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Consider a firm that uses two imputs, capital K and labor L, to produce output using the production function f(K, L) = KaL b . The output price is p and the input prices are w for labor and r for capital. The firm is constrained by a law that says that it must use exactly the same number of units of both inputs. Find the optimal input combination that maximizes firm’s profits.
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- 2. A firm faces the production function 1 Q = f(K,L) = 80 [0, 4K-0,25 +0,4L-0,25] 0.25. It can buy the inputs K and L at prices per unit of 5 TL and 2 TL respectively. What combination of L and K should be used to maximize output if its input budget is constrained to 150 TL?A firm employs labor and capital by paying $40 per unit of labor employed and $200 per hour to rent a unit of capital. The production function is given by: Q=60L-2L^2+180K-3K^2, where Q is total output. Determine the firm's optimal combination of capital (K) and labor (L)?Hannah and Sam run Moretown Makeovers, a home remodeling business. The number of square feet they can remodel in a week is described by the Cobb-Douglas production function Q=F(L,K) Q=10L^0.5K^0.5,where L is their number of workers and K is units of capital. The wage rate is $250 per week and a unit of capital costs $250 per week. Suppose that when initially producing 100 square feet a week, they use 10 units of capital.a. What is their short-run cost of remodeling 1,000 square feet per week? Instructions: Enter your answer as a whole number. $ b. What is their long-run cost of remodeling 1,000 square feet per week? Instructions: Enter your answer as a whole number. $
- Suppose that a firm has production function F(L, K) = L^2/3 K^1/3 for producing widgets, thewage rate for labor is w = $400, and the rental rate of capital is r = $25.a) Suppose that the firm has received an order for Q = 120 units of output. Neatly specify this firm’s costminimization problem, using the particulars associated with this problem.b) Give two equations that an interior solution satisfies, tailoring your equations to the particulars of thisproblem.c) Solve the two equations for the firm’s optimal choice. Show your work.d) Determine this firm’s minimum cost of producing 120 units.e) Now suppose that the firm’s production goal is left as the variable Q. Come up with the firm’s costfunction C(Q). Show your work.A profit-maximizing, perfectly competitive firm produces a single output from two inputs. The logarithm of the maximized profit function is given by: Int (p,r,w) = a, + a, In w+ a, ln r + a, In p where w, r, and p denote the price of labor, the price of capital, and the price of the firm's output, respectively. a) Derive the supply function and the demand functions for capital and labor. b) What restrictions can you place on the parameters a1, a2, and a3?Suppose in the long-rung the production function of a competitive firm is Q=f(L,K)= L1/5 K 1/2 , where L is the amount of labor and K is the amount of capital. The cost per unit of labor is w and the cost of capital is r, which is the interest rate. The price per unit of output is p. Derive unconditional factor demand functions.
- Suppose that widgets can be produced using two different production techniques, A and B. The following table provides the total input requirements for each of five different total output levels. Q = 1 Tech. K L A 4 1 B 1 3 Total Cost Q = 2 K L 2 5 6 2 Q=1 $6 L K Q = 3 Assuming that the price of labor (PL) is $2 and the price of capital (PK) is $1, calculate the total cost of production for each of the five levels of output using the optimal (least-cost) technology at each level. K 9 4 Q = 4 L K L 4 12 7 11 8 8 5 12 7 15 To do this, complete the table below by calculating the total cost of production, filling in the missing values using the optimal (least-cost) technology at each level. (Enter your responses as whole numbers.) Q = 5 $27 Q=2 $ 10 Q = 5 K L Q = 3 $ 17 Q = 4 $ 26 How many labor hours (units of labor) would be employed at each level of output? How many machine hours (units of capital)? To answer this, complete the table below for the units of labor and units of capital that…Catalina Films produces video shorts using digital editing equipment (K) and editors (L). The firm has the production function Q(K, L)=KxL, where Q is the hours of edited footage. The wage is $25, and the rental rate of capital is $50. The firm wants to produce 3,000 units of output at the lowest possible cost.a) Find the marginal product of each input.b) Determine whether the production function exhibits diminishing marginal product to each input.c) Find the marginal rate of technical substitution(MRTSLK)d) How does MRTSLK change as more L, is used holding output constant.e) Find the least costly combination of labor and capital to produce 3000 unitsConsider a firm that produces widgets according to the following Cobb-Douglas production function: Q = A * L^α * K^β where: Q is the quantity of output, L is the quantity of labor, K is the quantity of capital, A is a scale parameter (total factor productivity), α and β are the output elasticities of labor and capital respectively. Given that A = 1, α = 0.6, β = 0.4, L = 16 and K = 9, a) Calculate the quantity of output Q. b) If the firm increases the quantity of labor (L) to 20 while keeping the quantity of capital (K) constant, what will be the new quantity of output?
- Solve the attahment.A firm operates in the short run with the production function Q = 10 K0.5 L 0.5, where Q is output, K is capital, and L is labor. The price of capital is $100 per unit, and the wage rate is $50 per unit of labor. How many units of capital and labor should the firm use to minimize its total cost while producing 400 units of output?Assume that the production function for competitive firm is given by F(L)=9L1/3, where L is the number of units of labour used in the production process. Suppose that the cost per unit of labor is $3 and the price of output is $9 per unit, how many units of labor will the firm hire?
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