# Constant Returns to Scale

When you walk into a business - whether it sells groceries, electronics, or cars - you might wonder: how does this business know how many employees to hire, or how many products does this business sell to cover its costs? One part of the production process that businesses consider is a term called "returns to scale," which aims to find the total output from what inputs like labor and capital may produce. In this article, we will learn the meaning of the term constant returns to scale, how it is calculated, examples of constant returns to scale, and more.

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## What is the Meaning of Constant Returns to Scale?

"Returns to scale" is a term that refers to how well a firm is producing its goods or services. Assessing returns to scale aims to increase production concerning other factors contributing to the firm's production of goods and services over time.

Most processes firms use to calculate production include labor and capital as major inputs. A firm can determine whether its returns to scale are increasing, decreasing, or constant in calculating the quantities of labor and capital. In this article, we will be focusing on constant returns to scale.

Constant returns to scale occur when an input increase, such as labor and capital, proportionally increases output.

It is important to remember that the law of return to scale contains assumptions to function. Therefore, when a firm increases its variable inputs of production, such as labor and capital, and it produces a constant increase or decrease in its outputs, the following assumptions hold:

• All inputs of production are variable.

• Outputs are measured in physical terms.

• The market is perfectly competitive.

• Technology is constant.

## What Causes Constant Returns to Scale?

The definition of constant returns to scale states that changes in the proportion of inputs in the production of goods will result in the same change in the proportion of outputs. For instance, if a firm increases its units of labor and capital by 10%, then the expected output in the production process would also increase by 10%. Therefore, the cause of constant returns to scale is the factor by which the production's input affects the output.

The table below will illustrate the difference between constant, increasing, and decreasing returns to scale to better understand constant returns to scale.

 Returns to scale Return Value Sample Returns to scale (Y) Inputs doubled (m=2) Q= output$\left(Y\right)×m=Q$ Constant X=1 Y=1 $\left(1\right)×2=2$ Increasing X>1 Y=3 $\left(3\right)×2=6$ Decreasing 0

Table 1. Returns to scale

## Constant Returns to Scale: Diagram

Let's see what constant returns to scale look like in the form of a diagram. We see in Figure 1 below the representation of constant returns to scale in relation to output and units of labor and capital. Keep reading below to see an example using this type of return to scale.

Figure 1. Constant returns to scale, StudySmarter Originals

In figure 1. above, the graph shows a linear relationship between units of input (labor/capital) and output; when inputs are doubled, the output is doubled.

Suppose there are 5 units of labor and capital. In that case, the output level is 20, where 5 units of labor or 5 units of capital can produce 20 units. When inputs are 10, the output level is 40; when inputs go to 20, the output level is 80. With each increase, both inputs and outputs double, so they can increase by an equal percentage. We can conclude that if a firm increases inputs using these types of returns to scale, the outputs will also increase by the same percentage.

## Constant Returns to Scale Formula

Now to calculate constant returns to scale, we'll use a simple formula to help us find what we're looking for. We'll see what happens when we increase all production inputs by what is called a multiplier. To demonstrate, let's call the multiplier m.

The multiplier (m) when referring to returns to scale is the value that the inputs are increased by.

Suppose our inputs are labor and capital, and we double each of them, which makes the multiplier (a) equal to 2. We want to know how this affects the output (Q). Will it be more than double, less than double, or exactly double? Constant returns to scale will show that when the inputs are increased by whichever multiplier (m), the output (Q) will increase by exactly the value of the multiplier (m). You can calculate it by determining whether an input's marginal product (MP) is increasing, decreasing, or constant.

Supposes there's a small business owner, Douglas, who sells corn, "cobb," as he calls it for short. Douglas currently allocates $30,000 to labor and$20,000 to capital, and his annual yield of cobb sells for $100,000. Douglas wants to expand his production by investing in his production process, so he doubles the allocated funds for labor and capital. He now spends$60,000 on labor and $40,000 on capital, which leads him to have an annual yield of$200,000. Douglas is experiencing constant returns to scale on investment in his "cobb."

In this example, the annual yield is total production or output (Q). The doubling of inputs is the multiplier (m).Now let's look at how you would solve this in a formula format.

Q = total production, or output

m = multiplier

K = capital (the money allocated to durable assets, machinery, land.

L = labor (resources allocated to labor process)

$Q=F\left(m×K;m×L\right)=m×F\left(K;L\right)$

The F(K) refers to the function of K. So$F\left(K;L\right)=aK+bL$where a & b are industry-specific modifiers for how much the industry responds to change in that input.

## Constant Returns to Scale: Example

Now let's look at another formula, or production function, to see whether we find constant returns to scale. Note that some textbooks use Q for quantity in the production function, while others use Y for output. These differences don't change the analysis, so you may use whichever.

Imagine we have the following equation:

$Q=2K+3L$

In this equation, 2 and 3 are coefficients of output based on how the particular industry responds to investment.

In a production process that mostly uses labor to create value, the formula will look like $Q=1.5K+6L$, this is because increases in labor will yield more output than capital.

To determine the returns to scale, we will begin by increasing both K and L by the multiplier m. Doing so will create a new production function in Q1.

$Q\mathit{}\mathit{}\mathit{=}\mathit{}\mathit{2}K\mathit{+}\mathit{}\mathit{3}L\phantom{\rule{0ex}{0ex}}To\mathit{}increase\mathit{}output\mathit{,}\mathit{}we\mathit{}multiple\mathit{}both\mathit{}labor\mathit{}\mathit{\left(}L\mathit{\right)}\mathit{}and\mathit{}capital\mathit{}\mathit{\left(}K\mathit{\right)}\mathit{}by\mathit{}the\mathit{}value\mathit{}they\mathit{}are\mathit{}being\mathit{}increased\mathit{,}\mathit{}the\mathit{}multiplier\mathit{}\mathit{\left(}m\mathit{\right)}\phantom{\rule{0ex}{0ex}}{Q}_{1}=2\left(K×m\right)+3\left(L×m\right)\phantom{\rule{0ex}{0ex}}Distributethecorrespondingcoefficents\left(2&3\right)\phantom{\rule{0ex}{0ex}}{Q}_{1}=\left(2×K×m\right)+\left(3×L×m\right)\phantom{\rule{0ex}{0ex}}Factorout\mathit{}the\mathit{}multiplier\left(m\right)\phantom{\rule{0ex}{0ex}}{Q}_{1}=m\left(2×K+3×L\right)orm\left(2K+3L\right)$

After factoring, we can replace (2 x K + 3 x L) with Q, as we were given that from the start (Q = 2K + 3L)

$Q1=mxQ$

And we will compare Q1 to Q. Since Q1 = m x Q, we note that by increasing all of our inputs by the multiplier m, we have increased production by exactly m. As a result, we have constant returns to scale.

A special type of production function is known as the Cobb-Douglas production function. Check out this deep dive below for more information.

### Cobb-Douglas Production Function

Similarly, constant returns to scale may be found in the Cobb-Douglas production function. This is a specific type of production function that also measures total productivity.

$Y=A{L}^{\alpha }{K}^{\beta }$

• Y = total production

• L = labor

• K = capital

• A = total factor productivity

• α and β are labor and capital-output elasticities, respectively. These values are constants determined by available technology.

If the sum of α and β equals one, then the production will show constant returns to scale. Depending on whether the sum is greater or less than one, there will be increasing or decreasing returns to scale.

## Constant Returns to Scale vs. Economies of Scale

Let's examine the difference between constant returns to scale and economies of scale. Constant returns to scale refer to the output reflected in equal proportion to the input. If an input increases by 20%, the output will also increase by 20%. The same case goes for a decrease in production. If an input decreases by 10 points, the output will decrease by the same number. The key to this type of return to scale is that the return on production is constant.

On the other hand, economies of scale are cost advantages that a firm experiences when it increases the level of production. For example, grocery stores may purchase their products in bulk to save money on the products they sell. These grocery stores can use their savings to offer discounts to their own customers. The stores then benefit from a reduced cost of their products.

## Constant Returns to Scale - Key takeaways

• "Returns to scale" is a term that refers to how well a firm is producing its goods or services.
• Constant returns to scale occur when an input increase, such as labor and capital, proportionally increases output.
• The starting equation to find constant returns to scale is
• The cause of constant returns to scale is the factor by which the production's input affects the output.
• Economies of scale are cost advantages a firm experiences when it increases production.

## References

1. Table 1. Returns to Scale, StudySmarter Originals

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

What does the term “constant returns to scale” mean?

Constant returns to scale occurs when an increase in inputs, such as labor and capital, proportionally increases output.

How do you find constant returns to scale?

You may find the constant returns to scale by using the equation: Q = F(m x K, m x L) = m x F(K,L).

What are the reasons for constant returns to scale?

Constant returns to scale may occur for simple production processes that don't react strongly to change. For example, if the process is a laborer completing a task, then double labor, two laborers can complete two tasks.

What is the difference between increasing returns to scale and constant returns to scale?

When inputs are increased, and output increases by a higher percentage, then there would be increasing returns to scale. Constant returns to scale takes place when there is a proportional increase in output as there are from inputs.

What is an example of constant returns to scale?

If a firm increases its labor and capital by 15%, then in order to receive constant returns to scale, the output from labor and capital also will have to increase by 15%.

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