## Returns to Scale

Returns to scale in microeconomics describe a production situation that occurs in the long run when the scale of production increases when all inputs used are variable, which affects the output level. A proportionate change in output results from a proportionate change in input. Returns to scale explain what happens to total output when all production inputs increase, assuming that technology is constant and the market is perfectly competitive.

Returns to scale is a term in economics that refers to a rate at which a change in output leads to a change in input. It is a long-run theory of production.

In the short run, the firm cannot build a new factory to increase its returns to scale because it takes time and money to build one. It can, however, hire additional workers to increase its short-run returns, but only up to a certain point due to the law of diminishing marginal returns. However after the factory is built, the firm can hire more workers. These changes in the firm's inputs (labour, land, etc.) can help increase the firm's output, known as returns to scale.

## Types of Returns to Scale

An increase in a firm's input can lead to various changes in output. These changes are called types of returns to scale. There are three types of returns to scale (Figure 1):

Increasing returns to scale

Constant returns to scale

Diminishing returns to scale

### Increasing Returns to Scale

Increasing returns to scale are the rate at which output increases when the factors of production are increased. If an increase in production labour and capital factors leads to a disproportionate increase in output, then a firm experiences increasing returns to scale. Increasing returns to scale occur in the long run.

Suppose a company increases its input by 10%, leading to a change in output of more than 10%, then it is said to have increasing returns to scale. A proportional change in production factors leads to a larger proportional change in output.

From Figure 2 above, if capital and labour are 1, the output level is 20, 1 unit of labour and 1 unit of capital can produce 20 units. When inputs are 10, the output level is 50, and so on. Although output increases by 30 units each time, the increase in output is still greater than the increase in input.Therefore, we can conclude that when labour and capital in this enterprise increase, output also increases by a larger proportion.

#### Constant returns to scale

When a proportionate change in output equals the proportionate change in output, we speak of constant returns to scale. A production function with constant returns to scale is linear or considered homogeneous. Constant returns to scale can occur for a firm over the long run.

From Figure 3 above, 1 unit of labour and 1 unit of capital can produce 20 units of output. 10 units of inputs produce 40 units of output. With each increase, both inputs and outputs double, so they can increase by the same percentage, 100%.From this, we can conclude that if the company increases inputs, outputs will also increase by the same percentage.

#### Decreasing returns to scale

Decreasing returns to scale is the rate at which output changes less than the proportionate change in input – a proportionate change in input is greater than the proportionate change in output. Diminishing or decreasing returns to scale occur when a firm's production function becomes less efficient as its size increases. An example of this is when a firm expands beyond its ability to be effectively managed.

Remember, diminishing returns to scale are not the same as diminishing marginal returns.

Figure 4 shows that the output of firm A is 10 when the inputs of labour and capital are 1. Increasing inputs by 1 additional unit increases output from 10 to 15. However, if inputs increase from 6 to 8 by 2 additional units, output increases from 18 to 18.6, less than the proportionate increase in inputs. So it is not an increase or a doubling, and it does not remain constant, but the increase in inputs results in a lower output.

#### Difference between diminishing returns to scale and diminishing marginal returns

Decreasing returns to scale are also known as increasing costs. It is a situation where a portion of the production process increases the factors of production, such as labour and capital. However, the output does not increase by the same or a higher proportion. Diminishing returns to scale occur in the long run.

Diminishing marginal returns is a law that states an increase in the factor of production causes a relatively smaller increase in output. It assumes that the factor of production, capital, is fixed, and the factor, labour, is variable. Diminishing returns to scale occur in the short run.

Read our explanation of the 'Law of Diminishing Returns' to learn more.

## Law and Assumptions of Returns to Scale

The law of return to scale assumes that when a firm increases its variable factors of production and experiences a constant increase or decrease in its outputs, the following assumptions are in place:

All factors of production are variable.

Outputs are measured in physical terms.

The market is perfectly competitive.

Technology is constant; it does not change over time.

Figure 5 should help you remember these assumptions.

## Formula and Calculation of Returns to Scale

The mathematically gifted among us may be interested in calculating returns to scale. Even if math is not your strong suit, the calculation is not difficult!

By calculating returns to scale, you can determine whether a company is increasing, decreasing, or holding constant its output as it increases its input. You must multiply each input function by a positive constant (m > 0). The result shows whether the production function is equal, increased or decreased to the positive constant.

You can calculate it by determining whether an input's marginal product (MP) is increasing, decreasing or constant.

### Constant returns to scale formula

The formula for constant returns to scale is: $Q=2K+3L$

When L = Labour

K = Capital

m = Multiplier

If we increase our inputs (K and L) by our multiplier, m, we get:

$Q=2(K\times m)+3(L\times m)\phantom{\rule{0ex}{0ex}}=2\times K\times m+3\times L\times m\phantom{\rule{0ex}{0ex}}=m(2K+3L)\phantom{\rule{0ex}{0ex}}=m\times Q\phantom{\rule{0ex}{0ex}}$

Remember that Q = 2K + 3L, so it can be simplified as m x Q.

An increase in inputs by the multiplier m will cause outputs to increase by m. Therefore, we can conclude that there are **constant returns to scale**.

**Increasing returns to scale formula**

The formula for increasing returns to scale is: $Q=0.5KL$

When L = Labour

K = Capital

m = Multiplier

Again, if we increase our inputs by our multiplier, we get:

$Q=0.5(K\times m)(L\times m)\phantom{\rule{0ex}{0ex}}=0.5\times K\times L\times {m}^{2}\phantom{\rule{0ex}{0ex}}=Q\times {m}^{2}\phantom{\rule{0ex}{0ex}}$

Remember that Q = 0.5KL and thus can be simplified as Q x m^{2}.

Since m > 1 then m^{2 }> 2, increasing inputs by the multiplier m will increase output by m^{2}. From this, we can conclude that returns to scale** increase for this output**.

#### Decreasing returns to scale formula

Finally, the formula for decreasing returns to scale is: $Q={K}^{0.3}{L}^{0.2}$

When L = Labour

K = Capital

m = Multiplier

If we increase both K and L by the multiplier, m, we get:

$Q={(K\times m)}^{0.3}{(L\times m)}^{0.2}\phantom{\rule{0ex}{0ex}}={K}^{0.3}{L}^{0.2}{m}^{0.5}\phantom{\rule{0ex}{0ex}}=Q\times {m}^{0.5}$

Remember that Q = K^{0.3}L^{0.2 }so it can be sampled to Q x m^{0.5}

Since m > 1 then m^{0.5 }< m, an increase in inputs by the multiplier m results in a decrease in outputs. We can conclude that returns to scale **decrease in this production.**

## Returns to Scale - Key Takeaways

- Returns to scale occur in the long run.
- Returns to scale are the rate at which output changes due to a change in inputs.
- Decreasing or diminishing returns to scale are different from diminishing marginal returns. The former occurs in the long run, the latter in the short run.
- A change in inputs can lead to three changes in the output: remaining constant, increasing, or decreasing.
- Labour and capital are variable factors of production in the long run
- When the production function is increased, internal and external economies of scale come into play, increasing returns to scale.
- When inputs are increased, division of labour and specialisation occur.

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

What is diminishing returns to scale?

Diminishing returns to scale occur when a proportionate change in input is greater than the proportionate change in output.

What is return to scale in economics?

Return to scale is a term in economics that explains the different reactions of output levels when a firm decides to increase its inputs.

What is the law of return to scale?

The law of return to scale assumes that when inputs are increased in a production function and the output changes, the following are in place:

All factors of production are variable.

Outputs are measured in physical terms.

The market is perfectly competitive.

Technology is constant; it does not change over time.

How do you calculate return to scale?

Return to scale is calculated by multiplying each input function by a multiplier. If the result is greater than the multiplier, it increases returns to scale. If the result is less than the multiplier, then the production function will result in decreasing return to scale. If the result is the same as the multiplier, then the production function will result in constant returns to scale.

What causes return to scale?

A firm’s decision to change its production function by increasing the variable factors of production causes a return to scale.

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