Q

=

f(L, K)

where: Q = quantity of production or output, L = quantity of labor input, and K = quantity of capital input. The letter "f" indicates a generic, as of yet unspecified, functional equation.

The analysis of short-run production is commonly performed at the introductory level with simple tables and graphs. These are useful abstraction methods for isolating and analyzing key aspects of short-run production. However, mathematical equations are another, often more powerful, method of abstract analysis. This is where the production function comes into play. Because a mathematical equation can extend beyond the two dimensions of a graph, it is possible to consider relations beyond just that for a variable input and total production.

If the production function takes the form of a specific equation (such as Q = 5L + 10K + 2LK), then a total product curve relating total product and the variable input can be plotted. However, to do so, one of the two inputs (L or K) must be designated as a variable input and one designated as a fixed input. For most types of production, labor is more readily changed than capital, so L is generally the variable input and K is usually the fixed input. A short-run total product curve can then be derived by "fixing" K at a particular value, then plotting the values of Q for alternative values of L.

While a great deal of economic insight into short-run production decisions of a firm and market supply curves can be analyzed with a simple graph, when economists begin using mathematical equations, such as the production function, Q = f(L, K), the possibilities are almost unlimited. A wide assortment of additional input variables can be added to this equation to make it, not only more sophisticated, but also more revealing.