Question

Given production function: Q=L^{3/5}K^{1/5}.

Where L is labor, K is capital, w is wage rate, and r is rental rate.

What kinds of returns to scale does your firm face?

Find cost minimizing level of L and K, and long run cost function.

Answer #1

Q = L^{3/5}K^{1/5}

(i) When both inputs are doubled, new production function becomes

Q1 = (2L)^{3/5}(2K)^{1/5} =
2^{3/5}2^{1/5} x L^{3/5}K^{1/5} =
2^{4/5} x Q

Q1/Q = 2^{4/5} < 2

Since doubling both inputs less than doubles output, there is decreasing returns to scale.

(ii) Cost is minimized when MPL/MPK = w/r

MPL =
Q/L
= (3/5) x K^{1/5} / L^{2/5}

MPK =
Q/K
= (1/5) x L^{3/5} / K^{1/5}

MPL/MPK = 3 x (K/L) = w/r

3K/L = w/r

K = wL/3r

Substituting in production function,

Q = L^{3/5}(wL/3r)^{1/5}

Q = L^{3/5}L^{1/5}(w/3r)^{1/5}

Q = L^{4/5}(w/3r)^{1/5}

L^{4/5} = Q x (3r/w)^{1/5}

**L =** [Q x (3r/w)^{1/5}]^{5/4} =
**Q ^{5/4} x (3r/w)^{1/4}**

**K =** [Q^{5/4} x (3r/w)^{1/4}] x
(w/3r) = **Q ^{5/4} x
(w/3r)^{3/4}**

**Total cost =** wL + rK = w x [Q^{5/4} x
(3r/w)^{1/4}] + r x [Q^{5/4} x
(w/3r)^{3/4}] = **Q ^{5/4} x [{w x
(3r/w)^{1/4}} + {r x (w/3r)^{3/4}]**

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