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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
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- 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)?Suppose the long-run production function for a competitive firm is f(L,K)= L 1/3 K 1/4 , 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. Fixed costs are zero. .a. Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. .b. Draw the conditional input demand functions for labor and capital in the L-y and K-y spaces. .c. Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? .d. Write down the formula and draw the graph of the average cost and marginal cost functions, as functions of y.Suppose the long-run production function for a competitive firm is f(x1,x2)= min {x1,2x2}. The cost per unit of the first input is w1 and the cost of the second input is w2. .a. Find the cheapest input bundle, i.e. amount of labor and capital, that yields the given output level of y. .b. Draw the conditional input demand functions for labor and capital in the x1-y and x2- y spaces. .c. Write down the formula and draw the graph of the firm’s total cost function as a function of y, using the conditional input demand functions. What is the relationship between the returns to production scale and the behavior of the total costs? .d. Write down the formula and draw the graph of the average cost function, as a function of y. .e. Write down the formula and draw the graph of the marginal cost function, as a function of y.
- Let y = f(x1, x2)=x11/2 + x1x2 be a firm’s production function, where x1≥0, x2≥0. Write down the firm’s production possibility set, and its input requirement set. Is this production function concave, quasi-concave? Is this production function homogenous? Find its returns to scale when x1=1, and x2=1.The Director of ABC Enterprise hires labour (L) and rents capital equipment (K) in a competitive market to produce mango juice. At the moment, the wage rate of labour is GH¢2 per hour and capital is rented at GH¢5 per hour. Also, the unit price of mango juice is GH¢0.75 and total cost of production is GH¢1,000. Suppose the firm’s production function (Q) follows a Cobb-Douglas specification given as: 0.5 0.5 ?=14? ? +10 Determine the optimal input usage and the maximum profit that ABC Enterprise would obtain at the optimal input levels.Suppose a firm’s conditional factor demand functions (the inputs needed to produce q at minimum cost with prices w and q) for labor and capital are given by L(w, q, r) = 4rq and K(w, q, r) = 4wq . What is its total cost function?
- 7. The production function for a price-taking firm is given by q = 2.5k0.4L0.4. What are the demand functions for labor 1(v,w.p) and capital k(v,w,p)? [Show your work] 8. The production function for a price-taking firm is given by q = 2.5k0.4L0.4. What is the supply function q(v,w,p)? [Show your work]Consider a firm that has production function f(L,K)= 3L2/3K1/3. What is the expression for this firm’s Marginal Product of capital? MPK(L,K)= 3L2/3/K1/3. MPK(L,K)= 3L2/3/K2/3. MPK(L,K)= L1/3/K1/3. MPK(L,K)= L2/3/K2/3. MPK(L,K)= 2L2/3/K1/3.Suppose that Zamboni Enterprises is the only company that sells zambonis (ice resurfacing machines). To produce the machines, the company hires assembly workers. Since these workers can work in many different companies, Zamboni Enterprises must pay them the market wage, which is equal to $6. The number of zambonis that the company produces, which is denoted by y, is proportional to the number of assembly workers that it hires, which are denoted by N; in particular, the production function is given by y=0.76N. The economywide demand for zambonis is given by the following demand function: y=2191-219p, where y is the number of zambonis that consumers are willing to purchase at price p. Given this market structure, how many assembly workers will Zamboni Enterprises choose to hire? How many zambonis will Zamboni Enterprises produce and sell?
- Suppose that Zamboni Enterprises is the only company that sells zambonis (ice resurfacing machines). To produce the machines, the company hires assembly workers. Since these workers can work in many different companies, Zamboni Enterprises must pay them the market wage, which is equal to $6. The number of zambonis that the company produces, which is denoted by y, is proportional to the number of assembly workers that it hires, which are denoted by N; in particular, the production function is given by y=0.76N. The economywide demand for zambonis is given by the following demand function: y=2191-219p, where y is the number of zambonis that consumers are willing to purchase at price p. Given this market structure, how many assembly workers will Zamboni Enterprises choose to hire? How many zambonis will Zamboni Enterprises produce and sell? What will be the price of a zamboni? If the market for zambonis were competitive, how many zambonis would be produced? If the market for…Suppose that Zamboni Enterprises is the only company that sells zambonis (ice resurfacing machines). To produce the machines, the company hires assembly workers. Since these workers can work in many different companies, Zamboni Enterprises must pay them the market wage, which is equal to $6. The number of zambonis that the company produces, which is denoted by y, is proportional to the number of assembly workers that it hires, which are denoted by N; in particular, the production function is given by y=0.76N. The economywide demand for zambonis is given by the following demand function: y=2191-219p, where y is the number of zambonis that consumers are willing to purchase at price p. If the market for zambonis were competitive, how many zambonis would be produced? If the market for zambonis were competitive, how many assembly workers would be hired? If the market for zambonis were competitive, at what price would zambonis be sold?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.