11. A firm has a production function of Y = $100 Log(K), where K is the amount of capital used in production. The slope of the production function is given by $100/K. Suppose the interest rate is 100%. What is the optimal level of capital? a. $1 b. $20 C. $50 d. $100 e. $200
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- If a firm has 5 units of capital, 10 units of labor, and the production function is Q = 4L + 3K what is the average product of capital? 3 4 11 45Suppose a firm has the following production function: Y = A[ln K + ln L]. A is a variable representing the overall productivity of capital and labor. A profit maximizing firm will choose labor and capital so that their marginal products are equal to their marginal costs. Assume that the marginal cost of labor is the wage rate, w, and the marginal cost of capital is the real interest rate, r. a. Find the profit maximizing level of K and L for the firm. b. Does the firm use more or less capital and labor when productivity rises? (Show this mathematically and explain). c. Compute the second derivative with respect to K d. Compute the second derivative with respect to L e. If you were to graph the production function with K on the horizontal axis (and holding L fixed), would the graph be concave or convex? f. What is the partial cross derivative of this production function. Interpret the economic meaning of the result.1) In connection with the Law of the Minimum and the Law of Diminishing Returns, describe how the production function model can be a helpful tool in managing an agricultural production system.
- The equation below is a production function,Q = (200)L + (100)K – (0.2)L2 – (0.1)K2 , where Q is output, L is labor and K is capital. What is the Marginal Product of capital of this function? 200 - (0.4)L - (0.2)L 100 - (0.2)K - (0.1)KFirm ABC has estimated that it has the following production function: Q = 1.5LK – 0.3L2 – 0.15K2 Labor (L) has a price of €60 and capital (K) has a price of €75. ABC wants to maximize the production with a total cost of €2025. (a) Determine the quantities of labor and capital the firm should use. (round results to 2 decimal places) (b) What quantity of output will be produced at the optimal combination of factors of production?Suppose that the production function takes the form X = min(10L, 5K) and that a competitive firm faces a wage rate of £60 per week and a weekly capital rental of £32. (a) How much must the firm spend to produce 100 units of output, and what is the average cost of production when X = 100? (b) What is the incremental cost of producing the 101st unit of output? (c) What happens to the cost of producing 100 units of output if the wage rate and the rental cost of capital rise by 25 per cent each? What happens to the average and marginal cost? (d) What happens to the cost of producing 100 units of output if the wage rate increases by £1, or if the cost of capital increases by £1?
- Economics Suppose that production for good X is characterized by the following production function, Q = 4K0.5L0.5, where K is the fixed input in the short run. If the per-unit rental rate of capital, r, is $12 and the per-unit wage, w, is $20, then the average total cost of using 25 units of capital and 49 units of labor is Multiple Choice $6.25. $9.14.Correct $10.07. incalculable since there is insufficient information to determine the average total costs.Question 33 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? AC(q)=24/q+1 AC(q)=114/q+9/8 AC(q)=24/q+3 AC(q)=88/q+1/12 AC(q)=24/q+qSuppose an automobile manufacturer is choosing between two production options. It can produce 100 cars with 200 workers and 50 machines, or it can produce 166 cars with 300 workers and 75 machines. Would you describe the manufacturer’s production function as exhibiting decreasing, constant, or increasing returns to scale? Explain.
- If a firm has 5 units of capital, 10 units of labor, and the production function is Q = 4L + 3K what is the Marginal product of capital? 3 4 11 45Please explain why the law of diminishing returns applies only in the short-term period.An electronics plant’s production function is Q = 5LK, where Q is its output rate, L is the amount of labor it uses per period of time and K is the amount of capital it uses per period of time. The Price of labor is $1 per unit of labor and the price of capital is $2 per unit of capital. a) What is the optimal combination of inputs the plant should use to produce 1000 units of output per period? b) Suppose that the price of labor increases to $2 per unit. What effect will this have on optimal L and K to produce Q=1000? c) Is this plant subject to decreasing returns to scale? Why or why not?