# Production

# Production

Aggregate production functions

The economic theory of production is derived from (*a*) the behavioral premise that the decisionmaking unit (the firm or the production manager) desires to minimize the total cost of producing any given output or specified combination of outputs, and (*b*) the postulate that normally there exist alternative techniques or processes of production, with different patterns of resource (labor, capital, materials, energy) consumption and, therefore, different associated costs. Cost minimization is normally assumed to be a separable but contributing part of the primary objective of maximizing profit. However, this assumption is not necessary, for the decision maker may have other primary objectives, such as maximizing sales or market share, but still be concerned with producing the scheduled level of production at minimum cost.

Production theory is often called marginal productivity theory, in reference to the decision rules that represent necessary conditions for the achievement of maximum profit. These classical rules state that for maximum profit (including minimum cost) the quantity of each resource input used in a process must be adjusted until the value of the product resulting from the last increment of the resource (the marginal product) is equal to the cost of the last increment of the resource. Thus, if *MP _{L}* is the marginal product of labor and P is the price of the product of labor, then P ·

*MP*is the value of the marginal product of labor. If, furthermore, the constant price or wage of labor is W, the decision rule for the employment of labor is to adjust its employment until P ·

_{L}*MP*= W. If P ·

_{L}*MP*the return from an additional unit of labor would exceed its cost, and profit could be increased by expanding employment. If P ·

_{L}> W,*MP*W, the return from an additional unit of labor would be less than its cost, and profit could be increased by contracting employment. The validity of this argument requires

_{L}<*MP*to be a decreasing (downward-sloping) function of employment. Mathematically, this condition is sufficient for a local maximum of profit.

_{L}The original formulation of production theory is associated most closely with the names of John B. Clark, Philip H. Wicksteed, Francis Y. Edgeworth, and Léon Walras, whose writings on the subject appeared between 1874 and 1896. Walras’s 1874, 1877, and 1889 treatments of production were based on the assumption of fixed technical coefficients (see below), with only a brief discussion of the case of variable coefficients. Walras’s statement of the marginal productivity theory does not appear until the third edition of his *Elements,* in 1896 (see 1874–1877). The first clear statements of the marginal productivity theory seem to have been made independently by Clark (1889), Edgeworth (1891–1921), and Wicksteed (1894). Wicksteed’s treatment was the most influential, and he is probably the one most often given credit as the originator of the theory.

For a fuller discussion of the history of the marginal productivity theory, the reader is referred to Stigler (1941) and to Walras (1874–1877), especially the extensive notes by the translator, William Jaffé. [*See also the biography of*Walras.]

## The production function

Traditional production theory postulates an input-output relationship, or production function, showing the quantity of output (or the quantities of various outputs) that can be produced as a function of the quantities of the various inputs consumed. In the case of one output and two inputs, if we let *y* be the output rate and *x*_{1} and *x*_{2} be the input rates, the production function can be written

The output might be kilowatt hours of electricity per year; *x*_{1} might be tons of coal per year; and *x*_{2} might be maintenance hours per year. Or again, *y* might be a chemical product, with *x*_{1} a raw material reactant and *x*_{2} a fuel, a second reactant, or a catalytic material.

Equation (1) represents what is called an *efficiency* frontier in the sense that it provides the largest *y* that can be produced for given *x*_{1} and *x*_{2} (or the least *x*_{1} required for given *y* and *x _{2},* etc.). There are some production decisions that can be made on purely technical grounds, without knowledge of resource prices. These decisions are usually called

*engineering decisions,*as opposed to

*economic decisions.*Thus, if a modification of the manner in which a process is performed allows the same output while permitting the quantity of at least one input to be reduced without an increase in the quantity of any other input, then a decision in favor of the modification can be made on engineering grounds alone, without any knowledge of prices. An action that saves on one input without altering any other requirement of a process will lower cost, regardless of the price of that input.

The production function in (1) presupposes that all such engineering decisions have been made. In constructing this function, all methods, techniques, or processes that require more of one input and no less of any other input are rejected. Once all such engineering decisions have been made, we are left with the best engineering technology. But this technology presents a multiplicity of input possibilities which have the characteristic that output cannot be maintained at a given level, when one input is reduced, unless some other input is increased. The choice among these remaining input combinations is an *economic decision,* in that the decision requires knowledge of input prices.

Briefly stated, economic decisions require knowledge of input prices and best engineering technology. Engineering decisions are concerned with best engineering technology and require technical knowledge of physical processes.

The characteristics of a three-variable production function are readily summarized graphically with an isoproduct map. An *isoproduct* (constant output) *contour* is the locus of all inputs (*x*_{1}*x*_{2}) that are required to produce a specified quantity of output. An *isoproduct map* is a family of such contours, each corresponding to a different level of output. Such a family of contours is shown in Figure 1 for outputs of 10, 20, 30, 40, and 50 units.

### Substitution

Figure 1 also illustrates the phenomenon of substitution: for any given level of output, say 30, if input 2 is decreased, more of input 1 is required. In addition to having a negative slope, each isoproduct contour is convex to the origin (mathematically, . This characteristic is often referred to as the principle of increasing marginal rate of substitution. For a given output, larger increments of *x*_{1} are required to substitute for each incremental decrease in *x*_{2}, and vice versa.

There is, of course, great variation in the extent of substitution among different productive processes. At one extreme is the case of perfect substitution, in which the isoproduct contours are downward-sloping straight lines. Thus, for some purposes it might make no difference whatever whether a certain mechanical part is made of aluminum or brass, and we would say that the two materials are perfect substitutes in that use. The production function would be *y* = *a*_{1}*x*_{1} + *a*_{2}*x*_{2}, where *a*_{1} and *a*_{2} are the number of parts that can be produced from a pound of aluminum and a pound of brass, respectively; *x*_{1} and *x*_{2} are the pounds of input of each metal; and *y* is the total output of parts, that is, *a*_{1}*x*_{1} aluminum units plus *a*_{1}*x*_{2} brass units.

At the opposite extreme is the case of limitational inputs or fixed proportions, in which each input requirement is rigidly proportional to the output produced, e.g., *x _{1}* = b

_{1}

*y*,

*x*

_{2}=

*x*

_{2}

*y,*where

*b*

_{1}and

*b*

_{2}are the constants of proportionality. In this case the production function can be written

*y*= min (

*x*

_{1}/

*b*

_{1},

*x*

_{2}/

*b*

_{2}), where

*x*

_{1}/

*b*

_{1}is the largest output producible from the amount of input 1 available, and

*x*

_{2}/

*b*

_{2}is the largest output producible from the amount of input 2 available. The smaller of these two outputs determines how much can be produced —output is limited by the resource that is in shortest supply.

Some inputs to a process may be perfect or imperfect substitutes, whereas others are limitational. Suppose, for example, that *x*_{1} and *x*_{2} are perfect

substitutes, *x*_{3} and *x*_{4} are limitational inputs, and *x*_{5} and *x*_{6} are continuously substitutable according to the relation *y* = *f*(x_{5}, *x*_{6}). Then the production function can be written

*y* = min {*a*1*x*1 + *a*_{2}*x*_{2}, *x*_{3}/*b*_{3}, *x*_{1}/*b*_{4}, *f*(*x*_{5}, *x*_{6})}.

Output is limited by the resource or resource combination that is in shortest supply.

### Diminishing returns

According to the law of *diminishing returns,* if one of the two inputs is held constant and the other input is increased, the marginal productivity of that input must eventually decline (mathematically, beyond some value of *x*_{1} we must have , and beyond some value of . In Figure 1 it is seen that when *x*_{2} = 12, 10 units of output can be produced with *x*_{1} = 3; 20 units of output can be produced with *x*_{1}*–* 5; and so on. As output increases in 10-unit increments, the quantity of *x*_{1} required increases, first at a decreasing rate and then at an increasing rate. Since *d*_{1}>*d*_{2}>*d*_{3}<*d*_{1}<*d*_{5}, it is seen that the phenomenon of diminishing returns to *x*_{1} for *x*_{2} = 12 is experienced when output rises above 30 units.

### Returns to scale

The study of diminishing returns is concerned with how output changes with changes in an input when other inputs are held constant. The study of *returns to scale* is concerned with how output changes when proportional simultaneous changes are made in *all* inputs and there is no generally accepted law or principle requiring returns to scale eventually to decline.

In Figure 1, returns to scale are represented by reference to the straight line *0R* through the origin. Observe that if we begin with any input combination, such as that represented by the point *P*(*x*_{1} = 4, *x*_{2}, = 2), then a proportional increase in *both* inputs must be on the line OR. For example, if both inputs are doubled, the new input combination is represented by the point *Q*. If, beginning at point *P*, both inputs are increased 3/2 times (to *x*_{1} = 6 and *x _{2}* = 3), it is seen that output increases by a factor of 2 (from 10 to 20). Over this range there are

*increasing*returns to scale. If, beginning at Q; both inputs are increased 3/2 times (to

*x*

_{1}= 12 and

*x*

_{2}= 6), output increases by a factor of 4/3 (from 30 to 40), showing

*decreasing*returns to scale. In general, if

*D*

_{1}>

*D*

_{2}>

*D*

_{3}>

*D*

_{4}, etc., the process shows

*increasing returns to scale*at all output levels. If

*D*

_{1}=

*D*

_{2}=

*D*

_{3}=

*D*

_{4}, etc., the process shows

*constant returns to scale*at all outputs. If

*D*

_{1}<

*D*

_{2}, <

*D*

_{3}<

*D*

_{4}, etc., the process shows

*decreasing returns to scale*for all outputs. In the example of Figure 1, the process shows increasing returns to scale at first, then constant returns to scale, and finally decreasing returns to scale.

The question of returns to scale can be discussed in terms of the mathematical concept of a homogeneous function. The production function *y* = *f*(*x*_{1}*x*_{2}) is said to be homogeneous of degree 1 if an increase in all inputs by the same proportion, say *α* increases output by the proportion *α* That is, the production function is homogeneous of degree 1 and therefore exhibits constant returns to scale for all outputs if

If the production function is not homogeneous of degree 1 and shows either increasing or decreasing returns to scale, then

If *β > α* we have increasing returns to scale, while if *β < α* we have decreasing returns to scale.

Individual processes may show increasing, constant, or decreasing returns to scale. Still other processes may, over different ranges of output, show all three kinds of returns. However, it is usually argued that it is inefficient to operate any process at a size that is in the range of decreasing returns. The output is more efficiently produced by building multiple facilities of a size determined by the output at which returns just begin to decrease.

### Capital in the production function

For purposes of production theory, the distinguishing feature of capital goods is that their presence, in the form of physical stocks, is required if production is to take place (Smith 1961). Only in special cases, as with machines that rotate or reciprocate and whose speed of operation is variable, can we obtain meaningful measures of service or utilization intensity. Even in these cases, utilization intensity cannot be varied independently of raw material and energy consumption, which represent current or flow inputs that appear in the production function. For these reasons, it is common to measure capital as a *stock,* as opposed to a *flow,* input resource. The significance of capital stock inputs is to be found in the fact that so much pipe, cable, concrete, etc., arranged in a certain way, must be present if output is to be produced. Furthermore, the level of output that can be produced varies directly with the physical quantities of the capital inputs present when production takes place. Such inputs are not consumed in any sense similar to the consuming of raw materials, which may become physically embodied in the object produced, or of energy. This concept of capital as a collection of assets whose presence is required for production applies also to nonphysical capital, such as knowledge, or “human capital,” as it is sometimes called. Indeed, knowledge, in terms of durability and relative nonconsumption in production, is the most capital-like of all such assets.

In introducing physical capital goods into production function analysis, it is important to recognize another feature of such goods. Capital goods are usually freely variable in size only in the process design stage. In this drawing board stage, one may consider any size plant or individual items of equipment that one pleases, but once the investment is made, the “amount of capital” can only be varied by replacement with smaller or larger units or by installing parallel facilities.

### Empirical examples

The earliest studies of empirical production functions were made by agricultural economists and agronomists concerned with experimental studies of soil, fertilizer, moisture, and seed productivity. Early studies were also conducted of the weight-gaining characteristics of different feeds and feed supplements for livestock. In these cases there is a firm biological basis for expecting meat and crop outputs to be dependably related to the combinations of feed, fertilizer nutrient, and care expended. One experiment (Heady 1957) in corn fertilization yielded the production function , where *x*_{1} is pounds of nitrogen applied per acre per year, *x*_{2} is pounds of phosphate applied per acre per year, and *y* is yield of corn in bushels per acre per year. Production functions for many other agricultural products have been determined: the production of alfalfa as a function of the application of potash and phosphate fertilizers; of pork as a function of the amounts of corn and soybean oilmeal; of milk as a function of the forage and grain concentrate used.

In engineering design the earliest known use of an explicit cost minimization procedure using essentially, if not quite literally, the concept of a production function, was in the determination of economic conductor size by Lord Kelvin. Writing in 1881, before Clark, Edgeworth, or Wicksteed, Kelvin summarized his derivation in a form later known by electrical engineers as Kelvin’s law: “The most economical size of the copper conductor for the electric transmission of energy, whether for the electric light or for the performance of mechanical work, would be found by comparing the annual interest of the money value of the copper with the money value of the energy lost annually in the heat generated in it by the electric current” (1882, p. 526). His solution provides the conductor size “which makes the two constituents of the loss equal.”

In this case, the principle giving rise to substitution between energy and conductor cable is the energy loss mechanism in all forms of energy transmission and transformation. Energy output, *y,* is input, *x*_{1} minus loss, *L*, in the production of heat, i.e., *y* = *x*_{1} – *L*, Losses vary directly with the square of output but inversely with conductor size, and therefore, for a given length transmission line, inversely with conductor weight (or volume), *X*_{2}. Hence, *L* = *ky*^{2}/*X*_{2}, where *k* is a constant depending upon a variety of factors, such as the length of the line and the resistivity of the cable. The production function can therefore be written *y* = *x*_{1} – *ky*^{2}/*X*_{2}, or, in explicit form, .

The production functions for numerous other engineering processes have been studied: gas transmission (Chenery 1949); heat transmission, steam power production, filtration processes, and batch reactor chemical processes (Smith 1961); and metal cutting (Davidson et al. 1958; Kurz & Manne 1963). These studies have been sufficiently diverse to show that the principle of input substitution has a firm base in a wide variety of scientific and empirical engineering laws. Sometimes the laws generate substitution representing a negligible part of the productive activity of some firms; in other cases the laws generate substitution among the principal inputs of the enterprise. Thus, in steam power production, the basic production function governing the process arises from the law of energy conservation and the empirical laws governing energy loss in boilers, turbines, and generators. At each stage in the transformation of energy—from fuel into steam, steam into mechanical energy, and mechanical into electrical energy—losses occur at a rate which is an increasing function of the output of the stage and a decreasing function of equipment (capital) size in the stage. The production function is *y* = *f*(*x*_{1}, *X*_{2}, *X*_{3}, *X*_{4}), where *y* is power output rate, *x*_{1} is fuel consumption rate, and *X*_{2}, *X*_{3}, and *X*_{4} are size measures of the boiler, turbine, and generator equipment, respectively.

### Linear programming

An entirely different mechanism of substitution sometimes occurs where a product can be produced by any of several distinct processes, each process being characterized by a fixed coefficient production function. In that case one input substitutes for another when one process is chosen instead of another.

Consider a product that can be produced by either of two processes, each of which requires two inputs—fuel and labor. The two processes might simply be two different kinds of equipment. Process 1 requires one man-hour per unit of output and 1.5 gallons of fuel per unit of output. If *y*_{1} is output, *x*_{11} labor input, and *x*_{21} fuel input, then the process is characterized by the equations *x*_{11} = *y*_{1}, *x*_{21} = 1.5*y*_{1}. If process 2 requires two man-hours of labor and one gallon of fuel for each unit of output, then process 2 is characterized by the conditions *x*_{12} = 2*y*_{2}, *x*_{22} = *y*_{2}.

Each of these processes taken separately exhibits isoproduct contours like those encountered in any process whose inputs are used in fixed proportions. Figure 2 shows an isoproduct contour (*IGI′* and *JHJ′*) for an output of 300 for each of the two processes. However, if the two processes can be operated in parallel combination at any desired levels of output, then it will be possible to achieve any of the input–output combinations on the line segment *GH.*

To see this, consider the two rays O*B* and O*P* (Figure 2), whose slopes represent the ratios at which the two inputs must be combined in each process. The scale laid off along each ray represents the output that would result if the indicated input combinations were employed in that process. For example, 300 units of output can be produced with process 1 when 300 man-hours of labor are expended and 450 gallons of fuel consumed. Similarly, 300 units of output can be produced with process 2 when 600 man-hours of labor and 300 gallons of fuel are used. But 300 units of output can also be produced by employing the two processes in combination. For example, we could produce 100 units of output with process 1 and 200 units of output with process 2. In Figure 2 the length of the “vector” 0A represents 100 units of output by process 1, while the projections of this vector on the horizontal and vertical axes give the amounts of the two inputs required, 100 and 150

units respectively. Similarly, the vector 0*D* represents 200 units of output by process 2, and its projection on the axes identifies the input requirements of the process at this output. Now, combining or adding the two processes together is geometrically equivalent to adding the vector 0A to 0*D* or, since 0*D* equals *AC,* to adding 0*A* to *AC,* giving 0*C*—a vector representing a total output of 300 units and whose projections on the axes give the total input requirements of the two processes together. Therefore, the point *C* represents a point on the isoproduct contour for an output of 300 units which is obtained by operating process 1 at an output level of 100 units and process 2 at an output level of 200 units. Thus, all the points on the line *GH* are achievable, and the isoproduct contour for 300 units is *IGHJ′.*

It is now possible to write a mathematical statement of the efficient production function for this two-process system. Since the individual process outputs are *y*_{1} and *y*_{2}, total output is *y*_{1} + *y*_{2}. Process 1 uses *x*_{11} = *y*_{1} units of labor, and process 2 uses *x*_{12} = 2*y*_{2} uses *x 12* = *2y*_{2} units of labor. Hence, if *x*_{1} is the total labor input, we must have *y*_{1} + 2*y*_{2} ≤ *x*_{1}, since not more than the available labor can be used. Similarly, if *x*_{2} is the fuel available for consumption, we must require 1.5*y*_{1} + *y*_{2} ≤ *x*_{2}. Mathematically, the production function relating *y, x*_{1}, and *x*_{2} is defined by the condition that *y* = max_{y1,y2}(*y*_{1} + *y*_{2}), subject to *y*_{1} + 2*y*_{2} ≤ *x*_{1} and 1.5*y*_{1} + *y*_{2} ≤ *x*_{2}, or *y*_{1} + 2*y*_{2} + *S*_{1} = *x*_{1} and 1.5*y*_{1} + *y*_{2} + *S*_{2} = *x*_{2}, where *S*_{1} ≥ 0 and *S*_{2} ≥ 0 are surplus unused quantities of the inputs. This maximum problem is a linear programming problem (Dorfman et al. 1958). From Figure 2 it can be seen that if *x*_{1} = 500 and *x*_{2} = 350, we would have *y* = 300, with *y*_{1} = 100 and *y*_{2} = 200. It is easily verified that *y*_{1} = 100, *y*_{2} = 200, *S*_{1} = *S*_{2} = 0 is a solution to the above maximum problem. Similarly, if *x*_{1} = 800 and *x*_{2} = 300, the solution to the linear programming problem is *y*_{1} = 0, *y*_{2} = 300, *S*_{1} = 200, *S*_{2} = 0.

The above reasoning can be extended to any number of processes and any number of inputs. For *m* processes and *n* inputs, the production function *y* = *f*(*x*_{1}, · · ·, *x*_{n}) is obtained by solving the following linear programming problem:

where *a*_{ij} is the quantity of the *j*th input required per unit of production in the *J*th process. [*See*Programming.]

## Cost and production theory

Given the production function for a process, say *y* = *f*(*x*_{1}, *x*_{2}), and the prices of the inputs, e.g., *w*_{1}, *w*_{2}, the problem in the theory of cost is to determine decision rules for minimizing the cost of producing any given output. That is, for any specified *y,* we want *C* = *w*_{1}*x*_{1} + *w*_{2}*x*_{2} to be a minimum subject to the requirement that *y* = *f*(*x*_{1}, *x*_{2}). Necessary conditions for *C* to be a minimum are that *ω*_{1} – *λ*(*∂f*/*∂x*_{1}) ≥ 0, where *x*_{1} = 0 if the inequality holds, and that *ω*_{2} – *λ*(*∂f*/*∂x*_{2}) ≥ 0, where *x*_{2} = 0 if the inequality holds. (Mathematically, the problem is to minimize the Lagrangian *ϕ* = *ω*_{1}*x*_{1} + ω_{2}*x*_{2} – *λ*[*f*(*x*_{1}, *x*_{2}) – *y*], where *λ* is the Lagrange multiplier.) The *λ* in these expressions is interpreted as marginal total cost (*λ* = *∂C/∂y*). Written in the form *ω*/(*∂f*/*∂x*_{1}) ≥ *λ* ≤ *ω*_{2}(*∂f*/*∂x*_{2}) these conditions require that no input be employed in an amount such that the marginal cost of an additional unit of that input—the price of the input divided by its marginal product, *ω*_{1}(*∂f*/*∂x*_{1})—is smaller than the marginal total cost for the process. If the marginal cost of any input to the process

exceeds marginal total cost, that input is not used (we have a “boundary” solution). Thus, for two inputs, if *ω*_{1}/(*∂f*/*∂x*_{1}) = *λ* = *ω*_{2}/(*∂f*/*∂x*_{2}), or *ω*_{1}/*ω*_{2} = (*∂f*/*∂x*_{1})/(*∂f*/*∂x*_{2}), then we have an interior tangency solution, such as point *P* in Figure 3. If, however, *ω*_{1}/(*∂f*/*∂x*_{1}) > λ = *ω*_{2}/(*∂f*/*∂x*_{2}) or *ω*_{2}/*ω*_{2} > (*∂f*/*∂x*_{1})/(*∂f*/*∂x*_{2}) we have *x*_{1} = 0 and a boundary solution, as illustrated by point Q in Figure 3.

The condition *ω*_{2}/*ω*_{2} = (*∂f*/*∂x*_{1})/(*∂f*/*∂x*_{2}) in combination with the production function *y* = *f*(*x*_{1}, *x*_{2}), defines the minimum cost input combination , as a function of the parameters *y*, *ω*_{1}, *ω*_{2}. Hence, total cost, , and marginal cost, *λ*^{0}, are determined as functions of output and input prices. Over-all profit, that is, revenue minus minimum cost, *R*(*y*) – *C*°(*y*), is a maximum when *∂R*/*∂y* ≤ *∂C/∂y*, where *y* = 0 if the inequality holds. That is, if marginal total cost, *∂C/∂y*, exceeds marginal revenue, *∂R /∂y, at any output, that output should be contracted until either the two are equal or y = 0, whichever occurs first. For example, in the case of two inputs, if it pays to use both inputs and to produce some output, the optimum input–output combination satisfies the conditions*

and

If it does not pay to produce, this fact is expressed by the inequality *∂R/∂y* < *∂C/∂y*

The above analysis extends readily to the case of *n* inputs. If all inputs are used and production is positive, the necessary conditions are

If, say, all but inputs 1 and 2 are used, the conditions are

If one of the inputs, such as 2 in the two-input example, is a capital input, the formal aspects of the analysis remain unchanged except for two considerations: (1) the problem of specifying the “price,” *w*_{2}, to be associated with the capital input, *X*_{2}, requires some analysis, and (2) the capital input, as noted above, is not freely variable in response to changing prices, once it has been installed.

Considering the first problem, the cost function is *C* = *w*_{1}*x*_{1} + *w*_{2}*X*_{2}. Note that because *X*_{2} is a stock variable, *ω*_{2} must have the dimension “dollars per unit per unit of time.” That is, capital cost must be expressed as a rental time cost for the use of capital, including maintenance, service, “depreciation,” and interest on investment. As a simple example, if the upkeep costs are independent of age, say, *m* dollars per unit per year; the life of capital fixed, say, *L* years; the rate of interest *i;* and the unit construction or purchase cost of capital *W*_{2}; then *ω*_{2} = *iW*_{2}/[1 – (1 + *i*)^{–L}] + *m.* The expression *i*/[1 – (1 + *i*)^{–L}] is the annual payment stream over *L* years that is necessary to return a loan of one dollar plus interest at 100*i* per cent per year.

The second problem can be illustrated as follows: Suppose the firm has purchased a capital good of size *X*̅_{2}, representing a minimum cost capital investment. Now suppose demand, and therefore the optimal output rate, increases. The new production function is , where *X̅*_{2} is fixed and x_{1}, x^{’}_{1} and X^{’}_{2} are variable. The variable X^{’}_{2} is the size of a second parallel facility that might be operated in conjunction with the old facility of size *X̅*_{2}. The current inputs *x*_{1} and x^{’}_{1} are, of course, variable in both the old and a potential new facility. Total cost, i.e., , is to be minimized, subject to the above production function, with respect to the choice of x_{1}, x^{’}_{1} and X^{’}_{2}. Necessary conditions can be written as before (Smith 1961), yielding the solution , if it does not pay to introduce a second facility; , if it pays to discard the original facility when the new one is added (replacement); or (3) , if it pays to add a new facility and operate it parallel to the old.

### Dynamics and uncertainty

The theory of cost and production has been extended to treat cases in which demand and, therefore, the output requirements of production are assumed to be (1) given functions of time, or (2) random variables with known probability density functions (Arrow et al. 1958; Modigliani & Hohn 1955; Smith 1961). These problems are often discussed under the title “inventory theory,” since changes in output requirements over time and demand uncertainty are the two conditions giving rise to the necessity of holding product inventories. Under dynamic production requirements, inventories provide a means of smoothing the production plan. If output requirements are known to rise seasonally, the increased marginal costs associated with high output can be avoided by producing in advance for inventory. Optimal smoothing of the production plan requires a balancing of direct production costs against inventory holding costs.

Under uncertain demand requirements, inventories are held as buffer stocks to meet temporary unpredictable increases in demand. Optimal buffer stocks require a balancing of production and salesloss costs (the costs of failing to meet demand) with the costs of storage. [*See*Inventories, *article on*inventory control theory.]

## Aggregate production functions

The production function for particular processes can in principle be, and often is in practice, derived directly from engineering laws and data. Logically, therefore, if the economist has an interest in labor and capital productivity at the industry, regional, or national level, the appropriate aggregate production function could be obtained by aggregating individual process functions. But such a procedure is hopelessly impractical. Instead, economists have postulated the form of the production function and then used aggregative data to estimate its parameters. The analytical forms used have necessarily been restricted by considerations of convenience in estimation. [*See*Production and cost analysis.]

Vernon L. Smith

[*Directly related are the entries*Capital; Economies of scale; Firm, theory of the; Input–output analysis.]

## BIBLIOGRAPHY

*A comprehensive survey of production theory can be found in* Walters 1963, *which contains an extensive bibliography.*

Arrow, Kenneth J.; Karlin, Samuel; and Scarf, Herbert 1958 *Studies in the Mathematical Theory of Inventory and Production.* Stanford Univ. Press.

Chenery, Hollis B. 1949 Engineering Production Functions *Quarterly Journal of Economics* 63: 507 – 531

Clark, John Bates 1889 The Possibility of a Scientific Law of Wages. Volume 4, pages 37–69 in American Economic Association, *Publications.* Baltimore: The Association.

Davidson, Ralph K.; Smith, Vernon L.; and Wiley, Jay W. (1958) 1962 *Economics: An Analytical Approach.* Rev. ed. Homewood, Ill.: Irwin.

Dorfman, Robert; Samuelson, Paul A.; and Solow, Robert M. 1958 *Linear Programming and Economic Analysis.* New York: McGraw-Hill.

Edgeworth, Francis Y. (1891–1921) 1963 *Papers Relating to Political Economy.* 3 vols. New York: Franklin.

Heady, Earl O. 1957 An Econometric Investigation of the Technology of Agricultural Production Functions. *Econometrica* 25:249–268.

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# Production

# Production

MARGINALIST THEORY AND ITS CRITICS

In political economy the term *production* refers not merely to technical processes but also to the motives and the way human actions are organized to bring the output to existence. Consequently, production is a social process, and a theory of production can be linked to a theory of social development and change. One of the characteristics of production is that it takes time. The Austrian capital theory, and specifically the work of Eugen von Böhm-Bawerk (1889), emphasized the time character of production stemming from the division of labor. Karl Marx emphasized the so-called “realization” problem in his critique of the Say’s Law. John Maynard Keynes extended this problem to the role of uncertainty and expectations in monetary production economy.

The idea that production takes time was common for the Physiocrats, but after Adam Smith (1776) put forward the importance of the division of labor for explaining industrial growth, this concept became central for the argument of roundaboutness of production. Increasing division of labor requires more capital goods, and the production of these lengthens the production process. John Rae (1834) argued that increased division of labor goes together with increased durability of capital and hence longer periods of time required for production. Böhm-Bawerk (1889) conceptualized a “production function” in which he made the level of output obtained per unit of capital a function of the degree of roundaboutness of the production method. He argued that more roundabout methods of production are more capital intensive and therefore more productive. Maxine Berg (1980) connected the shift from concern with the division of labor to fixed capital formation to the social conflict in the early 1830s, which was prompted by technological development and by the so-called “machinery question”—the consequences of implementing radically new techniques and forces of production at the beginning of the nineteenth century.

## SCOPE OF PRODUCTION THEORY

With the emergence of the marginalist approach in economics in the nineteenth century, relative prices were put forward as the driving force for change, and the scope of production theory in this tradition became conflated with a theory of exchange. As a contrast, Marx, Thorstein Veblen, Keynes, and Dudley Dillard, among others, focused on “monetary production,” where money is not merely “the great wheel of circulation,” as Adam Smith characterized it.

Marx viewed the relations of exchange as a manifestation of the relations in production, which determine the social, political, and spiritual aspects of life (Marx 1859, p. 100). Although it appears as if exchange, or circulation of commodities, dominates production in capitalist economies, production is the causal force in classical political economy. This is not to say that there is no interaction between exchange and production. As Smith argued in the *Wealth of Nations* (1776), the division of labor, or production, is limited by the extent of the market (or expanding possibilities for exchange). The point is that the consequence of reducing production relations to the exchange and circulation of commodities explains away the contradictions and conflicts of capitalist production. Marx in *Capital*, Volume I, made this argument in his critique of Say’s Law (which holds that supply creates its own demand, and there is no possibility of the economy functioning below full employment), and referred to this tradition in economic analysis as “apologetic economy” (p. 114). Not only was production reduced to relations of circulation in this approach, but also circulation was explained as merely the barter of commodities, which is not characteristic for capitalist economies.

In his 1933 article “Monetary Theory of Production” and in the *General Theory* (1936) Keynes added a distinct component to the circuit theory of capitalist production—production of money by means of money. That component was liquidity preference. Keynes argued that money, with its special properties (low or zero elasticity of production and substitution) is crucial for understanding the changes and direction of output and employment at the macroeconomic level. Thus, he reaffirmed the importance of Marx’s realization problem for understanding a capitalist system.

Keynes recognized that because the decisions to undertake investment had to be made before the results were known, and because the future returns from capital assets would be uncertain, the confidence of expectations in the occurrence of future events becomes important for the occurrence of current investment. As a measure for this confidence Keynes proposed the concept of “liquidity preference” to replace the traditional quantity theory where the rate of interest is determined by “real” factors such as the productivity of capital and thriftiness. As a measure of investors’ confidence in their expectations for the occurrence of future events, liquidity preference determines the price that will be paid to possess money today.

## MONETARY THEORY OF PRODUCTION

In a monetary production economy the “return” from holding money comes from its liquidity premium (Keynes [1936] 1964, p. 227). Each asset has an expected total return (an own-rate of interest) composed of *q* – *c* + *l* + *a*, where *q* is the expected income from employing the asset in production; *c* is the carrying cost; *l* is the liquidity premium; and *a* is the expected capital gains (appreciation or depreciation). Physical capital will have a return comprised mainly of the yield it is expected to generate from employing it in production. Carrying costs are insignificant for liquid assets, but they would be large for physical capital that depreciates over time. The liquidity premium has two roles: first, protection from future uncertain conditions (expressed through increased liquidity preference); and second, opportunity for profiting from future uncertain conditions (expressed through animal spirits). Expectations about the returns from new investment are compared to those of existing capital, financial assets, and money. The interest rate on money competes with the expected return from employing capital in production—that is, with the marginal efficiency of capital. A situation in which the expected returns of assets are equal, so “that there is nothing to choose in the way of advantage between the alternative” (Keynes [1936] 1964, p. 228), is defined in Keynes’s analysis as equilibrium—a state of rest, not market clearing. In equilibrium the interest rate on money would be equal to the marginal efficiency of capital. But Keynes notes that this does not indicate at what level the equality will be effective. The expected return from holding money as a store of value could be in “equilibrium” (state of rest) with the expected returns from all existing assets at an income below full employment.

Because money is something that cannot be produced, and demand for it cannot be readily choked off, Keynes came to the conclusion that unemployment develops “because people want the moon”—the object of their desire, money, cannot be produced (Keynes [1936] 1964, p. 235). Thus, Keynes’s monetary theory of production explains unemployment not merely through a realization problem (as other economists employing the concept of the circuit do) but through the nature of money and its relation to production and distribution. In the tradition of classical political economy, Keynes’s monetary theory of production emphasized conflict. However, the marginalist schools based on “real-wage” systems demonstrated a harmony of cooperative exchange and equal status of the various revenue shares in production. This is due to the construction of functional relations between quantity consumed and utility on the demand side, and between quantity produced and cost on the supply side, and the resulting symmetry between consumption and production in the marginalist theory (Bharadwaj 1984).

The proposition that capitalists and workers have symmetrical roles in production was put forward by means of conceptualizing their payments as remuneration for their “services” and contribution to production. Profits are viewed as symmetrical to wages as remuneration for the sacrifices of capitalists and as reward for their “waiting,” in the same manner that wages reward labor efforts. Consequently, the distinction between wages and profits is blurred as these are merely returns for homogeneous “factors of production.” The revenue shares are not qualitatively different, as the distinction between the various factors is also obscured. Overall, the shift toward the centrality of individuals’ self-interest and its role in determining relative prices of commodities put forward a conception of an inherently just, value-free mechanism of distribution and production decisions.

## MARGINALIST THEORY AND ITS CRITICS

The analytical construct of supply and demand as an explanation of production and distribution was a departure from classical political economy, and it refocused economic theory onto exchange under competition. Increasing returns created problems for the assumption of competition in the neoclassical system, which was pointed out by Piero Sraffa (1926). The possibility for decreasing overhead fixed costs when output increases provides the theoretical possibility for a monopoly, which violates the assumption of competition. The theory of monopolistic competition emerged as an attempt to solve this problem. The theory of diminishing returns is grounded in the supposed technical fact that there is a decreasing productivity of the successive portions of a constant factor of production. The understanding that if more and more inputs are applied to a given piece of land the average output inevitably diminishes has been extended to other factors of production, as well as to the theory of consumption in the form of diminishing marginal utility. The latter is explained by “human nature” rather than technical conditions. Indeed, Sraffa (1926) argued that the same is also valid for diminishing returns, as the producer is the one who ranks the alternative combinations of resources according to their returns. If decreasing returns are not a technical fact, then the producer’s technical choices would not be ranked independently of distribution.

Sraffa critiqued the marginalist theory of production in *Production of Commodities by Means of Commodities* (1960) on the grounds that diminishing returns to a variable factor presupposes a possibility of substitution, which is problematic when there is heterogeneity of inputs that are also produced. Joan Robinson in “The Production Function and the Theory of Capital” (1953) also pointed out that substitution between capital and labor inputs does not take place within the production process in the way that marginalist theory suggests. Capital inputs are not simply added to or subtracted from other inputs used in production following changes of relative prices. In the marginalist tradition, relative prices then are interpreted as indicative of relative scarcities, instead of being linked to the production process (Roncaglia 1978, p. 92).

A production schema depicts the principal flows of produced goods in the technically required sequence. Such schema has a corresponding quantity model that refers to a precise system of production equations where the level of final demand determines the level of output, intermediate inputs, and labor inputs. The production schema, together with the quantity schema and the pricing model—referring to a precise system of pricing equations—form the price-quantity monetary production model of the economy as a whole (Leontief 1951; Lowe 1976; Pasinetti 1977; Lee 1998).

In the marginalist tradition the profit, often conflated with the interest rate, is presented as the price of a particular commodity—capital, which is subject to the functioning of the supply and demand mechanism. Thus, an increase in the price of capital would bring about an increase in the supply, and a decrease in the demand for this “commodity.” The problem that Sraffa identified and Robinson discussed is that this reasoning presupposes a measure of capital that does not depend on the distribution of income between wages and profits. Sraffa (1960) showed that such a measure cannot exist. Consequently, the linking of variations in output to variations in the quantity of capital and labor utilized in production is undermined. Under these circumstances, profit and wage rates cannot be determined on marginalist principles. Consequently, this critique of capital disputes the harmonious vision of production and distribution characteristic of the neoclassical exchange-based approach, and brings back the issue of conflict within the monetary theory of production.

**SEE ALSO** *Capital; Economics, Keynesian; Economics, Neoclassical; Economics, Post Keynesian; Expectations; Keynes, John Maynard; Labor; Marginalism; Marx, Karl; Profitability; Robinson, Joan; Sraffa, Piero; Technological Progress, Economic Growth; Veblen, Thorstein*

## BIBLIOGRAPHY

Berg, Maxine. 1980. *The Machinery Question and the Making of Political Economy, 1815-1848*. Cambridge, U.K.: Cambridge University Press.

Bharadwaj, Krishna. 1986. *Classical Political Economy and Rise to Dominance of Supply and Demand Theories*. Andhra Pradesh: Universities Press, India.

Böhm-Bawerk, Eugen von. [1889] 1891. *The Positive Theory of Capital*. London: Macmillan.

Dillard, Dudley. 1980. A Monetary Theory of Production. *Journal of Economic Issues* 16 (2): 255–275.

Keynes, John Maynard. [1933] 1973. A Monetary Theory of Production. *The Collected Writings*, vol. 7. London and Basingstoke, U.K.: Macmillan.

Keynes, John Maynard. [1936] 1964. *The General Theory of Employment, Interest, and Money*. New York: Harcourt, Brace, Jovanovich.

Lee, Fred. 1998. *Post Keynesian Price Theory*. Cambridge, U.K.: Cambridge University Press.

Leontief, Wassily. 1951. *The Structure of American Economy, 1919–1939*. White Plains, NY: International Arts and Sciences Press.

Lowe, Adolf. 1976. *The Path of Economic Growth*. Cambridge, U.K.: Cambridge University Press.

Marx, Karl. [1859] 1999. *A Contribution to the Critique of Political Economy*, ed. Maurice Dobb. New York: International Publishers.

Marx, Karl. [1867] 1977. *Capital*. New York: Vintage Books.

Pasinetti, Luigi. 1977. *Lectures on the Theory of Production*. New York: Columbia University Press.

Rae, John. 1834. *Statement of Some New Principles on the Subject of Political Economy*. Boston: Hilliard Gray and Company.

Robinson, Joan. 1953. The Production Function and the Theory of Capital. *Review of Economic Studies* 21: 81–106.

Roncaglia, Alessandro. 1978. The Sraffian Contribution. In *A Guide to Post-Keynesian Economics*, ed. Alfred Eichner, 87–99. White Plains, NY: M. E. Sharpe.

Smith, Adam. [1776] 1976. *An Inquiry into the Nature and Causes of the Wealth of Nations*. Oxford: Clarendon Press.

Sraffa, Piero. 1926. Increasing Returns and the Representative Firm. *Economic Journal* 36 (144): 535–550.

Sraffa, Piero. 1960. *Production of Commodities by Means of Commodities*. Cambridge, U.K.: Cambridge University Press.

Veblen, Thorstein. 1904. *The Theory of Business Enterprise*. New York: Scribner’s.

*Zdravka Todorova*

# production

pro·duc·tion
/ prəˈdəkshən; prō-/
•
n.
1.
the action of making or manufacturing from components or raw materials, or the process of being so manufactured:
*the production of chemical weapons*
*it is no longer in production.*
∎
the harvesting or refinement of something natural:

*nonintensive methods of food production.*∎ the total amount of something that is manufactured, harvested, or refined:

*steel production had peaked in 1974.*∎ the creation or formation of something as part of a physical, biological, or chemical process:

*excess production of collagen by the liver.*∎ [as adj.] denoting a car or other vehicle that has been manufactured in large numbers. 2. the process of or financial and administrative management involved in making a movie, play, or record:

*the movie was still*| [as adj.]

**in production***a production company.*∎ a movie, play, or record, esp. when viewed in terms of its making or staging:

*this production updates the play and sets it in the sixties.*∎ [in sing.] the overall sound of a musical recording; the way a record is produced:

*the record's production is gloriously relaxed.*PHRASES: make a production of do (something) in an unnecessarily elaborate or complicated way.

# Production

# PRODUCTION

Until the transformation of the constitutional law of economic regulation, beginning in 1937, "production" described economic activities that the Supreme Court regarded as local or intrastate in character and therefore beyond Congress's power to regulate under the commerce clause. In 1895 the Court ruled in united states v. e. c. knight co. that every form of production and matters related to it were stages of economic activity that preceded the buying, selling, and transportation of goods among the states. Manufacturing, mining, agriculture, domestic fisheries, stock raising, and labor had only an "indirect" effect upon commerce, by judicial definition. Because commerce came after production the United States had no constitutional authority to extend the sherman antitrust act to monopolies in production, nor could it control the trade practices of poultry dealers, or regulate agricultural production, or fix the maximum hours and minimum wages of miners. In united states v. darby (1941) the Court sustained the constitutionality of the fair labor standards act, which applied to workers engaged in production of goods for sale in interstate commerce, in the next year the Court in wickard v. filburn (1942) ended any remaining vestiges of the doctrine that Congress could not regulate production. The Court ruled that, although certain economic activities are local or intrastate, the commerce clause extends Congress's power to them if they affect commerce, making their regulation an appropriate way of governing commerce among the states.

Leonard W. Levy

(1986)

# production

**production** **1.** The total mass of organic matter that is manufactured in an ecosystem during a certain period of time. Net production is the yield of the producers and consumers and is the amount of living matter in the ecosystem.**2.** In energy-flow studies, that part of the assimilated food or energy which is retained and incorporated in the biomass of the organism, but excluding the reproductive bodies released by the organism. This may be regarded as growth. In energy-flow measurements, production is expressed as energy per unit time, per unit area.**3.** See primary productivity.

# production

**production** The transformation of resources, which include time and effort, into goods and services. Resources are understood always to be too scarce to provide for all wants and needs, so there is an emphasis on the efficiency of production, or productivity. Similarly, the costs of choosing certain goods and services are not measured by the money spent on them, but by the opportunity cost of the alternative uses of the available resources. See also GROSS NATIONAL PRODUCT.

# production

**production** **1.** In energy-flow studies, that part of the assimilated food or energy which is retained and incorporated in the biomass of the organism, but excluding the reproductive bodies released by the organism. This may be regarded as growth. In energy-flow measurements, production is expressed as energy per unit time, per unit area.

**2.** See PRIMARY PRODUCTIVITY.

# production

**production** The total mass of organic food that is manufactured in an ecosystem during a certain period of time. It is the net yield of the producers and consumers and determines the amount of living matter in the ecosystem.

# production

**production** In economics, methods by which wealth is produced. It is one of the basic principles of economics. The factors of production are land, labour and capital. Land in this sense includes natural resources, such as minerals, and the wealth of the sea, such as fish.

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**productivity**