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
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- If two painters can paint 200 square feet of wall in an hour, and three painters can paint 275 square feet, what is the marginal product of the third painter?which of the following statements is true?A) The MP of labor is always negative at short runB) The MP of labor is increasing at short run because the stock of capital fixedC) Adding more labor will produce progressively greater increments of outputD) The MP of labor after an increase the beginning declining while remaining positive at short runDuring the 1920s, Charles Cobb and Paul Dougkas modeled totalproduction output P (of a firm, company or entire economy) as a function of labourhours involved x and capital invested y (each includes the monetary worth of allbuildings and equipment). The Cobb-Douglas production function isP(x, y) = k x^α y^(1−α),where k ad α are constants representative of a particular firm or economy.(1) Show that doubling of both labor and capital results in doubling productionP.(2) Suppose a particular firm has the production function for k = 120 andα =3/4. Assume each unit of labor costs $250 and each unit of capital costs$400, and that the total expense for all costs cannot exceed $100,000. Findthe maximum production level for the firm.(3) Compute the production level at x = 64 and y = 81. If the capital isdecreased by 3 units, by how many units the labor should be adjusted toso that there is not change in production level.
- Consider a firm that employs capital and labour and uses the productiontechnology Q = min[K, L] (think of taxi cabs: you need exactly one car and one driver per taxi).Factor prices are r for capital and w for labour.a) Draw the isoquant for Q = 100.b) Does this production function exhibit increasing, constant, or decreasing returns toscale? Show/explain.c) What is the optimal relationship between K and L? What is the input demand for Labour(L as a function of Output) and the input demand for Capital?d) Derive the long-run cost function: (eg C = wL + rK where L and K are optimal choices).e) What is the long-run marginal and average cost function with r = 4 and w = 4?f) What is the cost of Q=100? Q= 200?g) Does this cost function exhibit increasing, constant, or decreasing economies of scale?Production Function. Consider the Cobb-Douglas production function discussed in class:F(K, L) = AK^1/3 L^2/3. Suppose that parameters are initially A = 1, K = 150, and L = 10. d) Suppose that the quantity of labor L doubles. Calculate Y, w, r, Y/L, and K/L. Com-ment on how and why these numbers changed relative to (c) and why they did so. E) Suppose that the quantity of capital K doubles as well. (So now both K and L aretwice their previous value). Calculate Y, w, r, Y/L, and K/L. Comment on how thesenumbers changed relative to both their initial values, and their values in (d).Can you put a solution to this? Instruction: (see other instructions below, for the submission of answer sheet) A. Complete the Column for MRS. Show your solution for each answer, Number each solution that corresponds to each unknown MRS. B. Graph the Labor-Capital function, making sure that Labor units are in the horizontal axis MRS = change in K/ change in L change in K = final K - initial K change in L = final L - initial L MRS = 26-30 /2-1 MRS = -4/1 MRS = - 4 Units of Labor Capital MRS 1 30 0 2 26 -4 3 22.5 (1) 4 19.5 (2) 5 17 (3) 6 15 (4) 7 13.5 (5) 8 12.5 (6) 9 12 (7) 10 12 (8)
- A producer has the following technology.y= 6K^(1/2)L^(1/2) a) Prove formally that the production function exhibits constant returns to scale (use “λ” argument). b) Find analytically MPL and MPK. Is MPL increasing, decreasing, or constant inL? Is MPK increasing, decreasing, or constant in K? c) Short-run: Given stock of capital ̄K= 1 find labor demand (formula) of a competitive firm. Find equilibrium real wage rate if labor supply is given by Ls= 9 (one number). d) Assume again ̄K= 1 and that Ls= 9. The government adopts a real minimum wage of wmin/p=(3/2). Find labor demand (one number) and the unemployment rate (one number). Please depict the equilibrium on a graph with the real wage on the vertical axis and labor on the horizontal axis, indicating the equilibrium quantity of labor, wage, and unemployment, as well as the relevant curves. e) Find the cost function given prices of inputs wK= 4 and wL= 1 (formula). Plot the cost function on a graph, indicating the slope of the cost…Make a table to complete AP and MP. Then, draw a suitable diagram to show the production theory in this case. Below is the table for the reference. Input X Input Y Total Product (TP/Q) Average Product (AP) Marginal Product (MP) 1 0 0 1 1 20 1 2 80 1 3 180 1 4 230 1 5 270 1 6 270 1 7 210A firm produces good Y with just 2 factors: Capital which is fixed in supply and labour which is variable. Identify the stages of production in the diagram and explain why the firm still hire labour even though it is in the range of diminishing returns. What is the number of workers after which diminishing marginal returns starts? Is this a short run or long run phenomenon? Labour (units) 1 2 3 4 5 6 7 8 9 Total Product (TP) in Units 8 15 24 30 35 37 38 38 36 Average Product (AP) in units 8 7.5 8 7.5 7 6.17 5.43 4.75 4 Marginal Product (MP) in units 8 7 9 6 5 2 1 0 -2
- 7 The Paunch Burger restaurant chain currently produces 250,000 units of output by using 2,400 units of capital and 520 units of labor. Paunch Burger’s marginal rate of technical substitution (MRTS) of labor for capital is 3.5. If Paunch Burger’s regional manager wants to produce the same amount of output but increase the company’s use of capital by 175 units, how many units of capital will Paunch Burger now need to use?Suppose the production data of XYZ company mentions labor (L) as a variable input and machine (K) as a fixed input. Complete the table as XYZ increases its number of laborers (L). L TP MP AP 1 100 100 100 2 220 120 110 3 270 1) _____ 90 4 270 2)_____ 3)_____ 6 4)_______ 5)_____ 50 8 180 6) ____ 7) ____When is the firm operating in the short run and when does it operate in the long run? and Discuss the factor(s) that determine how many workers the firm wishes to hire in the long run.