Question

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 MPK= L and MPL= 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?

Answer #1

Q = KL

When K = 16 and Q = 32,

16L = 32

So, L = 32/16 = 2

Thus, L = 2 when K is fixed at 16.

Cost of this combination, C = factor price of L*(L) + factor price
of K*(K) = 2*2 + 4*16 = 4+64 = 68

Least cost combination of K and L occurs at the point where,
MRTS = factor price of L/factor price of K

(MRTS = MPL/MPK = K/L)

So, K/L = 2/4 = 1/2

So, L = 2K

Now, KL = Q

So, K(2K) = 32 (as L = 2K)

So, 2K^{2} = 32

So, K^{2} = 32/2 = 16

So, K = 4

And, L = 2K = 2*4 = 8

Cost of this combination, C' = factor price of L*(L) + factor price
of K*(K) = 2*8 + 4*4 = 16+16 = 32

Thus, savings = C - C' = 68 - 32 = 36

Thus, firm can save $36.

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