Question

# Given production function: Q=L3/5K1/5. Where L is labor, K is capital, w is wage rate, and...

Given production function: Q=L3/5K1/5.

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.

Q = L3/5K1/5

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

Q1 = (2L)3/5(2K)1/5 = 23/521/5 x L3/5K1/5 = 24/5 x Q

Q1/Q = 24/5 < 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 K1/5 / L2/5

MPK = Q/ K = (1/5) x L3/5 / K1/5

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

3K/L = w/r

K = wL/3r

Substituting in production function,

Q = L3/5(wL/3r)1/5

Q = L3/5L1/5(w/3r)1/5

Q = L4/5(w/3r)1/5

L4/5 = Q x (3r/w)1/5

L = [Q x (3r/w)1/5]5/4 = Q5/4 x (3r/w)1/4

K = [Q5/4 x (3r/w)1/4] x (w/3r) = Q5/4 x (w/3r)3/4

Total cost = wL + rK = w x [Q5/4 x (3r/w)1/4] + r x [Q5/4 x (w/3r)3/4] = Q5/4 x [{w x (3r/w)1/4} + {r x (w/3r)3/4]

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