# 4.10 Scaling Production in the Long Run: “Returns to Scale”

Finally, let’s consider what happens when you scale *all* inputs proportionally. For simplicity, let’s ask the question: if we double all inputs, what happens to the output: does it also double, or less than double, or more than double? In other words, does output scale proportionally to inputs?

Mathematically, the way we can analyze this is by comparing the amount that’s produced with the input combination $(L,K)$ and the combination $(2L,2K)$. For example, with a linear production function \(f(L,K) = 2L + 4K\) we have \(f(2L, 2K) = 2(2L) + 4(2K) = 4L + 8K\) We can then ask ourselves: is this equal to, greater than, or less than twice the original input? Well, twice the original input is \(2f(L,K) = 2[2L + 4K] = 4L + 8K\) So in this case, doubling all inputs exactly doubles the output.

Now consider the production function $f(L,K) = LK$. For this function,
\(f(2L,2K) = (2L)(2K) = 4LK\)
This is four times the original output of $LK$; so doubling all inputs *more than* doubles the output.

More formally, we say that for any $t > 1$, the production function exhibits:

**Decreasing returns to scale**(DRS) if $f(tL, tK) < tf(L,K)$**Constant returns to scale**(CRS) if $f(tL, tK) = tf(L,K)$**Increasing returns to scale**(IRS) if $f(tL, tK) > tf(L,K)$

The example of doubling all inputs is just this method with $t = 2$ for mathematical simplicity.

## Returns to Scale and Cobb-Douglas Production

Let’s look at this more generally for a Cobb-Douglas production function of the form \(f(L,K) = AL^aK^b\) If we double all inputs, we have \(f(2L, 2K) = A(2L)^a(2K)^b = 2^{a+b}AL^aK^b = 2^{a+b}f(L,K)\) Since $2^1 = 2$, this means the function is decreasing returns to scale if $a + b < 1$; constant returns to scale if $a + b = 1$; and increasing returns to scale if $a + b > 1$.

You can use the following graph to examine how doubling just labor or doubling both labor and capital affect output for a Cobb-Douglas production function. The solid blue isoquant shows $q = f(L,K)$; the dashed blue isoquant shows double that quantity, $\hat q = 2f(L,K)$:

Notice that if $a + b = 1$, doubling both inputs *exactly doubles* the output: that is, $f(2L, 2K) = 2f(L,K)$, and the point $(2L,2K)$ lies *along* the dashed isoquant for $2f(L,K)$. If $a + b < 1$, then doubling both inputs *less than doubles* output, so $(2L,2K)$ lies *below* the isoquant for $2f(L,K)$. Likewise, if $a + b > 1$, then doubling both inputs *more than doubles* output, so $(2L,2K)$ lies *above* the isoquant for $2f(L,K)$

## The real world

In the real world, production functions may scale differently at different levels of input use: many startups may have increasing returns to scale when they are small and expanding can allow them to achieve greater efficiencies, but start experiencing a loss of efficiency as they grow and become more corporate in nature. This is just one more example of a case in which we’ll start out by looking at mathematically simple but unrealistic functions to get our start in economic modeling; if you continue in your studies of economics, you’ll see more interesting and realistic mathematical structures.