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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 non-monetary, 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).
In its most general mathematical form, a production function is expressed as:
There are several ways of specifying this function. One is as an additive production function:
Another is as a Cobb-Douglas production function (multiplicative):
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 Xs (inputs) is fixed. In the long run all factor inputs are variable at the discresion of management.
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.).
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.
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