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## Decreasing Returns to Scale Explanation

The explanation regarding decreasing returns to scale is made simpler once we understand returns to scale.

**Returns to scale** is the rate at which output changes from some change in input.

**Decreasing returns to scale** occur when the output produced by a firm increases by a proportion *less* than the increase in inputs.

Inputs in this instance are labor and capital, for example.

Let's go over a quick example so we can better visualize decreasing returns to scale.

Suppose that you own a business that produces computers.

In year 1, you have 50 employees and 20 machines producing 500 computers a year — fantastic! Now let's say that in year 2, you now have 150 employees and 60 machines producing 1,000 computers a year.

As you can see, your inputs, labor and capital, tripled from year 1 to year 2. However, your output of computers did not triple, it only doubled.

You might think that tripling your inputs will lead to a proportional increase in output, but this is not always the case. As such, this quick example provides an insight into the decreasing returns to scale that businesses may endure.

**Decreasing Returns to Scale **occur when the output increases by a proportion less than the increase in inputs.

**Returns to Scale** are the rate at which output changes from some change in input.

## Decreasing Returns to Scale Example

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

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:

$Q=L+K\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Where:\phantom{\rule{0ex}{0ex}}Q=Output\phantom{\rule{0ex}{0ex}}L=Labor\phantom{\rule{0ex}{0ex}}K=Capital$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.

The constant can be a variable or any number you choose to test the output!

## 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:

$Q=5{L}^{1/3}{K}^{1/3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Where:\phantom{\rule{0ex}{0ex}}Q=Output\phantom{\rule{0ex}{0ex}}L=Labor\phantom{\rule{0ex}{0ex}}K=Capital$

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.

$Q\text{'}=5{\left(3L\right)}^{1/3}{\left(3K\right)}^{1/3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}DistributeExponents:\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Q\text{'}=5\times {3}^{1/3}\times {L}^{1/3}\times {3}^{1/3}\times {K}^{1/3}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Factoroutthe{3}^{1/3}:\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Q\text{'}={3}^{1/3}\times {3}^{1/3}(5{L}^{1/3}{K}^{1/3})\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Q\text{'}={3}^{2/3}(5{L}^{1/3}{K}^{1/3})\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Q\text{'}={3}^{2/3}Q\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

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

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

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!

**Increasing Returns to Scale **occurs when the output increases by a greater proportion than the increase in inputs.

### 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.

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?

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:

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.

**Diseconomies of Scale **occur when the long-run average total cost increases as output increases.

## Decreasing Returns to Scale - Key takeaways

- Decreasing returns to scale occurs when the output increases by a smaller proportion than the increase in inputs.
- Returns to scale is the rate at which output changes due to some change in input.
- Increasing returns to scale occurs when the output increases by a greater proportion than the increase in input.
- The formula generally used to calculate returns to scale is the following: Q = L + K
- Decreasing and increasing returns to scale occur on opposite ends of the LRATC curve.

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##### Frequently Asked Questions about Decreasing Returns to Scale

What is decreasing returns to scale?

Decreasing returns to scale is when the output increases by a smaller proportion than the inputs.

How do you calculate decreasing return to scale?

You calculate decreasing returns to scale with the following formula: Q = L + K

What are the causes of decreasing returns to scale?

The causes of decreasing returns to scale happen when a firm is paying more for its inputs than they are producing products.

What happens to cost in decreasing returns to scale?

Cost tends to rise in decreasing returns to scale.

What is the formula for finding the decreasing returns to scale?

The following formula is used to find decreasing returns to scale: Q = L + K

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