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Book III, Chapter IX: The Value of Complementary Goods - Eugen von Böhm-Bawerk, The Positive Theory of Capital 
The Positive Theory of Capital, trans. William A. Smart (London: Macmillan and Co., 1891).
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Book III, Chapter IX
The Value of Complementary Goods
It very often occurs that, in order to obtain an economic utility, several goods require to co-operate in such a way that, if one good falls out of its place, the utility cannot be obtained, or cannot be completely obtained. Goods whose uses thus supplement each other we may follow Menger in calling Complementary goods. Thus, for instance, paper, pen and ink, needle and thread, cart and horse, bow and arrow, right and left hand gloves, and so on, are complementary goods. This complementary character obtains generally, indeed almost universally, among productive goods.
It is easy to see that the intimate co-relation of complementary goods—the co-relation in which they afford this utility—will be reflected in the formation of their value. This leads to a number of peculiarities, all, however, occurring within the limits of the universal law of marginal utility. In stating these we must distinguish between the value which belongs to the complete group, and that which belongs to individual members of it.
The total value of the complete group adapts itself, as a rule, to the amount of the marginal utility which it is capable of affording as a group. If, for instance, three goods, A, B, and C, form a complementary group, and if the smallest utility economically obtainable by the joint employment of these three goods amounts to a value of a hundred, the three goods A, B, and C taken together will be worth a hundred.
The only exception to this rule occurs in those cases where, on the general principles with which we are now familiar, the value of a good is to be measured, not by the immediate marginal utility of its own class, but by the marginal utility of other classes of goods drawn on to serve as substitutes. In the special case under consideration this will occur if every individual member of the complete group is replaceable by purchase, or production, or even by taking a substitute out of some other isolated employment, and if, at the same time, the total sum of the utility which the substituted goods would otherwise (in isolation) have had is less than the marginal utility they afford as combined. If the latter, for instance, amounts to 100, while the substitutionary value, the value of the three members individually, is only 20, 30, and 40—that is in all 90,—the thing that depends on the group of three is not the obtaining of the combined utility of 100—which is, in any case, assured by the substitutionary goods—but only the obtaining of the smaller utility, the 90, which fails of its provision when the members are taken away and become substitutes in the group. Since, however, in such cases the complementary character has, properly speaking, no influence on the formation of value, and the value is simply determined according to the ordinary laws already familiar to us, we need not give any separate consideration to this. In what follows, then, I shall give particular attention only to the normal case, where the marginal utility attainable by goods in joint employment is, at the same time, the true marginal utility.
As was before remarked, this marginal utility, first of all, determines the united value of the whole group. But in the manner in which this total value is divided out among the single members of the group, considerable differences emerge, varying with the casuistical peculiarity of the case.
First, if none of the members admits of any use other than the joint use, and if, at the same time, no one member which co-operates towards the joint utility can be replaced, then one single member has the full total value of the group, and the other members are entirely valueless. Suppose, for instance, I pay five shillings for a pair of gloves, five shillings is the total value of the pair. If I lose one of the gloves I lose the whole utility, and, with it, the whole value of the pair; and the remaining glove has no value. Of course either of the two gloves equally admits of either valuation, and it is simple circumstances that decide which of them is to rank as all, and which as nothing—the glove needed to complete the pair, or the useless single glove. Cases of this kind are relatively scarce in practical life.
Second, and more common, is the case where the individual members of the group can afford another, though a less utility, outside of their joint employment. Here the value of the single member does not lie between everything and nothing, but between the amount of the marginal utility which it is capable of affording in isolation as minimum, and the amount of the joint marginal utility, after deducting the isolated marginal utility of the other members, as maximum. Suppose, for instance, that three goods, A, B, and C, in co-operation afford a marginal utility of 100; that A by itself has a marginal utility of 10, B by itself of 20, and C by itself of 30; the value of A is determined as follows. If a merchant owns this good by itself he can get from it only its isolated marginal utility of 10, and the value of the good, accordingly, is only 10. But suppose he owns the whole group, and is asked to sell or give away the good A out of that group, what he has to consider is that, with the good A he can get a marginal utility of 100; without it, only the smaller (isolated) utility of the goods B and C, that is 20 + 30 = 50; and that, accordingly, on the having or losing of the good A depends a difference in value of 50. As complement of the group it is, therefore, worth 100-(20+30)=50; as an isolated good it is worth only 10.23 Here the difference in value is not so extreme as in the first case, but still it is very considerable.
Third, and more common still, is the case where some individual members of the group are not only employed for other purposes, but are, at the same time, replaceable by other goods of the same kind. For instance, building ground, bricks, beams, and labour are complementary goods in the building of a house. But if a few carts of bricks, intended for the building, go astray in transit, or some of the labourers engaged for the job refuse to work, in normal circumstances this does not in the least hinder the obtaining of the joint utility—the built house. The labourers and materials are simply replaced by others. The consequences as regards the formation of value are as follows:—
Now, of course, it is not likely that any one individual, within the limits of his own economy, will possess a thousand goods of one kind, and a thousand different opportunities of employing them. But, all the same, the efficiency of the influences just described is in no wise annulled; it is only the scene of their operation that is changed, from individual economy to the market, and that in the following way. Individuals buy what they require, and sell their surpluses in the market. Here, then, all the stocks of goods and all the opportunities of employing them over the entire field covered by the market, come together. And now—exactly as before—everything depends on whether, in the market, commodities and opportunities of employing them are scarce or not. If the commodity is very scarce, it makes a very considerable difference in the determination of price whether we approach the particular good as buyer or as seller. For instance, suppose, as before, that there are only three similar goods, and three buyers each wishing to acquire just one such good, with the view of using it in employments that will yield 50, 20, and 10. Then, if one of these goods be withdrawn from the market to serve in a complementary employment, the two remaining goods are bought for the employments indicated by 50 and 20, and—according to laws which will be explained in next book—the purchase price must be fixed between 10 and 20, say at 15. But if now the complementary employment fails, and the third good also is thrown on the market, it must—if it is to find a sale at all—fall to the buyer who can get 10 by employing it, and the result is that the market price is in all cases fixed below the level of 10. Here, then, the price—and the subjective exchange value based on it—varies not inconsiderably.
If, on the other hand, there are a thousand similar goods offered, and a thousand buyers demand them, evidently it will not make the smallest difference to the market price whether there appears a thousand and first buyer, or a thousand and first seller; the good obtains a price and value independently of whether it finds a place in the single complementary employment or not.
Thus, under the assumptions now laid down, the value of the replaceable members is fixed at a certain level independently of their concrete complementary employment, and this value they have when we distribute out the total value of the group among its individual members. The distribution, then, will be made thus: of the total value of the whole group—which is determined by the marginal utility of the joint employment—this fixed value is previously assigned to the replaceable members, and the remainder—which varies according to the amount of the marginal utility—is reckoned to the nonreplaceable members as their individual value. To use our old illustration again; say that the joint marginal utility amounts to 100, and that the members A and B have a fixed "substitution" value of 10 and 20 respectively, 70 must be reckoned the individual value of the nonreplaceable good C; or, say that the marginal utility of the group amounts to 120, the individual value of C will be 90.25
Of the three cases we have discussed the last mentioned is by far the most common in practical life, and, accordingly, in the great majority of cases, the value of complementary goods is determined according to the latter formula. The most important application of it is in the distribution of the product among the various productive powers co-operating in producing it. Almost every product is the result of the co-operation of a group of complementary goods consisting of uses of ground, labour, fixed and floating capital. Of the complementary members the great majority are marketable commodities, and replaceable at will; as, for instance, the labour of wage-earners, the raw materials, fuel, tools, etc. Only a few of them are non-replaceable, or not easily replaceable; as, for instance, the land on which the peasant works, the mine, the railway lines, the factory walls, the activity of the undertaker himself with his peculiar and high qualifications, and so on. It is easy to see, therefore, that here we have exactly those casuistical circumstances in which the foregoing formula of distribution obtains, and, as a fact, it is acted upon in practical life in the most accurate way. In actual business the "costs" are first deducted from the total return. If we look closer, however, we shall see that what is deducted is not all the costs—for, if so, the use of ground, or the undertaker's activity, as both valuable goods, would come under costs—but only the expenditure for the replaceable means of production with a given substitution value, viz. the wage of labour, raw materials, wear and tear of tools, etc. The remainder, under the name of "net return," is ascribed to the non-replaceable member or members: the peasant calculates it to his land, the mine-owner to his mine, the manufacturer to his factory, the merchant to his undertaking activity.
If the joint returns increase, it would not occur to anybody to ascribe the surplus to the replaceable members; it is always the ground or the mine that "produces more." And, similarly, if the joint returns decrease, nobody would credit the "costs" with the reduced amount; the deficiency also is conceived as exclusively due to the diminished productiveness of the ground or the mine. And this is entirely logical and correct: on goods replaceable at any moment only the fixed substitution value is actually dependent; the entire remainder of the joint amount of utility obtainable depends on the goods that cannot be replaced.
The theory of the value of complementary goods is the key which will solve one of the most important and difficult problems of political economy—the problem of the distribution of goods as made in the present state of society, where competition is more or less free and prices are determined by free contract. All products come into existence through the cooperation of the three complementary "factors of production," labour, land, and capital. Now our theory, in showing how much of the joint product may economically26 be considered as due to each of these, and what share of the total value may, accordingly, be assigned to each of them, lays down, at the same time, the most decisive basis for determining the amount of remuneration which each of the three factors obtains. And thus although, as we know, capital as "factor of production" does not exactly coincide with capital as "source of income," yet this gives us at least a rough indication of the way in which the amount of the three branches of income—wage, rent, and interest—is determined.
It does not indeed do this quite directly. That quota which the workers receive, and that other quota which the owners of the co-operating ground receive, is directly identical with wage and rent. But the quota which falls to the co-operation of capital is not interest—as, in theories of distribution, economists have repeatedly assumed ever since the days of Say with fatal precipitation. It is, first, the gross remuneration for the co-operation of capital; and, out of this, interest is got, like a kernel out of a shell, because, and to the extent that, something remains over after deducting from the gross remuneration the value of the worn-out capital. To explain how this is so is a problem in itself. To make it quite clear by an illustration, suppose that a commodity, produced by the co-operation of all three factors, is worth £100. The law of complementary goods will carry us thus far; it will enable us to determine that the share of labour (the labour directly employed in the production) amounts to, say, £20, that of ground to £10, that of capital to £70. But it does not tell us what, or how much, of that £70 remains over net, as interest, after deduction of the wear and tear of capital. On the contrary, the law of complementary goods in itself would rather lead us to the conclusion that nothing remains over. For, according to it, it would be most natural to assume that the capital, to the co-operation of which the return of $70 is ascribed, and which has been consumed in obtaining that return, had already been valued at the entire £70; and, if this were the case, the return to capital would naturally be entirely absorbed by the wear and tear of the capital. That this is not the case is, so to speak, an internal mattera matter which plays its part inside the gross share of capital determined by the law of complementary goods, and is the object of an independent problem, the peculiar problem of Interest. But before we can discuss interest there is still a great deal to be explained.27
[23.]Of course in this case also the peculiarities of the case decide which member is to be valued as the completing member of the group, and which as simply the isolated piece. If, for instance, the owner of the complete group is asked to sell the good A, he will value it as completing member, and the other goods B and C as isolated pieces. But if he is asked to sell C, he will value it as completing member at 100-(10+20)=70, while A and B will be valued as isolated pieces.
[24.]To put it concretely: although a load of bricks were absolutely indispensable to finish a house, the load could never obtain any higher value than that determined by the marginal utility of bricks generally; that is, as determined by all the uses to which bricks generally are put.—W. S.
[25.]If C also were replaceable by a substitute of less value the case mentioned on p. 170 would emerge, and the marginal utility of the joint use would not determine the value of the complementary group.
[26.]Not physically. It would, in most cases, be absolutely impossible to calculate the physical share—how could one be supposed to distinguish what percentage the material and what percentage the artist had contributed physically to the making of a statue?—but it is also a matter of no importance. On the other hand it is, in most cases, quite easy to determine what share of the utility, or of the value, would have to be done without if one were not in possession of a definite individual factor, and this quota, conditioned by the possession of one factor, I call its economical share in the total product.
[27.]The confusion, so common in economic literature, between the gross share assigned to the co-operation of capital (Rohzins) and net interest, has been fully discussed in my Capital and Interest (see the criticism of Lauderdale, p. 146; of Carey, p. 155; of Strasburger, p. 175; of Say, p. 189, etc.) It will not be expected of me to give a complete theory of distribution in the passing, as it were. I purposely refrain from going deeper into the subject than is necessary for my special task, the development of the Interest theory. And for this it is sufficient to sketch only in the broadest lines the principles which limit the gross share of capital, as against the shares of labour and uses of land that co-operate with it: our special task will be to lay down what is the state of the case as regards the gross share of capital. Moreover I hope that on this question of the shares allotted to the various factors, which I am compelled to treat in a very cursory way, the eagerly expected work of Wieser will very shortly shed a clear light. (Wieser's Der Natürliche Werth, Vienna, 1889, appeared while this was passing through the press.—W. S.)