Question

Suppose the economyís production function is

Y = K0:3L0:7

Show the the production function slopes upward from left to right. Show that the production function features diminishing marginal product of labor. When constant return to scale is assumed, derive the per worker produc- tion function.

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Answer 1

Y = K^0.3L^0.7

The production function slopes upward where there are diminishing returns to MRTS

MPL = dY/dL = 0.7*(K/L)^0.3

MPK = dY/dK = 0.3*(L/K)^0.7

MRTS = MPL/MPK = ( 0.7*(K/L)^0.3) / (0.3*(L/K)^0.7)

MRTS = 2.33*(K/L)

Differentiating MRTS wrt L

dMRTS/dL = -2.33(K/L^2)

Since, it is negative so production function slopes upward from left to right.

MPL = 0.7*(K/L)^0.3

dMPL / dL = -0.21*K^0.3L^-1.3 < 0

So, MPL shows diminishing marginal product of labor

Per worker production function, y = Y/L

y = (K^0.3L^0.7) / L

y = (K/L)0.3

y = k^0.3 , where k = K/L