1. Suppose that a producer has the following production function: -13 Where Q is output, and L and K are man-hours and machine-hour the two inputs used in the production process. IB) (8 Points) Determine the cost-minimize ratio of inputs where capital costs $1 per machine-hour and labor costs $4 per lab-hour.
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- Suppose you run a factory whose short-run production function is given by f (L) = 180L² – 3L³. (a) Find mathematical expressions for the total product of labor, the average product of labor and the marginal product of labor. (b) On two separate axes, graph TPL, APL, and M PL. The diagrams do not need to be scale, but be sure they are internally consistent. In other words, make sure that all relevant relationships between the three lines are correct. If you have trouble figuring out what the graph looks like, feel free to use a graphing calculator or an online graphing tool to find the basic shape. Find the quantity of labor associated with each of the following points and label it on your diagram: (i) the maximum value of TPL, (ii) the maximum value of APL, (iii) the maximum value of MPL. (c) For what values of L does the production function exhibit each of the following: i. increasing marginal labor productivity ii. diminishing marginal labor productivity iii. negative marginal labor…1. Assume a daily production function for a firm is Q= min(3L, 4K) a, If L = 100 and K = 100 what quantity is produced? Explain why. b. What would be the cost-minimizing labor and capital combination for this firm to produce Q = 1200? (show your work) Labor = Capital = At this level of production from part b (Q-1200), if the wage rate is: w- $30 and the rental rate of capital is: r= $100, what is the total cost of production? (show your calculations) C.The following is a production function. 50,000- 45,000- 40,000- Draw a graph of marginal product as a function of labor. 35,000- 30,000- 25,000- Total output (Q) 20,000- 15,000- 10,000 ng 5,000- 0+ 0 200 100 300 Units of labor (L) 400 L Q 1.) Using the line drawing tool, graph the marginal product curve from 0 to 100 units of labor. Label this line 'MP Segment 1" 2.) Using the line drawing tool, graph the marginal product curve from 100 to 300 units of labor. Label this line 'MPS Note: Carefully follow the instructions above and only draw the required objects. Does this graph exhibit diminishing returns? Explain your answer. ○ A. Yes, it does exhibit diminishing returns, because the marginal product of labor decreases. B. No, it does not exhibit diminishing returns, because the marginal product of labor is zero. ○ C. No, it does not exhibit diminishing returns, because the marginal product of labor is increasing. ○ D. Yes, it does exhibit diminishing returns, because the marginal…
- 5. Selpats produces pens with the production function f(k, l) = 200vkl. The price of capital is wk = 40 and the price of labor is wL = 10. (i) Determine the optimal input bundle for producing y pens in the long run and find the long-run cost function. (ii) Determine the optimal input bundle for producing y = 10,000 pens in the long run. (iii) Now suppose that capital is fixed at k = 25 in the short run. Find the short-run total cost function and the short-run marginal cost function. Explain the relationship between the short-run and long-run marginal costs when y = 10,000. (iv) Suppose that the price of output is p. Which value of y maximizes short-run profit? р.A 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.Suppose a firm's production function is Q=200L +5L2-0.08L³. Which of the following is the Average Production of Labor (APL)? Oa. 200+10L-0.24L² Ob. 10L-0.24L² Oc. 5L-0.08L² Od. 200+5L-0.08L²
- 1. Given the production function: q = 2KL, (a) Fill in the table below which has information on the short-run production function, the marginal product of labor (MPL), and the average product of labor (APL) if capital is held fixed at 3 units in the short-run. Labor Quantity of Output (MPL) (APL) 0 1 4 9 16 25 (b) Plot the short-run production function. (c) Plot the marginal and average products of labor on a new graph. (d) Does the law of diminishing marginal returns apply? If so, at what level of labor do diminishing marginal returns set in? (e) What is the relationship between the marginal product of labor (MPL) and the average product of labor (APL)? (f) If the amount of capital increased to 6 units, what would happen to your graphs in part C)?***PLEASE NOTE - An answer is NOT needed for parts A, B and C; these are included to assist with answering part D. Only an answer for part D is required, but it is derived from the previous answers*** Given: A farmer raises peaches using land (K) and labor (L), and has an output of ?(?,?)= ?0.5?0.5 bushels of apples. a. Find several input combinations that give the farmer 6 bushels of apples. Sketch the associated isoquant on a graph, with L on the x-axis and K on the y-axis. b. In the short run, the farmer only has 4 units of land. What is his short-run production function? Graph it for values of L from 0 to 16, with L on the x-axis and output on the y-axis. What is the name of the slope of this curve? c. Assuming the farmer still only has 4 units of land, how much extra output does he get from adding 1 extra unit of labor if he is already using only 1 unit of labor? How much extra output does he get from adding 1 extra unit of labor if he is already using 4 units of labor?…2. Consider a Cobb-Douglas production function with three inputs. K is capital (the number of machines), L is labor (the number of workers), and H is human capital (the number of college degrees among the workers). The production function Y = K2/6 L3/6 H1/6 a) Derive an expression for the marginal product of labor. How does an increase in the amount of human capital affect the marginal product of labor? (Hint: The marginal product of labor MPL is found by differentiating the production function (Y) with respect to labor (L)) b) Derive an expression for the marginal product of capital. How does an increase in the amount of human capital affect the marginal product of capital? (Hint: The marginal product of capital MPK is found by differentiating the production function (Y) with respect to capital (K)).
- Q)solve it correctly The marginal products of capital (MPK) and labor (MPL) are, respectively, MPK = 2000 units; MPL = 1500 units. The input prices are: PK = $10/unit; and PL = $150/unit. To minimize production costs, the firm should A. increase both capital and labor B. decrease both capital and labor C. increase capital; decrease labor D. decrease capital; increase labor E. do nothing; costs are minimized7. The unit price of production input K is 8 euros and the unit price of production input L is 10 euros. What is the minimum cost at which a company could produce 20 units of product Y if the company's production function is Y = K / 4 + L?30. The production function is f(L, M) = 4L1/2 M¹2, where L is the number of units of labor and M is the number of machines used. If the cost of labor is $49 per unit and the cost of machines is $16 per unit, then what is the total cost of producing 5 units of output?