Question

1. Consider the following production function:

Y=F(A,L,K)=A(K^α)(L^(1-α))

where α < 1.

a. Derive the Marginal Product of Labor(MPL).

b. Show that this production function

exhibit diminishing MPL.

c. Derive the Marginal Production of Technology (MPA).

d. Does this production function exhibit diminishing MPA? Prove or disprove

Answer 1

(a) Y= F(A,L,K) = A K^a L^1-a

For calculating MPL , take the first partial derivative of Y with respect to L, we get:

MPL = (1-a) A K^a L^1-a-1 = (1-a) AK^a L^-a.

(b) To see whether the production function exhibit diminishing MPL or not , take the second order partial derivative with respect to L, we get:
= (-a)(1-a) AK^a L^-a-1 < 0   

Hence, It shows that the production function exhibit diminishing returns to factor L.

(c) To calculate MPA , take the partial derivative of Y with respect to A , we get:
MPA = K^a L^1-a

(d) To see whether the production function exhibit diminishing MPA or not, take the second order partial derivative of Y with respect to A, we get:

= 0 .

This implies that the production function exhibits constant returns to factor A.