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In microeconomics and macroeconomics, a production function is a function that specifies the output of a firm, an industry, or an entire economy for all combinations of inputs. This function is an assumed technological relationship, based on the current state of engineering knowledge; it does not represent the result of economic choices, but rather is an externally given entity that influences economic decision-making. Almost all economic theories presuppose a production function, either on the firm level or the aggregate level. In this sense, the production function is one of the key concepts of mainstreamneoclassical theories. Some non-mainstream economists, however, reject the very concept of an aggregate production function.
Concept of production functions
In micro-economics, a production function is a function that specifies the output of a firm for all combinations of inputs. A meta-production function (sometimes metaproduction function) compares the practice of the existing entities converting inputs into output 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.clarification needed In either case, the maximum output of a technologically-determined production process is a mathematical function of one or more inputs. 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 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 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. A decision frame in which one or more inputs are held constant may be used; for example, (physical) capital may be assumed to be fixed (constant) in the short run, and labour and possibly other inputs such as raw materials variable, while in the long run, the quantities of both capital and the other factors that may be chosen by the firm are variable. In the long run, the firm may even have 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 reflected in the function. But 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. Moreover, production functions do not ordinarily model the business processes, either, ignoring the role of management. (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.
Specifying the production function
A production function can be expressed in functional form as the right side of
- Q = f(X_1,X_2,X_3,...,X_n)
- Q = quantity of output
- X_1,X_2,X_3,...,X_n = quantities of factor inputs (such as capital, labour, land or raw materials).
If Q is a scalar, 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:
- 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:
- Q = aX_1^b X_2^c \cdot \cdot \cdot .
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
- Q = \min (aX_1, \ \ bX_2, \ \ ...).
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 (a, b, c, ...) 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.
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 sin
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