Cost Function
A cost function expresses the minimum cost of producing a given level of output as a function of input prices and of the quantity produced. It is a central construct in microeconomics, where it summarises the technological and market conditions faced by a firm and provides the basis for the analysis of supply, of pricing, and of profit.
Definition
Formally, the cost function relates the minimum cost of producing output q to the prices of inputs and to q itself. It is derived from the production function, which describes the maximum output obtainable from given inputs, by choosing the combination of inputs that minimises cost for each level of output.
The cost function is therefore a consequence of both technology and input prices. A change in technology alters the relationship between inputs and output and thereby shifts the cost function. A change in input prices alters the relative cost of inputs and thereby changes the least-cost combination.
Short Run and Long Run
In the short run, some inputs are fixed, so the firm can vary its output only by changing the variable inputs. The short-run cost function distinguishes fixed cost, which does not vary with output, from variable cost, which does. The sum of the two is total cost.
In the long run, all inputs are variable, and the firm can adjust all of them. The long-run cost function is the lower envelope of the short-run cost functions, since the firm can choose the scale of all its inputs. The long-run average cost curve is therefore flatter than the short-run average cost curves, which it envelopes.
Cost Concepts
Several cost concepts are derived from the cost function. Average cost is total cost divided by output. Marginal cost is the change in total cost from producing one additional unit. Fixed cost is that part of total cost that does not vary with output in the short run.
The relationship between average cost and marginal cost is central to the analysis of supply. When marginal cost is below average cost, average cost is falling; when marginal cost is above average cost, average cost is rising. Marginal cost equals average cost at the minimum of the average cost curve.
Applications
The cost function underlies the analysis of supply, since a competitive firm supplies output up to the point at which price equals marginal cost. It underlies the analysis of pricing under imperfect competition, where the firm equates marginal revenue and marginal cost. It underlies the analysis of entry and exit, since firms enter when price exceeds minimum average cost and exit when it falls below.
Cost functions are also used in empirical work, where they are estimated from data on firms and used to assess efficiency, economies of scale, and the effects of policy. The estimation of cost functions requires assumptions about functional form and about the nature of the data, and it is subject to the usual problems of identification and of measurement.
Functional Forms
The empirical and theoretical analysis of cost functions relies on specific functional forms. The Cobb-Douglas cost function implies constant elasticities of cost with respect to output and input prices, and it is convenient for estimation. The translog cost function, which is a second-order approximation to an arbitrary cost function, allows the elasticities to vary and is widely used in applied work.
The choice of functional form affects the results. A function that imposes constant returns to scale will not reveal economies of scale even if they exist, while a more flexible form may capture them but at the cost of additional parameters and of greater demands on the data. The researcher must therefore balance flexibility against the requirements of identification.
The estimation of cost functions also raises issues of simultaneity, since input prices and output are determined jointly with cost. Instrumental variables and other techniques are used to address these issues, though the available instruments are often weak and the results sensitive to the specification.
Economies of Scope and Scale
The cost function provides the basis for the analysis of economies of scale and of scope. Economies of scale exist when average cost falls as output rises, so that the long-run average cost curve is downward-sloping. Economies of scope exist when the cost of producing several products jointly is less than the cost of producing them separately.
The sources of economies of scale include the spreading of fixed costs over more output, the specialisation of labour and of capital, and the use of more efficient technologies. The sources of economies of scope include the sharing of inputs and of distribution networks, and the reuse of knowledge and of brand reputation.
The presence of economies of scale and of scope has implications for market structure. Where they are large relative to the size of the market, the industry tends towards concentration, and competition may be difficult to sustain. Where they are small, the industry may support many firms, and competition may be vigorous.
Cost Functions and Regulation
Cost functions are used in the regulation of natural monopolies and of other firms with market power. Regulators estimate the cost function of the regulated firm to determine the prices that the firm may charge and to assess whether the firm is operating efficiently.
The use of cost functions in regulation raises issues of information and of incentives. The regulator typically knows less about the cost conditions of the firm than the firm itself, and the firm has an incentive to overstate its costs. Mechanisms such as price caps and yardstick competition are designed to mitigate these problems, by tying the firm's revenue to external benchmarks rather than to its reported costs.
The estimation of cost functions for regulatory purposes also requires attention to the quality of service and to the range of outputs that the firm produces. A firm that reduces costs by lowering quality may appear efficient according to a narrow cost measure, while in fact imposing costs on consumers.
Dynamic Considerations
The cost function is often analysed in a static setting, but in reality costs evolve over time. Learning by doing reduces costs as firms accumulate experience, and technological progress shifts the cost function downward. These dynamic effects mean that the cost function estimated from historical data may not describe the costs that a firm will face in the future.
The treatment of adjustment costs is also important. Firms may face costs of changing their inputs, particularly their capital stock, so that the adjustment to a new equilibrium is gradual. Recognising these costs enriches the analysis of investment and of the response of firms to changes in prices and in demand.
Summary
The cost function is a central tool of microeconomics, summarising the relationship between output, input prices, and cost. Its estimation and its interpretation require attention to functional form, to the nature of the data, and to the dynamic and institutional context in which firms operate.