Information about Production Function
In microeconomics, a production function asserts that the maximum output of a technologically-determined production process is a mathematical production of input factors of production. Considering 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.
In a general mathematical form, a production function can be expressed as:
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:
and 
) 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.
Quadratic Production Function
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.).
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 overutilized. 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
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.
is said to be homogeneous of degree n, if given any positive constant 
, 
. When 
, the function exhibits increasing returns, and decreasing returns when 
. 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.
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.
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:
- :where:
- ::
quantity of output
- ::
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:
- :where and are parameters that are determined empirically.
and ) 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.
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.)
alt text
Quadratic Production Function
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 overutilized. 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.)
alt text
Shifting a Production Function
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.
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 is said to be homogeneous of degree n, if given any positive constant , . When , the function exhibits increasing returns, and decreasing returns when . 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.
See also
- Distribution (economics)
- Production theory basics
- Production, costs, and pricing
- Production possibility frontier
- Microeconomics
References and external links
- A brief history of production functions
- A further description of production functions
- 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.
Microeconomics (or price theory) is a branch of economics that studies how individuals, households, and firms make decisions to allocate limited resources,[1] typically in markets where goods or services are being bought and sold.
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In economics, factors of production are resources used in the production of goods and services, including land, labor, and capital.
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Land, labor, and capital
Resource in economics distinguish among such factors of production as:..... Click the link for more information.
In economics, x-efficiency is the effectiveness with which a given set of inputs are used to produce outputs. If a firm is producing the maximum output it can, given the resources it employs, such as men and machinery, and the best technology available, it is said to be x-efficient.
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Allocative efficiency is the market condition whereby resources are allocated in a way that maximizes the net benefit attained through their use. Allocative efficiency refers to a situation in which the limited resources of a country are allocated in accordance with the wishes of
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The term business model describes a broad range of informal and formal models that are used by enterprises to represent various aspects of business, such as operational processes, organizational structures, and financial forecasts.
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In microeconomics, Production is simply the conversion of inputs into outputs. It is an economic process that uses resources to create a commodity that is suitable for exchange. This can include manufacturing, storing, shipping, and packaging.
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In economics, the marginal product or marginal physical product is the extra output produced by one more unit of an input (for instance, the difference in output when a firm's labour is increased from five to six units).
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In economics, the Cobb-Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. It was proposed by Knut Wicksell (1851-1926), and tested against statistical evidence by Paul Douglas and Charles Cobb in 1928.
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Constant Elasticity of Substitution (CES) production function introduced by Arrow, Chenery, Minhas, and Solow, (1961), is:
where
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where
- = Output
- = Factor productivity
- = Share parameter
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In microeconomics, Production is simply the conversion of inputs into outputs. It is an economic process that uses resources to create a commodity that is suitable for exchange. This can include manufacturing, storing, shipping, and packaging.
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In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. Examples are given by homogeneous polynomials.
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Macroeconomics is a branch of economics that deals with the performance, structure, and behavior of a national economy as a whole.[1] Macroeconomists seek to understand the determinants of aggregate trends in an economy with particular focus on national income,
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The Cambridge capital controversy refers to a 1960s debate in economics concerning the nature and role of capital goods (or means of production). The name arises because of the location of the those most involved in the controversy: the debate was largely between economists such as
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Distribution in economics refers to the way total output or income is distributed among individuals or among the factors of production (labor, land, and capital) (Samuelson and Nordhaus, 2001, p. 762).
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In microeconomics, Production is simply the conversion of inputs into outputs. It is an economic process that uses resources to create a commodity that is suitable for exchange. This can include manufacturing, storing, shipping, and packaging.
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In microeconomics, production is the act of making things, in particular the act of making products that will be traded or sold commercially. Production decisions concentrate on what goods to produce, how to produce them, the costs
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In economics, a production possibilities frontier (PPF) or “transformation curve” is a graph that shows the different quantities of two goods that an economy (or agent) could efficiently produce with limited productive resources.
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Microeconomics (or price theory) is a branch of economics that studies how individuals, households, and firms make decisions to allocate limited resources,[1] typically in markets where goods or services are being bought and sold.
..... Click the link for more information.
..... Click the link for more information.
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