1 Production Functions and Returns

Inputs, output, and product

A production function describes the maximum output a firm can produce from specified inputs, given its current technology and production methods. A common representation is $Q=f(L,K)$, where $Q$ is output, $L$ is labor, and $K$ is capital, such as buildings, machines, and equipment. Firms may also use land, energy, raw materials, and other inputs.

This function describes a technical relationship between inputs and output. By itself, it does not show the cost of inputs or the price of the product. For example, a bakery combines workers, ovens, ingredients, and preparation methods to make bread.

Total product is the amount produced with a given combination of inputs. Marginal product is the additional output from using one more unit of an input while holding other inputs constant. For labor, $MP_L=\frac{\Delta Q}{\Delta L}$.

In the bakery example, daily output is $0$ loaves with $0$ workers, $10$ with $1$ worker, $24$ with $2$, $36$ with $3$, and $45$ with $4$. The successive marginal products are $10$, $14$, $12$, and $9$ loaves per added worker. Thus, the second worker adds $14$ loaves, while the fourth adds $9$. Marginal product is the change in total product between successive input levels; on a graph, it corresponds to the slope of the total-product curve.

Short-run and long-run choices

In the short run, at least one input is fixed. This is not a particular number of days or months: the period's length depends on how quickly a firm can adjust its inputs. A bakery may quickly change staffing and ingredient orders but need much longer to expand its building or install another oven.

With capital fixed, short-run production can be represented as $Q=f(L,\bar K)$, where $\bar K$ indicates that capital cannot be changed during the period considered. A firm can raise output by increasing variable inputs, but the fixed input can eventually limit how effectively those additions are used.

In the long run, all inputs can be varied. The bakery can change its labor, number of ovens, workspace, and other inputs. This distinction matters because a firm has more choices about how to produce in the long run; it does not mean every input must actually change.

How returns change

When at least one input is fixed, the law of diminishing marginal returns says that adding more units of a variable input will eventually reduce that input's marginal product, other things unchanged. In a bakery, early hires may allow specialization and raise marginal product. Later, with the same ovens and workspace, workers may have to wait or compete for equipment, causing marginal product to fall.

Diminishing marginal product does not necessarily mean total output falls. As long as marginal product remains positive, total product continues to rise, but at a decreasing rate. Total product falls only if marginal product becomes negative.

The initial rise in marginal product is sometimes called increasing marginal returns: additional units of the variable input contribute more output than earlier units. This can occur when workers specialize or use fixed equipment more fully. Increasing marginal returns may be followed by diminishing marginal returns as the fixed input becomes a constraint.

Returns to scale is a related but distinct long-run concept. It describes how output changes when all inputs increase in the same proportion. If doubling labor, capital, and other inputs more than doubles output, the firm has increasing returns to scale. If output doubles, it has constant returns to scale; if output less than doubles, it has decreasing returns to scale. Returns to scale concern changing all inputs, whereas diminishing marginal returns concern changing one input while holding others fixed.

From productivity to cost

Production helps explain short-run costs. If a firm pays a constant wage $w$ and labor is its only variable input, producing an additional unit of output requires approximately $\frac{1}{MP_L}$ additional units of labor. Under these conditions, marginal cost is

Here, $MC$ is marginal cost and $MP_L$ is labor's marginal product. When labor's marginal product rises, marginal cost tends to fall; when marginal product diminishes, marginal cost tends to rise. This relationship assumes the wage and other relevant conditions are unchanged and links the production process to the shape of cost curves.