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Derive the long-run TC, MC, and AC from the production function Q= (L^a)(K^b) assuming that the prices of K and L (r,w) are both 1
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- Craig and Javad run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's longrun production function is Q = 4KL, where Q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. The weekly cost function for the paper plant is C = 20K + 4L, where C is the total weekly cost. (a) What ratio of capital to labor minimizes Craig and Javad's total costs? (b) How much capital and labor will Craig and Javad need to rent and hire in order to produce 1,000 reams of paper each week? (c) How much will hiring these inputs cost them?A firm uses labor (L) and capital (K) to produce output (q) according to the function: q= L ².K In the short-run, the firm's level of capital is fixed at one unit (K 1). Assume the firm has already paid a fixed cost of $1 (F = 1) for their one unit of capital. In addition, assume that each unit of labor must be paid a wage of $1 (w = 1). If the firm can sell each unit of output at a price of $6 ( p = 6), answer the following two questions: A) What is the firm's profit maximizing level of output in the short-run? Profit maximizing q 3 = B) What is the maximum profit the firm can earn in the short-run? Maximum profit 8 = dollarsConsider 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.
- Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1, and m = 2, where m is the price per unit of materials.a) Suppose that the firm is required to produce Q units of output. Show how the cost - minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity of capital depends on the quantity Q. Show how the cost - minimizing quantity of materials depends on the quantity Q. b) Find the equation of the firms long-run total cost curve.c) Find the equation of the firms long-run average cost curve.d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50 units (ie., K 50). Show how the cost- minimizing quantity of labor depends on the quantity Q. Show how the cost- minimizing quantity of materials depends on the quantity Q. e)…Consider a purely competitive firm that has two variable inputs L (labor hour) and K (machine) for production. The price of product is $p. The production function is Q (K; L) = 4L^1/4 K^1/4 . Assume that the hourly wage of workers is fixed at $w and the price per machine is $r Write out the optimal inputs quantities, L and K, as a function of parameters, p, w, and rConsider a firm that produces widgets according to the following Cobb-Douglas production function: Q = A * L^α * K^β where: Q is the quantity of output, L is the quantity of labor, K is the quantity of capital, A is a scale parameter (total factor productivity), α and β are the output elasticities of labor and capital respectively. Given that A = 1, α = 0.6, β = 0.4, L = 16 and K = 9, a) Calculate the quantity of output Q. b) If the firm increases the quantity of labor (L) to 20 while keeping the quantity of capital (K) constant, what will be the new quantity of output?
- Consider the production function q=√L+8K^(3). Starting from the input combination (3,6), does the production function exhibit increasing, constant or decreasing returns to scale if inputs double?Consider the following production function when K is fixed. (This is a description of the figure: it shows a two-axis graph; in the horizontal axis we measure labor and in the vertical axis we measure meals; the graph of the production function is a line that intersects the vertical axis at a positive amount; this graph is a line with positive slope and passes through the point (4,300)). Can we say that the production function satisfies the law of decreasing marginal returns of labor?True FalseA firm operates in the short run with the production function Q = 10 K0.5 L 0.5, where Q is output, K is capital, and L is labor. The price of capital is $100 per unit, and the wage rate is $50 per unit of labor. How many units of capital and labor should the firm use to minimize its total cost while producing 400 units of output?
- The production function for a product is given by q= 10K^(1/2)L^(1/2) where K is capital, and L is labor and q is output d) Now suppose w =30 and r = 120. What is the minimum cost of producing q=1000. (You must show your work by clearly writing the equations that you use to derive the cost minimizing levels of L and K.) e) Now suppose that the firm is in the short run and cannot vary the amount of capital. That is, it must use the same amount of capital as in part d). However, the firm wants to produce 1200 units of output. How much labor should it use to minimize its cost and what is the minimum cost of producing q =1200?Consider a production function Q = (K1/2) + (L1/2) + (M1/2) where L, K, M represent labor, capital, and material. Input prices for L, K, M are 1, 1, and 1. The firm wants to produce 12 units of output. a) What is the total cost if K = 4? b) What is the total cost if K = 4 and L = 9?Given the production function y = ( ip - Kp)1/p , what is the technical rate of substitution, the elasticity of substitution, and the returns to scale when p = 0.5?
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