Consider the following production functions: 1. Y = AK23L3 2. Y = AK + BL 3. Y = (AK)4L3/4 4. Y = AH2L For each of the production functions listed above: а. Determine whether the function exhibits CRS, diminishing returns to physical capital (or human capital, when applicable), and diminishing returns to labor. b. Check whether it satisfies the Inada conditions.
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- 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?Explain in details and stepsSuppose 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?
- Consider the production function Y = K1/2 N12 a. Compute output when K = 49 and N = 81 %3D b. If both capital and labor double, what happens to output? c. Is this production function characterized by constant returns to scale? Explain. d. Write this production function as a relation between output per worker and capital per worker. e. Let K/N = 4. What is Y/N? Now double K/N to 8. Does Y/N double as a result? f. Does the relation between output per worker and capital per worker exhibit constant returns to scale? g. Is your answer to f. the same as your answer to c.? Why or why not?Suppose the production function for widgets is given by q=kl -0.8k²-0.21² where q represents the annual quantity of widgets produced, k represents annual capital input, and I represents annual labor input. Suppose k = 10; at what level of labor input does this average productivity reach maximum? (please put your answer in numerical values with no comma or decimal place). How many widgets are produced at that point? (please put your answer in numerical values with no comma or decimal place). If k = 10, at what level of labor input does MPL = 0? Suppose capital inputs were increased to k = 20, at what level of labor input does this average productivity reach maximum? widgets are produced at that point? (please put your answer in numerical values with no comma or decimal place). If k = 20, at what level of labor input does MPL = 0? answer in numerical values with no comma or decimal place). Does the widget production function exhibit constant, increasing, or decreasing returns to scale?…The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C)=²1C²2 where co. ₁, and care constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume co5, c, 0.25, and c₂-0.75. Assume sach unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, devalop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. L, CZO € 70,000 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0S LS 3,000 and 0 s Cs 1,000. Round your answers to the nearest integer when necessary.) units at (L. C)=(
- In economics and econometrics, the Cobb-Douglas production function is a particular functional form of ne production function, widely used to represent the technological relationship between the amounts of two r more inputs (particularly physical capital and labor) and the amount of output that can be produced by nose inputs. The function they used to model production is defined by, P(L, K) = 6LªK!-a where P is the total production (the monetary value of all goods produced in a year), L is the amount f labor (the total number of person-hours worked in a year), and K is the amount of capital invested (the onetary worth of all machinery, equipment, and buildings). Its domain is {(L, k)|L > 0, K > 0} because L nd K represent labor and capital and are therefore never negative. Show that the Cobb-Douglas production function can be written as P P(L, K) = 6LªK1-a → In K L In b+ a ln KConsider 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.) Denote by APL(L,K)=f(L,K)/L the so-called average product of labor (here f is the production function of the firm). From the graph we know that for the corresponding K: APL(L1,K)=MPL(L1,K) APL(L2,K)>MPL(L2,K)…1. Suppose the production function for trucks is given by: q = kl + 6l² – - 13 3 where q represents the weekly quantity of trucks produced, k represents weekly capital input, and I represents weekly labor input. a. Suppose k = 45; at what level of labor input does this average productivity reach a maximum? How many trucks are produced at that point? b. Again assuming that k = 45, at what level of labor input does the total production reach a maximum? How many trucks are produced at that point?
- 1) Suppose the prdduction function for widgets is given by: q= KL – 0.8K² – 0.2L?. a.) Suppose K=10, at what level of labor input does this AP1,reach a maximum and how many widgets are pIdduced at that point? At what level of labor input does MP1=0 and how many widgets are produced at that point? Graph the TP1, AP1 and MP1 curves. b.) Suppose capital inputs were increased to K=20. How would your answers to parts (a) change? c.) Does the widget production function exhibit constant, increasing, or decreasing retums to scale?Please no written by hand and no image Suppose that the production function is given by Y=AK0.4N0.6. What is the percentage change in output if both capital and labor rise by 42%? Write the answer in percent terms with up to two decimals (e.g., 10.22 for 10.22%, or 2.33 for 2.33%).Suppose the production function for widgets is given by q=KL+6L²-0.1L³ where q represents the annual quantity of widgets produced, K represents annual capital input and L represents annual labor input. A) Suppose K=10. At what level of labor input does average product of labor reach a maxiumum? How many widgets are produced at that point? B) Again assuming that K=10, at what level of labor input does MPL=0? C)Determine and show whether the production process exhibits law of diminishing returns.