The production function is a statement of the relation between the volumes of the inputs and the outputs of a production process. Its form has implications for the concept of economic equilibrium and it is widely used in the construction of economic models.
Returns to scale - interpretation
The relation between the inputs to a production process and its output is conventionally termed the "returns to scale" of that process, but economists have placed a restricted interptetation upon that term. Alfred Marshal interpreted it to mean what happens when producers make the most efficient possible use of existing technology . (That restriction limits the possibility of empirical verification: it excludes the use of a time series of observations because of the possible intervention of changes of technology, and other sources of data are hard to find.) He also drew a distinction between what happens in the short run, before the production manager is able to correct an imbalance between the quantities of labour and capital; and the long run during which the optimum proportion of capital to labour can be restored. Economists have also reasoned that to apply the concept to a single production unit would be to overlook possible interactions with competitors who might be assumed to be bidding for the same input resources, and that consequently its net effect may be apparent only at industry level and above. In the absence of convincing empirical evidence, the subject has usually been approached by postulating plausible relationships and adopting those that prove useful in a wider context.
Diminishing returns and economic equilibrium
Diminishing returns is often taken to mean a progressive reduction in the increment of output produced by an increment of input that develops after a certain point - implying that proportionality is preserved up to that point. For the purposes of the law of supply and demand, however, it is necessary to assume that there is no proportional interval, and that the diminishing process is always at work. The diminishing returns hypotheses is intuitively convincing when one input is increased while the other is held constant, and is in accord with experience in agricultural production and elsewhere. But that state of affairs may be assumed to be confined in practice to a short-term expansion in production, and the law of supply and demand depends upon the assumption that it is universal. Its validity for longer-term output expansions depends upon the argument that the resulting increase in the demands for inputs is at the expense of other users, who may be expected to bid up their price.
The above analysis does not exclude the possibility that tendencies to increasing returns to scale are also present , but it has been generally assumed that, with few exceptions, the diminishing returns tendency predominates. In recent decades, however, the exceptions have come to assume increasing importance . The "learning-curve effect"  often enables the costs of repeated skilled operations to be progressively reduced and the "customer lock-in effect" often enables the early introducer of new technology to gain a protracted advantage that may be further reinforced by learning . In network industries such as telecommunications and internet-based enterprises, the unit value of the product rises rapidly with the number of links provided  (one of the founders of the internet has even suggested that the value of a network rises in proportion to the square of the number of its users ). Under those circumstances, the conditions necessary for the establishment of the equilibrium of the industries affected are absent.
Modeling the production function
The returns to scale concept lacks the precision necessary for quantification, and economists have sought to represent plausible versions of it by algebraic equations. The Cobb-Douglas function, which relates output to the product of the inputs, after each has been raised to a constant power is the best-known result, although it had earlier antecedents and has had later elaborations: , . Its widespread acceptance probably stems from the claim by its creators to have used it successfully to predict the shares of US manufacturing output that went to capital and labour during the period 1899 to 1922 . By varying its parameters, the Cobb-Douglas function can be made to represent diminishing, constant or increasing returns to scale. It is confined to processes for which the elasticity of substitution of labour for capital is unity, but less restricted functions have developed which allow for variable elasticities of substitution.
Qualifications and objections
Both the law of diminishing returns and the Cobb-Douglas function are presented in most textbooks as established theory, most economists take them for granted, and they have been widely used in the development of economic theory. However, serious objections have been raised by a number of eminent economists. One of them has claimed the law of diminishing returns to be logically inconsistent , and another referred to the mathematical statement of the production function as "a powerful instrument of miseducation". . Some of the objections that have been raised are summarised on the tutorials page.
- See Alfred Marshall Principles of Economics chapter XIII, Macmillan 1964
- See Alfred Marshall Principles of Economics Book V chapter XIII, Macmillan 1964
- Brian Arthur: "Increasing Returns and the New World of Business", Harvard Business Review July-August 1996
- See the tutorials page.
- Brian Arthur "Competing Technologies, Increasing Returns and Lock-in by Historical Events" Economic Journal September 1989
- Oz Shy: The Economics of Network Industries, Cambridge University Press, 2001
- Metcalfe's Law: a definition by Whatis.com
- see S K Mishra: A brief history of the Production Function, SSRN Working Paper
- C W Cobb and P H Douglas: "A Theory of Production", American Economic Review, December 1928
- Piero Sraffa: "The Laws of Return Under Competitive Conditions", The Economic Journal December 1926
- Joan Robinson: Collected Economic Papers, Blackwell 1960