Production function
One important purpose of the production function is to address allocative efficiency in the use of factor inputs in production and the resulting distribution of income to those factors, while abstracting away from the technological problems of achieving technical efficiency, as an engineer or professional manager might understand it.
[1] In macroeconomics, aggregate production functions are estimated to create a framework in which to distinguish how much of economic growth to attribute to changes in factor allocation (e.g. the accumulation of physical capital) and how much to attribute to advancing technology.
The production function, therefore, describes a boundary or frontier representing the limit of output obtainable from each feasible combination of input.
Alternatively, a production function can be defined as the specification of the minimum input requirements needed to produce designated quantities of output.
Assuming that maximum output is obtained from given inputs allows economists to abstract away from technological and managerial problems associated with realizing such a technical maximum, and to focus exclusively on the problem of allocative efficiency, associated with the economic choice of how much of a factor input to use, or the degree to which one factor may be substituted for another.
In the decision frame of a firm making economic choices regarding production—how much of each factor input to use to produce how much output—and facing market prices for output and inputs, the production function represents the possibilities afforded by an exogenous technology.
The production function is central to the marginalist focus of neoclassical economics, its definition of efficiency as allocative efficiency, its analysis of how market prices can govern the achievement of allocative efficiency in a decentralized economy, and an analysis of the distribution of income, which attributes factor income to the marginal product of factor input.
are the quantities of factor inputs (such as capital, labour, land or raw materials).
Moysan and Senouci (2016) provide an analytical formula for all 2-input, neoclassical production functions.
A typical (quadratic) production function is shown in the following diagram under the assumption of a single variable input (or fixed ratios of inputs so they can be treated as a single variable).
As additional units of the input are employed, output increases but at a decreasing rate.
Point B is just tangent to the steepest ray from the origin hence the average physical product is at a maximum.
Beyond point B, mathematical necessity requires that the marginal curve must be below the average curve (See production theory basics for further explanation and Sickles and Zelenyuk (2019) for more extensive discussions of various production functions, their generalizations and estimations).
To simplify the interpretation of a production function, it is common to divide its range into 3 stages.
In Stage 2, output increases at a decreasing rate, and the average and marginal physical product both decline.
The output per unit of both the fixed and the variable input declines throughout this stage.
By definition, in the long run the firm can change its scale of operations by adjusting the level of inputs that are fixed in the short run, thereby shifting the production function upward as plotted against the variable input.
If fixed inputs are lumpy, adjustments to the scale of operations may be more significant than what is required to merely balance production capacity with demand.
By reducing the amount of fixed capital inputs, the production function will shift down.
A linearly homogeneous production function with inputs capital and labour has the properties that the marginal and average physical products of both capital and labour can be expressed as functions of the capital-labour ratio alone.
Moreover, in this case, if each input is paid at a rate equal to its marginal product, the firm's revenues will be exactly exhausted and there will be no excess economic profit.
[5]: pp.412–414 Homothetic functions are functions whose marginal technical rate of substitution (the slope of the isoquant, a curve drawn through the set of points in say labour-capital space at which the same quantity of output is produced for varying combinations of the inputs) is homogeneous of degree zero.
[6] During the 1950s, '60s, and '70s there was a lively debate about the theoretical soundness of production functions (see the Capital controversy).
[7] According to the argument, it is impossible to conceive of capital in such a way that its quantity is independent of the rates of interest and wages.
Nevertheless, Anwar Shaikh has demonstrated that they also have no empirical relevance, as long as the alleged good fit comes from an accounting identity, not from any underlying laws of production/distribution.
When Robert Solow and Joseph Stiglitz attempted to develop a more realistic production function by including natural resources, they did it in a manner economist Nicholas Georgescu-Roegen criticized as a "conjuring trick": Solow and Stiglitz had failed to take into account the laws of thermodynamics, since their variant allowed man-made capital to be a complete substitute for natural resources.
Neither Solow nor Stiglitz reacted to Georgescu-Roegen's criticism, despite an invitation to do so in the September 1997 issue of the journal Ecological Economics.
[2][9]: 127–136 [3][10] Georgescu-Roegen can be understood as criticizing Solow and Stiglitz's approach to mathematically modelling factors of production.
[11][12] However, as discussed in more-recent work, this approach does not accurately model the mechanism by which energy affects production processes.
The practical application of production functions is obtained by valuing the physical outputs and inputs by their prices.
