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Fundamentals of Human Action

Mises Daily: Tuesday, February 21, 2012 by

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The Law of Returns

We have concluded that the value of each unit of any good is equal to its marginal utility at any point in time, and that this value is determined by the relation between the actor's scale of wants and the stock of goods available. We know that there are two types of goods: consumers' goods, which directly serve human wants, and producers' goods, which aid in the process of production eventually to produce consumers' goods. It is clear that the utility of a consumers' good is the end directly served. The utility of a producers' good is its contribution in producing consumers' goods. With value imputed backward from ends to consumers' goods through the various orders of producers' goods, the utility of any producers' good is its contribution to its product — the lower-stage producers' good or the consumers' good.

As has been discussed above, the very fact of the necessity of producing consumers' goods implies a scarcity of factors of production. If factors of production at each stage were not scarce, then there would be unlimited quantities available of factors of the next lower stage. Similarly, it was concluded that at each stage of production, the product must be produced by more than one scarce higher-order factor of production. If only one factor were necessary for the process, then the process itself would not be necessary, and consumers' goods would be available in unlimited abundance. Thus, at each stage of production, the produced goods must have been produced with the aid of more than one factor. These factors cooperate in the production process and are termed complementary factors.

Factors of production are available as units of a homogeneous supply, just as are consumers' goods. On what principles will an actor evaluate a unit of a factor of production? He will evaluate a unit of supply on the basis of the least importantly valued product which he would have to forgo were he deprived of the unit factor. In other words, he will evaluate each unit of a factor as equal to the satisfactions provided by its marginal unit — in this case, the utility of its marginal product. The marginal product is the product forgone by a loss of the marginal unit, and its value is determined either by its marginal product in the next stage of production, or, if it is a consumers' good, by the utility of the end it satisfies. Thus, the value assigned to a unit of a factor of production is equal to the value of its marginal product, or its marginal productivity.

Since man wishes to satisfy as many of his ends as possible, and in the shortest possible time, it follows that he will strive for the maximum product from given units of factors at each stage of production. As long as the goods are composed of homogeneous units, their quantity can be measured in terms of these units, and the actor can know when they are in greater or lesser supply. Thus, whereas value and utility cannot be measured or subject to addition, subtraction, etc., quantities of homogeneous units of a supply can be measured. A man knows how many horses or cows he has, and he knows that four horses are twice the quantity of two horses.

Assume that a product P (which can be a producers' good or a consumers' good) is produced by three complementary factors, X, Y, and Z. These are all higher-order producers' goods. Since supplies of goods are quantitatively definable, and since in nature quantitative causes lead to quantitatively observable effects, we are always in a position to say that: a quantities of X, combined with b quantities of Y, and c quantities of Z, lead to p quantities of the product P.

Now let us assume that we hold the quantitative amounts b and c unchanged. The amounts a and therefore p are free to vary. The value of a yielding the maximum p/a, i.e., the maximum average return of product to the facto, is called the optimum amount of X. The law of returns states that with the quantity of complementary factors held constant, there always exists some optimum amount of the varying factor. As the amount of the varying factor decreases or increases from the optimum, p/a, the average unit product declines. The quantitative extent of that decline depends on the concrete conditions of each case. As the supply of the varying factor increases, just below this optimum, the average return of product to the varying factor is increasing; after the optimum it is decreasing. These may be called states of increasing returns and decreasing returns to the factor, with the maximum return at the optimum point.

The law that such an optimum must exist can be proved by contemplating the implications of the contrary. If there were no optimum, the average product would increase indefinitely as the quantity of the factor X increased. (It could not increase indefinitely as the quantity decreases, since the product will be zero when the quantity of the factor is zero.) But if p/a can always be increased merely by increasing a, this means that any desired quantity of P could be secured by merely increasing the supply of X. This would mean that the proportionate supply of factors Y and Z can be ever so small; any decrease in their supply can always be compensated to increase production by increasing the supply of X. This would signify that factor X is perfectly substitutable for factors Y and Z and that the scarcity of the latter factors would not be a matter of concern to the actor so long as factor X was available in abundance. But a lack of concern for their scarcity means that Y and Z would no longer be scarce factors. Only one scarce factor, X, would remain. But we have seen that there must be more than one factor at each stage of production. Accordingly, the very existence of various factors of production implies that the average return of product to each factor must have some maximum, or optimum, value.

In some cases, the optimum amount of a factor may be the only amount that can effectively cooperate in the production process. Thus, by a known chemical formula, it may require precisely two parts of hydrogen and one part of oxygen to produce one unit of water. If the supply of oxygen is fixed at one unit, then any supply of hydrogen under two parts will produce no product at all, and all parts beyond two of hydrogen will be quite useless. Not only will the combination of two hydrogen and one oxygen be the optimum combination, but it will be the only amount of hydrogen that will be at all useful in the production process.

The relationship between average product and marginal product to a varying factor may be seen in the hypothetical example illustrated in table 1. Here is a hypothetical picture of the returns to a varying factor, with other factors fixed. The average unit product increases until it reaches a peak of eight at five units of X. This is the optimum point for the varying factor. The marginal product is the increase in total product provided by the marginal unit. At any given supply of units of factor X, a loss of one unit will entail a loss of total product equal to the marginal product.

Table 1
FACTOR
Y
b UNITS
FACTOR
X
a UNITS
TOTAL
PRODUCT
p UNITS
AVERAGE
UNIT
PRODUCT
p/a
MARGINAL
PRODUCT
Δp/Δa
3
3
3
3
3
3
3
3
0
1
2
3
4
5
6
7
0
4
10
18
30
40
45
49
0
4
5
6
7.5
8
7.5
7

4
6
8
12
10
5
4

Thus, if the supply of X is increased from three units to four units, total product is increased from 18 to 30 units, and this increase is the marginal product of X with a supply of four units. Similarly, if the supply is cut from four units to three units, the total product must be cut from 30 to 18 units, and thus the marginal product is 12.

It is evident that the amount of X that will yield the optimum of average product is not necessarily the amount that maximizes the marginal product of the factor. Often the marginal product reaches its peak before the average product. The relationship that always holds mathematically between the average and the marginal product of a factor is that as the average product increases (increasing returns), the marginal product is greater than the average product. Conversely, as the average product declines (diminishing returns), the marginal product is less than the average product.[1]

It follows that when the average product is at a maximum, it equals the marginal product.

It is clear that, with one varying factor, it is easy for the actor to set the proportion of factors to yield the optimum return for the factor. But how can the actor set an optimum combination of factors if all of them can be varied in their supply? If one combination of quantities of X, Y, and Z yields an optimum return for X, and another combination yields an optimum return for Y, etc., how is the actor to determine which combination to choose? Since he cannot quantitatively compare units of X with units of Y or Z, how can he determine the optimum proportion of factors? This is a fundamental problem for human action, and its methods of solution will be treated in subsequent chapters.

Convertibility and Valuation

Factors of production are valued in accordance with their anticipated contribution in the eventual production of consumers' goods. Factors, however, differ in the degree of their specifity, i.e., the variety of consumers' goods in the production of which they can be of service. Certain goods are completely specific — are useful in producing only one consumers' good. Thus, when, in past ages, extracts from the mandrake weed were considered useful in healing ills, the mandrake weed was a completely specific factor of production — it was useful purely for this purpose. When the ideas of people changed, and the mandrake was considered worthless, the weed lost its value completely. Other producers' goods may be relatively nonspecific and capable of being used in a wide variety of employments. They could never be perfectly nonspecific — equally useful in all production of consumers' goods — for in that case they would be general conditions of welfare available in unlimited abundance for all purposes. There would be no need to economize them. Scarce factors, however, including the relatively nonspecific ones, must be employed in their most urgent uses. Just as a supply of consumers' goods will go first toward satisfying the most urgent wants, then to the next most urgent wants, etc., so a supply of factors will be allocated by actors first to the most urgent uses in producing consumers' goods, then to the next most urgent uses, etc. The loss of a unit of a supply of a factor will entail the loss of the least urgent of the presently satisfied uses.

[product:10535]

The less specific a factor is, the more convertible it is from one use to another. The mandrake weed lost its value because it could not be converted to other uses. Factors such as iron or wood, however, are convertible into a wide variety of uses. If one type of consumers' good falls into disuse, iron output can be shifted from that to another line of production. On the other hand, once the iron ore has been transformed into a machine, it becomes less easily convertible and often completely specific to the product. When factors lose a large part of their value as a result of a decline in the value of the consumers' good, they will, if possible, be converted to another use of greater value. If, despite the decline in the value of the product, there is no better use to which the factor can be converted, it will stay in that line of product or cease being used altogether if the consumers' good no longer has value.

For example, suppose that cigars suddenly lose their value as consumers' goods; they are no longer desired. Those cigar machines which are not usable in any other capacity will become, valueless. Tobacco leaves, however, will lose some of their value, but may be convertible to uses such as cigarette production with little loss of value. (A loss of all desire for tobacco, however, will result in a far wider loss in the value of the factors, although part of the land may be salvaged by shifting from tobacco to the production of cotton.)

Suppose, on the other hand, that some time after cigars lose their value this commodity returns to public favor and regains its former value. The cigar machines, which had been rendered valueless, now recoup their great loss in value. On the other hand, the tobacco leaves, land, etc., which had shifted from cigars to other uses will reshift into the production of cigars. These factors will gain in value, but their gain, as was their previous loss, will be less than the gain of the completely specific factor. These are examples of a general law that a change in the value of the product causes a greater change in the value of the specific factors than in that of the relatively nonspecific factors.

To further illustrate the relation between convertibility and valuation, let us assume that complementary factors 10X, 5Y, and 8Z produce a supply of 20P. First, suppose that each of these factors is completely specific and that none of the supply of the factors can be replaced by other units. Then, if the supply of one of the factors is lost (say 10X), the entire product is lost, and the other factors become valueless. In that case, the supply of that factor which must be given up or lost equals in value the value of the entire product — 20P, while the other factors have a zero value. An example of production with purely specific factors is a pair of shoes; the prospect of a loss of one shoe is valued at the value of the entire pair, while the other shoe becomes valueless in case of a loss. Thus, jointly, factors 10X, 5Y, and 8Z produce a product that is valued, say, as rank 11 on the actor's value scale. Lose the supply of one of the factors, and the other complementary factors become completely valueless.

Now, let us assume, secondly, that each of the factors is nonspecific: that 10X can be used in another line of production that will yield a product, say, ranked 21st on the value scale; that 5Y in another use will yield a product ranked 15th on the actor's value scale; and that 8Z can be used to yield a product ranked 30th. In that case, the loss of 10X would mean that instead of satisfying a want of rank 11, the units of Y and Z would be shifted to their next most valuable use, and wants ranked 15th and 30th would be satisfied instead. We know that the actor preferred the satisfaction of a want ranked 11th to the satisfaction of wants ranked 15th and 30th; otherwise the factors would not have been engaged in producing P in the first place. But now the loss of value is far from total, since the other factors can still yield a return in other uses.

Convertible factors will be allocated among different lines of production according to the same principles as consumers' goods are allocated among the ends they can serve. Each unit of supply will be allocated to satisfy the most urgent of the not yet satisfied wants, i.e., where the value of its marginal product is the highest. A loss of a unit of the factor will deprive the actor of only the least important of the presently satisfied uses, i.e., that use in which the value of the marginal product is the lowest. This choice is analogous to that involved in previous examples comparing the marginal utility of one good with the marginal utility of another. This lowest-ranked marginal product may be considered the value of the marginal product of any unit of the factor, with all uses taken into account. Thus, in the above case, suppose that X is a convertible factor in a myriad of different uses. If one unit of X has a marginal product of say, 3P, a marginal product in another use of 2Q, 5R, etc., the actor ranks the values of these marginal products of X on his value scale. Suppose that he ranks them in this order: 4S, 3P, 2Q, 5R. In that case, suppose he is faced with the loss of one unit of X. He will give up the use of a unit of X in production of R, where the marginal product is ranked lowest. Even if the loss takes place in the production of P, he will not give up 3P, but shift a unit of X from the less valuable use R and give up 5R. Thus, just as the actor gave up the use of a horse in pleasure riding and not in wagon-pulling by shifting from the former to the latter use, so the actor who (for example) loses a cord of wood intended for building a house will give up a cord intended for a service less valuable to him — say, building a sled. Thus, the value of the marginal product of a unit of a factor will be equal to its value in its marginal use, i.e., that use served by the stock of the factor whose marginal product is ranked lowest on his value scale.

We now can see further why, in cases where products are made with specific and convertible factors, the general law holds that the value of convertible factors changes less than that of specific factors in response to a change in the value of P or in the conditions of its production. The value of a unit of a convertible factor is set, not by the conditions of its employment in one type of product, but by the value of its marginal product when all its uses are taken into consideration. Since a specific factor is usable in only one line of production, its unit value is set as equal to the value of the marginal product in that line of production alone. Hence, in the process of valuation, the specific factors are far more responsive to conditions in any given process of production than are the nonspecific factors.[2]

As with the problem of optimum proportions, the process of value imputation from consumers' good to factors raises a great many problems which will be discussed in later chapters. Since one product cannot be measured against other products, and units of different factors cannot be compared with one another, how can value be imputed when, as in a modern economy, the structure of production is very complex, with myriads of products and with convertible and inconvertible factors? It will be seen that value imputation is easy for isolated Crusoe-type actors, but that special conditions are needed to enable the value-imputing process, as well as the factor-allocating process, to take place in a complex economy. In particular, the various units of products and factors (not the values, of course) must be made commensurable and comparable.

Labor versus Leisure

Setting aside the problem of allocating production along the most desired lines and of measuring one product against another, it is evident that every man desires to maximize his production of consumers' goods per unit of time. He tries to satisfy as many of his important ends as possible, and at the earliest possible time. But in order to increase the production of his consumers' goods, he must relieve the scarcity of the scarce factors of production; he must increase the available supply of these scarce factors. The nature-given factors are limited by his environment and therefore cannot be increased. This leaves him with the choice of increasing his supply of capital goods or of increasing his expenditure of labor.

It might be asserted that another way of increasing his production is to improve his technical knowledge of how to produce the desired goods — to improve his recipes. A recipe, however, can only set outer limits on his increases in production; the actual increases can be accomplished solely by an increase in the supply of productive factors. Thus, suppose that Robinson Crusoe lands, without equipment, on a desert island. He may be a competent engineer and have full knowledge of the necessary processes involved in constructing a mansion for himself. But without the necessary supply of factors available, this knowledge could not suffice to construct the mansion.

One method, then, by which man may increase his production per unit of time is by increasing his expenditure of labor. In the first place, however, the possibilities for this expansion are strictly limited — by the number of people in existence at any time and by the number of hours in the day. Secondly, it is limited by the ability of each laborer, and this ability tends to vary. And, finally, there is a third limitation on the supply of labor: whether or not the work is directly satisfying in itself, labor always involves the forgoing of leisure, a desirable good.[3]

We can conceive of a world in which leisure is not desired and labor is merely a useful scarce factor to be economized. In such a world, the total supply of available labor would be equal to the total quantity of labor that men would be capable of expending. Everyone would be eager to work to the maximum of capacity, since increased work would lead to increased production of desired consumers' goods. All time not required for maintaining and preserving the capacity to work would be spent in labor.[4] Such a situation could conceivably exist, and an economic analysis could be worked out on that basis. We know from empirical observation, however, that such a situation is very rare for human action. For almost all actors, leisure is a consumers' good, to be weighed in the balance against the prospect of acquiring other consumers' goods, including possible satisfaction from the effort itself. The more a man labors, the less leisure he can enjoy. Increased labor therefore reduces the available supply of leisure and the utility that it affords. Consequently, "people work only when they value the return of labor higher than the decrease in satisfaction brought about by the curtailment of leisure."[5] It is possible that included in this "return" of satisfaction yielded by labor may be satisfaction in the labor itself, in the voluntary expenditure of energy on a productive task. When such satisfactions from labor do not exist, then simply the expected value of the product yielded by the effort will be weighed against the disutility involved in giving up leisure — the utility of the leisure forgone. Where labor does provide intrinsic satisfactions, the utility of the product yielded will include the utility provided by the effort itself. As the quantity of effort increases, however, the utility of the satisfactions provided by labor itself declines, and the utility of the successive units of the final product declines as well. Both the marginal utility of the final product and the marginal utility of labor-satisfaction decline with an increase in their quantity, because both goods follow the universal law of marginal utility.

In considering an expenditure of his labor, man not only takes into account which are the most valuable ends it can serve (as he does with all other factors), these ends possibly including the satisfaction derived from productive labor itself, but he also weighs the prospect of abstaining from the expenditure of labor in order to obtain the consumers' good, leisure. Leisure, like any other good, is subject to the law of marginal utility. The first unit of leisure satisfies a most urgently felt desire; the next unit serves a less highly valued end; the third unit a still less highly valued end, etc. The marginal utility of leisure decreases as the supply increases, and this utility is equal to the value of the end that would have to be forgone with the loss of the unit of leisure. But in that case, the marginal disutility of work (in terms of leisure forgone) increases with every increase in the amount of labor performed.

In some cases, labor itself may be positively disagreeable, not only because of the leisure forgone, but also because of specific conditions attached to the particular labor that the actor finds disagreeable. In these cases, the marginal disutility of labor includes both the disutility due to these conditions and the disutility due to leisure forgone. The painful aspects of labor, like the forgoing of leisure, are endured for the sake of the yield of the final product. The addition of the element of disagreeableness in certain types of labor may reinforce and certainly does not counteract the increasing marginal disutility imposed by the cumulation of leisure forgone as the time spent in labor increases.

Thus, for each person and type of labor performed, the balancing of the marginal utility of the product of prospective units of effort as against the marginal disutility of effort will include the satisfaction or dissatisfaction with the work itself, in addition to the evaluation of the final product and of the leisure forgone. The labor itself may provide positive satisfaction, positive pain or dissatisfaction, or it may be neutral. In cases where the labor itself provides positive satisfactions, however, these are intertwined with and cannot be separated from the prospect of obtaining the final product. Deprived of the final product, man will consider his labor senseless and useless, and the labor itself will no longer bring positive satisfactions. Those activities which are engaged in purely for their own sake are not labor but are pure play, consumers' goods in themselves. Play, as a consumers' good, is subject to the law of marginal utility as are all goods, and the time spent in play will be balanced against the utility to be derived from other obtainable goods.[6]

In the expenditure of any hour of labor, therefore, man weighs the disutility of the labor involved (including the leisure forgone plus any dissatisfaction stemming from the work itself) against the utility of the contribution he will make in that hour to the production of desired goods (including future goods and any pleasure in the work itself), i.e., with the value of his marginal product. In each hour he will expend his effort toward producing that good whose marginal product is highest on his value scale. If he must give up an hour of labor, he will give up a unit of that good whose marginal utility is lowest on his value scale. At each point he will balance the utility of the product on his value scale against the disutility of further work. We know that a man's marginal utility of goods provided by effort will decline as his expenditure of effort increases. On the other hand, with each new expenditure of effort, the marginal disutility of the effort continues to increase. Therefore, a man will expend his labor as long as the marginal utility of the return exceeds the marginal disutility of the labor effort. A man will stop work when the marginal disutility of labor is greater than the marginal utility of the increased goods provided by the effort.[7]

Then, as his consumption of leisure increases, the marginal utility of leisure will decline, while the marginal utility of the goods forgone increases, until finally the utility of the marginal products forgone becomes greater than the marginal utility of leisure, and the actor will resume labor again.

This analysis of the laws of labor effort has been deduced from the implications of the action axiom and the assumption of leisure as a consumers' good.

Notes

[1] For algebraic proof, see George J. Stigler, The Theory of Price (New York: Macmillan & Co., 1946), pp. 44–45.

[2] For further reading on this subject, see Böhm-Bawerk, Positive Theory of Capital, pp. 170–88; and Hayek, Counter-Revolution of Science, pp. 32–33.

[3] This is the first proposition in this chapter that has not been deduced from the axiom of action. It is a subsidiary assumption, based on empirical observation of actual human behavior. It is not deducible from human action because its contrary is conceivable, although not generally existing. On the other hand, the assumptions above of quantitative relations of cause and effect were logically implicit in the action axiom, since knowledge of definite cause-and-effect relations is necessary to any decision to act.

[4] Cf. Mises, Human Action, p. 131.

[5] Ibid., p. 132.

[6] Leisure is the amount of time not spent in labor, and play may be considered as one of the forms that leisure may take in yielding satisfaction. On labor and play, cf. Frank A. Fetter, Economic Principles (New York: The Century Co., 1915), pp. 171–77, 191, 197–206.

[7] Cf. L. Albert Hahn, Common Sense Economics (New York: Abelard-Schuman, 1956), pp. 1 ff.