1 Do the following Production Functions exhibit increasing c

1. Do the following Production Functions exhibit increasing, constant or decreasing returns to scale in K and L. (That is if K and L are changed). Demonstrate in each case carefully (Y-Output or GDP, K-Capital and L-Labor) (a) Y = K1/2 L1/2 (b) Y = K1/3 L2/3 (c) Y = K2/3 L2/3 (d) Y = 6K + L (e) Y = A K1/2 L1/2 (Where A is some positive fixed constant) (Where A is some positive fixed constant) Note: In parts (e) and (f) above ‘A\' is a fixed constant or what we call a para,neter. It does not change. Whereas K and L are the inputs in the production process and what we call or consider to be the variables in the system

Solution

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

Now, by doubling the inputs K and L , we get,

= (2K)^1/2 (2L)^1/2

= 2^{1/2 +1/2} K^1/2 L^1/2

= 2 K^1/2 L^1/2 = 2Y

This implies constant returns to scale.

(b) Y= K^1/3 L^2/3

Now, by doubling the inputs K and L ,we get:

= (2K^1/3) (2L)^2/3

= 2^{1/3 +2/3} K^1/3 L^2/3

= 2 K^1/3 L^2/3 = 2Y

This implies constant returns to scale.

(c) Y = K^2/3 L^2/3

Now, by doubling the inputs K and L , we get:

= (2K)^2/3 (2L)^2/3

= 2^2/3+2/3 K^2/3 L^2/3

= 2^4/3 K^2/3 L^2/3 > 2Y

It implies increasing returns to scale.

(d) Y = 6K + L

Now, by doubling the inputs K and L , we get:

= 6(2K) + 2L

= 2 (6K + L ) = 2 Y

It implies constant returns to scale.

(e) Y = AK^1/2L^1/2

Now, by doubling the inputs K and L, we get:
= A (2K)^1/2(2L)^1/2

= 2^{1/2 +1/2} A K^1/2 L^1/2

= 2 AK^1/2 L^1/2 = 2Y

It implies constant returns to scale.

(f) Y= K^1/3 L ^2/3 + A

Now, by doubling the inputs K and L , we get

= (2K)^1/3 (2L)^2/3 + A

= 2^{1/3 +2/3} K^1/3 L^2/3 + A

= 2 K^1/3 L^2/3 + A < 2Y = 2 K^1/3L^2/3 + 2A

This implies decreasing returns to scale.