**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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Write My Essay For MeCurrently 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 =*2

*K*and (2)

*Q = LK*.

(1) and (2) imply that *Q = 2K ^{2}*, or

*32 = 2K*, and thus

^{2}*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.

**A firm operates with the production function Q = K**^{2}L. 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.**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?****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?****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?**

- 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 = K
^{2}(80) => K = 10.

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

- 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 = K^{2} / 200 = 10^{2} / 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.

- 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/K^{2} = 400 / 200 => K = L

(2) Production Requirement: K^{2}L = 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.

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