3. Assume the production function is Q = 4K0.5L0.5, and K = 4 in the short run. Find the marginal product of labor. Using derivatives, show that the production faces diminishing marginal return to labor. If the market wage is w, express the optimal number of labor (L) in terms of w. Are they positively or negatively related, and why?
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- A firm can manufacture a product according to the production function Q = 2(K)1/4 (L)3/4 where K represents capital equipment and L is labor. The company has already spent $10,000 on the 4 units capital. a. Please show the expression for the average product of labor, APL and the marginal product of labor, MPL. If workers at the firm are paid a competitive wage of $100 and the product is sold for $200 each, what is optimal level of labor usage and what is the maximum profit?A firm's production function is given by 1/2 1/2 y = 10x1x2 where x₁ indicates the input labor and x₂ indicates the input capital. (a) Compute the partial derivatives of the production function. Provide an economic interpretation of each derivative. (b) When an amount of capital available is fixed by x2 = 9, compute the par- tial derivative with respect to ₁ and sketch its graph. Determine whether the derivative is monotone increasing or decreasing, and provide an inter- pretation.doubt EXERCISE 11.6 Suppose capital and labour are perfect complements in a one- to-one ratio. That is, suppose that Q = min (L, K). Currently, the wage is w=5 and the rental rate is r = 10. What is the minimum cost and method of produc ing Q = 20 units of output? Suppose the wage rises to w' = 20. If we keep total cost the same, what level of output can now be produced and what method of production (input mix) is used? PVER
- Suppose the Cobb-Douglas production function is given as: , where b0 = level of technology, b1 = proportion with which labor is used in production, b2 = proportion with which capital is used in production. Then, a. Find marginal product of factors (L and K). b. Compute marginal rate of substitutions (MRSLK and MRSKL). c. Find the elasticity of substitution for this production function. d. Compute elasticity of output with respect to labor and capital.Consider the following production function for shirts: q = v6 L3/4K/4, where L is worker-hours, and K is sewing machine-hours. a. Compute the marginal products of labor and capital, the average product of labor, and the marginal rate of technical substitution of labor for capital (i.e. how many units of capital are needed to make up for the loss of one unit of labor)? b. Are there diminishing returns to labor (that is, does the marginal product of labor decrease when labor L increases)? What about to capital? Is there diminishing marginal rate of technical substitution (MRTS)? с.Answer 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?
- A firm can manufacture a product according to the production function Q = 2(K)1/2 (L)1/2 where K represents capital equipment and L is labor. The company has already spent $10,000 on the 4 units capital. a. Please show the expression for the average product of labor, APL and the marginal product of labor, MPL. b. If workers at the firm are paid a competitive wage of $100 and the product is sold for $200 each, what is optimal level of labor usage and what is the maximum profit?A firm has the production function, Q=15L^1/3 K^2/3 where ? is output, ? is labour units and ? is capital units. Calculate and interpret the degree of homogeneity. If labour and capital units increase by 5%, what is the percentage change in output?Suppose the two factors are perfect complements and the production function is given by y=min { 13x1,2x2} Where x1=300, X2=50 What will be level of production and draw corresponding iso quant?
- Assume that Donnell Corp. is currently producing 500 units of output per period, using 25 units of labor and 20 units of capital. Values for the marginal product of each input and the prices of the inputs are as follows: MPK = 100, MPL = 200, w = 2, and r = 3. Given the information above, which of the following is true? a. The firm is currently using the optimal levels of capital and labor. b. The firm should increase labor and reduce capital usage. c. The firm is not using the optimal levels of capital and labor, and it is impossible to determine the optimal levels from the given information.If a firm's production function is Leontief and the wage rate goes up, the: Multiple Choice firm must use more capital in order to minimize the cost of producing a given level of output. firm must use more labor in order to minimize the cost of producing a given level of output. firm must use less labor in order to minimize the cost of producing a given level of output. cost minimizing combination of capital and labor does not change.Consider 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.