The Production Function

bidders are entrepreneurs.An individual may,of course,be both a laborer and entrepreneur in the same production process.Nevertheless,the entrepreneur is unique in that it is through his or her initiative and enterprise that the other factors of production are organized in the first place.

The entrepreneur is distinguished from the “hired hand”in a number of other ways.The entrepreneur,for example,makes the nonroutine decisions for the firm.The entrepreneur determines the firm’s organizational objectives.The entrepreneur is an innovator,constantly on the alert for the best (least expensive) production techniques.The entrepreneur also endeavors to ascertain the wants and needs of the public to be able to introduce new and better products,and to discontinue production of goods and services that are no longer in demand.The entrepreneur is a risk taker in the expectation of earning a profit.

It should be pointed out once again that it is assumed that the expectation of earning profit is the motivating incentive for all commercial activities in a free economy.Without the profit motive,it is assumed that entrepreneurs would not incur risk the financial risks associated with combining productive resources to produce final goods and services for sale in the market.In the absence of the profit motive,the production of goods and services in quantities that are most in demand by society would not occur.Moreover,whenever the profit incentive is diluted—say,through a tax on profits—fewer goods and services will be produced by the private sector in exchange for an increase in goods and services produced by the public sector.Whether society is better off or worse off will ultimately depend on which sector is most responsive to society’s needs and preferences and which sector is most efficient in the utilization of productive resources in the delivery of those goods and services.The disappointing economic experiences of centrally planned economies in the communist countries of eastern Europe and Asia in the latter half of the twentieth century, compared with that of the essentially free market economies of western Europe,North America,and Japan often convincing testimony to the power of the profit motive in promoting the general economic well-being of society at large.To quote from Adam Smith:“By pursuing his own interest he (the entrepreneur) frequently promotes that of society more effectively than when he really intends to promote it.”1

THE PRODUCTION FUNCTION

The technological relationship that describes the process whereby factors of production are efficiently transformed into goods and services is

1 Adam Smith, TheWealthofNations.(New York:Modern Library,1937) p.423.(Originally published in 1776.)

called the production function.Mathematically,a production function utilizing capital,labor,and land inputs may be written as

(5.1)

where K is capital, L is labor,and M is land.

The production function defines the maximum rate of output per unit of time obtainable from a given set of productive inputs.Obviously,some firms organize inputs inefficiently,thereby producing less than the maximum possible level of goods and services.In a competitive market environment, however,firms that do not adopt the most efficient (cost-effective) production technology will not prosper and may,if fact,be forced to curtail operations as their market position is undermined by more aggressive competitors.In free-market economies there is a tendency for more efficient operations to drive less efficient firms out of an industry.

For the sake of pedagogical,theoretical,and graphical convenience,let us assume that all productive inputs may be classified as either labor (L) or capital goods (K).Equation (5.1) may therefore be rewritten as

(5.2)

In most cases,labor and capital may be substituted,albeit in varying degrees.As we will see,the precise manner in which these inputs are combined will depend on the relative rental price and marginal productivity of the factor of production.A firm that operates efficiently will choose that combination of productive inputs that minimizes the total cost of producing a given level of output.Equivalently,a firm that operates efficiently will choose that combination of productive inputs that will maximize production subject to a given operating budget.

It should be noted that the production function signifies the technological relationship between inputs and outputs.It is an engineering concept, devoid of any economic content.The production function defines the maximum output obtainable from a particular combination of inputs.Input prices must be combined with the production function to determine which of the many possible combinations of inputs is the most desirable given the firm’s objectives,such as profit maximization.

Q fKLM = ( ) , ,

Q fKL = ( ) ,

THE COBB–DOUGLAS PRODUCTION FUNCTION

Production functions take many forms.In fact,there is a unique production function associated with each and every production process.In practice,it may not be possible to precisely define the mathematical relationship between the output of a good or service and a set of productive inputs.In spite of this,it is possible to approximate a firm’s production function.Perhaps the most widely used functional form of the production process for empirical and instructional purposes is the Cobb–Douglas pro-

duction function.The appeal of the Cobb–Douglas production function stems from certain desirable mathematical properties.The general form of the Cobb–Douglas production function for the two-input case may be written

(5.3) where A, a and b are known parameters and K and L represent the explanatory variables capital and labor,respecitvely.It is further assumed that 0 £ (a, b) £ 1.

Consider,for example,the following empirical Cobb–Douglas production function:

(5.4)

Table 5.1,which summarizes the different output levels associated with alternative combinations of labor and capital usage,illustrates three important relationships that highlight the desirable mathematical properties of the Cobb–Douglas production function.These are the relationships of substitutability of inputs and returns to scale,and the law of diminishing marginal product.Although these properties of production functions are discussed at greater length later,a brief discussion of their significance in terms of the example illustrated in Table 5.1 is useful here.

Substitutability

When a given level of output is generated,factors of production may or may not be substitutable for each other.Table 5.1 illustrates the substitutability of labor and capital for the production function summarized in Equation (5.4).It can be seen,for example,that to produce 122 units of output,labor and capital may be combined in (row,column) combinations of (3,8),(4,6),(6,4),and (8,3).

Q AKL = a b

Q K L = 25 05 05 . .

TABLE 5.1 Substitutability

Labor

Capital 1 2 3 4 5 6 7 8

1 25 35 43 50 56 61 66 71 2 35 50 61 71 79 87 94 100 3 43 61 75 87 97 106 115 122 4 50 71 87 100 112 122 132 141 5 56 79 97 112 125 137 148 158 6 61 87 106 122 137 150 162 173 7 66 94 115 132 148 162 175 187 8 71 100 122 141 158 173 187 200

The Cobb–Douglas production function also illustrates the fundamental economic problem of constrained optimization.Although there are theoretically an infinite number of output levels possible,the firm is typically subject to a predetermined operating budget that limits the amount of productive resources that can be acquired by the firm.Because of this restriction,an increase in the use of one factor of production requires that less of some other factor be employed.The degree of substitutability of inputs is important because it suggests that managers are able to alter the input mix required to produce a given level of output in response to changes in input prices.

Returns to Scale

Suppose that output is described as a function of capital and labor. Suppose that capital and labor are multiplied by some scalar.If output increases by that same scalar,the term constant returns to scale (CRTS) opplies.In Table 5.1 we observe that when labor and capital inputs are raised by a factor of 2,as when labor and capital are doubled from 2 units to 4 units,then output is raised by the same scalar (i.e.,output increases from 50 to 100 units).If capital and labor are multiplied by a scalar and output increases by a multiple greater than the scalar,the condition is referred to as increasing returns to scale (IRTS).Finally,if capital and labor are multiplied by a scalar and output increases by a multiple less than the scalar,the condition is referred to as decreasing returns to scale (DRTS).

Law of Diminishing Marginal Product

Finally,the Cobb–Douglas production function exhibits a very important technological relationship—the law of diminishing marginal product (also referred to as the law of diminishing returns).This law,sometimes referred to as the second fundamental law of economics,states that when at least one productive input is held fixed while at least one other productive resource is increased,output will also increase but by successively smaller increments.The law of diminishing marginal product is a short-run production concept.As will be discussed in the next section,the “short run”in production refers to that period of time during which at least one factor of production is held fixed in amount.Mathematically,the law of diminishing marginal product requires that the first partial derivative of the production function with respect to a variable input be positive and the second partial derivative negative.In Table 5.1,for example,when capital is held constant at K = 1,and labor is successively increased from L = 1 to L = 6,output increases from 25 to 61 units,but successive marginal increments are 10,8, 7,6,and 5.

It was noted earlier that the production function is characterized as efficient (i.e.,the most output obtainable from a given level of input usage). Efficient utilization of productive inputs is characterized by production pos-