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We will take a cobb Douglass production function for solow swan model. In the long run both labour and capital will vary.
Production function
Y = f ( A, K, L)
Y= K^a L^ b
a+b=1
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- The Cobb-Douglas production function is given by Y = AK" L-. Here a is a given parameter that satisfies 0< a< 1. The marginal product of capital is and shows Douglas production function displays and the level of output returns to capital. The Cobb- per worker can be written as a function of the level of Oa: diminishing marginal; constant returns to scale; technology a: diminishing marginal: increasing returns to scale: capital per worker ai diminishing marginal; constant returns to scale; capital per worker (1 - a) : increasing marginal; decreasing returns to scale; capital per workerIf the production function describing an economy is Y= 20 KL23, then the share of income going to capital: O is about 33 percent. O depends on the state of technology. is about 67 percent. depends on the quantities of labor and capital.Consider the following production functions: 1. Y = AK2/3L1/3 2. Y = AK + BL 3. Y = (AK)34L3/4 4. Y = AH2L 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.
- Suppose the production function for widgets is given byq = kl - 0.8k2 - 0.2l2where q represents the annual quantity of widgets produced, k represents annual capital input, and l represents annual labor input.a. Suppose k = 10; graph the total and average productivity of labor curves. At what level of labor input does this average productivity reach a maximum? How many widgets are produced at that point?b. Again assuming that k = 10, graph the MPl curve. At what level of labor input does MPl = 0?c. Suppose capital inputs were increased to k = 20. How would your answers to parts (a) and (b) change?d. Does the widget production function exhibit constant, increasing, or decreasing returns to scale?In your "toast" production function, you used your labor and a toaster as capital. Keeping capital constant, i.e., with only one toaster if you keep adding the labor, i.e., bring your friends to help you with making toasts, which of the following might happen? Group of answer choices Marginal returns to labor will be a constant. Marginal returns to labor will keep decreasing and, after a point, it will become negative. Marginal returns to labor will keep increasing. Marginal returns to labor will keep increasing and then be a constant.Suppose the production function is given by O = 5K - 2L What is the marginal product of the 6th capital when 5 units of capital and 10 units of labor are employed? A5 b. 4 C. 11 d. 45
- Prof. Smith and Prof. Jones are going to produce a new textbook. The production function for the book is: ?=?1/2?1/2?= is the number of pages in the finished book?= is the number of working hours spent by Smith?= is the number of working hours spent by JonesSmith's labor is valued at 3 TL per working hour and Jones's labor is valued at 12 TL per working hour. After having spent 900 hours preparing the first draft, Smith cannot contribute any more to the book. Jones will revise the Smith's draft to complete the book.a) How many hours will Jones have to spend to produce a finished book of 300 pages?b) What is the marginal cost of the 300th page of the finished book?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)Suppose the production function is given by Q=K1/2L1/2, and that Q=30 and K = 36. How much labor isemployed by the firm?A. 49 B.6 C.36 D.25
- The concept of diminishing returns to a factor of production applies not only to capital but to labor as well. Prior to the Industrial Revolution, there was no sustained growth in living standards. Draw a graph to illustrate the relationship between population and output produced, where population is measured on the horizontal axis, both pre- and post-Industrial Revolution. 1.) Using the 3-point curve drawing tool, draw the production curve showing diminishing returns to labor. Label your curve 'Pre.' 2.) Using the 3-point curve drawing tool, draw the production curve after the Industrial Revolution. Label your curve 'Post. Carefully follow the instructions above and only draw the required objects. aConsider the following diagram of the per-worker production function (output per worker as a function of capital per worker) A. Production function Capital equipment per worker ($ thousands) Which 2 of the following statements are correct? Select one or more: а. The higher capital per worker is, the higher the marginal product of capital is. b. The diagram is consistent with the diminishing marginal product of capital. O c. When workers have less capital, an increase in capital per worker has much more impact than when they already have a lot. O d. The slope of a tangent to the curve shows the marginal product of labour Output per worker ($ thousands)1. Assume that production technology is defined as Q = AL K³ where Q is the quantity produced, L is the labor, K is the capital, and A is the productivity parameter. The output Q needs to be at Q = Qº. Find the marginal rate of substitution of capital for labor, MRS = |OK/OL, needed to keep the output Qº using the implicit function theorem.