3. Consider the production function: Y = F(K, LE) = Kº4 (LE) 0.6 in which output (Y) depends on capital (K) and the number of effective workers (LE), while the number of effective workers depends on the amount of workers (L) and the efficiency of each worker (E). (a) Derive the per effective-worker production function: y = f(k), where y Y LE K & k = LE
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- (1) Suppose the production function for widgets is given by q = kl – 0.8k? –0.272 where q is the annual quantity of widgets produced, k is the annual capital input and 1 is the annual labor input. (a) Suppose k-10; Find the average productivity of labor. At what level of labor input does the average productivity reach a maximum? (b) Suppose k-10; Find the marginal productivity of labor. At what level of labor input does the marginal productivity reach a maximum? Is the average product equal to the marginal product at the point where average product reaches its maximum? Please showL it.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)=(Suppose that production function of certain devices is determined by: q = k * L - 0.8 k2 - 0.2 L2 Where q represents annual quantity of devices produced, k represents annual quantity of capital factor, and L represents annual quantity of labor factor. (a) Suppose k = 10; draw a graph with average and total labor productivity. What is level of labor (L) factor at which average productivity reaches a maximum? How many devices are produced at that point? (b) Assuming again that k = 10, plot Marginal Product (MP) curve. At what level do we have that labor (L) factor is MP = 0? (c) Suppose capital (k) factor were to increase to k = 20. How would your previous answers change?
- Suppose the production function for widgets is given by KL – 0.5K2 – 0.1 L2 , where q represents the annual quantity of widgets produced, K represents annual capital input, and L represents annual labor input. (a). Suppose K=5; what is the average productivity of labor (Average product of Labor, MPL) (b). Suppose K=10; at what level of labor input does the total output reach the maximum?3(a) A firm's production function is initially Q=VKL. Over time, the production function changes to Q=KL^2 (Assume that L21 and K21). Verify that this change represents technological progress.Based on the notations and assumptions used in class, which of the following is/are TRUE regarding the production function y = ka, where 0 < a <1? (i) It is expressed in per capita form. (ii) The function exhibits constant returns to scale. (iii) Production per worker depends on the amount of capital per worker. O a. (i), (ii), and (iii) O b. Only (i) O c. Only (iii) O d. Only (ii)
- (b) Prove that the per-worker production function has the following properties: f'(k) ≥ 0, lim f'(k)= +∞, and k-0 lim f'(k) (c) Assume that the production factors receive their respective marginal products. Derive the expressions for the wage rate, W, and the rental rate on capital, r + 8, when technology is Cobb-Douglas. What happens to (W,r) as k 0 and as k→ ∞o? Derive the expression for the factor price frontier (FPF), i.e. the expression linking W and r, and illustrate it graphically. Show what happens to the FPF if general productivity increases. = 0..Let a firm's production function Q (L)=12L^2-1/20L^3 (i.e. output as TL orthe amount of production depends on the workforce, L, as Q is the number of people with a single production input). This functionlet be the foreground 0 ≤ L ≤ 200 for. Accordingly,A) What is the number of labour force (L*) that will make output (Q) the highest?B) what is the number of workers (L**) who will make the maximum amount of output per employee(i.e. Q (L)/L)?c) in both cases the output of the L* and L** values you find and the output per employee at the mosthow can you be sure it's high? Show me.Draw a production function that shows the relationship between capital per worker and output per worker assuming there is diminishing returns to capital. Illustrate the diminishing returns to capital by showing the change in output per worker when capital is increased by 1 unit at two different levels of capital per worker.
- 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 workerBased on the notations and assumptions used in class, which of the following is/are TRU regarding the production function y=k, where 0 < a <1? (i) It is expressed in per capita form. (ii) The function exhibits constant returns to scale. (iii) Production per worker depends on the amount of capital per worker. O a. (i), (ii), and (iii) O b. Only (1) O c. Only (iii) Od. Only (ii)Production Function. Consider the Cobb-Douglas production function discussed in class:F(K, L) = AK1/3 L2/3. Suppose that parameters are initially A = 1, K = 150, and L = 10. b) State the equation relating outper per worker and capital per worker. Calculate outputper worker and capital per worker in this case and verify that they satisfy this equation.
