To understand short and long run cost functions, it is important to understand the concept of cost. A cost is the value of inputs that are used to produce output. Total cost (TC) is the total cost of producing a given level of output and is divided into total fixed cost (TFC) and total variable cost (TVC). Total fixed cost does not change with the level of output and total variable cost does change with the level of output.
A cost function C(q) is a function that shows what the minimum cost for producing q units of output is. Labour is denoted as (L) and Capital as (K). So, with w as the cost of labour per unit and r as the cost per unit of capital, the production cost is:
w * L + r * K
Total cost can be divided into fixed cost, which is independent of quantity, and variable cost, which is dependent on quantity:
C(q) = FC + VC(q)
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.