The act of production is the transformation of raw substances (including human labor itself) into other forms that are distinguished by their more desirable functional, locational, or temporal characteristics. Production of tangible goods encompasses gathering, extracting, refining, combining, assembling, packaging, transporting, and distributing activities. Production of services includes performance activities as well.
Efficiency is the name of the economizing game. Efficiency is served in either of two ways: by producing a larger volume of output from a given amount of inputs, or by employing smaller amounts of the inputs in the production of a target output. Experience has demonstrated that in both cases organized human effort is typically more efficient than is isolated effort.
The essential entrepreneurial functions are the perception of an unfilled market demand and the assumption of risk in organizing a productive process to exploit the market potential. The three crucial managerial problems are (1) to select an appropriate technology for implementation of production, (2) to discern the right volume of output to meet the market demand, and (3) to choose the optimal combination of inputs to produce the target level of output. Associated with these three broad problem areas are a myriad of detailed functions ranging from choosing the site for the production facility, procuring inputs and scheduling production, and packaging and distributing the final product.
Real-world production processes may be quite detailed and complex. Our task in this chapter shall be to discern general production decision criteria that can be transferred to specific production situations. What we are about is learning how to think about production problems rather than what to do in specific situations.
a. What is an appropriate technology for producing the desired output?The first two are entrepreneurial decisions; the last two are managerial in nature.
b. What size plant should be constructed to implement the selected technology?
c. At what volume of output (or rate of production) should the constructed plant be operated?
d. What are the appropriate quantities of input to combine to produce the target output.
These fundamental production questions will not always be addressed in the same sequence. Some of them must be confronted simultaneously. The perceived market demand ultimately constrains the answer to the volume-of-output question and suggests a response to the size-of-plant question. The target production volume may then limit the eligible range of technologies. We shall defer consideration of the volume-of-output question to Chapters 9 and 10. Assuming that the target volume is known, we shall address the technology question later in this chapter. Our immediate task is to confront question (d), the appropriate quantities of inputs to be combined to produce output.
then a generalized production function can be represented as,
where Q is the volume of output.
But equation (1) is an incomplete specification of the production function. There are two possible ways to complete it. One is to further specify additional aspects of production that are represented as implicit inputs:
Here the symbols T, E, and M stand for technology, entrepreneurship, and managerial capacity. They are grouped together and separated from the list of physical inputs by the semicolon because they are not per se physical inputs. Rather, they both enable and constrain the combination of the physical inputs in the production of outputs.
In recognition that T, E, and M are not physical inputs, some analysts prefer the following representation of the production function:
This representation clearly indicates the relationship between physical inputs and output in equation (3), but signifies with equation (4) that the production function itself is a function of other conditions, i.e., technology, entrepreneurship, and managerial capacity.
While these are abstract production function representations, a production function that is specific to a particular production process might include in its input listing various types of labor, materials, and capital. An example of a production function to which almost anyone can relate is any kitchen recipe for the preparation of an edible dish. Boiled down to essentials, a production function is nothing more than a list of ingredients together with instructions for combining them in the preparation of a desired output.
where the slash is the conventional indication that all items in the input list appearing to the right of it (in this case, "all other variables") are assumed constant.
Four hypothesized shapes for the three-dimensional production surfaces are illustrated in Figure 9-1. The four panels show, respectively, the following output patterns as the quantities of the inputs are increased:
(a) output increases at an increasing rate;There are of course other possible patterns that might be imagined, including an absolute decrease of output consequent upon employing more of the inputs. The output-decrease possibility is a logical extension of patterns (c) and (d) for excessively large increments to the inputs.
(b) output increases at a constant rate;
(c) output increases at a decreasing rate; and
(d) output increases at varying rates in the sequence of increasing, constant, and decreasing rates of increase.
Not all of these hypothesized shapes are plausible representations of real-world production relationships. Panels (a) and (b) illustrate two patterns that are thought to be inconsistent with observed production phenomena over any but the shortest ranges of the inputs. It is thought that most real-world production processes exhibit the behaviors represented by panel (c), or possibly the generalized shape of panel (d) which incorporates over limited input ranges all three of the other hypothesized shapes.
In the analysis of how input variation affects output, given a selected technology, economists distinguish three situations: (a) a single input is changed vis-a-vis fixed quantities of all other inputs; (b) all inputs are changed (positively or negatively) by the same proportion (greater or lesser than 100 percent); or (c) inputs are changed in varying proportions vis-a-vis each other. The first case describes the analysis of returns a variable input; the second describes returns to scale; the third describes the general situation, variable proportions, inferences about which may be drawn from an analysis of the first two situations. We shall defer consideration of returns to scale and variable proportions to a later section of this chapter.
The realm of returns to a variable input permits us to distinguish the short run from the long run. In the long run, all inputs are presumed to be variable. The analysis of returns to scale thus belongs to the long-run. A change of a single input, given fixed quantities of other inputs, is then clearly an analysis of the short run. The short run can be described as the period of time during which at least one of the inputs cannot be changed. The duration of the short run is until the yet-unchanged input can be changed. In the real world, production decision makers may plan for the long-run changes that they intend to make, but all decisions are made in short-run settings, even the decisions to make long-run changes. In this sense, then, the freedom of the decision maker to vary inputs is constrained by the temporal setting.
It is tempting to identify capital as the input class that typically is fixed in the short run, but we must recognize that this concept is not descriptive of all real-world situations. An example of this caveat consists in the family-owned business (a farm or a commercial establishment) where the labor force is the fixed input (mom, pop, children, cousins, etc.). The relevant input question in the short run is how much land or capital equipment to use (rent, buy), not how much labor to employ.
We now have in place all of the conceptual tools so that we can begin the analysis of production in the short run. Our objective is to identify the relevant criteria that may be used to guide production decision making by rational and perceptive production decision makers. Because of its behavioral inclusiveness, we shall adopt the generalized shape of the production surface illustrated in Figure 9-1, panel (d). It is reproduced in an enlarged format in Figure 9-2. We should note two essential caveats before proceeding with the analysis. First, many real-world production processes may satisfactorily be modeled with the simpler linear or second-order shapes illustrated in panels (b) and (c) of Figure 9-1. Second, the smooth, continuous surface illustrated in Figure 9-2 is only an heroic representation of what certainly are discontinuous real-world relationships. In fact, no more than a few points on or near such a surface may be observable for any real-world production process.
The problem for the production decision maker is to choose an appropriate amount of labor to employ with capital input K1. The analysis may be conducted by assuming alternate labor-employment decisions that follow the path across the floor of the diagram from L1 through L2 and L3. Theoretically, any other quantities of labor along this path might have been chosen; these are simply a few representative quantities. But, the real-world production process might be characterized by a few, discrete labor-quantity choices, such as L1 or L3.
As the labor employed with capital K1 is increased from L1 toward L4, output increases along the path on the surface from Q1 to Q2, Q3, and Q4. Given the adopted shape of this production surface, it is apparent that over the labor input range from L1 to L2, output increases at an increasing rate (the surface is concave upward) from Q1 to Q2. Point Q2 in the surface path is near what mathematicians would call the inflection point, i.e., where a curve changes concavity, in this case from being concave upward to being concave downward. As the labor input is further increased from L2 to L3, output continues to increase to Q3, but at a decreasing rate of increase. Further increases of the labor input from L3 to L4 yield additional output, also at a decreasing rate over the Q3 to Q4 range. It should be clear that the DQ3 output increment is smaller than the DQ2 output increment. This phenomenon of output increasing at a decreasing rate continues some beyond Q4, and until the output path peaks around Q5 and turns downward.
The labor input range from K1 to L2 is described by economists as the increasing returns range. It is thought to be an early or temporary phenomenon in the production process, and may not be observable in most real-world production situations. This range is missing entirely in the surface illustrated in panel (c) of Figure 9-1.
The principle of diminishing returns is thought to govern all real-world production processes. Diminishing returns may not be evident in the very early stages of production characterized by low levels of labor employment, but it becomes obvious as progressively more labor is employed. It is simply implausible to believe and unreasonable to expect that output can continue to increase at increasing or even constant rates forever as the labor input is progressively increased vis-a-vis a given plant size. This physical relationship was recognized earliest in agricultural settings and subsequently in engineering situations. It has been adopted by economists as the fundamental behavioral premise in the explanation of input-output relationships. Although diminishing returns are rarely subject to direct examination or empirical testing, the essential truth of the principle may be verified by the logical process of reduction to absurdity.
As an example, consider a typical peasant farm in South Asia, perhaps 15 acres in size, equipped with a fixed amount of capital equipment including a yoke of oxen and a wooden plow with metal tip, and perhaps two or three other digging or cultivating implements. The peasant farmer by himself can exact some volume of agricultural production from the 15 acres. It is likely that the farmer and his son, working together, can produce more than twice what the farmer alone could produce (the range of increasing returns). As successive additional workers (usually family members) are employed on the farm, output can be expected to continue to increase, but eventually at a decreasing rate, and ultimately to actually decrease if too much labor is employed. In case the reader is skeptical of this conclusion, we invite him to think about the possibilities of employing 5 workers on the 15 acres, then 10 workers, 15, 20, 50, 100, 1000, 1 million workers on 15 acres. Is there any doubt that the principle of diminishing returns has to be true and applicable (at least, eventually) to every real-world production process?
Employing analytical techniques first noted in Chapter 6, we can now trace out in panel (b) of Figure 9-3 the average product (AP) and marginal product (MP) curves that correspond to the TP curve in panel (a). The average product of labor may be defined and computed as the amount of output, Q, divided by the quantity of labor, L, employed in its production, given all other inputs, i.e.,
for the ith amount of labor employed. For example, the average product of the L2 volume of labor employed is Q2/L2. With this concept in mind, we should be able to discern the behavior of the labor AP curve by observing the slopes of rays drawn from the origin to successive points on the TP curve. This is so because the slope of the ray drawn to a point on the TP curve is the hypotenuse of a right triangle formed by the horizontal axis and a vertical erected from the labor quantity point on the axis to the TP curve. Then, the trigonometric tangent of the angle so formed is the ratio of the opposite to the adjacent sides of the triangle, e.g., Q2/L2, which we have already defined as the average product of the L2 quantity of labor. In panel (a) of Figure 9-3, rays to the successive points along the TP curve have progressively steeper slopes until Q3 is reached, beyond which the rays become shallower of slope. Thus we can draw the AP curve in panel (b) as rising from the origin to a peak at the L3 quantity of labor, beyond which it falls back toward the horizontal axis. The vertical axis units in panel (b) have been expanded relative to those in panel (a) so that the behavior of AP can be made quite obvious. The relationships among the computed average products for the points along the TP curve illustrated in Figure 9-3 are:
The average product of the variable input is relatively easy to measure; the only information required is the amount of output and the corresponding quantity of the input required to produce the output. Because it is easy to measure, the AP of the variable input is a tempting criterion for production decision making. However, economists usually reject it in favor of the MP of the variable input. The average product of labor does find usefulness in aggregate production settings where it is commonly referred to as the output per capita of the labor force. The downward-sloping range of the aggregate AP of labor curve has been compared to the hypothesized subsistence level of income in neo-Malthusian studies.
The marginal product of labor (MPL) may be defined as the ratio of an increment of output (DQ) divided by the smallest possible increment of labor (DL), i.e.,
This definition should be suggestive of an application of the calculus: the first derivative of the TP function with respect to the labor input, i.e., dQ/dL, may be used as a measure of MPL. If there are other inputs present, a partial derivative must be computed. The MPL measures the rate of change of TP, and can be illustrated graphically as the slope of a tangent to the TP curve at a selected point.
This concept provides the means for discerning the behavior of the MPL. Tangents have been drawn to each of the representative points on the TP curve illustrated in panel (a) of Figure 9-3. As the labor input is increased, and output consequently increased, the slopes of the tangents become progressively steeper. The maximum steepness is reached at point Q2, beyond which they become shallower until a zero slope is reached at Q5. Beyond Q5 the slopes of tangents to points on the TP curve are negative. Thus, we draw the MP of labor curve as rising from the origin to a peak at the L2 level of labor input, then falling until it is zero at the L5 level of labor input, beyond which it is below the horizontal axis.
Since the APL and the MPL curves are superimposed in panel (b) of Figure 9-3, we may note the corresponding behaviors of the two curves. Over the initial range of labor input, MPL rises much faster than does APL. For example, at Q1 the slope of the tangent is steeper than the slope of the ray to Q1. MPL remains greater than APL even after the peak of the MPL curve is reached. MPL decreases and passes through the peak of the APL curve at the labor input level L3. Here, the tangent to the TP curve at Q3 is coincidental with a ray from the origin to Q3. For all labor input levels greater than L3, MPL is both decreasing and less than APL. The slope of the tangent at Q4 is shallower than the slope of the ray drawn to Q4.
For reasons that shall become apparent in subsequent discussion, economists advocate the use of MP as a production decision criterion. However, the MP is a more abstract concept than is the AP, and the true marginal product of a variable input is substantially more difficult to measure and compute than is the average product. In order to compute the marginal product, the equation of the TP function must be known or estimated before the derivative can be computed. The equation of a function can be guessed at, "eyeball" fashion, but the generally accepted means of equation estimation is statistical regression analysis of historical data for the Q and L variables. The data for the regression analysis may have been generated by experimental procedures or captured by observing natural production processes in operation. These procedures are troublesome, time-consuming, and costly. It is no wonder that real-world production decision makers have an aversion to using the MP as a production decision criterion.
Economists make a distinction between the marginal product and the incremental product (IP) of a variable input. While the true MP may be measured as the limit of the ratio DQ/DL as DL approaches zero, the incremental product may be measured as the ratio, DQ/DL, for any measurable DL. With reference to Figure C3-3, panel (a), the incremental product of labor over the L2 to L3 input range can be computed as
In this sense, the incremental product may be measured as the slope of a chord connecting any two points on the TP curve. This is so because the chord forms the hypotenuse of a right triangle drawn to the two points.
Although the true marginal product of a variable input is troublesome to compute, the incremental product is relatively easy to compute. The only data needed are the quantities of output and labor input for the two points on the TP curve. It is important to note two caveats in this regard. First, two observed production points may not be on the same TP curve if one or more of the other (than L) determinants of output have changed (i.e., there may be an identification problem). If the two points happen to be on different TP curves, the computed IP ratio will over- or understate the IP that might have been computed for points on the same TP curve. Second, even if the two observed points are on the same TP curve, the computed IP will over- or understate the true MP computed by differentiation for a point at either end of the chord connecting the two points.
Where does this leave us? MP is the ideal production decision criterion (we shall substantiate this point shortly), but it is troublesome to compute. IP is in practical terms both measurable and computable with relative ease, but it is likely to over- or understate the true MP. The production decision maker is urged to go to the trouble to compute MP if it is not too costly to do so; otherwise, the IP may be computed and used as a decision criterion, subject to the recognition that it is only an approximation to the true MP of the variable input.
Any point like Q2 on the production surface lies on both a TP curve for labor as variable input and a TP curve for capital as variable input. Thus we have drawn line segments cross-wise to the path of the slice at each of the selected points along the labor-variable TP curve. These cross-wise line segments represent tangents to the surface, the slopes of which measure the marginal products of capital. Even though the quantity of capital does not change, we should be able to compute for every different amount of labor employed both the MP of labor and the MP of capital.
How do the marginal products of labor and capital vary with respect to each other? Figure 9-6 illustrates a TP curve for labor as a variable input, assuming one unit of installed capital. In this illustration, the TP curve is carried to the point where so much labor has been employed that output has fallen to zero. For purpose of illustration, we assume that the horizontal axis between the origin and the point at which TP returns to zero can be divided into ten equal labor units. In panel (b) of Figure 9-6 we represent the explicit labor-variable MP and AP curves that correspond to the TP curve, and further plot selected points along implicit MP and AP curves for capital as if it were the variable input. Along the horizontal scale we note that even though the labor input increases explicitly from zero to ten units, the capital input remains constant at one unit.
In Figure 9-6 we may examine the results of explicitly decreasing the labor input from ten toward zero units. When this happens, the quantity of capital employed per unit of labor implicitly increases from 1/10 of a unit to 1/9 of a unit, then to 1/8, 1/7, and so on. Conversely, when the quantity of labor is explicitly increased from one unit to two units, the quantity of capital per unit of labor is implicitly decreased from 1/1 unit to 1/2 unit. Recognition that the relative quantity of each fixed input does implicitly change consequent upon an explicit change of a variable input is essential to an understanding of how the marginal products change vis-a-vis each other.
As may be seen in panel (b) of Figure 9-6, over the initial range of labor input, the MP of labor increases to a peak, then begins to decrease; but the MP of capital is negative. Economists identify this range as the Stage I of production for labor, but the Stage III of production for capital. Likewise, moving from right-to-left as the quantity of capital is implicitly increased, the MP of capital rises to a peak and begins to decrease; but the MP of labor is negative. Economists identify this range as the Stage I of production for capital, but the Stage III of production for labor. A rational and perceptive production decision maker would not choose to operate in either Stage I or Stage III. In labor's Stage I (capital's Stage III), labor is underutilized while capital is overutilized (implied by the negative marginal productivity of capital). In labor's Stage III (capital's Stage I), labor is overutilized (implied by the negative marginal productivity of capital) while capital is underutilized.
We may thus identify Stage II (common to both inputs) as the relevant range of production. Within Stage II the marginal products of both inputs are positive, although they vary in opposite directions. The boundaries for Stage II for each input are found at the point of zero MP for the other input, which coincidentally correspond to the intersections of the MP and AP curves for each of the inputs.
Some very important behavioral relationships may now be noted. Within production Stage II, as either input is increased in quantity, its MP decreases, but the MP of the other input (which explicitly does not vary) increases. Conversely, if the quantity of either input is decreased, its MP will increase, but the MP of its complement(s) may be expected to decrease. Knowledge of this relationship enables us to answer a very important question for the production manager: how can the (marginal) productivity of labor be increased? Two answers are possible: either by decreasing the labor input, or by providing labor with more capital equipment. The former response is typically regarded as a short-run adjustment, the latter as a long-run change.
An alternate version of this relationship may be expressed as:
This very important relationship is known as the Equimarginal Principle. Stated as an operational decision criterion, the optimal combination of labor and capital is the one for which the marginal product per dollar's worth of labor is equal to the marginal product per dollar's worth of capital. But this is really not a decision criterion; rather, it is an equilibrium condition. The practical decision criterion might better be stated:
Following this decision criterion, if the MP per dollar's worth of capital is greater than the MP per dollar's worth of labor, the production decision maker should employ more capital and/or less labor. By so doing, the MP of capital will fall and the MP of labor will rise, thereby tending to bring about an equality of the marginal product per dollar's worth of the two inputs. We leave it to the reader to explore alternate possibilities. In Figure 9-6, if the price of a unit of capital is twice the price of a unit of labor, then approximately 5.2 units of labor should be employed to operate the one unit of capital since at 5.2 units of labor the MP of capital is approximately twice the MP of labor.
A variant of the Equimarginal Principle can be specified to enable discovery of the input combination that maximizes profits. To do so, we must define a new concept, marginal revenue product (MRP). The MRP of a resource is the addition to the firm's total revenue due to selling the additional output resulting from using one more unit of the resource. In a purely competitive context (as defined in Chapter 12), MRP can be computed as the marginal product of the resource multiplied by (evaluated at) the price per unit of the product (i.e., MRPR=MPR x P). In an imperfectly competitive situation, because the firm's product demand curve slopes downward from left to right, the MRP is computed as the marginal product of the resource multiplied by the marginal revenue resulting from selling the additional product, or
We note in Chapter 12 that MR=P in the purely competitive context, so specification (13) is good for both the purely competitive and imperfectly competitive contexts. The schedule of MRP values for different quantities of the resource used by a firm constitutes its demand curve for the resource, DR=MRPR, as illustrated in Figure 9-7.
If the firm purchases the resource in a competitive market, the resource price, PR, is the firm's marginal factor cost (MFC); the firm's MFCR curve would be horizontal and coincident with its resource supply curve at the level of PR. But if the resource is bought in an imperfectly competitive market, the MFC of employing an additional unit of the resource is greater than its price. This is so because the firm has to offer successively higher prices to purchase larger quantities of the resource, and the higher prices normally apply to all units of the resource purchased, not just the additional units. In an imperfectly competitive resource market the resource supply curve slopes upward. The marginal factor cost schedule also slopes upward and lies above the resource supply curve as illustrated in Figure 9-7.
These resource market relationships enable specification of the criterion for finding the resource utilization level for any single resource that maximizes the firm's profits, given all other resources:
At any resource utilization level for which MRPR > MFCR, e.g., Q1, the firm can by using one more unit of the resource add more to its total revenue when the additional output is sold than will be added to its total costs by employing the resource unit. Or, if MFCR > MRPR, e.g., at Q2, the firm should reduce its utilization of that resource. This will decrease it total revenues and total costs, but total costs will fall by more than total revenue will decrease. The firm should stop making adjustments to its level of utilization of the resource as soon as there is nothing more to be gained, e.g., at Q3, where
When this equality is attained, profit is maximized. This relationship can be rearranged as
If any inequality emerges anywhere in this string of relationships, an adjustment to the resource input combination needs to be made. Such an inequality may emerge as technologies, product prices, or resource supply conditions vary. For example, suppose that one of the resources, L, becomes more abundant and its price to the firm falls. This will decrease MFCL, the denominator of the first ratio, thereby causing the value of the ratio to increase. In this scenario the firm should adjust its input combination to use more of the resource L, or less or of other resources. Using more of L will have the dual effect of decreasing its marginal productivity and bidding its market price up. Both effects will cause the ratio MRPL/ MFCL to fall until the string of equalities is reestablished.
Two points need reiteration. First, the resource input combination resulting in the equalities of MRP/MFC across all of the inputs and equal to unity (1) will be the profit maximizing resource input combination. Second, the profit maximizing input combination found by equating the MRP/MFC ratios will not necessarily (in fact, not likely) coincide with the cost minimizing combination of resource inputs that is found by equating the MP/P ratios across all inputs.
We shall examine a generalized third-order equation of a production function with a single variable input, L, assuming all other inputs constant,
a graphic display of which is illustrated in the top panel of Figure 9-8. In this representation, the power 3 in the third term on the right side of this function makes it a third-order equation. The constant in this equation is missing, implicitly having a value of zero, so that the graphic depiction of the production function passes through the origin. The signs on the coefficients of L2 and L3 indicate that as L increases from the origin, Q at first increases at an increasing rate, then eventually increases at a decreasing rate (the range of diminishing returns) until Q reaches a maximum value, beyond which Q decreases. The average and marginal functions to the Q function are displayed in the bottom panel of Figure 9-8.
Suppose that the value of the coefficient of the L3 term decreases from -0.05 to -0.045 so that the equation of the production function becomes
Figure 9-9 illustrates graphic depictions of the total, average, and marginal functions. It is apparent that all of the curves have shifted upward and outward from the origin consequent upon the change in the coefficient of the L3 term.
As we have noted in Chapters 8 and 9, the loci of the firm's production function curves may change either because wear or weathering (i.e., depreciation) results in capital consumption, or because the management of the firm implements changes in the technologies employed in the firm's production processes. Capital consumption may be expected to shift the product curves downward and to the right. If the technology changes are output-increasing, they will shift the product curves upward. If they are input-saving they will shift the product curves to the left; input-using changes will shift the product curves to the right.
The management of an organization may gather output data via its production and inventory accounting systems; its research staff may perform regression analyses upon the data to estimate the parameters of its production functions. With this information in hand, the management of the firm may employ the SIMMOD applet to model the equations of its total, average, and marginal product functions. When any of these functions change either by deliberate actions of the management or due to matters beyond the control of the management, it may analyze the effects of such changes in SIMMOD by making appropriate changes to the respective functions as illustrated above.
Readers may access the SIMMOD Java applet at http://www.dickstanford.com/SIMMOD/Simmod3g.html to experiment with parameter changes.