Question

1. Consider the following production functions. In each case determine if: • the function is Cobb Douglas (Y = AKαL 1−α). If the function is Cobb Douglas, what is the value of the parameter α? • Do capital and labor exhibit diminishing returns. Explain your thinking using algebra / calculus /a graph etc.

(a) F(K, L) = 2√ K + 15√ L

(b) F(K, L) = 5K + 3L

(c) F(K, L) = K0.5L 0.5

(d) F(K, L) = √ K2 + L2

Answer 1

(a) Y= 2K^1/2 + 15L^1/2

No, the given function is not Cobb douglas.

MPK= 1/2 (2)K^-1/2 = K^-1/2

Now, take the derivative of MPK w.r.t K ,we get :

= (-1/2) K^-3/2 <0.

This implies that the function exhibits diminishing returns to capital.

Similarly, MPL = 1/2( 15)L^-1/2 = (15/2)L^-1/2

Now, take the derivative of MPL w.r.t L, we get:

= (15/2)(-1/2) L^-3/2 = (-15/4) L^-3/2 <0.

This implies that the function exhibits diminishing returns to labor.

(b) Y=5K + 3L

No,this production function is not Cobb douglas production function.

MPK = 5

MPL = 3

This production function exhibits constant returns to capital and labor.

(c) Y= K^0.5 L^0.5

Yes, this production function is Cobb douglas production function.

And the parameter a=0.5.

MPK = 0.5 K^-0.5

Now,take the derivative of MPK w.r.t K ,we get :

= (0.5)(-0.5) K^-1.5

= -0.25 K^-1.5 <0.

This implies that the production function exhibits diminishing returns to capital.

MPL = 0.5 L^-0.5

Now,take the derivative of MPL w.r.t L,we get :

= (0.5)(-0.5) L^-1.5

= -0.25 L^-1.5 <0.

This implies that the production function exhibits diminishing returns to labor.

(d) Y= (K² + L²) ^1/2

No, this production function is not the Cobb douglas form.

MPK = (1/2) (K² + L²) 2K = K³ + KL²

Now, take the derivative of MPK w,r,t K ,we get :

=3K² + L² >0.

This implies that the production function exhibits increasing returns to capital.

Similalrly, MPL = (1/2) (K² + L²) 2L = LK² + L³

Now, take the derivative of MPL w,r,t L ,we get :

= K² + 3L² >0.

This implies that the production function exhibits increasing returns to labor.