A firm can vary all productive resources in the long run, but at least one resource is fixed in the short run. Linkages between inputs and outputs are formalized in production functions.
Production functions summarize relationships between combinations of inputs and the maximum outputs that each combination can produce.
Output = f(inputs) is an example of a production function, and is read as "output is a function f of inputs." The function f summarizes how current technology translates various combinations of inputs into specific amounts of output. In this context, technology encompasses current knowledge about production techniques, as well as such things as government regulations, weather, and the laws of physics and chemistry.
Production functions are commonly written q = f(K, L) where q equals output, K equals capital services, and L equals labor services used per production period. (For simplicity, land and entrepreneurship are ignored for now.) Suppose that production engineers indicate that 1,000 swimsuits can be sewn daily using 600 machine hours (75 sewing machines per 8-hour shift) and 800 labor hours (100 workers each 8-hour shift). The function f summarizes a production relationship of this type. Technological advances boosting productivity 50 percent would require switching from the f production function to, say, g. Now, q= g(K, L), and 600 machine hours plus 800 labor hours yield 1,500 swimsuits. Complete production functions identify output possibilities in the long run, when a firm can vary all resources. In the short run, however, at least one resource is fixed.