Cost functions are derived functions. They are derived from the production function, which describes the available efficient methods of production at any one time.
Economic theory distinguishes between short-run costs and long-run costs. Short-run costs are the costs over a period during which some factors of production (usually capital equipment and management) are fixed.
The long-run costs are the costs over a period long enough to permit the change of all factors of production. In the long run all factors become variable. Both in the short run and in the long run, total cost is a multivariable function, that is, total cost is determined by many factors.
Symbolically we may write the long-run cost function as:
C = ƒ(X, T, Pf)
and the short-run cost function as
C = ƒ(X, T, Pf, K)
where C = total cost
X = output
T — technology
Pf = prices of factors
K = fixed factor(s)
Graphically, costs are shown on two-dimensional diagrams. Such curves imply that cost is a function of output, C = ƒ(X), ceteris paribus. The clause ceteris paribus implies that all other factors which determine costs are constant. If these factors do change, their effect on costs is shown graphically by a shift of the cost curve. This is the reason why determinants of cost, other than output, are called shift factors. Mathematically there is no difference between the various determinants of costs.
The distinction between movements along the cost curve (when output changes) and shifts of the curve (when the other determinants change) is convenient only pedagogically, because it allows the use of two-dimensional diagrams. But it can be misleading when studying the determinants of costs. It is important to remember that if the cost curve shifts, this does not imply that the cost function is indeterminate.
The factor ‘technology’ is itself a multidimensional factor, determined by the physical quantities of factor inputs, the quality of the factor inputs, the efficiency of the entrepreneur, both in organizing the physical side of the production (technical efficiency of the entrepreneur), and in making the correct economic choice of techniques (economic efficiency of the entrepreneur).
Thus, any change in these determinants (e.g., the introduction of a better method of organisation of production, the application of an educational programme to the existing labour) will shift the production function, and hence will result in a shift of the cost curve. Similarly the improvement of raw materials or the improvement in the use of the same raw materials will lead to a shift downwards of the cost function.
The short-run costs are the costs at which the firm operates in any one period. The long-run costs are planning costs or ex ante costs, in that they present the optimal possibilities for expansion of the output and thus help the entrepreneur plan his future activities. Before an investment is decided the entrepreneur is in a long-run situation, in the sense that he can choose any one of a wide range of alternative investments, defined by the state of technology. After the investment decision is taken and funds are tied up in fixed-capital equipment, the entrepreneur operates under short-run conditions he is on a short-run cost curve.
A distinction is necessary between internal (to the firm) economies of scale and external economies. The internal economies are built into the shape of the long-run cost curve, because they accrue to the firm from its own action as it expands the level of its output. The external economies arise outside the firm, from improvement (or deterioration) of the environment in which the firm operates.
Such economies external to the firm may be realized from actions of other firms in the same or in another industry. The important characteristic of such economies is that they are independent of the actions of the firm, they are external to it. Their effect is a change in the prices of the factors employed by the firm (or in a reduction in the amount of inputs per unit of output), and thus cause a shift of the cost curves, both the short-run and the long-run.
In summary, while the internal economies of scale relate only to the long-run and are built into the shape of the long-run cost curve, the external economies affect the position of the cost curves both the short-run and the long-run cost curves will shift if external economies affect the prices of the factors and/or the production function.
Any point on a cost curve shows the minimum cost at which a certain level of output may be produced. This is the optimality implied by the points of a cost curve. Usually the above optimality is associated with the long-run cost curve. However, a similar concept may be applied to the short-run, given the plant of the firm in any one period.