Long - run factor demand A price taking firm (p= 100) chooses amounts of labor a

Question

Long - run factor demand A price taking firm (p= 100) chooses amounts of labor and capital that maximizes its profits. It has a decreasing returns to scale Cobb - Douglass production function: q(L, K) = L^1/6 K^1/8. The wage rate and rental rate of capital are 10 (w = r = 10). (a) Set up the profit function. (b) Provide the two profit maximization conditions for this case (no need to solve for L*,K*). (c) With an increase in price, p, would we expect to see a greater increase in labor or capital?

Explanation / Answer

(a)

Q = L1/6K1/8

Total cost, TC = wL + rK = 10L + 10K

Profit function is:

Profit = P x Q - TC

= (100 x L1/6K1/8 ) - 10L - 10K

(b) The two profit-maximization conditions are:

(i) MRTS (L,K) = MPL / MPK, and

(ii) MPL / MPK = w / r

(c)

Here, MPL = dQ / dL = (1/6) x K1/8 / L5/6

MPK = dQ / dK = (1/8) x L1/6 / K7/8

So, MPL / MPK = (8 / 6) x (K / L) = 1.33 (K / L) > 1

Hence, as price inceases, labor increases more than capital.

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