The production function simply states the quantity of output (q) that a firm can produce as a function of the quantity of inputs to production. There can be a number of different inputs to production, i.e. "factors of production," but they are generally designated as either capital or labor. (Technically, land is a third category of factors of production, but it's not generally included in the production function except in the context of a land-intensive business.) The particular functional form of the production function (i.e. the specific definition of f) depends on the specific technology and production processes that a firm uses.

### The Production Function

In the short run, the amount of capital that a factory uses is generally thought to be fixed. (The reasoning is that firms must commit to a particular size of factory, office, etc. and can't easily change these decisions without a long planning period.) Therefore, the quantity of labor (L) is the only input in the short-run production function. In the long run, on the other hand, a firm has the planning horizon necessary to change not only the number of workers but the amount of capital as well, since it can move to a different size factory, office, etc. Therefore, the long-run production function has two inputs that be changed- capital (K) and labor (L). Both cases are shown in the diagram above.

Note that the quantity of labor can take on a number of different units- worker-hours, worker-days, etc. The amount of capital is somewhat ambiguous in terms of units, since not all capital is equivalent, and no one wants to count a hammer the same as a forklift, for example. Therefore, the units that are appropriate for the quantity of capital will depend on the specific business and production function.

### The Production Function in the Short Run

Because there is only one input (labor) to the short-run production function, it's pretty straightforward to depict the short-run production function graphically. As shown in the above diagram, the short-run production function puts the quantity of labor (L) on the horizontal axis (since it's the independent variable) and the quantity of output (q) on the vertical axis (since it's the dependent variable).

The short-run production function has two notable features. First, the curve starts at the origin, which represents the observation that the quantity of output pretty much has to be zero if the firm hires zero workers. (With zero workers, there isn't even a guy to flip a switch to turn on the machines!) Second, the production function gets flatter as the amount of labor increases, resulting in a shape that is curved downward. Short-run production functions typically exhibit a shape like this due to the phenomenon of diminishing marginal product of labor.

In general, the short-run production function slopes upwards, but it is possible for it to slope downwards if adding a worker causes him to get in everyone else's way enough such that output decreases as a result.

### The Production Function in the Long Run

Because it has two inputs, the long-run production function is a bit more challenging to draw. One mathematical solution would be to construct a three-dimensional graph, but that is actually more complicated than is necessary. Instead, economists visualize the long-run production function on a 2-dimensional diagram by making the inputs to the production function the axes of the graph, as shown above. Technically, it doesn't matter which input goes on which axis, but it is typical to put capital (K) on the vertical axis and labor (L) on the horizontal axis.

You can think of this graph as a topographical map of quantity, with each line on the graph representing a particular quantity of output. (This may seem like a familiar concept if you have already studied indifference curves) In fact, each line on this graph is called an "isoquant" curve, so even the term itself has its roots in "same" and "quantity." (These curves are also crucial to the principle of cost minimization.)

Why is each output quantity represented by a line and not just by a point? In the long run, there are often a number of different ways to get a particular quantity of output. If one were making sweaters, for example, one could choose to either hire a bunch of knitting grandmas or rent some mechanized knitting looms. Both approaches would make sweaters perfectly fine, but the first approach entails a lot of labor and not much capital (i.e. is labor intensive), while the second requires a lot of capital but not much labor (i.e. is capital intensive). On the graph, the labor-heavy processes are represented by the points toward the bottom right of the curves, and the capital heavy processes are represented by the points toward the upper left of the curves.

In general, curves that are further away from the origin correspond to larger quantities of output. (In the diagram above, this implies that q_{3} is greater than q_{2}, which is greater than q_{1}.) This is simply because curves that are further away from the origin are using more of both capital and labor in each production configuration. It is typical (but not necessary) for the curves to be shaped like the ones above, as this shape reflects the tradeoffs between capital and labor that are present in many production processes.