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9. Some Fallacies Relating to Utility
A doctrine commonly held by writers on utility is that the consumer acts so as to bring the marginal utility that any good has for him into equality with the price of that good. To understand this thesis, let us examine the preference scale of Mr. Jones in contemplating the purchase of one or more suits (and we shall assume that each suit is of the same quality—the same “good”). Suppose his value scale is as follows:
And suppose also that the market price is 2.9 grains per suit. Jones will buy not one or three, but two, suits. He will buy up to the last unit at which the diminishing marginal utility that the suit has for him exceeds the increasing marginal utility of money.36 This is obvious. Now, if a writer couches the exposition in terms of highly divisible goods, such as butter, and in terms of small units of money, such as pennies, it is easy to leap unthinkingly to the conclusion that the consumer for each good will act in such a way as to equalize, at the market price, the marginal utility of the sum of money and the marginal utility of the good. It should be clear, however, that there is never any such “equalization.” In the case of the suit, the rank of the second suit is still considerably above the rank of the 2.9 grains. So there is no equalization. Even in the case of the most divisible of goods, there will still be a difference in rank, not an equalization, between the two utilities. A man may buy 11 ounces of butter at 10 cents an ounce, until there is nothing ranking between the 11th ounce and the 10 cents on his utility scale; yet there is still no equality, but a difference in rank, with the last ounce bought ranking higher than the last sum of money spent. Of course, the consumer tries to spend his money so as to bring the two as close as possible, but they can never be equal.
Furthermore, the marginal utility of each particular good, after the purchases are made, differs in rank from that of every other. Thus, let us take one grain of gold as the monetary unit under consideration. Let us say that the given market-prices of various goods are as follows:
eggs — 1 dozen per grain;
butter — 1 pound per grain;
bread — 1 loaf per grain;
candy — 1 bar per grain.
Now each individual will purchase each commodity until the last point at which the marginal utility of the unit exceeds the marginal utility of a grain of gold. For one man, this might mean the purchase of five pounds of butter, three loaves of bread, two bars of candy, etc. This would mean that either a sixth pound of butter or a fourth loaf of bread would have a lower marginal utility than a grain of gold forgone. However, the marginal utility of each good will still differ in rank from that of every other and will not be equal to that of any other.
Another, even more curious doctrine holds that in equilibrium the ratio of the marginal utilities of the various goods equals the ratio of their prices. Without entering in detail into the manner by which these writers arrive at this conclusion, we can see its absurdity clearly, since utilities are not quantities and therefore cannot be divided.
These fallacies stem from a related one: the idea that an individual will act so as to equalize the marginal utility that any good will have in each of its uses. Applied to money, this would imply that the marginal utility of a unit of money is equal for each field of expenditure for each person. This is incorrect, as we have just seen that the marginal utilities of the various goods are not equalized. Successive units of a good are allocated to the most desired end, then to the next most desired satisfaction, etc. If there are several uses for the good, each one involving many possible units, the marginal utility of a unit in each use continues to decline as the supply increases. As goods are purchased, the marginal utility of each good purchased diminishes, and a man may allocate his money first to one use, then to another, and then to the first use again. However, in no case is there any equalization of marginal utilities.
The dogma of the equalization of marginal utilities may best be illustrated in the following passage from perhaps the originator of this line of argument:
Let s be the whole stock of some commodity, and let it be capable of two distinct uses. Then we may represent the two quantities appropriated to these uses by xl and y1, it being a condition that xl plus y1 equal s. The person may be conceived as successively expending small quantities of the commodity; now it is the inevitable tendency of human nature to choose that course which appears to offer the greatest advantage at the moment. Hence, when the person remains satisfied with the distribution he has made, it follows that no alteration would yield him more pleasure; which amounts to saying that an increment of commodity would yield exactly as much utility in one use as in another. Let Δu1, Δu2, be the increments of utility, which might arise respectively from consuming an increment of commodity in the two different ways. When the distribution is completed, we ought to have Δu1 = Δu2 ... The same reasoning ... will evidently apply to any two uses, and hence to all uses simultaneously, so that we obtain a series of equations less numerous by a unit than the number of ways of using the commodity. The general result is that the commodity, if consumed by a perfectly wise being, must be consumed with a maximum production of utility.37
The chief errors here consist in conceiving utility as a certain quantity, a definite function of an increment in the commodity, and in treating the problem in terms of infinitely small steps. Both procedures are fallacious. Utilities are not quantities, but ranks, and the successive amounts of a commodity that are used are always discrete units, not infinitely small ones. If the units are discrete, then the rank of each unit differs from that of every other, and there can be no equalization.
Many errors in discussions of utility stem from an assumption that it is some sort of quantity, measurable at least in principle. When we refer to a consumer's “maximization” of utility, for example, we are not referring to a definite stock or quantity of something to be maximized. We refer to the highest-ranking position on the individual's value scale. Similarly, it is the assumption of the infinitely small, added to the belief in utility as a quantity, that leads to the error of treating marginal utility as the mathematical derivative of the integral “total utility” of several units of a good. Actually, there is no such relation, and there is no such thing as “total utility,” only the marginal utility of a larger-sized unit. The size of the unit depends on its relevance to the particular action.38
This illustrates one of the grave dangers of the mathematical method in economics, since this method carries with it the bias of the assumption of continuity, or the infinitely small step. Most writers on economics consider this assumption a harmless, but potentially very useful, fiction, and point to its great success in the field of physics. They overlook the enormous differences between the world of physics and the world of human action. The problem is not simply one of acquiring the microscopic measuring tools that physics has developed. The crucial difference is that physics deals with inanimate objects that move but do not act. The movements of these objects can be investigated as being governed by precise, quantitatively determinate laws, well expressed in terms of mathematical functions. Since these laws precisely describe definite paths of movement, there is no harm at all in introducing simplified assumptions of continuity and infinitely small steps.
Human beings, however, do not move in such fashion, but act purposefully, applying means to the attainment of ends. Investigating causes of human action, then, is radically different from investigating the laws of motion of physical objects. In particular, human beings act on the basis of things that are relevant to their action. The human being cannot see the infinitely small step; it therefore has no meaning to him and no relevance to his action. Thus, if one ounce of a good is the smallest unit that human beings will bother distinguishing, then the ounce is the basic unit, and we cannot simply assume infinite continuity in terms of small fractions of an ounce.
The key problem in utility theory, neglected by the mathematical writers, has been the size of the unit. Under the assumption of mathematical continuity, this is not a problem at all; it could hardly be when the mathematically conceived unit is infinitely small and therefore literally sizeless. In a praxeological analysis of human action, however, this becomes a basic question. The relevant size of the unit varies according to the particular situation, and in each of these situations this relevant unit becomes the marginal unit. There is none but a simple ordinal relation among the utilities of the variously sized units.
The tendency to treat problems of human action in terms of equality of utility and of infinitely small steps is also apparent in recent writings on “indifference maps.” Almost the entire edifice of contemporary mathematical economics in consumption theory has been built on the “indifference” assumption. Its basis is the treatment of large-sized classes of combinations of two goods, between which the individual is indifferent in his valuations. Furthermore, the differences between them are infinitely small, so that smooth lines and tangents can be drawn. The crucial fallacy is that “indifference” cannot be a basis for action. If a man were really indifferent between two alternatives, he could not make any choice between them, and therefore the choice could not be revealed in action. We are interested in analyzing human action. Any action demonstrates choice based on preference: preference for one alternative over others. There is therefore no role for the concept of indifference in economics or in any other praxeological science. If it is a matter of indifference for a man whether he uses 5.1 or 5.2 ounces of butter for example, because the unit is too small for him to take into consideration, then there will be no occasion for him to act on this alternative. He will use butter in ounce units, instead of tenths of an ounce. For the same reason, there are no infinitely small steps in human action. Steps are only those that are significant to human beings; hence, they will always be finite and discrete.
The error in reasoning on the basis of “indifference” is the failure to appreciate the fact that a problem important in the field of psychology may have no significance in the realm of praxeology, to which economics belongs. Psychology deals with the problem of how or why the individual forms value scales, and for this question it is relevant to consider whether the individual is decisive or inclined to be “indifferent” between various alternatives. Praxeology, however, is a logical science based on the existence of action per se; it is interested in explaining and interpreting real action in its universal sense rather than in its concrete content. Its discussion of value scales is therefore a deduction from the nature of human action and not a speculative essay on the internal workings of the mind. It is consequently irrelevant for praxeology whether a man, in having to decide between alternatives A and B, makes a choice firmly and decisively, or whether he decides by tossing a coin. This is a problem for psychology; praxeology is concerned only with the fact that he chooses, for example, A rather than B, and that therefore A ranked higher in his preference scale than B. Utility theory is not concerned with psychology or the internal operations of the mind, but is part of a separate science based on the logical consequences of the simple existence of action.
Neither is praxeology based on behaviorist psychology. In fact, in so far as praxeology touches on psychology, its principles are the reverse of those of behaviorism. As we have seen, far from simply observing action in the same way as we observe and record the movements of stones, praxeology is based on a fundamental distinction between human action and the motion of inorganic matter, namely, that human action is motivated toward the achievement of certain ends. Means and resources are used for the achievement of these ends. Far from leaving mind out of the picture, praxeology rests fundamentally on the basic axiom of action, action caused and put into effect by human minds. However, praxeology is not concerned with the content of these ends, the manner of arriving at them, or their order; it is concerned with analysis of the logical implications of the existence of these ends.
Some writers, in their artificial separation of value scales from real action, have actually gone to the length of attempting to discover people's indifference maps by means of questionnaires. These attempts, besides being open to the stricture that indifference is not praxeologically valid, fail to realize that value scales can and do change continually and that therefore such questionnaires have no relevance to the business of economics. Economics is interested not in value scales professed in response to questionnaires, but in the values implied by real action. As Ludwig von Mises states, with regard to all attempts to separate value scales from action:
... the scale of value is nothing but a constructed tool of thought. The scale of value manifests itself only in real acting; it can be discerned only from the observation of real acting. It is therefore impermissible to contrast it with real acting and to use it as a yardstick for the appraisal of real actions.39
Since indifference is not relevant to human action, it follows that two alternatives for choice cannot be ranked equally on an individual's value scale. If they are really ranked equally, then they cannot be alternatives for choice, and are therefore not relevant to action. Hence, not only are alternatives ranked ordinally on every man's value scale, but they are ranked without ties; i.e., every alternative has a different rank.
The famous illustration used by the indifference theorists to demonstrate the relevance of indifference to human action is the case of Buridan's ass. This is the fable of the ass who stands, hungry, equidistant from two equally attractive bales of hay, or, thirsty, equidistant from two water holes. Since the two bales or water holes are equally attractive in every way, the ass can choose neither one and must therefore starve. This example is supposed to prove the great relevance of indifference to action and to be an indication of the way that indifference is revealed in action. Compounding confusion, Schumpeter refers to this ass as “perfectly rational.40
In the first place, it is of course difficult to conceive of an ass or a person that could be less rational. He is confronted not with two choices, but with three, the third being to starve where he is. Even on the indifferentists’ own grounds, this third choice will be ranked lower than the other two on the actor's value scale. He will not choose starvation.
If both the left and right water holes are equally attractive, and he can find no reason for preferring one or the other, the ass or the man will allow pure chance, such as a flip of a coin, to decide on either one. But on one he must and will decide. Again, we are interested in preference as revealed through choice and not in the psychology of preferences. If the flipped coin indicated the left water hole, then the left water hole was finally placed higher on the actor's value scale, as was revealed when he went toward it. Far from being a proof of the importance of indifference, the case of Buridan's ass is an excellent demonstration of the fact that indifference can play no part whatever in an analysis of human action.
Another way of attempting a justification of the indifference analysis is to suppose that a man, Jones, chooses each of two alternatives A and B about 50 percent of the time, upon repeated opportunities. This shifting is alleged to be a demonstration that Jones is really indifferent as between the two alternatives. Yet what is the reasonable inference? Clearly, that in some cases, A was preferred to B on Jones’ value scale, and that in the others, the positions were shifted so that B was preferred to A. In no case was there indifference between the two alternatives. The shift of choice indicates a shift in the preference scale, and not indifference on a constant value scale. Of course, if we were dealing with psychology, we could enter into a discussion of intensities of preferences and opine that the man, with respect to his underlying personality, was relatively indifferent rather than intensely biased, as between the two alternatives. But in praxeology we are not interested in the concrete content of his value scales nor in his underlying personality. We are interested in value scales as revealed through choice.
- 36. We are omitting possible shifts in rank resulting from the increasing utility of money, which would only complicate matters unduly.
- 37. W. Stanley Jevons, The Theory of Political Economy (3rd ed.; London: Macmillan & Co., 1888), pp. 59–60.
- 38. See Appendix A below, “The Diminishing Marginal Utility of Money,” and Rothbard, “Toward a Reconstruction of Utility and Welfare Economics.”
- 39. Mises, Human Action, p. 102. Dr. Bernardelli justly says:
If someone asks me in abstracto whether my love for my country is greater than my desire for freedom, I am somewhat at a loss how to answer, but actually having to make a choice between a trip in my country and the danger of losing my freedom, the order of intensities of my desire becomes only too determinate. (Harro F. Bernardelli, “What has Philosophy to Contribute to the Social Sciences, and to Economics in Particular?” Economica, November, 1936, p. 451) Also see our discussion of “consumer surplus” in section 4 above.
- 40. Schumpeter, History of Economic Analysis, pp. 94 n. and 1064.