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

A function exhibit constant returns to scale If f(tX,tY) = t*f(X,Y) for all t > 1

A function exhibit increasing returns to scale If f(tX,tY) > t*f(X,Y) for all t > 1

A function exhibit decreasing returns to scale If f(tX,tY) < t*f(X,Y) for all t > 1

Here Y = F(K,L) = K^0.5L^0.5 => F(tK,tL) = (tK)^0.5(tL)^0.5 = tK^0.5L^0.5 = tF(,L)

=> F(tK,tL) = tK^0.5L^0.5. Hence using above information this function exhibit Constant returns to scale.

Marginal productivity of Capital (MPK) is given by:

Hence, As amount of K(i.e. capital) increases Marginal Productivity of Capital will decrease.

Marginal productivity of Labor (MPL) is given by:

which is negative

Hence, As amount of L(i.e. Labor) Increases Marginal Productivity of Labor will decrease.

Hence, the marginal products of capital and labor are both declining.

From above :

which is greater than 0

=> As K increases Marginal Productivity of Labor will also increase.

Hence, Capital and Labor are increasing in the input of the other factor.