Question

4. (4 points) Suppose the Average Product of Capital for a production technology is AP, = 5K /412 . What sort of returns to scale does this production function exhibit?

I understand that the picture has the answer to it. However, I'm wanting help understanding why that's the answer/how they got to the answer. How did we go from APk=(5K^(-1/4)L^(1/2)) to Q=5k^(3/4)L^(1/2)? Why do you then proceed to add the exponents together and essentially ignore the other parts of the problem completely (as to arrive at the answer of 5/4)? Furthermore, how did they get the "n" looking thing in the parenthesis before the "K" and "L" ? What does it do? How did it help figure out the answer that came after that? Why is it ultimately an increasing returns to scale?

Answer 1

To find out the production function from the average product of capital, we multiply AP_k by K.

So, K(AP_k) = K(5K^(-1/4)L^(1/2))

This gives us Q=5k^(3/4)L^(1/2)

Also, there are two ways to find out whether the returns to scale are increasing, decreasing, or constant.

Lets say that a general cobb douglas function is Q = AL^αK^β

In our case, α=3/4, β=1/2

α+β = 3/4 + 1/2 = 5/4. Since this is greater than 1, it represents increasing returns to scale. Had it been lesser than 1, it would mean decreasing returns to scale, and equal to 1 would mean constant returns to scale.

Another way is to multiply K and L separately by lambda ($\lambda$). The 'n' is actually lambda.

So multiplying K and L by lambda and then looking at the result will give us the answer. If the power $\lambda$ is raised to is greater than 1, then it means increasing returns to scale, as in the given solution.