# Returns to Scale

Returns to scale tell us how production changes in response to an increase in all inputs in the long run. An industry can exhibit constant returns to scale, increasing returns to scale or decreasing returns to scale.

Study of whether efficiency increases with increase in all factors of production is important for both businesses and policy-makers. It tells businesses about their optimal production level and it lets policy-makers determine whether the industry will consist of large number of small producers or a small number of large producers.

Law of diminishing returns tells us what happens when one input increases while other inputs stay the same. It is most relevant in the short-run i.e. time scale in which at least one factor of production is constant. In the long run, all factors of production can be changed, and it is then when the returns to scale become relevant.

There are three possibilities for total production function when all inputs increase: (a) increase at increasing rate, (b) increase at a fixed rate or (c) increase at a decreasing rate. These three possibilities result in three forms of returns to scale.

Before we define each type, let’s look at the Cobb-Douglas production function:

$$ \text{Q}=\text{A}\times \text{L}^\text{a}\times \text{K}^\text{b} $$

Where Q is the total product, L and K are the units of labor and capital respectively, and A, a and b are constants.

## Constant Returns to Scale

Constant returns to scale mean that total product changes proportionately with increase in all inputs. In other words, the percentage increase in total product under the constant returns to scale is the same as the percentage increase in all inputs.

If the sum of a and b in the Cobb-Douglas production function equals 1, it represents constant returns to scale.

### Examples

Constant returns to scale prevail in very small businesses. For example, let’s consider a car wash in which one car wash takes 30 minutes. If there is one wash space (hydraulic jack) and two workers running two 8-hour shifts, total product would be 32. If there are two wash spaces and four workers i.e. when inputs double, total washes possible rise to 64 (=4 × 8 × 60/30) and so on.

## Increasing Returns to Scale

In industries subject to increasing returns to scale, a 1% increase in total inputs will result in a more than 1% increase in total product i.e. total product increases at a rate higher than the rate in which all inputs increase. Increasing returns to scale are also referred to as economies of scale.

You must be wondering whether it is not against the law of diminishing returns. It is not because the law of diminishing returns is applicable only in short-run for only a change in one input but the returns to scale determine change in total product in response to changes in all inputs. Causes of increasing returns to scale include specialization of labor, synergies, etc.

Sum of a and b in the Cobb-Douglas production function is higher than 1 in case of increasing returns to scale.

### Examples

Industries that exhibit increasing returns to scale typically have small number of large firms. Because there are advantages to production at high level, large companies are at considerable advantage as compared to small firms. Airplane producers, large express shipping companies, telecommunication companies, etc. exhibit increasing returns to scale.

## Decreasing Returns to Scale

In case of decreasing returns to scale, total product increases at a rate lower than the rate of increase in inputs. In other words, additional investment generates progressively less and less additional production. If the sum of a and b in the Cobb-Douglas production function is less than 1, it represents decreasing returns to scale. Decreasing returns to scale are also referred to as diseconomies of scale.

### Examples

Examples of industries that exhibit decreasing returns to scale include companies engaged in exploration of natural resources (because it becomes increasingly difficult to extract as easier low-hanging minerals are extracted), companies where complexity results in higher risk of failure such as power distribution, etc.

Returns to scale can be graphed using isoquants.

Written by Obaidullah Jan, ACA, CFA and last modified on