Cost Function
The cost function expresses the minimum cost of producing a given level of output as a function of input prices and output, given the available technology. It is a central construct in microeconomics and managerial economics, summarising the cost side of the firm's optimisation problem and providing the foundation for analysis of supply, pricing, and market structure.
Derivation from Production
The cost function is derived from the production function through cost minimisation. Given the desired output level and input prices, the firm chooses the combination of inputs that achieves that output at least cost. The solution yields conditional input demand functions, and evaluating those demands at prevailing prices gives the cost function.
The dual relationship between production and cost means that the properties of one are mirrored in the other. If the production function exhibits constant returns to scale, the cost function is linear in output. If it exhibits decreasing returns, marginal cost rises with output. Duality makes it possible to estimate technology from observed cost data and to move freely between the two representations.
Short Run and Long Run
The distinction between the short run and the long run is central. In the short run some inputs, typically capital, are fixed, so the firm can vary only variable inputs. Short-run cost therefore includes fixed and variable components. In the long run all inputs are variable, and the firm can choose the optimal scale and input mix. The long-run cost function is the lower envelope of short-run cost functions, each corresponding to a different level of the fixed input.
This envelope relationship gives rise to the distinction between economies and diseconomies of scale. When long-run average cost falls as output increases, production exhibits economies of scale; when it rises, diseconomies of scale dominate; the output at which average cost is minimised defines the minimum efficient scale.
Properties and Measurement
A well-behaved cost function is non-decreasing in input prices and output, homogeneous of degree one in input prices, and concave in input prices. These properties follow from cost minimisation and are used to test the consistency of empirical estimates with theory. Estimation typically employs flexible functional forms, such as the translog, that do not impose restrictive assumptions on the structure of technology.
Cost functions are used in regulation to benchmark efficiency, in antitrust to assess economies of scale, and in corporate planning to evaluate pricing and investment decisions. They also inform the analysis of natural monopoly, where long-run average cost falls over the relevant range of demand.
Relationship to Supply
Marginal cost, the derivative of the cost function with respect to output, determines the firm's supply decision in competitive markets. Average variable cost defines the shutdown point in the short run, and long-run average cost determines entry and exit. The cost function therefore links the technology available to the firm with the market supply curve and, ultimately, with market equilibrium.
Flexible Functional Forms
Empirical implementation of cost functions requires functional forms that are flexible enough to capture the features of interest without imposing unwarranted restrictions. The translog cost function, a second-order approximation to any cost function, is widely used because it permits variable elasticities of substitution and does not impose constant returns to scale or homotheticity. Other forms include the generalised Leontief and the normalised quadratic.
Estimation is typically carried out jointly with the cost share equations derived from Shephard's lemma, which improves efficiency. Care must be taken to impose the theoretical restrictions of homogeneity and symmetry, and to address endogeneity of output and input prices.
Economies of Scope and Multi-product Firms
For firms producing multiple products, the cost function can be used to measure economies of scope, the cost savings from producing goods jointly rather than separately. If the cost of producing two products together is less than the sum of the costs of producing them separately, scope economies exist. These economies arise from shared inputs, joint production, and the spreading of fixed costs.
Multi-product cost functions are used to analyse the behaviour of banks, utilities, hospitals, and transport operators, where joint production is common and where policy questions about consolidation and pricing depend on the structure of costs.
Cost Functions in Regulation
Regulators use cost functions to benchmark the efficiency of regulated firms, setting allowed revenues by reference to an efficient cost frontier. Techniques such as stochastic frontier analysis and data envelopment analysis estimate the frontier and measure the distance of individual firms from it. The approach is central to incentive regulation, where the regulator seeks to reward efficiency while ensuring that firms recover reasonable costs.
The use of cost benchmarking raises questions about the comparability of firms, the treatment of environmental and quality differences, and the risk of setting targets that are too demanding or too lax. These issues are prominent in the regulation of network industries.
Dynamics and Adjustment Costs
Standard cost functions describe static optimisation, but firms adjust inputs gradually because of adjustment costs. Incorporating adjustment costs produces dynamic cost functions in which current costs depend on the path of adjustment as well as on current output and prices. These models are used to analyse investment behaviour, employment dynamics, and the response of firms to shocks.