### Returns to Scale

In the short run, a firm's growth potential is usually characterized by the firm's marginal product of labor, i.e. the additional output that a firm can generate when one more unit of labor is added. This is done in part because economists generally assume that, in the short run, the amount of capital in a firm (i.e. the size of a factory and so on) is fixed, in which case labor is the only input to production that can be increased. In the long run, however, firms have the flexibility to choose both the amount of capital and the amount of labor that they want to employ- in other words, the firm can choose a particular *scale of production*. Therefore, it's important to understand whether a firm gains or loses efficiency in its production processes as it grows in scale.

In the long run, companies and production processes can exhibit various forms of *returns to scale*- increasing returns to scale, decreasing returns to scale, or constant returns to scale. Returns to scale are determined by analyzing the firm's long-run production function, which gives output quantity as a function of the amount of capital (K) and the amount of labor (L) that the firm uses, as shown above. Let's discuss each of the possibilities in turn.

### Increasing Returns to Scale

Put simply, increasing returns to scale occur when a firm's output more than scales in comparison to its inputs. For example, a firm exhibits increasing returns to scale if its output more than doubles when all of its inputs are doubled. This relationship is shown by the first expression above. Equivalently, one could say that increasing returns to scale occur when it requires less than double the number of inputs in order to produce twice as much output.

It wasn't necessary to scale all inputs by a factor of 2 in the example above, since the increasing returns to scale definition holds for any proportional increase in all inputs. This is shown by the second expression above, where a more general multiplier of a (where a is greater than 1) is used in place of the number 2.

A firm or production process could exhibit increasing returns to scale if, for instance, the larger amount of capital and labor enables the capital and labor to specialize more effectively than it could in a smaller operation. It's often assumed that companies always enjoy increasing returns to scale, but, as we'll see shortly, this isn't always the case!

### Decreasing Returns to Scale

Decreasing returns to scale occur when a firm's output less than scales in comparison to its inputs. For example, a firm exhibits decreasing returns to scale if its output less than doubles when all of its inputs are doubled. This relationship is shown by the first expression above. Equivalently, one could say that decreasing returns to scale occur when it requires more than double the quantity of inputs in order to produce twice as much output.

It wasn't necessary to scale all inputs by a factor of 2 in the example above, since the decreasing returns to scale definition holds for any proportional increase in all inputs. This is shown by the second expression above, where a more general multiplier of a (where a is greater than 1) is used in place of the number 2.

Common examples of decreasing returns to scale are found in many agricultural and natural resource extraction industries. In these industries, it's often the case that increasing output gets more and more difficult as the operation grows in scale- quite literally because of the concept of going for the "low-hanging fruit" first!

### Constant Returns to Scale

Constant returns to scale occur when a firm's output exactly scales in comparison to its inputs. For example, a firm exhibits constant returns to scale if its output exactly doubles when all of its inputs are doubled. This relationship is shown by the first expression above. Equivalently, one could say that increasing returns to scale occur when it requires exactly double the number of inputs in order to produce twice as much output.

It wasn't necessary to scale all inputs by a factor of 2 in the example above since the constant returns to scale definition holds for any proportional increase in all inputs. This is shown by the second expression above, where a more general multiplier of a (where a is greater than 1) is used in place of the number 2.

Firms that exhibit constant returns to scale often do so because, in order to expand, the firm essentially just replicates existing processes rather than reorganizing the use of capital and labor. In this way, you can envision constant returns to scale as a company expanding by building a second factory that looks and functions exactly like the existing one.

### Returns to Scale Versus Marginal Product

It's important to keep in mind that marginal product and returns to scale are not the same concept and need not go in the same direction. This is because the marginal product is calculated by adding one unit of either labor or capital and keeping the other input the same, whereas returns to scale refer to what happens when all inputs to production are scaled up. This distinction is shown in the figure above.

It is generally true that most production processes start to exhibit decreasing marginal product of labor and capital pretty quickly as quantity increases, but this doesn't mean that the firm also exhibits decreasing returns to scale. In fact, it's quite common and perfectly reasonable to observe decreasing marginal products and increasing returns to scale simultaneously.

### Returns to Scale Versus Economies of Scale

Although it's fairly common to see the concepts of returns to scale and economies of scale used interchangeably, they are not, in fact, one and the same. As you've seen here, the analysis of returns to scale looks directly at the production function and doesn't consider the cost of any of the inputs, or factors of production. On the other hand, the analysis of economies of scale considers how the cost of production scales with the quantity of output produced.

That said, returns to scale and economies of scale exhibit equivalence when procuring more units of labor and capital doesn't affect their prices. In this case, the following similarities hold:

- Increasing returns to scale happen when economies of scale are present, and vice versa.
- Decreasing returns to scale happen when diseconomies of scale are present, and vice versa.

On the other hand, when procuring more labor and capital results in either driving the price up or receiving volume discounts, one of the following possibilities could result:

- If buying more inputs increases the prices of the inputs, increasing or constant returns to scale could result in diseconomies of scale.
- If buying more inputs decreases the prices of the inputs, decreasing or constant returns to scale could result in economies of scale.

Note the use of the word "could" in the statements above- in these cases, the relationship between returns to scale and economies of scale depends on where the tradeoff between the change in the price of the inputs and the changes in production efficiency falls.