Question

Suppose the aggregate production function is given by Y = K0.5L0.5. Does it have increasing, decreasing or constant returns to scale? Show that the marginal products of capital and labour are declining. Show that they are increasing in the input of the other factor.

Answer 1

Y=K^0.5L^0.5

Let us say inputs are changed to x times. i.e. New xK and L is xL

Y=(xK)^0.5(xL)^0.5

Y=x^(0.5+0.5) K^0.5L^0.5

Y=x*K^0.5L^0.5

We find that output has also changed to xY. It means production function exhibits constant returns to scale.

Let us find Marginal Products of K and L

We find that marginal product of labor i.e. MPL decreases as L increases. So, Marginal product of labor is declining.

We find that marginal product of capital i.e. MPK decreases as K increases. So, Marginal product of capital is declining.

We observe that MPL i.e. marginal product of labor increases as other input K increases. So, MPL is increasing with other input increase.

We observe that MPK i.e. marginal product of capital increases as other input L increases. So, MPK is increasing with other input increase.