2. Assume the production function Y = K\alpha L(1-\alpha). Assume that the long-run supply curves for labor and capital are vertical. a. Draw a graph showing the effect of an exogenous decreases in the supply of capital on the rental price of capital. Label all axes and curves. b. Draw a graph showing the effect of an exogenous decrease in the supply of capital on the real wage. Label all axes and curves.
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- 2. Consider the production function given by Y= altbK. The prices of labor and capital are wLand wK, respectively. a. Find the conditional input demand for L and K b. Find the cost function C. How will an increase in wk affect the labor demand in a? Show graphically d. If K is fixed at 10, find the conditional labor demandCoosider the production function characterizing garri plant as: Q= 20L + 60k - L^2 - K^2 If total outlay for both capital, K =N50, and a Labour, L= N20, is N4,600. Find: a. The optimal quantity of K* and L* to be employed. b. Calculate the maximum possible output of garri that the plant can produce.Q1. Suppose we are given the constant returns-to-scale CES production function q = [k + l]1/ where k represents capital and l represents labora. a. Show that MPk = (q/k)1 and MPl = (q/l)1 . b. Show that RTS = (k/l)1 ; use this to show that elasticity of substitution between labor and capital= 1/(1 – ). c. Determine the output elasticities for k and l; and show that their sum equals 1.Note: Output elasticity measures the response of change in q to a change in any input. Elasticity of output wrt k is eq,k = %q/%k = (q/k)*(k/q) or (q/k)*(k/q) or lnq/lnkSimilarly for elasticity of output wrt l, eq,ld. Prove that q/l = (q/l) and hence that ln(q/l) = ln(q/l)
- The wage rate is $20 per hour and the cost of capital is $30 per hour. At the current output, the firm's MPL is 6 units and MPK is 8 units. In order to maximize profit without changing the production level, the firm should Select one: OA. keep its current level of labour and capital inputs. В. rent more capital and hire less labour c. hire less labour and rent less capital O D. rent less capital and hire more labourTrue or False: The shape of the production function reflects the law of increasing marginal returns. O True O FalseThere are three factors of production, capital, K skilled labour Le and unskilled labour Ly. Which of the following production functions exhibits constant returns to scale in the production inputs K, L5 and Ly? C a. ab 4 ab 2(ab+3e) Y= Ka+6lab+c) La+2(3ab +c)+4c a+2(ab+3c)+4ab O b. All of these answers are correct. ac Y= Ka+b+bc L, a+b+bc L. a+b+bc O d. blb-e) b-e Y= K*+b?-L,+b?-L a+b- b>c Oe. 1-a a(1-a) 1-a 0Q1. Suppose we are given the constant returns-to-scale CES production function q = [k + l]1/ where krepresents capital and l represents labora. Show that MPk = (q/k)1 and MPl = (q/l)1 .b. Show that RTS = (k/l)1 ; use this to show that elasticity of substitution between labor and capital= 1/(1 – ).c. Determine the output elasticities for k and l; and show that their sum equals 1.Note: Output elasticity measures the response of change in q to a change in any input.Elasticity of output wrt k is eq,k = %q/%k = (q/k)*(k/q) or (q/k)*(k/q) or lnq/lnkSimilarly for elasticity of output wrt l, eq,ld. Prove that q/l = (q/l) and hence that ln(q/l) = ln(q/l)Q2. Suppose the production of airframes is characterized by a CobbDouglas production function: Q =LK. The marginal products for this production function are MPL = K and MPK = L. Suppose the price oflabor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital if the manufacturer wants to…If capital and labour are perfect complements then the marginal products of capital and labour are undefined A True Faise Question 16 A firm uses 10 units of labour and 20 units of capital to produce 10 units of output. The marginal product of labour is 0.5. If there are constant returns to scale the marginal product of labour must be 0.25 True B False Question 17 Afirm uses 10 units of labour and 30 units of capital to produce 10 units of output. The marginal product of labour is 05. If there are constant returns to scale the marginal product of labout must be 0.25 A True False2. Given Q = AK LB: Q = 2 = 2,000, A = 100, α = 0.6,ß = 0.8, w = 50, r = 80 a. Does this production function exhibit increasing, decreasing, or constant returns to scale? Explain b. Find the firm's marginal rate of technical substitution of labor for capital. Provide an economic interpretation. c. Solve for the cost-minimizing combination of K and L, and the resulting minimum cost d. If output increases to 4,000, what will be the new cost-minimizing values of K, L, and C?2. A firm has a production function of q = 0.25KL0.5 and in the short-run capital is fixed at 100 (K = 100). %3D The rental rate of capital = = $100. r The wage rate = w = = $25 а. Write the equation for the short-run production function. b. What is the firm's short-run demand for labor (L as a function of output)? What are the equations for the firm's short-run total costs (TC as a function of q), short-run average total costs (ATC as a function of q), and short-run marginal costs (MC as a function of q)? а.Suppose that the production function is What is the average product of labor AP₁, holding capital fixed at Â? O A. APL =q/L. B. AP₁ = L-0.50 0.50 APL = 0.50L-0.500.50 D. Both a and b. E. All of the above. q=L0.50 0.50sitier the production model studied in Chapter 4. Final output in the economy is pro- duced using capital K and labor L. The production function is: Y = ĀK'/³L^/5 Assume that the supply of all inputs are exogenous and equal to L and K. Perfectly compet- itive firms are price-takers and choose how much capital and labor to demand by maximizing profits. Let w and r denote the wage and rental rate of one unit of labor and capital respec- tively. (a) What are the 5 endogenous variables in this model? Also list the exogenous variables and the parameters and provide a brief explanation of each. (b) Show the first order conditions for labor and capital. Explain the intuition for why these conditions are optimal for a firm. Are there diminishing returns to these inputs in this model? ( (c) Show that the solution for output per capita can be written as y = Ak/5. Observed GDP per capita is 3 in this economy. Capital per person is 32. If Ā = 1, what does this model predict for GDP per person?…SEE MORE QUESTIONS