Factor Returns and Scale Returns

[Q:]Consider the production function Q = K^aL^b.

Find conditions on a and b so that the function exhibits decreasing returns to each factor, but increasing returns to scale.

[A:] Thanks for your great question!

Recall that in the article Increasing, Decreasing, and Constant Returns to Scale that we can easily answer these questions by simply doubling the necessary factors and doing some simple substitutions.

Increasing returns to scale would be when we double all factors, and production more that doubles. In our example we have two factors K and L, so we'll double K and L and see what happens:

Q = K^aL^b

Now lets double all our factors, and call this new production function Q'

Q' = (2K)^a(2L)^b

Rearranging leads to:

Q' = 2^a+bK^aL^b

Now we can substitute back in our original production function, Q:

Q' = 2^a+bQ

To get Q' > 2Q, we need 2^(a+b) > 2. This occurs when a + b > 1.

As long as a+b>1, we will have increasing returns to scale. We also need decreasing returns to scale in each factor. Decreasing returns for each factor occurs when we double only one factor, and the output less than doubles. Try it first for K: Q = K^aL^b

Now lets double K, and call this new production function Q'

Q' = (2K)^aL^b

Rearranging leads to:

Q' = 2^aK^aL^b

Now we can substitute back in our original production function, Q:

Q' = 2^aQ

To get 2Q > Q' (since we want decreasing returns for this factor), we need 2 > 2^a. This occurs when 1 > a.

Similarly for L: Q = K^aL^b

Now lets double L, and call this new production function Q'

Q' = K^a(2L)^b

Rearranging leads to:

Q' = 2^bK^aL^b

Now we can substitute back in our original production function, Q:

Q' = 2^bQ

To get 2Q > Q' (since we want decreasing returns for this factor), we need 2 > 2^a. This occurs when 1 > b.

So there are you conditions. You need a+b > 1, 1 > a, 1 > b. By doubling factors, we can easily create conditions where we have increasing returns to scale overall, but decreasing returns to scale in each factor.

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