In economics, a production function relates physical output of a production process to physical inputs or factors of production. The production function is one of the key concepts of mainstream neoclassical theories, used to define marginal product and to distinguish allocative efficiency, the defining focus of economics. The primary 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.
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 capital) and how much to attribute to advancing technology. Some non-mainstream economists, however, reject the very concept of an aggregate production function.
Concept of production functions 
In general, economic output is not a (mathematical) function of input, because any given set of inputs can be used to produce a range of outputs. To satisfy the mathematical definition of a function, a production function is customarily assumed to specify the maximum output obtainable from a given set of inputs. 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, given available technology.) By assuming that the maximum output, which is technologically feasible, from a given set of inputs, is obtained, economists are abstracting away from technological, engineering and managerial problems associated with realizing such a technical maximum, 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 production function, itself, the relationship of output to inputs is non-monetary; that is, a production function relates physical inputs to physical outputs, and prices and costs are not reflected in the function.
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. Under certain assumptions, the production function can be used to derive a marginal product for each factor. The profit-maximizing firm in perfect competition (taking output and input prices as given) will choose to add input right up to the point where the marginal cost of additional input matches the marginal product in additional output. This implies an ideal division of the income generated from output into an income due to each input factor of production, equal to the marginal product of each input.
The inputs to the production function are commonly termed factors of production and may represent primary factors, which are stocks. Classically, the primary factors of production were Land, Labor and Capital. Primary factors do not become part of the output product, nor are the primary factors, themselves, transformed in the production process. The production function, as a theoretical construct, may be abstracting away from the secondary factors and intermediate products consumed in a production process.
The production function is not a full model of the production process: it deliberately abstracts from inherent aspects of physical production processes that some would argue are essential, including error, entropy or waste, and the consumption of energy or the co-production of pollution. Moreover, production functions do not ordinarily model the business processes, either, ignoring the role of strategic and operational business management. (For a primer on the fundamental elements of microeconomic production theory, see production theory basics).
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. The firm is assumed to be making allocative choices concerning how much of each input factor to use and how much output to produce, given the cost (purchase price) of each factor, the selling price of the output, and the technological determinants represented by the production function.
Specifying the production function 
A production function can be expressed in a functional form as the right side of
- quantity of output
- quantities of factor inputs (such as capital, labour, land or raw materials).
If Q is not a matrix (i.e. a scalar, a vector, or even a diagonal matrix), then this form does not encompass joint production, which is a production process that has multiple co-products. On the other hand, if f maps from Rn to Rk then it is a joint production function expressing the determination of k different types of output based on the joint usage of the specified quantities of the n inputs.
One formulation, unlikely to be relevant in practice, is as a linear function:
- where and are parameters that are determined empirically.
Another is as a Cobb-Douglas production function:
The Leontief production function applies to situations in which inputs must be used in fixed proportions; starting from those proportions, if usage of one input is increased without another being increased, output will not change. This production function is given by
Other forms include the constant elasticity of substitution production function (CES), which is a generalized form of the Cobb-Douglas function, and the quadratic production function. The best form of the equation to use and the values of the parameters () vary from company to company and industry to industry. In a short run production function at least one of the 's (inputs) is fixed. In the long run all factor inputs are variable at the discretion of management.
Production function as a graph 
Any of these equations can be plotted on a graph. 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 the can be treated as a single variable). All points above the production function are unobtainable with current technology, all points below are technically feasible, and all points on the function show the maximum quantity of output obtainable at the specified level of usage of the input. From the origin, through points A, B, and C, the production function is rising,ve beyond point X. From point A to point C, the firm is experiencing positive but decreasing marginal returns to the variable input. As additional units of the input are employed, output increases but at a decreasing rate. Point B is the point beyond which there are diminishing average returns, as shown by the declining slope of the average physical product curve (APP) beyond point Y. 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.).
Stages of production 
To simplify the interpretation of a production function, it is common to divide its range into 3 stages. In Stage 1 (from the origin to point B) the variable input is being used with increasing output per unit, the latter reaching a maximum at point B (since the average physical product is at its maximum at that point). Because the output per unit of the variable input is improving throughout stage 1, a price-taking firm will always operate beyond this stage.
In Stage 2, output increases at a decreasing rate, and the average and marginal physical product are declining. However, the average product of fixed inputs (not shown) is still rising, because output is rising while fixed input usage is constant. In this stage, the employment of additional variable inputs increases the output per unit of fixed input but decreases the output per unit of the variable input. The optimum input/output combination for the price-taking firm will be in stage 2, although a firm facing a downward-sloped demand curve might find it most profitable to operate in Stage 1. In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over-utilized in the sense that their presence on the margin obstructs the production process rather than enhancing it. The output per unit of both the fixed and the variable input declines throughout this stage. At the boundary between stage 2 and stage 3, the highest possible output is being obtained from the fixed input.
Shifting a production function 
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. For example, you may only need to increase production by a million units per year to keep up with demand, but the production equipment upgrades that are available may involve increasing productive capacity by 2 million units per year.
If a firm is operating at a profit-maximizing level in stage one, it might, in the long run, choose to reduce its scale of operations (by selling capital equipment). By reducing the amount of fixed capital inputs, the production function will shift down. The beginning of stage 2 shifts from B1 to B2. The (unchanged) profit-maximizing output level will now be in stage 2.
Homogeneous and homothetic production functions 
There are two special classes of production functions that are often analyzed. The production function is said to be homogeneous of degree n, if given any positive constant , . If , the function exhibits increasing returns to scale, and it exhibits decreasing returns to scale if . If it is homogeneous of degree 1, it exhibits constant returns to scale. The presence of increasing returns means that a one percent increase in the usage levels of all inputs would result in a greater than one percent increase in output; the presence of decreasing returns means that it would result in a less than one percent increase in output. Constant returns to scale is the in-between case. In the Cobb-Douglas production function referred to above, returns to scale are increasing if , decreasing if , and constant if .
If a production function is homogeneous of degree one, it is sometimes called "linearly homogeneous". 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.: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. Due to this, along rays coming from the origin, the slopes of the isoquants will be the same. Homothetic functions are of the form where is a monotonically increasing function (the derivative of is positive ()), and the function is a homogeneous function of any degree.
Aggregate production functions 
In macroeconomics, aggregate production functions for whole nations are sometimes constructed. In theory they are the summation of all the production functions of individual producers; however there are methodological problems associated with aggregate production functions, and economists have debated extensively whether the concept is valid.
Criticisms of production functions 
There are two major criticisms of the standard form of the production function. On the history of production functions, see Mishra (2007).
On the concept of capital 
During the 1950s, '60s, and '70s there was a lively debate about the theoretical soundness of production functions. (See the Capital controversy.) Although the criticism was directed primarily at aggregate production functions, microeconomic production functions were also put under scrutiny. The debate began in 1953 when Joan Robinson criticized the way the factor input capital was measured and how the notion of factor proportions had distracted economists.
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. The problem is that this independence is a precondition of constructing an isoquant. Further, the slope of the isoquant helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.
On the empirical relevance 
As a result of the criticism on their weak theoretical grounds, it has been claimed that empirical results firmly support the use of neoclassical well behaved aggregate production functions. Nevertheless, Anwar Shaikh has demonstrated that they also have no empirical relevance, as long as alleged good fit outcomes from an accounting identity, not from any underlying laws of production/distribution.
Natural resources 
Often natural resources are omitted from production functions. When Solow and Stiglitz sought to make the production function more realistic by adding in natural resources, they did it in a manner that economist Georgescu-Roegen criticized as a "conjuring trick" that failed to address the laws of thermodynamics, since their variant allows capital and labour to be infinitely substituted for natural resources. Neither Solow nor Stiglitz addressed his criticism, despite an invitation to do so in the September 1997 issue of the journal Ecological Economics. For more recent retrospectives, see Cohen and Harcourt  and Ayres-Warr (2009).
See also 
- Distribution (economics)
- Production theory basics
- Production, costs, and pricing
- Production possibility frontier
- Productivity model
- A list of production functions
- Daly, H (1997). "Forum on Georgescu-Roegen versus Solow/Stiglitz". Ecological Economics 22 (3): 261–306. doi:10.1016/S0921-8009(97)00080-3.
- Cohen, A.J. and Harcourt, G.C. (2003) "Retrospectives: Whatever Happened to the Cambridge Capital Theory Controversies?" Journal of Economic Perspectives, 17(1), pp.199-214.
- Chiang, Alpha C. (1984) Fundamental Methods of Mathematical Economics, third edition, McGraw-Hill.
- S.K. Mishra (2007) A Brief History of Production Functions, Working Paper Series, Social Science Research Network (SSRN) http://www.scribd.com/doc/417083/A-Brief-History-of-Production-Functions
- Shaikh, A. (1974) "Laws of Production and Laws of Algebra: The Humbug Production Function" The Review of Economics and Statistics, 56(1), pp.115-120.
- Robert U. Ayres and Benjamin Warr, The Economic Growth Engine: How useful work creates material prosperity, 2009. ISBN 978-1-84844-182-8
- A brief history of production functions
- A further description of production functions
- Anatomy of Cobb-Douglas Type Production Functions in 3D
- Anatomy of CES Type Production Functions in 3D
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- Moroney, J. R. (1967) Cobb-Douglass production functions and returns to scale in US manufacturing industry, Western Economic Journal, vol 6, no 1, December 1967, pp 39–51.
- Pearl, D. and Enos, J. (1975) Engineering production functions and technological progress, The Journal of Industrial Economics, vol 24, September 1975, pp 55–72.
- Robinson, J. (1953) The production function and the theory of capital, Review of Economic Studies, vol XXI, 1953, pp. 81–106
- Anwar Shaikh, "Laws of Production and Laws of Algebra: The Humbug Production Function", in The Review of Economics and Statistics, Volume 56(1), February 1974, p. 115-120. http://homepage.newschool.edu/~AShaikh/humbug.pdf
- Anwar Shaikh, "Laws of Production and Laws of Algebra—Humbug II", in Growth, Profits and Property ed. by Edward J. Nell. Cambridge, Cambridge University Press, 1980. http://homepage.newschool.edu/~AShaikh/humbug2.pdf
- Anwar Shaikh, "Nonlinear Dynamics and Pseudo-Production Functions", published?, 2008. http://homepage.newschool.edu/~AShaikh/Nonlinear%20Dynamics%20and%20Pseudo-Production%20Functions.pdf
- Shephard, R (1970) Theory of cost and production functions, Princeton University Press, Princeton NJ.
- Thompson, A. (1981) Economics of the firm, Theory and practice, 3rd edition, Prentice Hall, Englewood Cliffs. ISBN 0-13-231423-1
- Elmer G. Wiens: Production Functions - Models of the Cobb-Douglas, C.E.S., Trans-Log, and Diewert Production Functions.