Note on short and long run cost function.

In the short run, at least one input is fixed and cost curves are defined as operating curves. In the short run, the level of output that correlates to the minimum average total cost is called the capacity of the firm. Since firms cannot change capital:

r * K = constant

When a firm produces less output than the minimum average total cost, it has excess capacity.

In the long run, all inputs are variables (so K and L are variable) and cost curves are defined as planning curves. The long-run average cost curve shows the lowest cost of producing at a certain level of output.

Short and long run cost functions are an integral part of mathematical economics and important to understanding and representing the role of technology in the production process.