# How to Calculate Returns to Scale

Returns to scale is a concept in economics to describe the rise in output as a result of an increase in inputs. This is particularly useful when seeking efficient production or maximizing profits by lowering production costs. If a company increases output in greater proportion than its increase in inputs, it has achieved increasing returns to scale, which often results as firms ramp up to larger production but don't need to increase some inputs (for example, management or physical plant) to achieve it. Conversely, as sometimes happens when companies grow too fast for management to effectively run and output drops proportionately to the increase in inputs, the firm is suffering from decreasing returns to scale. Although the calculation for returns to scale may appear intimidating, the process is relatively easy and requires only basic algebra.

## Input and Output

A company's returns to scale is determined by the level of input relative to the level of output produced. Production efficiency is achieved by using less input to achieve the same level of output. Production, or output, is often portrayed in equations as the letter Q or Y. Capital and labor, represented in equations as K and L respectively, are the input mechanisms used for production. The balance of input and output can thus be represented by the equation Q = K+L.

## The Multiplier

The multiplier determines the rate of increase in production scale, and thus the cost of production. The multiplier is added to the production equation as the letter m or x. When including an additional production scale, the equation now reads Q' = mK+mL, because capital and labor must be increased in order to increase output. For example, an m of 1.1 signifies that the cost of production has increased by 10 percent.

## Q Prime

To compare current production with potential production, solve for Q prime and compare the results with your initial production level Q. For example, if you have three machines for production and a labor force of only four employees, your initial Q was equal to 3 K and 4 L. You want to know how much production you can achieve with an increase of m inputs. Your current production equation would thus be Q = 3K+4L. Your potential production, or Q prime, would be represented as Q' = 3(K_m)+4(L_m). Once solved, compare Q' with Q to understand how your output will be affected once input is increased by m amount.

## Solving the Calculation

Simplify the equation by removing common factors and do the same to both sides of the equation so that the equation reads Q_m = m(3K+4L). As a result, Q_m = Q', meaning that in this example, by increasing our input by m, production has also increased by m. This is known as a constant returns to scale. When production has produced less than m, this is known as a decreasing returns to scale. Finally, when increasing input by m results in a return that proves to be greater than m, the company has achieved increasing returns to scale.