### The Production Function

Economists use the production function to describe the relationship between inputs (i.e. factors of production) such as capital and labor and the quantity of output that a firm can produce. The production function can take either of two forms- in the short run version, the amount of capital (you can think of this as the size of the factory) as is taken as given and the amount of labor (i.e. workers) is the only parameter in the function. In the long run, however, both the amount of labor and the amount of capital can be varied, resulting in two parameters to the production function.

It's important to remember that the amount of capital is represented by K and the amount of labor is represented by L. q refers to the quantity of output that is produced.

### Average Product

Sometimes it's helpful to quantify output per worker or output per unit of capital rather than focusing on the total quantity of output produced.

The *average product of labor* gives a general measure of output per worker, and it is calculated by dividing total output (q) by the number of workers used to produce that output (L). Similarly, the *average product of capital* gives a general measure of output per unit of capital, and it calculated by dividing total output (q) by the amount of capital used to produce that output (K).

Average product of labor and average product of capital are generally referred to as AP_{L} and AP_{K}, respectively, as shown above. Average product of labor and average product of capital can be thought of as measures of labor and capital productivity, respectively.

### Average Product and the Production Function

The relationship between the average product of labor and total output can be shown on the short-run production function. For a given quantity of labor, the average product of labor is the slope of a line that goes from the origin to the point on the production function that corresponds to that quantity of labor. This is shown in the diagram above.

The reason that this relationship holds is that the slope of a line is equal to the vertical change (i.e. the change in the y-axis variable) divided by the horizontal change (i.e. the change in the x-axis variable) between two points on the line. In this case, the vertical change is q minus zero, since the line starts at the origin, and the horizontal change is L minus zero. This gives a slope of q/L, as expected.

One could visualize the average product of capital in the same way if the short-run production function were drawn as a function of capital (holding the quantity of labor constant) rather than as a function of labor.

### Marginal Product

Sometimes it's helpful to calculate the contribution to the output of the last worker or the last unit of capital rather than looking at the average output over all workers or capital. To do this, economists use *marginal product of labor* and *marginal product of capital*.

Mathematically, the marginal product of labor is just the change in output caused by a change in the amount of labor divided by that change in the amount of labor. Similarly, the marginal product of capital is the change in output caused by a change in the amount of capital divided by that change in the amount of capital.

Marginal product of labor and marginal product of capital are defined as functions of the quantities of labor and capital, respectively, and the formulas above would correspond to the marginal product of labor at L_{2} and a marginal product of capital at K_{2}. When defined this way, marginal products are interpreted as the incremental output produced by the last unit of labor used or the last unit of capital used. In some cases, however, marginal product might be defined as the incremental output that would be produced by the next unit of labor or next unit of capital. It should be clear from context which interpretation is being used.

### Marginal Product Relates to Changing One Input at a Time

Particularly when analyzing the marginal product of labor or capital, in the long run, it's important to remember that, for example, the marginal product or labor is the extra output from one additional unit of labor, *all else held constant*. In other words, the amount of capital is held constant when calculating marginal product of labor. Conversely, the marginal product of capital is the extra output from one additional unit of capital, holding the amount of labor constant.

This property illustrated by the diagram above and is particularly helpful to think about when comparing the concept of marginal product to the concept of returns to scale.

### Marginal Product as the Derivative of Total Output

For those who are particularly mathematically inclined (or whose economics courses use calculus!), it's helpful to note that, for very small changes in labor and capital, marginal product of labor is the derivative of output quantity with respect to the quantity of labor, and marginal product of capital is the derivative of output quantity with respect to the quantity of capital. In the case of the long-run production function, which has multiple inputs, the marginal products are the partial derivatives of output quantity, as noted above.

### Marginal Product and the Production Function

The relationship between the marginal product of labor and total output can be shown on the short-run production function. For a given quantity of labor, the marginal product of labor is the slope of a line that is tangent to the point on the production function that corresponds to that quantity of labor. This is shown in the diagram above. (Technically this is true only for very small changes in the amount of labor and doesn't apply perfectly to discrete changes in the quantity of labor, but it's still helpful as an illustrative concept.)

One could visualize the marginal product of capital in the same way if the short-run production function were drawn as a function of capital (holding the quantity of labor constant) rather than as a function of labor.

### Diminishing Marginal Product

It's almost universally true that a production function will eventually show what is known as *diminishing marginal product of labor*. In other words, most production processes are such that they will reach a point where each additional worker brought in will not add as much to output as the one that came before. Therefore, the production function will reach a point where the marginal product of labor decreases as the quantity of labor used increases.

This is illustrated by the production function above. As noted earlier, the marginal product of labor is depicted by the slope of a line tangent to the production function at a given quantity, and these lines will get flatter as the quantity of labor increases as long as a production function has the general shape of the one depicted above.

In order to see why the diminishing marginal product of labor is so prevalent, consider a bunch of cooks working in a restaurant kitchen. The first guy is going to have a high marginal product since he can run around and use as many parts of the kitchen as he can handle. As more workers are added, however, the amount of capital available is more of a limiting factor, and eventually, more cooks won't lead to much extra output because they can only use the kitchen when another cook leaves to take a smoke break! It's even theoretically possible for a worker to have a negative marginal product, perhaps if his introduction into the kitchen just puts him in everyone else's why and inhibits their productivity!

Production functions also typically exhibit *diminishing marginal product of capital* or the phenomenon that production functions reach a point where each additional unit of capital is not as useful as the one that came before. One need only think about how useful a 10th computer would be for a worker in order to understand why this pattern tends to occur.