Decreasing Returns to Scale

Fig. 1 - Engineering Working with Computers

Decreasing Returns to Scale Example

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

Fig. 2 - Decreasing Returns to Scale

The graph above shows the long-run average total cost curve (LRATC) for a business. Since we are focusing on decreasing returns to scale, we should direct our attention to points B and C. Why might that be the case?

Viewing the graph from left to right, the LRATC curve slopes downward, flattens out, and slopes upward at the end. Decreasing returns to scale requires that the output (quantity) increases by a smaller proportion than the increase of inputs (costs).

Knowing this information, we can see why points B and C should be of focus for us — this is where the firm is increasing their costs as they are increasing output. Visually, we can also see that the curve is getting steeper which implies that costs are increasing while the output increase is slowing down!

Decreasing Returns to Scale Formula

The returns to scale formula will be useful in determining whether a firm is experiencing decreasing returns to scale. The general formula we can use for determining decreasing returns to scale is the following: Q = L + K

We will plug in values for the inputs to see what the corresponding effect on output will be.

Let's go over the equation in detail:

What does the formula above tell us? Q is output, L is labor, and K is capital. To determine the returns to scale, we need to know how much of each input is being used. Once we know the inputs that we are given, we can find the output by multiplying the inputs by some constant of our choosing.

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

Decreasing Returns to Scale Calculation

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

The function of the firm's output is the following:

$$\begin{gathered} Q=5L^{1/3}{K}^{1/3} \\ Where: \\ Q=Output \\ L=Labor \\ K=Capital \end{gathered}$$

The equation given is our starting point for our calculation.

Next, we will use some constant, say 3, to determine the change in output from the production inputs — labor and capital.

$$\begin{gathered} Q\text{'}=5{\left(3L\right)}^{1/3}{\left(3K\right)}^{1/3} \\ DistributeExponents: \\ Q\text{'}=5\times 3^{1/3}\times L^{1/3}\times 3^{1/3}\times K^{1/3} \\ Factoroutthe{3}^{1/3}: \\ Q\text{'}=3^{1/3}\times 3^{1/3}(5L^{1/3}{K}^{1/3}) \\ Q\text{'}=3^{2/3}(5L^{1/3}{K}^{1/3}) \\ Q\text{'}=3^{2/3}Q \end{gathered}$$

Is there anything you notice about the numbers in the parenthesis?

Correct! It is the same as our initial equation: $Q=5L^{1/3}{K}^{1/3}$

Therefore, we know that the value inside the parenthesis is equal to Q, our output.

We can now say that our output increased by $3^{2/3}$ which equals 2.08. Recall that our initial constant was 3 for our inputs. Therefore, the output increased by a smaller proportion than the input increased by, indicating decreasing returns to scale!

Decreasing Returns to Scale Versus Increasing Returns to Scale

It is important to recognize the difference between decreasing returns to scale and increasing returns to scale. Let's begin our analysis by looking at the concepts between the two.

Decreasing Returns to Scale Versus Increasing Returns to Scale: Concept

Recall the definition of decreasing returns to scale: when the output increases by a smaller proportion than the increase in inputs. A firm is increasing the cost of its inputs (labor and capital) by a greater amount than they are increasing their production as a result. Generally, this is a bad position to be in!

Let's now go over the definition of increasing returns to scale: when the output increases by a greater proportion than the increase in inputs. A firm is increasing the cost of its inputs (labor and capital) by a smaller amount than they are increasing their production as a result. Generally, this is a favorable position to be in!

Decreasing Returns to Scale Versus Increasing Returns to Scale: Graphs

Let's now look at the differences between decreasing and increasing returns to scale on a graph.

Fig. 3 - Decreasing Returns to Scale

What does the graph above tell us? Recall the definition for decreasing returns to scale: when the output increases by a smaller proportion than the increase in inputs. The right-hand side of the LRATC curve visually portrays this definition — output is increasing by a smaller proportion than the cost shown by the upward sloping curve.

Where are increasing returns to scale portrayed in this graph?

Fig. 4 - Increasing Returns to Scale

What does the graph above tell us? Recall the definition of increasing returns to scale: when output increases by a greater proportion than the increase in inputs. The left-hand side of the LRATC curve visually portrays this definition — output is increasing by a greater proportion than the cost shown by a downward sloping curve. This is increasing returns to scale!

Decreasing Returns to Scale versus Diseconomies of Scale

Decreasing returns to scale and diseconomies of scale are closely related, but it's important to recognize the differences between the two. Recall that decreasing returns to scale is when the output increases by a smaller proportion than the increase in inputs. Diseconomies of Scale is when long-run average total costs increase while output increases.

If a firm is undergoing decreasing returns to scale, then the firm will likely be going through diseconomies of scale. Let's look at a firm's long-run average total cost curve to understand this relationship:

Fig. 5 - Decreasing Returns to Scale and Diseconomies of Scale

The graph above gives us a good visualization of why decreasing returns to scale and diseconomies 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, then slopes upward to the right of point B. During the upward slope, the cost for the firm is increasing as the quantity being produced increases — this is the exact definition of diseconomies of scale!

Recall: diseconomies of scale is when the long-run average total cost increases as output increases.

But what about decreasing returns to scale?

Decreasing returns to scale is when outputs increase by a smaller proportion than inputs. Generally, if a firm has diseconomies of scale, then they will likely have decreasing returns to scale as well.