Returns to Scale

Returns to Scale

The law of variable proportions, which we studied earlier, applies when only one input changes while others stay fixed. That situation is a short-run phenomenon. But what happens in the long run, when all inputs can be changed? The firm can then choose any combination of inputs, including scaling its entire operation up or down. The effect of such a proportional change in all inputs on the level of output is called returns to scale.

The Basic Idea: Scaling All Inputs

Consider a production function $q = f(x_1, x_2)$, where $q$ is the quantity of output produced using $x_1$ units of factor 1 and $x_2$ units of factor 2. Now suppose the firm decides to increase the employment of both factors by the same proportion. Let that proportion be $t$, where $t > 1$. So the new input levels are $t x_1$ and $t x_2$.

The question is: what happens to output? The new output level is $f(t x_1, t x_2)$. We compare this to the original output $q = f(x_1, x_2)$. There are three possible outcomes.

Three Types of Returns to Scale

1. Constant Returns to Scale (CRS)

If a proportional increase in all inputs leads to an increase in output by the same proportion, the production function exhibits constant returns to scale.

Mathematically, for any $t > 1$:

For example, if all inputs are doubled ($t = 2$), output also doubles. If inputs are tripled ($t = 3$), output triples. This is the benchmark case — the production process is perfectly replicable. If you build a second identical factory and staff it identically, you get exactly twice the output.

2. Increasing Returns to Scale (IRS)

If a proportional increase in all inputs leads to an increase in output by a larger proportion, the production function exhibits increasing returns to scale.

Mathematically, for any $t > 1$:

For example, if all inputs are doubled ($t = 2$), output more than doubles. This can happen due to specialisation, better division of labour, or the ability to use more efficient large-scale machinery that is not feasible at a small scale.

3. Decreasing Returns to Scale (DRS)

If a proportional increase in all inputs leads to an increase in output by a smaller proportion, the production function exhibits decreasing returns to scale.

Mathematically, for any $t > 1$:

For example, if all inputs are doubled ($t = 2$), output less than doubles. This often arises from coordination and management difficulties that grow as the firm becomes very large — it becomes harder to communicate, monitor, and organise effectively.

A Numerical Illustration

Suppose a production process uses two inputs. Initially, the firm uses 10 units of labour and 5 units of capital, producing 100 units of output. Now the firm doubles all inputs: it uses 20 units of labour and 10 units of capital.

• If the new output is 200 units, the production function exhibits CRS.
• If the new output is 250 units (more than 200), the production function exhibits IRS.
• If the new output is 150 units (less than 200), the production function exhibits DRS.

The textbook uses the same logic: "if the output gets doubled, the production function exhibits CRS. If output is less than doubled, then DRS holds, and if it is more than doubled, then IRS holds."