of P = $100. It also pays its workers a wage of w = $10. a. How many workers would this firm hire to maximize its profit if it only has labor costs and no
- A profit-maximizing competitive firm uses just one input, x. Its production function is q = 4x^{1/2}. The price of output is $28 and the factor price is $7. What is the amount of the factor that the firm demands?
- A profit-maximizing competitive firm uses just one input, x. Its production function is q = 4x^{1/2}. The price of output is $28 and the factor price is $7. The amount of the factor that the firm demands is A. 8. B. 16. C. 64. D. 60. E. None of the above.
- A firm has the production function q = 10L^{0.5}K^{0.5}. The price of labor is w = 10 and the price of capital is r = 20. Derive the short-run marginal and average variable cost curves when K is fixed.
- Suppose a firm with a production function given by Q=4K^0.25L^0.75 produces 100 units of output. The firm pays a wage of $30 per units and pays a rental rate of capital $10 per unit. The minimum cost
- Assume a perfectly competitive market. The selling price of a firm is $9 and the profit-maximizing output q* is 80. Producing 80 units cost $800. a. What is the MR for this firm? b. Calculate profi
- Assume that a firm's marginal cost function is MC = 3q and its marginal revenue function is MR = 120 - 3q. At the output level of 18 units, in order to maximize profit, this firm should: a. increase its output by 2 units b. decrease its output by 2 units
- Suppose that the production function is Q = L^{2 / 3} K^{1 / 2}. a. What is the average product of labour, holding capital fixed? b. What is the marginal product of labour? c. Determine whether the production function exhibits diminishing marginal product
- Suppose a firm's cost function is given by C (q) = 2 + 12 q + 3 q^2 and the demand for its product is given by p = 20 - q (the firm can affect the price of its product). What is the firm's maximized profit? (a) 5. (b) 2. (c) 15. (d) 4.
- Capital and labor, denoted by K and L restively, are used to produce output denoted by q, so firm's production function is equal to q = min {K, 2L} The price of capital is $5 and the price of labor is
- A firms production function is given be Q=f(L,K)=L^1/2K^1/2. The prices of labor (w) and of capital (r) are $2.5/unit and $5/unit, respectively. Suppose that the firm has a capital employment of 25 un
- Suppose that a firm's production function is q = 10 L^0.5 K^0.5. This means that the marginal rate of technical substitution is K/L.The cost of a unit of labor is $20 and the cost of a unit of capital is $80. The firm wants to produce 130 units of output.
- Suppose that the firm's production function is given by Q = 10K(L)^1/3. The firm's capital is fixed at K. What amount of labor will the firm hire to solve its short-run cost-minimization problem?
- Suppose a perfectly competitive firm has a cost function described by TC = 200 Q + Q^2 + 225 Each firm's marginal revenue is $240. a. Find the profit maximizing level of output. b. Is this a short-run or long-run situation? c. Assuming that this firm's to
- A firm's product function is Q = 5L^{0.5}K^{0.5}. Labor costs $40 per unit and capital costs $10 per unit. K = 16 in the short run. Suppose the production of a firm is Q = 5 + 2K + L. Which of the following statements is correct? A. The firm's production
- If a firm has a production function f(k,l) = 3k^{0.3}l^{0.5} where r = 8, w = 9, and the price of output = 17, What is profit maximizing level of capital? For this question both the level of capital
- A firm has a demand function P = 200 - Q and cost function: AC=MC=20 and a potential entrant has a cost function: AC=MC=30. a. Determine the optimal price, quantity and economic profit for the firm
- Suppose a firm produces an output measured in units Q. The cost of producing Q units is given by the cost function C(Q) = aQ^2 + bQ + c, where you can assume a 0,b 0,c 0. In Economics we also think about cost per unit (average cost) given by: AC(Q) = C(
- Consider a firm with the production function f(L,K) = L^{0.5}K^{0.5}. The wage rate and rental rate on capital are w and r, respectively. a. Use the Lagrangian for cost minimization to do derive the long-run cost function for this firm. b. Suppose the
- Suppose a firm's production function is q=2K(L+1) where q is output, K is capital input, and L is labor. (a) Confirm whether the iso-quant curve is convex. (b) Suppose that the price of output is p
- Suppose a firm with a production function Q = KL is producing 125 units of output by using 5 workers (L) and 25 units of capital (K). The wage rate (W) per worker is $10 and the rental rate (price) pe
- Suppose a firm's production function is Q = 2K0.5L0.5. If the level of capital is fixed at 25 units, then what is the firm's short-run production function?
- Suppose the cost function is a multivariate of the type: where Q is output and K, capital; L, Labor; and M, materials are inputs. 2.1 Find the marginal products of the inputs.