Consider an economy with production function given by Y = AK0.5 10.5 where A is the total factor productivity (TFP), K is the capital stock and L is the labor input. For simplicity assume capital is fixed and equal to 1. Assume A=200. a. Write the firm's problem of choosing labor demand. Derive the demand for labor as a function of the real wage.
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- A firm's production function is: q = 20L1/2K1/2 where q is the firm's total product, L is the quantity of labor employed, and K is the quantity of capital employed. The price of labor is $25 per unit and the price of capital is $100 per unit. a. What is the equation for the marginal product of labor? b. What is the equation for the marginal product of capital? c. Given the price of labor is $25 per unit and the price of capital is $100 per unit, what is the cost-minimizing combination of capital and labor that can produce 800 units of output?2. Consider the production function given by Y= altbK. The prices of labor and capital are wLand wK, respectively. a. Find the conditional input demand for L and K b. Find the cost function C. How will an increase in wk affect the labor demand in a? Show graphically d. If K is fixed at 10, find the conditional labor demandProduction Function and Input Demand Consider the production function Q = K + √L. Derive the inputdemand curves for L and K, as a function of the output Q, inputprices w (price of labor services) and r (price of capitalservices). Does the production function exhibit IRS, CRS orDRS?
- Consider a firm that produces output using a typical production function as discussed in the course, and that hires workers in a competitive labor market. 1. Explain what determines the firm's choice of its labor input. Use a suitable figure. Discuss how changes in the capital stock and in the wage affect the firm's choice of its labor input. Suppose that the firm uses a Cobb-Douglas production function Y = K"N-«. In this case, the marginal product of labor is given by (1 – a)K“N-«. The parameter a is 2/3, and the firm uses one unit of capital. 2. Calculate (show your steps) how the firm's labor input changes if the wage increases by 1%. 3. Calculate how the firm's labor input changes if the wage is unchanged, but the firm decides to double its capital stock.Consider the following production function: q = 9LK + 6L² – Assuming capital is measured on the vertical axis and labor is measured on the horizontal axis, determine the value of the marginal rate of technical substitution when K= 30 and L= 10. MRTS = -. (Enter a numeric response using a real number rounded to two decimal places.) 20 tv MacBook Air F6 FB 10 F3 23 2$ & з 4 6 7 8 { [ E Y U P D F G J K > C V N M command option - .. .- BConsider a firm for which production depends on two normal inputs, labor and capital, with prices w and r,respectively. Initially, the firm faces market prices of w=$5 and r=$15. Assume the firm has a cost budget of$1,500.a. Using the isoquant-isocost model, graphically show the optimal level of employment for this firm in thelong run.b. Suppose the government now imposes a minimum wage of $10 for workers. Using the same graph aspart a, graphically show the impact of the minimum wage on the optimal level of employment in thelong run.c. Refer to the initial situation described in part a. Now suppose a new innovation causes the price ofcapital to fall to $10. Using a new isoquant-isocost model, graphically show how this change impacts theoptimal levels of employment and capital in the long run. Clearly identify the resulting scale andsubstitution effects caused by the lower cost of capital. Could you please answer this question using the graphs please.
- Short Answer Q1 A firm's production function is estimated as Q = 400K + 80L -0.2K 2-0.4L? where K is capital and Lis labor. If the firm is using 200 units of K and 50 units of L, solve for: The marginal product of capital MPK The marginal product of labor MPL Don't just give your answers for MPK and MPL, show your computations If the price of Kis $70 per unit and the price of L is $14 per unit, is this an optimal production choice for the firm? Explain why it is or is not.Consider the following production function that depends only on labor:Q = 4L + 12L² - 6L³ 1. Compute the APL (average product of labor). 2. Compute the MPL (marginal product of labor). 3. What is the value of L* at which APL is the highest? 4. For L > L*, which one is bigger, APL or MPL? How about when L < L* and L = L*? 5. Draw APL and MPL on the y-axis as a function of L on the x-axis. Label the point of the intersection of APL and MPL.A firm uses capital and labour to produce widgets . In the short run capital is fixed ,while labour is variable . The short-run production is X = -L^3 +24l^2 +240L where X is the number of widgets produced in per week,and L is the number of workers employed .Each worker works a 40-hourweek. The wage rate is $12 per hour. a. Calculate the ranges of variables for L over which the firm is in stage 1,2 and 3 b. What is the minimum product price at which the firm will operate in the short run? c. The product price, over which the firm has no control is such that the firm's maximum posible pure profit $ 1096 per week. In order to achieve that level of profit it must employ 16 workers .How much is the firm's total cost?.
- During 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 the following production function: q = 32LK + 10L2 - Assume capital is fixed at K= 25. At what level of employment does the marginal product of labor equal zero? The marginal product of labor equals zero when (Enter a numeric response using an integer.) Beyond a labor input of 40, the marginal product of labor is 20 étv MacBook Air DI DD F3 F5 F6 F7 FB F10 FW 2# $ & 3 4 5 8 9 E R Y P D F G н J K C V B N M command VSuppose that the production function is Yt = AtKt^(1/2) Lt^(1/2) where Yt is output, Kt is the stock of capital and Lt is amount of labor firms hire. Assume that Kt = 100, At = 2. (a) Firms in this economy maximize their profits, given by revenue net of labor costs: Yt −WtLt . Derive the firm’s labor demand curve. (b) Workers in this economy maximize their utility, given by U(Ct , Nt) = log(Ct) − Nt , where Ct is consumption and Nt is the amount of labor workers supply. Their budget constraint is Ct = WtNt . Derive the workers’ labor supply curve. (c) Calculate the equilibrium wage rate and employment in this economy using the expressions for labor demand and supply you derived above. Also calculate the total amount of output the economy produces. (d) Suppose that capital doubles, to Kt = 200. Calculate what happens to the wage rate, employment and output. Illustrate the effects of this increase in capital graphically, using a labor demand/supply diagram.