Question

Given the production function z = F(K, L) = (K^0.6 + L^0.8)^2 and K>0 and L>0.

Prove that the function is strictly increasing in capital K and in labor L. Provide an economic implication.

Answer 1

The production function z = F(K, L) = (K^0.6 + L^0.8)^2 and K>0 and L>0.

Differentiating the function with respect to K,

1.2(K^0.6 + L^0.8)/K^0.4 - equation 1

Since K>0 and L>0, equation 1 is positive. Thus, the function is strictly increasing in capital K.

Similarly, differentiating the function with respect to L,

1.6(K^0.6 + L^0.8)/L^0.2 -equation 2

Since L>0 and K>0, equation 2 is positive. Thus, the function is strictly increasing in labor L.

ECONOMIC IMPLICATION:

Keeping everything else constant, increasing the amount of capital will lead to a higher level of output. The same is true for additional labor employed. In other words, the marginal productivity of capital and labor is positive.