Question

Part C: Production and Production Costs 4)The estimated production function of Rosa ’s noise cancelation and isola - tion device in Liechtenstein is Q = 2.83 L1.52 K 0.82 , where L is labor and K is capital. A study estimated that the production function for housing in Liecht - enstein is Q = 1.38 L 0.144 M 0.856 , where L is land and M is an aggregate of all other factors used in the production which we call materials. Another study estimated that the production function for supermarkets in Liechtenstein is Q = L0.23 K 0.10 M 0.66 , where L is labor, K is capital, and M is materials. Show and explain, with the help of algebra, if each of these production functions have decreasing, constant or increasing returns to scale.

Answer 1

Q= 2.83 L^1.52 K^0.82

Double the inputs K and L , we get:

= 2.83(2L)^1.52 (2K)^0.82

= 2.83 (2)^1.52+0.82 ( L^1.52 K^0.82)

= 2^2.34 [  2.83 L^1.52 K^0.82] > 2Q

Because increasing the inputs K and L , increase the amount by more than that amount . This implies that production function exhibits increasing returns to scale.

Q= 1.38 L^0.144 M^0.856

Double the inputs M and L ,we get :

= 1.38 (2L)^0.144 (2M)^0.856

= 1.38 (2)^0.144+0.856 ( L^0.144 M^0.856)

= 2 [ 1.38 L^0.144 M^0.856] = 2Q

Because increasing the inputs M and L , increase the output by the same amount. This implies that the production function exhibits constant returns to scale.

Q=L^0.23 K^0.10 M^0.66

Double the inputs K , L and M , we get :

= (2L)^0.23 (2K)^0.10 (2M)^0.66

= 2^{0.23+0.10+0.66} (L^0.23 K^0.10 M^0.66)

= 2^0.99 (L^0.23 K^0.10 M^0.66) < 2Q

Because increasing the inputs K , L and M , increase the output by less than that amount . This implies that the production function exhibits decreasing return to scale.