Production function
From Academic Kids

In microeconomics, a production function expresses the relationship between an organization's inputs and its outputs. It indicates, in mathematical or graphical form, what outputs can be obtained from various amounts and combinations of factor inputs. In particular it shows the maximum possible amount of output that can be produced per unit of time with all combinations of factor inputs, given current factor endowments and the state of available technology. Unique production functions can be constructed for every production technology.
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. This is just a reformulation of the definition above.
The relationship is nonmonetary, that is, a production function relates physical inputs to physical outputs. Prices and costs are not considered. (For a primer on the fundamental elements of physical production theory, see production theory basics).
The production function as an equation
In its most general mathematical form, a production function is expressed as:
 Q= f(X_{1},X_{2},X_{3}...)
 where:
 Q= quantity of output
 X_{1}, X_{2}, X_{3}, etc.= factor inputs (such as capital, labour, raw materials, land, technology, or management)
 where:
There are several ways of specifying this function. One is as an additive production 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 CobbDouglas 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 CobbDouglas 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 Xs (inputs) is fixed. In the long run all factor inputs are variable at the discresion 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.
Stages_of_production_small.png
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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 shortrun 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.
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(X_{1},X_{2}) is said to be homogeneous of degree n, if given any positive constant k, f(kX_{1},kX_{2})=k^{n}f(X_{1},X_{2}). 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 a special class of homogeneous function in which the marginal rate of technical substitution is constant along the function.
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 complained about 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 isoproduct curve. Further, the slope of the isoproduct curve helps determine relative factor prices, but the curve cannot be constructed (and its slope measured) unless the prices are known beforehand.
See also:
 Production theory basics
 Production, costs, and pricing
 Production possibility frontier
 Microeconomics
References
 A further description of production functions (http://cepa.newschool.edu/het/essays/product/prodfunc.htm)
 Heathfield, D. F. (1971) Production Functions, Macmillan studies in economics, Macmillan Press, New York.
 Moroney, J. R. (1967) CobbDouglass production functions and returns to scale in US manufacturing industry, Western Economic Journal, vol 6, no 1, December 1967, pp 3951.
 Pearl, D. and Enos, J. (1975) Engineering production functions and technological progress, The Journal of Industrial Economics, vol 24, September 1975, pp 5572.
 Robinson, J. (1953) The production function and the theory of capital, Review of Economic Studies, vol XXI, 1953, pp. 81106
 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 0132314231