Econ. 410Spring 2008Tauchen/BiglaiserPractice Problems on Long Run and Short Run CostsUnless otherwise mentioned, assume that a firm’s technology satisfies the usual assumptions. Specifically, the marginal product for each input is positive and satisfies diminishing marginal returns. Also, the production function satisfies diminishing marginal rate of technical substitution. As usual, w denotes the price of a unit of labor and r denotes the price of a unit of capital. 1. The graph below shows the isoquants for output levels q=10, 20, 30, and 40 units of output with the isoquants for higher outputs farther from the origin. The price of labor is $2 and the price of capital is $8.a. Construct the isocost lines and identify the cost-minimizing input combinations for producing 10, 20, 30, and 40 units of output (approximately). What is the marginal rate of technical substitution between labor and capital at each of the cost-minimizing input combinations? b. What is the minimum cost at which the firm can produce 10 units, 20 units, 30 units, and 40 units of output?c. What is the LRAC for producing 10, 20, 30, and 40 units of output?d. Is the isoquant graph below consistent with diminishing marginal returns to labor? (This question is from the production function section of the course.)0123450 2 4 6 8 10 12 14 16Workers/DayUnits Capital/Day2. Suppose now that the firm’s capital is fixed at the optimal level of capital for producing 30 units of output. The input prices are again w=$2 and r=$8. What is the short-run cost of producing 10, 20, 30, and 40 units of output with the fixed capital stock?3. Graph the LRAC from question #1 and the short-run ATC for a firm that has the amount of capital that is optimal for producing 30 units of output.4. The graph below shows the same isoquant map as for the previous problems. The isoquants are for output levels q=10, 20, 30, and 40 units of output with the isoquants for higher outputs farther from the origin. Construct the isocost line that allows you to identify the cost minimizing input combination for producing 20 units of output with the input prices of w=$2 and r=$8. What is the cost associated with thisisocost line? Let’s refer to this cost as C2.Now suppose that the price of labor increases to $8. Construct the isocost line for the new input price andfor cost C2 . Is it possible to produce 20 units of output with any of the input combinations on this isocostline? Does the cost or producing 20 units of output necessarily increase as the price of one input increases (assuming that the firm uses both inputs)? 0123450 2 4 6 8 10 12 14 16Workers/DayUnits Capital/Day5. In the previous problem, you used a specific isoquant map and numerical values for the input prices to show that the cost of producing 20 units of output increases if the price of one of the inputs increases (assuming that the firm uses both inputs). Use the same line of reasoning to show that the cost of producing any output level increases if one of the input prices increases (assuming that the firm uses both inputs). 6. The graph below shows the same isoquant map as for problem #1. a. Show the expansion path for w=$1 and r=$8.b. Show the expansion path for w=$4 and r=$32.c. Show the expansion path for w=$2 and r=$1.d. Provide an intuitive explanation for the relationship between the expansion paths for a. and b.0123450 2 4 6 8 10 12 14 16Workers/DayUnits Capital/Day7. A firm’s production function is Q=K1/3 L1/3 . The price of a unit of capital is r and of a unit of labor is w. a. Determine the cost-minimizing input combination for producing Q units of output. [Note: The equation for the cost-minimizing amount of capital is also referred to as the output constrained capital demand function. Similarly, the equation for the cost-minimizing amount of labor is also referred to as the output constrained input demand function.]b. Determine the firm’s long run cost function. Also determine the firm’s long run average and marginal cost curves8. A firm selects its inputs to minimize the cost of producing its output. Recently, the firm increased eachof its inputs by 10 percent. As a result, its output increased from 100 units to 120 units. Indicate whether each of the following statements is true, false, or uncertain. Explain your answer.a. At q=100, there are increasing returns to scale.b. At q=100, the LRAC curve is downward sloping.c. At q=100, LRMC exceeds LRAC. 9. A firm uses two inputs – capital and labor. When w=$10 and r=$10, the cost-minimizing input combination for producing 100 units of output is 20 units of labor and 20 units of capital. When w=$8 and r=$10, the cost-minimizing bundle for 100 units of output is 28 units of labor and 19 units of capital. Indicate whether each of the following statements is true, false, or uncertain. Explain your answer.a. When w=$10 and r=$10, the LRAC for producing 100 units of output is $4.b. The output expansion path is downward sloping.c. When w=$5 and r=$5, the LRAC for producing 100 units of output is $2. 10. The practice problem web page provides a link to both this homework assignment and the accompanying excel file that allows you to investigate the relationship between long-run and short-run average cost curves. When you open the excel file, you will see the LRAC curve for a production function of the form Q=Ka Lb . The excel graph also shows three short-run average total cost curves. Along any short-run average total cost curve, the amount of capital is held constant. For the average total cost curve labeled K1, the amountof capital is held constant at the optimal amount of capital for producing 1 unit of output. For the averagetotal cost curve labeled K5, the amount of capital is held constant at the optimal amount of capital for producing 5 units of output. For the average total cost curve labeled K15, the amount of capital is held constant at the optimal amount of capital for producing 15 units of output. You can use the excel file to determine the effects of changes in the technology or in the input prices on the average cost curves. When you open the file, the values of a and b are set at a=.8 and b=.2. The input prices are both set to $2. You can type different values for the exponents and/or input prices in the yellowcells on the second line of the excel file. The new average cost curves appear after you press the “enter” key. a. When the