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):

1. Increasing returns to scale

2. Constant returns to scale

3. 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.

Figure 2. Increasing returns to scale, StudySmarter

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.

Figure 3. Constant Returns to Scale, StudySmarter

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.

Figure 4. Decreasing returns to scale, StudySmarter

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.

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:

1. All factors of production are variable.

2. Outputs are measured in physical terms.

3. The market is perfectly competitive.

4. 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:

$$\begin{gathered} Q=2(K\times m)+3(L\times m) \\ =2\times K\times m+3\times L\times m \\ =m(2K+3L) \\ =m\times Q \end{gathered}$$

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:

$$\begin{gathered} Q=0.5(K\times m)(L\times m) \\ =0.5\times K\times L\times m^2 \\ =Q\times m^2 \end{gathered}$$

Since m > 1 then m² > 2, increasing inputs by the multiplier m will increase output by m². 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:

$$\begin{gathered} Q={(K\times m)}^{0.3}{(L\times m)}^{0.2} \\ =K^{0.3}{L}^{0.2}{m}^{0.5} \\ =Q\times m^{0.5} \end{gathered}$$

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.