Suppose we are given a production function: q = [alP + (1 – a)kPlē a. (1– a)-1 and MP= a(-)-1 Please show that MPk= b. Please derive the RTS c. Please derive the elasticity of substitution o. d. Does this production function exhibit constant, increasing, or decreasing returns to scale?
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- Q5 A firm's production function is Q is equal to 5L2Ka) Find out the MPL and MPb) Does this production function exhibit constant, increasing, or decreasingreturns to scale?c) What is the marginal rate of technical substitution of L for K for thisproduction function Q6.Suppose the firm sells its output according to the following demand scheduleMarginalTotal Product Total Revenue eLabor Product Price Revenue Product$3.50192c2.80182.30291.80391.65471.5021.4053Fill in the remaining two columns of the table. How many workers wilhired at a wage of $72Take a Cobb-Douglas production function, find its Elasticities w.r.t output and Elasticity of Substitution with the help of partial derivatives.Suppose a firm has a production function with two inputs, capital (K) and labor (L). The production function takes the form: Q = L2K2. Further, let the wage be given by w, the rental rate of capital be given by r, and suppose that the firm wishes to produce Q0 units of output. Determine the elasticity of substitution for this production function. Explain your answer. Determine the returns to scale for the production function. Solve for the long-run optimal input demand functions for capital and labor as a function of exogenous variables only. Derive an expression for long-run total cost as a function of the exogenous variables. Let w=16, r=25, and Q0 = 10,000. Solve for the long-run cost-minimizing input combination.
- 1a) Consider the production function Q = 0 + 6L + 5L2 - .2L3. The range of labor covering Stage II of production is ____ to ____. You can use Excel spreadsheet or otherwise to answer this question. b) A firm’s production function is Q = 600*L -1.0*L2. The firm is currently employing 83 units of labor. What is the elasticity of production? You can use a excel spreadsheet or otherwise to answer this question c) A firm’s LRTC = 600Q - .5Q2 + .001Q3. At what level of output does the firm experience minimum efficient scale? Use either Excel Spreadsheet, Excel Solver or OtherwiseAnswer 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?Consider a Cobb-Douglas production function:f(l, k) = Alα k1−α,where A is the total factor of productivity (a constant greater than 1), 0 < α < 1, lrepresentslabor, and k represents capital. The following sub-questions will guide you through showing thatthe elasticity of substitution is constant.a) Find the marginal product of labor. Verify that this production function exhibits diminishingmarginal productivity of labor. b) Find the marginal product of capital. Verify that this production function exhibits diminishingmarginal productivity of capital. c) Find the marginal rate of technical substitution. Write your answer as MRT S = . . . d) In part (C), you should’ve found the MRTS as a function of the input ratio, kl. Take theabsolute value of both sides and solve for the input ratio, so that the expression gives theinput ratio as a function of MRTS (i.e. kl = . . .). Take the log of both sides, then take thederivative with respect to the log of MRTS. Is the elasticity of…
- A firm engaged in the manufacture of RTWS faces the short-run production function Q = 250L - 5L², where L is the number of units of labor and Q is the number of RTWs produced annually. d.) How many RTWS can be produced by the firm in a year if there are 10 units of labor? e.) Compute the marginal product of the 40th unit of labor. f.) How many RTWs can be produced by the firm in a year if there are 40 units of labor? g.) Sketch the graph of the production function.Consider a production function for an economy with 2 factors of production L, K: Y=10(KL)1/2 where Y is real output, L is labour, K is capital. In this economy the factors of production are in fixed supply with L = 10, K = 10. What type of returns to scale does this production function exhibit. Demonstrate by example. If the economy is competitive what is the total income that will go to the owners of capital?The Long-run production function is given by; Y = 180 L0.8 K1.8Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt)a) Calculate Marginal Product of Labour (MPL) and Marginal Product of Capital (MPK), if L=12 and K=20 .b) Derive the equation for Isoquent and graphically show it by assuming L= 10, 15, 20 25 and 30. c) Determine factor intensity and returns to scale of this production function. d) Prove that the elasticity of labour is 0.8 and elasticity of capital is 1.8 .
- The Long-run production function is given by; Y = 180 L0.8 K1.8Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt)a) Calculate Marginal Product of Labour (MPL) and Marginal Product of Capital (MPK), if L=12 and K=20 b) Derive the equation for Isoquent and graphically show it by assuming L= 10, 15, 20 25 and 30. c) Determine factor intensity and returns to scale of this production function. d) Prove that the elasticity of labour is 0.8 and elasticity of capital is 1.8 a) According to the Wall Street Journal, Mitsubishi Motors recently announced a major restructuring plan in an attempt to reverse declining global sales. Suppose that as part of the restructuring planMitsubishi conducts an analysis of how labour and capital are used in its production process. Prior to restructuring Mitsubishi’s marginal rate of technical substitution is 0.15 ( in absolute value). Tohire workers. Suppose that Mitsubishi must pay the competitive hourly wage of US$ 15. In the study of…The Long-run production function is given by; Y = 180 L0.8 K1.8Where, Y = Output (mt/day), L = Labour (hours/mt) K = Capita (Rs/mt)a) Calculate Marginal Product of Labour (MPL) and Marginal Product of Capital (MPK), if L=12 and K=20 b) Derive the equation for Isoquent and graphically show it by assuming L= 10, 15, 20 25 and 30. c) Determine factor intensity and returns to scale of this production function. d) Prove that the elasticity of labour is 0.8 and elasticity of capital is 1.8 a)According to the Wall Street Journal, Mitsubishi Motors recently announced a major restructuring plan in an attempt to reverse declining global sales. Suppose that as part of the restructuring plan Mitsubishi conducts an analysis of how labour and capital are used in its production process. Prior to restructuring Mitsubishi’s marginal rate of technical substitution is 0.15 ( in absolute value). To hire workers. Suppose that Mitsubishi must pay the competitive hourly wage of US$ 15. In the study of…A firm has two variable factors and a productionfunction f(x1, x2) = 6x1/21X21/3. The price of its output is 3, the price of factor 1 is 3, and the priceof factor 2 is 2.– What is the optimal production output level?– What is the maximum profit-level?
