Can a production function with two factors exhibit increasing returns to scale while at the same time have diminishing returns to each factor?

Question

What will be an ideal response?

Answer 1

The answer is yes. Look at the following production function: Q = S0.5 A0.6. Here, we have Q for the quantity of auto parts produced. S and A are the amounts of steel and aluminum used in production.
For checking if the production function exhibits increasing returns to scale, first set S = A = 1 unit. Then total auto parts produced are Q = (1)0.5 (1)0.6 = 1 unit.
Now double the inputs and set S = A = 2 units.
Now check for the new output: New Q = (2)0.5 (2)0.6 = (2)1.1> 2 units.
So here we see that doubling both inputs more than doubles the output. This reflects increasing returns to scale.
Now we want to see if each factor exhibits diminishing returns.
To see if S exhibits diminishing returns, set A = 1 unit. Increase S from 1 to 2 units and check the output level:
A = 1, S = 1 and so Q = (1)0.5 (1)0.6 = 1 unit.
A = 1. S = 2 and so Q = (2)0.5 (1)0.6 = 1.41 units.
As S increases by 1 unit, Q increases only by 0.41 units. So S exhibits decreasing returns. Do the same exercise for A and you will see a similar result:
S = 1, A = 1 and so Q = (1)0.5 (1)0.6 = 1 unit.
S = 1. A= 2 and so Q = (1)0.5 (2)0.6 = 1.51 units.
As A increases by 1 unit, Q increases only by 0.51 units. So A exhibits decreasing returns as well.