Increasing Returns to Scale

Increasing Returns to Scale Example

Let's look at an example of increasing returns to scale on a graph.

Fig 1. - Increasing Returns to Scale

What does the graph in Figure 1 above tell us? The graph above shows the long-run average total cost curve for a business, and the LRATC is the long-run average total cost curve. For our study of increasing returns to scale, it's best to direct our attention to points A and B. Let's go over why that is.

Viewing the graph from left to right, the long-run average total cost curve is downward sloping and decreasing while the quantity being produced is increasing. Increasing returns to scale is predicated on the output (quantity) increasing by a larger proportion than the increase of inputs (costs). Knowing this, we can see why points A and B should be of focus for us — this is where the firm is able to increase output while costs are still going down.

However, at point B directly, there are no increasing returns to scale since the flat part of the LRATC curve means that outputs and costs are equal. At point B there are constant returns to scale, and to the right of point B there are decreasing returns to scale!

Increasing Returns to Scale Formula

Understanding the returns to scale formula will help us determine whether a firm has increasing returns to scale. The formula for finding increasing returns to scale is plugging the values for inputs to calculate a corresponding increase in output using a function such as this one: Q = L + K.

Let's look at the equation that is commonly used to figure out the returns to scale for a firm:

$$\begin{gathered} Q=L+K \\ Where: \\ Q=Output \\ L=Labor \\ K=Capital \end{gathered}$$

What does the formula above tell us? Q is output, L is labor, and K is capital. To get the returns to scale for a firm, we need to know how much of each input is being used — labor and capital. After knowing the inputs, we can find out what the output is by using a constant to multiply each input by.

For increasing returns to scale, we are looking for an output that increases by a larger proportion than the increase in inputs. If the increase in output is the same or less than the inputs, then we do not have increasing returns to scale.

Increasing Returns to Scale Calculation

Let's look at an example of increasing returns to scale calculation.

Let's say that a function of the firm's output is:

$$\begin{gathered} Q=4L^2+K^2 \\ Where: \\ Q=Output \\ L=Labor \\ K=Capital \end{gathered}$$

With this equation, we have our starting point to begin our calculation.

Next, we have to use a constant to find the change in output resulting from the increase in production inputs - labor and capital. Let's say that the firm increases the amount of these inputs five fold.

$$\begin{gathered} Q\text{'}=4{\left(5L\right)}^2+{\left(5K\right)}^2 \\ Distributeexponents: \\ Q\text{'}=4\times 5^2\times L^2+5^2\times K^2 \\ Factoroutthe{5}^2: \\ Q\text{'}=5^2(4L^2+K^2) \\ Q\text{'}=25(4L^2+K^2) \\ Q\text{'}=25Q \end{gathered}$$

What do you notice about the numbers in the parenthesis? They are the exact same as the initial equation that told us what Q was equal to. Therefore, we can say that the value inside the parenthesis is Q.

We can now say that our output, Q, increased 25 times based on the increase in inputs. Since the output increased by a larger proportion than the input, we have increasing returns to scale!

Increasing Returns to Scale vs Economies of Scale

Increasing returns to scale and economies of scale are closely related, but not exactly the same thing. Recall that increasing returns to scale occur when output increases by a larger proportion than the increase in input. Economies of Scale, on the other hand, are when the long-run average total cost declines as output rises.

Chances are if a firm has economies of scale they also have increasing returns to scale and vice versa. Let's look at a firm's long-run average total cost curve for a better look:

Fig 2. - Increasing Returns to Scale and Economies of Scale

The graph in Figure 2 above us gives us a good visualization of why increasing returns to scale and economies of scale are closely related. Looking at the graph from left to right, we can see that the LRATC (long-run average total cost) curve is downward sloping up to point B on the graph. During this slope, the cost for the firm is decreasing as the quantity being produced increases — this is the exact definition of economies of scale! Recall: economies of scale is when the long-run average total cost decreases as output increases.

But what about increasing returns to scale?

Increasing returns to scale is when outputs increase by a greater proportion than inputs. Generally, if a firm has economies of scale then they will likely have increasing returns to scale as well.