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- 1. Consider the production function Q=K³/ª L/4 Further suppose that capital is fixed at 16 units. Answer the following questions: a. Find Q when L =16 units. What is the output (Q) when L increases to 81 units? e b. Find the average product of labor when L when L increases 81 units? 16 units. What is the average product of labor %3D c. Find the marginal product of labor when L 16 units. What is the marginal product of labor when L increases 81 units? d. Show that doubling all the inputs doubles the output.Question 19 Consider the following graph of a production function when capital is constant. (The following is a description of the figure: it shows a two-axis graph; the horizontal axis measures labor and the vertical axis measures output; for a K fixed, the graph shows that maximal production that the firm can achieve with different levels of labor; the graph starts at cero production for zero labor; then it is increasing in all of its range; three levels of labor are shown as reference; there are L1, L2, and L3; they are related as follows L1<L2<L3; the graph is convex from 0 to L1, that is, its slope is increasing; the graph is concave from L1 on, that is, its slope is decreasing; the line that is tangent to the curve at L2, passes through the origin of the graph.) From the graph we know that for the corresponding K: MPL(L2,K)=MPL(L3,K) MPL(L1,K)>MPL(L2,K) MPL(L1,K)=MPL(L2,K) MPL(L1,K)<MPL(L2,K) MPL(L3,K)>MPL(L2,K)PS3 (2) Question: What is the marginal product of capital? Given: * The marginal product of labor in the production of computer chips is 50 chips per hour. * The marginal rate of technical substitution of hours of labor for hours of machine capital is 1/4. * MRTS MPL/MPK =
- Suppose the production function is given by a = 3k - 41. what is the average product of capital when 10 units of capital and 10 units of labor are employed? A. 3 B. 4 C. 7 D. 452. 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)).Some economists believe that the US. economy as a whole can be modeled with the following production function, called the Cobb-Douglas production function: Y = AK¹/32/3 where Y is the amount of output K is the amount of capital, L is the amount of labor, and A is a parameter that measures the state of technology. For this production function, the marginal product of labor is MPL = (2/3) A(K/L)¹/³. Suppose that the price of output P is 2, A is 3, K is 1,000,000, and L is 1/100. The labor market is competitive, so labor is paid the value of its marginal product. a. Calculate the amount of output produced Y and the dollar value of output PY. b. Calculate the wage W and the real wage W/P. (Note: The wage is labor compensation measured in dollars, whereas the real wage is labor compensation measured in units of output)
- 5. A company that produces auto parts has a production function of Q = 300 L-75 K-5. %3D If L= 250 and K= 25, what is the output of auto parts? If L increases to 350, what is the output of auto parts? What does the change in output show regarding labor as a factor of production?22. Consider the following production function: Q = 8K³/4L¹/², with capital (K) and labour (L) as the two inputs. (a) Find the marginal products of capital and labour and comment on whether they reflect increasing, constant or decreasing marginal returns. What type of returns to scale does the production function exhibit? (b) What are the average products of capital and labour and what happens to them as inputs increase? How do the marginal and average products relate to each other? Provide an intuitive explanation as to whether this relationship is as you would expect. (c) Assuming that the prices of Labour and Capital are w and r respectively, find the optimal conditional input demands of capital and labour. (d) Suppose that the firm must keep output at the original level, Qo, but that the price of labour, w, falls. By drawing a diagram, illustrate and explain how the firm's optimal inputs would change and what this would mean for the firm's demand curve for labour.Suppose that you are the Production Manager of Mini Me, Inc., and you have estimated the relationship between total product (or output) and labor for Prime action figures to be: TABLE 1 0 45 100 170 230 260 275 280 280 270 Total Product Labor (In Man Hours) 0 10 15 20 25 30 35 40 45 50 Assuming that there is one fixed input (machine) and one variable input (labor) in making the action figures (1 point per item): 1.Compute the marginal product of labor (round-off to the nearest whole number).
- Consider the following production functions: 1. Y = AK2/3L1/3 2. Y = AK + BL 3. Y = (AK)3/4L3/4 4. Y = AH²L %3D For each of the production functions listed above: Determine whether the function exhibits CRS, diminishing returns to physical capital (or а. human capital, when applicable), and diminishing returns to labor. b. Check whether it satisfies the Inada conditions. c. Compute the per capita production function.Question 24 Consider the following production function when K is fixed. (This is a description of the figure: it shows a two-axis graph; in the horizontal axis we measure labor and in the vertical axis we measure output, in this case articles as in the production function of a newspaper; the graph of the production function, for the given level of capital K fixed, is composed of two line segments; the first goes from the origin to the point (10,30); the second, starting from 10 on, is a horizontal line; it is also shown that the production for five units of labor is 15). Can we say that the production function satisfies the law of decreasing marginal returns of labor?True FalseIn Example 6.4, wheat is produced according to the production function: | = 100 (K°-$L05). Beginning with a capital input of 4 and a labor input of 49, show that the marginal product of labor and the marginal product of capital are both decreasing. (Round your responses to two decimal places.) The MPK at 5 units of capital is 156.52. The MPK at 6 units of capital is 142.89 The MP, at 50 units of labor is 14.14. The MP, at 51 units of labor is 14. Does this production function exhibit increasing, decreasing, or constant returns to scale? A. Decreasing returns to scale because the inputs exhibit diminishing marginal returns. B. Increasing returns to scale because the inputs exhibit diminishing marginal returns. C. Constant returns to scale because a proportionate increase in all inputs results in the same proportionate increase in output.