Question

1: Do the following functions exhibit increasing, constant, or decreasing returns to scale? What happens to the marginal product of each individual factor as that factor is increased, and the other factor is held constant?

a. q = (L 0.5 + K 0.5) 2   b. q = 3LK2 c. q=L0.3 K 0.6

Answer 1

a. Constant returns to scale. Decreasing marginal product of labour and decreasing marginal product of capital.

Reason-

q = (L^0.5 + K^0.5)^2

Increase each input by λ

( (λL)^0.5 + (λK)^0.5)^2

= ((λ)^0.5(L^0.5 + K^0.5))^2

=λ * ((L^0.5 + K^0.5)^2)

=λ *q

So when inputs are multiplied by λ, output is also multiplied by λ. Hence it is constant returns to scale.

Marginal product of labour= dq/dL= 2*(L^0.5 + K^0.5)*0.5L^(-0.5) = 1+(√K/√L)

When K is constant and L increases, MPL decreases.

Similarly MPK= 1+(√L/√K)

When L is constant and K increases, MPK decreases.

B. increasing returns to scale.

The marginal product of labor is constant and the marginal product of capital is increasing.

Reason-

q = 3LK^2

Increase each input by λ

q' = 3λL(λK^2)

q' = 3λLλ^2(K^2)

q ' = λ^3*3LK^2

q'= λ^3*q

Since output increases by λ^3 when input increases byλ. It is increasing returns to scale.

MPL= dq/dl=3K^2

When K is constant MPL is constant.

MPK= dq/dk= 6LK^2.

When L is constant, MPK increases.

c) constant returns to scale. Decreasing marginal product of labour and decreasing marginal product of capital.

Reason-

q= L^0.3* K^0.6

Increase each input by λ

q'= (λL)^0.3* (λK)^0.6

q' = λ*(L^0.3* K^0.6)

q'=λ*q

When input increases by λ, output increses by λ.

Hence it is a constant returns to scale.

MPL= dq/dL= 0.3L^(-0.7)*K^0.6

When K is constant and L increases , MPL decreases.

MPk= dq/dk= 0.6K^(-0.4)*L^0.4

When L is constant and L increases , MPK decreases