Front Page Titles (by Subject) Failure of the Equations of Exchange. - The Theory of Political Economy
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Failure of the Equations of Exchange. - William Stanley Jevons, The Theory of Political Economy 
The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.
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Failure of the Equations of Exchange.
Cases constantly occur in which equations of the kind set forth in the preceding pages fail to hold true, or lead to impossible results. Such failure may indicate that no exchange at all takes place, but it may also have a different meaning.
In the first case, it may happen that the commodity possessed by A has a high degree of utility to A, and a low degree to B, and that vice versâ B's commodity has a high degree of utility to B and less to A. This difference of utility might exist to such an extent, that though B were to receive very little of A's commodity, yet the final degree of utility to him would be less than that of his own commodity, of which he enjoys much more. In such a case no benefit can arise from exchange, and no exchange will consequently take place. This failure of exchange will be indicated by a failure of the equations.
It may also happen that the whole quantities of commodity possessed are exchanged, and yet the equations fail. A may have so low a desire for consuming his own commodity, that the very last increment of it has less degree of utility to him than a small addition to the commodity received in exchange. The same state of things might happen to exist with B as regards his commodity: under these circumstances the whole possessions of one might be exchanged for the whole of the other, and the ratio of exchange would of course be defined by the ratio of these quantities. Yet each party might desire the last increment of the commodity received more than he desires the last increment of that given, so that the equations would fail to be true. This case will hardly occur practically in international trade, since two nations usually trade in many commodities, a fact which would alter the conditions.
Again, the equations of exchange will fail to be possible when the commodity or useful article possessed on one or both sides is indivisible. We have always assumed hitherto that more or less of a commodity may be had, down to infinitely small quantities. This is approximately true of all ordinary trade, especially international trade between great industrial nations. Any one sack of corn or any one bar of iron is practically infinitesimal compared with the quantities exchanged by America and England; and even one cargo or parcel of corn or iron is a small fraction of the whole. But, in exceptional cases, even international trade might involve indivisible articles. We might conceive the British Government giving the Koh-i-noor diamond to the Khedive of Egypt in exchange for Pompey's Pillar, in which case it would certainly not answer the purpose to break up one article or the other.1 When an island or portion of territory is transferred from one possessor to another, it is often necessary to take the whole, or none. America, in purchasing Alaska from Russia, would hardly have consented to purchase less than the whole. In every sale of a house, factory, or other building, it is usually impracticable to make any division without greatly lessening the utility of the whole. In all such cases our equations must fail to exist, because we cannot contemplate the existence of an increment or a decrement to an indivisible article.
Suppose, for example, that A and B each possess a book: they cannot break up the books, and must therefore exchange them entire, if at all. Under what conditions will they do so? Plainly on the condition that each makes a gain of utility by so doing. Here we deal not with the final degree of utility depending on an infinitesimal quantity, but on the whole utility of the complete article. Now let us assign the symbols as follows:—
Then the conditions of exchange are simply
We might indeed theoretically contemplate the case where the utilities were exactly equal on one side; thus
B would then be wholly indifferent to the exchange, and I do not see any means of deciding whether he would or would not consent to it. But we need hardly consider the case, as it could seldom practically occur. Were the utilities exactly equal on both sides in respect to both objects, there would obviously be no motive to exchange. Again, the slightest loss of utility on either side would be a complete bar to the transaction, because we are not supposing, at present, that any other commodities are in possession so as to allow of separate inducements, or that any other motives than such as arise out of simple desire of one's own convenience enter into the question.
A much more difficult problem arises when we suppose an indivisible article exchanged for a divisible commodity. When Russia sold Alaska this was a practically indivisible thing; but it was bought with money of which more or less might be given to indefinitely small quantities. A bargain of this kind is exceedingly common; indeed it occurs in the case of every house, mansion, estate, factory, ship, or other complete whole, which is sold for money. Our former equations of exchange certainly fail, for they involve increments of commodity on both sides. The theory seems to give a very unsatisfactory answer, for the problem proves to be, within certain limits, indeterminate.
Let X be the indivisible article; u1 its utility to its possessor A, and u2 its utility to B. Let y be the quantity of commodity given for it, a commodity which is supposed to be divisible ad infinitum; let v1 be the total utility of y to A, and v2 its total utility to B. Then it is quite evident that, in order to give rise to exchange, v1 must be greater than u1, and u2 must be greater than v2; that is, there must, as before, be a gain of utility on each side. The quantity y must not be so great then as to deprive B of gain, nor so small as to deprive A of gain. The following is an extract from Mr. Thornton's work which exactly expresses the problem:—
"There are two opposite extremes—one above which the price of a commodity cannot rise, the other below which it cannot fall. The upper of these limits is marked by the utility, real or supposed, of the commodity to the customer; the lower, of its utility to the dealer. No one will give for a commodity a quantity of money or money's worth, which, in his opinion, would be of more use to him than the commodity itself. No one will take for a commodity a quantity of money or of anything else which he thinks would be of less use to himself than the commodity. The price eventually given and taken may be either at one of the opposite extremes, or may be anywhere intermediate between them."1
Three distinct cases might occur, which can best be illustrated by a concrete example. Suppose we can read the thoughts of the parties in the sale of a house. If A says £1200 is the least price which will satisfy him, and B holds that £800 is the highest price which it will be profitable for him to give, no exchange can possibly take place. If A should find £1000 to be his lowest limit, while B happens to name the same sum for his highest limit, the transaction can be closed, and the price will be exactly defined. But supposing, finally, that A is really willing to sell at £900, and B is prepared to buy at £1100, in what manner can we theoretically determine the price? I see no mode of solving the question. Any price between £900 and £1100 will leave a profit on each side, and both parties will lose if they do not come to terms. I conceive that such a transaction must be settled upon other than strictly economical grounds. The result of the bargain will greatly depend upon the comparative amount of knowledge of each other's positions and needs which either bargainer may possess or manage to obtain in the course of the transaction. Thus the power of reading another man's thoughts is of high importance in business, and the art of bargaining mainly consists in the buyer ascertaining the lowest price at which the seller is willing to part with his object, without disclosing if possible the highest price which he, the seller, is willing to give. The disposition and force of character of the parties, their comparative persistency, their adroitness and experience in business, or it may be feelings of justice or of kindliness, will also influence the decision. These are motives more or less extraneous to a theory of Economics, and yet they appear necessary considerations in this problem. It may be that indeterminate bargains of this kind are best arranged by an arbitrator or third party.
The equations of exchange may fail again when commodities are divisible, but not to infinitely small quantities. There is always, in retail trade, a convenient unit below which we do not descend in purchases. Paper may be bought in quires, or even in packets, which it may not be desirable to break up. Wine cannot be bought from the wine merchant in less than a bottle at a time. In all such cases exchange cannot, theoretically speaking, be perfectly adjusted, because it will be infinitely improbable that an integral number of units will precisely verify the equations of exchange. In a large proportion of cases, indeed, the unit may be so small compared with the whole quantities exchanged as practically to be infinitely small. But suppose that a person be buying ink which is only to be had, under the circumstances, in one shilling bottles. If one bottle be not quite enough, how will he decide whether to take a second or not? Clearly by estimating the aggregate utility of the bottle of ink compared with the shilling. If there be an excess, he will certainly purchase it, and proceed to consider whether a third be desirable or not.
This case might be illustrated by Fig. VI., in which the spaces o q1, p1q2, p2q3, etc., represent the total utilities of successive bottles of ink; while the equal spaces o r1, p1r2, etc., represent the total utilities of successive shillings, which we may assume to be practically invariable. There is no doubt that three bottles will be
purchased, but the fourth will not be purchased unless the mixtilinear figure p3q3q4p4 exceed in area the rectangle p3r3r4p4.
Cases of this kind are similar to those treated in pp. 120-124, where the things exchanged are indivisible, except that the question of exchange or no exchange occurs over and over again with respect to each successive unit, and is decided in respect to each by the excess of the total utility of the unit to be received over the total utility of that to be given. There is indeed perfect harmony between the cases where equations can and where they cannot be established; for we have only to imagine the indivisible units of commodity to be indefinitely lessened in size to enable us to pass gradually down to the case where equality of the increments of utility is ultimately established.
[]Since the above was written the value of Cleopatra's Needle has actually formed the subject of decision in the Admiralty Court, in connection with the award of salvage. The fact, however, is that in the absence of any act of exchange concerning such an object, the notion of value is not applicable at all. At the best the value assigned, namely £25,000, is a mere fiction arbitrarily invented to represent what might conceivably be given for such an object if there were a purchaser. It is, moreover, curious that since the first edition was printed Russia has actually made an exchange of islands with Japan.
[]Thornton On Labour; its Wrongful Claims and Rightful Dues (1869), p. 58.