Question

Q=100K^0.7L^0.4

1. Find the marginal product of labor
2. Find the marginal product of capital
3. Is there diminishing marginal rate of technical substitution? Explain.
4. Does the production function exhibit constant, increasing, or decreasing returns to scale.

Answer 1

a) MPL = 100 * 0.4 K^0.7 L^-0.6

= 40 K^0.7 L^-0.6

b) MPK = 100 * 0.7 K^-0.3 L^0.4

= 70 K^-0.3 L^0.4

c) MRTS = MPL / MPK

= 40 K^0.7 L^-0.6 / 70 K^-0.3 L^0.4

= 4K/7L

Now dMRTS/dL = -(4K/7)L^-2 which is less than 0

This suggests that as L increases MRTS declines. Hence we have diminishing MRTS

d) We have increasing returns. Multiply inputs by a fraction 'a'

New Q = 100(aK)^0.7 (bK)^0.4

= 100a^0.7 a^0.4 K^0.7 L^0.4

= a^1.1*100K^0.7 L^-0.6

= a^1.1 old Q

Since new Q in increased by a fraction greater than 'a', we have increasing returns to scale