(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 A = 1, what does this model predict for GDP per person? Does this model over-predict or under-predict GDP? Show your working. (d) If the model is to explain the observed value of y, what must TFP (A) be equal to? Briefly explain why TFP might vary across countries.-
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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 demandsitier 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?…Answer the Constrained Optimization: Cobb-Douglas Production Function:3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product of Capital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combination of K and L that the firm should use to produce the maximum possible output. Do not solve the problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it faces
- Consider the following production function: x = f0.K) = A e°k where xis the output e is the labour input, kis the capital input, and A, bare positive constants. (a) Set up the cost minimization problem and solve for the first order conditions using the Lagrange Method. Let wbe the wage rate and rthe rental rate of capital. (b) Using your answer in (a), find how much labour and capital would the firm use to produce x outputs, given wage rate wand rental rate of capital ? what are these functions called? (e) Using your answer in (b), find the minimum cost it takes to produce xoutputs, What is this function called?Please let me know seftuon A) if the input price would rise or fallAnswer the Constrained Optimization: Cobb-Douglas Production Function:1. Based from the factor shares of the two inputs, what will happen to the number of output ifit the firm decides to triple both the amount of labor and capital?2. State the optimization problem of the firm.3. Solve for the formulas of the Marginal Product of Labor (MPL), and Marginal product ofCapital (MPK)4. Using your knowledge of the tangency condition in Producer’s theory, find the combinationof K and L that the firm should use to produce the maximum possible output. Do not solvethe problem using the Lagrangian method.Note: The tangency conditions just states that the slope of the production function must beequal to the slope of the isocost function.5. What is the maximum possible output that the firm could earn given the constraint it faces?
- Coosider 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.Suppose there are two inputs for production, labor and capital. The firm’s production process isdefined by the following production functiony=f(L,K).Howdoweinterpretthefirm’smarginalrateof technical substitution? a)How many units of capital the firm would have to give up in order to attain one more unit oflabor, such that the firm maintains the same cost level b)How many units of capital the firm would have to give up in order to attain one more unit oflabor, such that the firm produces one more unit of output c)How many units of capital the firm would have to give up in order to attain one more unit oflabor, such that the firm maintains the same level of production d)a) and b) are correct e)a) and c) are correctTrue or False: The shape of the production function reflects the law of increasing marginal returns. O True O False
- 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 False(04.Suppose we are given (he constant returns to scale (CES) production function q= { ßa + aa }1/a a. Show that the marginal product of Beta is (g/B) and marginal product of Alpha (q/a)** b. Show that RTS = (2/B); I lence or otherwise shov that the ò =-1/(1-a) 1-aFill in the missing parts in the following tables to illustrate the behavior of production and costs in the short-run; then determine the optimal level of production according to the number of inputs you are intending to use. Output Units Land Square Meters Capital Egyptian Pounds Labor Units 0 1000 1600 2100 2500 2600 Output FC VC TC MCL MPL AVC AFC ATC 0 1000 1600 2100 2500 2600 -Hint Consider land and capital as fixed factors, while labor units as variable factors. The cost of land is L.E20 per square meter. The interest rate on capital is 10%. The average wages per unit of labor is L.E1500.