Question

For the following production functions, find the returns to scales.

1. F(K,L)=K^0.3L^0.7

2. F(K,L)=2K+L

3. F(K,L)=KL

4. F(K,L)=K^0.2L^0.3

An explanation on how to do this, would be appreciated!

Answer 1

Returns to Scale is a long run concept. If we change all the inputs by a certain proportion say , then output also changes by a certain proportion. How much the output changes is what we consider in the returns to scale concept. It has either increasing returns to scale, constant returns to scale or decreasing returns to scale. Increasing returns to scale occurs when the output increases by a larger proportion than the increase in inputs, constant returns to scale occurs when the output increases by the same proportion of the increase in inputs, and decreasing returns to scale occurs when the output increases by a proportion that is less than the proportional increase in input.

1.  F(K,L)=K^0.3L^0.7

Let us change K and L by proportion and hence we have,

F(K,L) = (K)^0.3(L)^0.7

= ^0.3+0.7(K^0.3 L^0.7)

= ¹(K^0.3 L^0.7)

Here, the production function generates Constant Returns to Scale because when we are changing the inputs by proportion, the output is also changing by the same proportion i.e. .

2. F(K,L)=2K+L

Let us change K and L by proportion and hence we have,

F(K,L) = 2K + L

= (2K+L)

Here, the production function generates Constant Returns to Scale because when we are changing the inputs by proportion, the output is also changing by the same proportion i.e. ​.

3. F(K,L)=KL

Let us change K and L by proportion and hence we have,

F(K,L) = (K)(L)

=  ²KL

Here, the production function generates Increasing Returns to Scale because when we are changing the inputs by proportion, the output is also changing by a greater proportion i.e. ​².   

4. F(K,L)=K^0.2L^0.3

Let us change K and L by proportion and hence we have,

F(K,L) = (K)^0.2(L)^0.3

= ^0.2+0.3(K^0.2 L^0.3)

= ^0.5(K^0.2 L^0.3)

Here, the production function generates Decreasing Returns to Scale because when we are changing the inputs by proportion, the output is also changing by a lesser proportion i.e. ​^0.5​.