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# production function

Equation that expresses the relationship between the quantities of productive factors (such as labour and capital) used and the amount of product obtained. It states the amount of product that can be obtained from every combination of factors, assuming that the most efficient available methods of production are used. The production function can thus measure the marginal productivity of a particular factor of production and determine the cheapest combination of productive factors that can be used to produce a given output.

In economics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs. A meta-production function (sometimes metaproduction function) compares the practice of the existing entities converting inputs X into output y to determine the most efficient practice production function of the existing entities, whether the most efficient feasible practice production or the most efficient actual practice production. In either case, the maximum output of a technologically-determined production process is a mathematical function of input factors of production. Put another way, given the set of all technically feasible combinations of output and inputs, only the combinations encompassing a maximum output for a specified set of inputs would constitute the production function. 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. It is usually presumed that unique production functions can be constructed for every production technology.

By assuming that the maximum output technologically possible from a given set of inputs is achieved, economists using a production function in analysis are abstracting away from the engineering and managerial problems inherently associated with a particular production process. The engineering and managerial problems of technical efficiency are assumed to be solved, so that analysis can focus on the problems of allocative efficiency. The firm is assumed to be making allocative choices concerning how much of each input factor to use, given the price of the factor and the technological determinants represented by the production function. A decision frame, in which one or more inputs are held constant, may be used; for example, capital may be assumed to be fixed or constant in the short run, and only labour variable, while in the long run, both capital and labour factors are variable, but the production function itself remains fixed, while in the very long run, the firm may face even a choice of technologies, represented by various, possible production functions.

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 considered. But, the production function is not a full model of the production process: it deliberately abstracts away from essential and inherent aspects of physical production processes, including error, entropy or waste. Moreover, production functions do not ordinarily model the business processes, either, ignoring the role of management, of sunk cost investments and the relation of fixed overhead to variable costs. (For a primer on the fundamental elements of microeconomic production theory, see production theory basics).

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. Under certain assumptions, the production function can be used to derive a marginal product for each factor, which implies an ideal division of the income generated from output into an income due to each input factor of production.

## The production function as an equation

There are several ways of specifying the production function.

In a general mathematical form, a production function can be expressed as:

$Q = f\left(X_1,X_2,X_3,...,X_n\right)$
where:
$Q =$ quantity of output
$X_1,X_2,X_3,...,X_n =$ factor inputs (such as capital, labour, land or raw materials). This general form does not encompass joint production, that is a production process, which has multiple co-products or outputs.

One way of specifying a production function is simply as a table of discrete outputs and input combinations, and not as a formula or equation at all. Using an equation usually implies continual variation of output with minute variation in inputs, which is simply not realistic. Fixed ratios of factors, as in the case of laborers and their tools, might imply that only discrete input combinations, and therefore, discrete maximum outputs, are of practical interest.

One formulation is as a linear function:

$Q=a+b X_1+c X_2+d X_3, ...$
where $a, b, c,$ and $d$ are parameters that are determined empirically.
Another is as a Cobb-Douglas production function (multiplicative):
$Q = aX_1^b X_2^c$
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 which is a specific type of additive function. The best form of the equation to use and the values of the parameters ($a, b, c,$ and $d$) vary from company to company and industry to industry. In a short run production function at least one of the $X$'s (inputs) is fixed. In the long run all factor inputs are variable at the discretion of management.

## The 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. 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 levels of inputs. From the origin, through points A, B, and C, the production function is rising, indicating that as additional units of inputs are used, the quantity of outputs also increases. Beyond point C, the employment of additional units of inputs produces no additional outputs, in fact, total output starts to decline. The variable inputs are being used too intensively (or to put it another way, the fixed inputs are under utilized). With too much variable input use relative to the available fixed inputs, the company is experiencing negative returns to variable inputs, and diminishing total returns. In the diagram this is illustrated by the negative marginal physical product curve (MPP) beyond point Z, and the declining production function beyond point C.

From the origin to point A, the firm is experiencing increasing returns to variable inputs. As additional inputs are employed, output increases at an increasing rate. Both marginal physical product (MPP) and average physical product (APP) is rising. The inflection point A, defines the point of diminishing marginal returns, as can be seen from the declining MPP curve beyond point X. From point A to point C, the firm is experiencing positive but decreasing returns to variable inputs. As additional inputs are employed, output increases but at a decreasing rate. Point B is the point of 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 an explanation.).

## The 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 efficiency, reaching a maximum at point B (since the average physical product is at its maximum at that point). The average physical product of fixed inputs will also be rising in this stage (not shown in the diagram). Because the efficiency of both fixed and variable inputs is improving throughout stage 1, a firm will always try to operate beyond this stage. In stage 1, fixed inputs are underutilized.

In Stage 2, output increases at a decreasing rate, and the average and marginal physical product is declining. However the average product of fixed inputs (not shown) is still rising. In this stage, the employment of additional variable inputs increase the efficiency of fixed inputs but decrease the efficiency of variable inputs. The optimum input/output combination will be in stage 2. Maximum production efficiency must fall somewhere in this stage. Note that this does not define the profit maximizing point. It takes no account of prices or demand. If demand for a product is low, the profit maximizing output could be in stage 1 even though the point of optimum efficiency is in stage 2.

In Stage 3, too much variable input is being used relative to the available fixed inputs: variable inputs are over utilized. Both the efficiency of variable inputs and the efficiency of fixed inputs decline through out this stage. At the boundary between stage 2 and stage 3, fixed input is being utilized most efficiently and short-run output is maximum.

## Shifting a production function

As noted above, it is possible for the profit maximizing output level to occur in any of the three stages. If profit maximization occurs in either stage 1 or stage 3, the firm will be operating at a technically inefficient point on its production function. In the short run it can try to alter demand by changing the price of the output or adjusting the level of promotional expenditure. In the long run the firm has more options available to it, most notably, adjusting its production processes so they better match the characteristics of demand. This usually involves changing the scale of operations by adjusting the level of fixed inputs. 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 production by 2 million units per year.

If a firm is operating (inefficiently) 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 and to the left. The beginning of stage 2 shifts from B1 to B2. The (unchanged) profit maximizing output level will now be in stage 2 and the firm will be operating more efficiently.

If a firm is operating (inefficiently) at a profit maximizing level in stage three, it might, in the long run, choose to increase its scale of operations (by investing in new capital equipment). By increasing the amount of fixed capital inputs, the production function will shift up and to the right. For example, if it takes 5 pounds of sugar to make 2 cookies, then the production function would be 5 to 2. It would also be 10 to 4. 4 cookies for every 10 pounds of sugar.

## Homogeneous and homothetic production functions

There are two special classes of production functions that are frequently mentioned in textbooks but are seldom seen in reality. The production function $Q=f\left(X_1,X_2\right)$ is said to be homogeneous of degree n, if given any positive constant $k$, $f\left(kX_1,kX_2\right)=k^n f\left(X_1, X_2\right)$. When $n>1$, the function exhibits increasing returns, and decreasing returns when $n<1$. When it is homogeneous of degree 1, it exhibits constant returns. Homothetic functions are functions whose marginal technical rate of substitution (slope of the isoquant) is homogeneous of degree zero. Due to this, along rays coming from the origin, the slope of the isoquants will be the same.

## Aggregate production functions

In macroeconomics, production functions for whole nations are sometimes constructed. In theory they are the summation of all the production functions of individual producers, however this is an impractical way of constructing them. There are also methodological problems associated with aggregate production functions.

## Criticisms of production functions

During the 1950s, 60s, and 70s there was a lively debate about the theoretical soundness of production functions. (See the Capital controversy.) Although most of 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 an abstract quantity of capital which is independent of the rates of interest and wages. The problem is that this independence is a precondition of constructing an iso-product curve. Further, the slope of the iso-product curve helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.

Neoclassical economists often omit natural resources 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 and would imply that to make a cake, all that is needed is a cook, a kitchen, and some non-zero amount of ingredients. The model is absurd in that it suggests that the size of the cake could be expanded indefinitely without extra ingredients. Neither Solow nor Stiglitz addressed his criticism, despite an invitation to do so in the September 1997 issue of the journal Ecological Economics. This may seem like a highly technical argument amongst competing schools of economists, but conceding the point would involve admitting that much of growth theory flounders on the rocks of biophysical limits and may help explain why economists have by and large failed to understand and anticipate emerging environmental problems such as climate change.

• 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
• Heathfield, D. F. (1971) Production Functions, Macmillan studies in economics, Macmillan Press, New York.
• 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
• 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.

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