A firm produces an output

A firm produces an output with the production function Q = KL, where Q is the number of units of output per hour when the firm uses K machines and hires L workers each hour. The marginal products for this production function are MP_K = L and MP_L = K. The factor price of K is 4 and the factor price of L is 2. The firm is currently using K = 16 and just enough L to produce Q = 32. How much could the firm save if it were to adjust K and L to produce 32 units in the least costly way possible?

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Currently the firm must be using L = Q/K = 32/16 = 2 units of labor. Let the factor prices of capital and labor be, respectively, r and w.

Its total expenditure is C = wL + rK = 2(2) + 4(16) = 68.

If it were to minimize cost, it would hire L and K so that (1) MP_K/r = MP_L/w, or L/4 = K/2, or L = 2K and (2) Q = LK.

(1) and (2) imply that Q = 2K², or 32 = 2K², and thus K = 4 and L = 8.

So Q = 32 can be produced efficiently with a cost of C = wL + rK = 2(8) + 4(4) = 32.

The firm could save 68 – 32 = 36 by producing efficiently.

3. A firm operates with the production function Q = K²L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day. The manager has been given a production target: Produce 8,000 units per day. She knows that the daily rental price of capital is $400 per unit. The wage rate paid to each worker is $200 day.
4. a) Currently the firm employs at 80 workers per day. What is the firm’s daily total cost if it rents just enough capital to produce at its target?
5. b) Compare the marginal product per dollar sent on K and on L when the firm operates at the input choice in part (a). What does this suggest about the way the firm might change its choice of K and L if it wants to reduce the total cost in meeting its target?
6. c) In the long run, how much K and L should the firm choose if it wants to minimize the cost of producing 8,000 units of output day? What will the total daily cost of production be?

80. a) Suppose that the firm is operating in the short run, with L = 80. To produce Q = 8000, how much K will it require? From the production function we observe that 8,000 = K² (80) =>  K = 10.

The total cost would be C = wL + rK = $200(80) + $400(10) = $2,000 per day.

80. b) Let’s examine the “bang for the buck” for K and L when K = 10 and L = 80.

For capital: MP_K / r =  2KL / 400 = 2(10)(80) / 400 = 4

For labor:  MP­_L / w = K² / 200 = 10² / 200 = 0.5

So the marginal product per dollar spent on capital exceeds that of labor. The firm would like to rent more capital and hire fewer workers.

1. c) Because the production function is Cobb-Douglas, we know that it has diminishing MRTS_L,K and that the isoquants do not intersect either the K or L axis. Thus the cost reducing basket (K,L) will be interior (with K > 0 and L > 0). To find the optimum, we use the two conditions:

(1) Tangency  condition:            MP_K / MP­_L = r / w  =>    2KL/K­­­² = 400 / 200   =>  K = L

(2) Production Requirement:      K²L = 8,000

Together equations (1) and (2) tell us that K = 20 and L = 20.

The total cost would be C = wL + rK = $200(20) + $400(20) = $12,000 per day.