The Theory of Political Economy

CHAPTER IV
THEORY OF EXCHANGE

Importance of Exchange in Economics.

EXCHANGE is so important a process in the maximising of utility and the saving of labour, that some economists have regarded their science as treating of this operation alone. Utility arises from commodities being brought in suitable quantities and at the proper times into the possession of persons needing them; and it is by exchange, more than any other means, that this is effected. Trade is not indeed the only method of economising: a single individual may gain in utility by a proper consumption of the stock in his possession. The best employment of labour and capital by a single person is also a question disconnected from that of exchange, and which must yet be treated in the science. But, with these exceptions, I am perfectly willing to agree with the high importance attributed to exchange.

It is impossible to have a correct idea of the science of Economics without a perfect comprehension of the Theory of Exchange; and I find it both possible and desirable to consider this subject before introducing any notions concerning labour or the production of commodities. In these words of J. S. Mill I thoroughly concur: "Almost every speculation respecting the economical interests of a society thus constituted, implies some theory of Value: the smallest error on that subject infects with corresponding error all our other conclusions; and anything vague or misty in our conception of it creates confusion and uncertainty in everything else." But when he proceeds to say, "Happily, there is nothing in the laws of Value which remains for the present or any future writer to clear up; the theory of the subject is complete"*65—he utters that which it would be rash to say of any of the sciences.

Ambiguity of the term Value.

I must, in the first place, point out the thoroughly ambiguous and unscientific character of the term value. Adam Smith noticed the extreme difference of meaning between value in use and value in exchange; and it is usual for writers on Economics to caution their readers against the confusion of thought to which they are liable. But I do not believe that either writers or readers can avoid the confusion so long as they use the word. In spite of the most acute feeling of the danger, I often detect myself using the word improperly; nor do I think that the best authors escape the danger.

Let us turn to Mill's definition of Exchange Value,*66 and we see at once the misleading power of the term. He tells us—"Value is a relative term. The value of a thing means the quantity of some other thing, or of things in general, which it exchanges for." Now, if there is any fact certain about exchange value, it is, that it means not an object at all, but a circumstance of an object. Value implies, in fact, a relation; but if so, it cannot possibly be some other thing. A student of Economics has no hope of ever being clear and correct in his ideas of the science if he thinks of value as at all a thing or an object, or even as anything which lies in a thing or object. Persons are thus led to speak of such a nonentity as intrinsic value. There are, doubtless, qualities inherent in such a substance as gold or iron which influence its value; but the word Value, so far as it can be correctly used, merely expresses the circumstance of its exchanging in a certain ratio for some other substance.

Value expresses Ratio of Exchange.

If a ton of pig-iron exchanges in a market for an ounce of standard gold, neither the iron is value nor the gold; nor is there value in the iron nor in the gold. The notion of value is concerned only in the fact or circumstance of one exchanging for the other. Thus it is scientifically incorrect to say that the value of the ton of iron is the ounce of gold: we thus convert value into a concrete thing; and it is, of course, equally incorrect to say that the value of the ounce of gold is the ton of iron. The more correct and safe expression is, that the value of the ton of iron is equal to the value of the ounce of gold, or that their values are as one to one.

Value in exchange expresses nothing but a ratio, and the term should not be used in any other sense. To speak simply of the value of an ounce of gold is as absurd as to speak of the ratio of the number seventeen. What is the ratio of the number seventeen? The question admits no answer, for there must be another number named in order to make a ratio; and the ratio will differ according to the number suggested. What is the value of iron compared with that of gold?—is an intelligible question. The answer consists in stating the ratio of the quantities exchanged.

Popular use of the term Value.

In the popular use of the word value no less than three distinct though connected meanings seem to be confused together. These may be described as

(1) Value in use;
(2) Esteem, or urgency of desire;
(3) Ratio of exchange.

Adam Smith, in the familiar passage already referred to, distinguished between the first and the third meanings. He said,*67 "The word value, it is to be observed, has two different meanings, and sometimes expresses the power of purchasing other goods which the possession of that object conveys. The one may be called 'value in use;' the other 'value in exchange.' The things which have the greatest value in use have frequently little or no value in exchange; and, on the contrary, those which have the greatest value in exchange have frequently little or no value in use. Nothing is more useful than water: but it will purchase scarce anything; scarce anything can be had in exchange for it. A diamond, on the contrary, has scarce any value in use; but a very great quantity of other goods may frequently be had in exchange for it."

It is sufficiently plain that, when Smith speaks of water as being highly useful and yet devoid of purchasing power, he means water in abundance, that is to say, water so abundantly supplied that it has exerted its full useful effect, or its total utility. Water, when it becomes very scarce, as in a dry desert, acquires exceedingly great purchasing power. Thus Smith evidently means by value in use, the total utility of a substance of which the degree of utility has sunk very low, because the want of such substance has been well nigh satisfied. By purchasing power he clearly means the ratio of exchange for other commodities. But here he fails to point out that the quantity of goods received in exchange depends just as much upon the nature of the goods received, as on the nature of those given for them. In exchange for a diamond we can get a great quantity of iron, or corn, or paving stones, or other commodity of which there is abundance; but we can get very few rubies, sapphires, or other precious stones. Silver is of high purchasing power compared with zinc, or lead, or iron, but of small purchasing power compared with gold, platinum, or iridium. Yet we might well say in any case that diamond and silver are things of high value. Thus I am led to think that the word value is often used in reality to mean intensity of desire or esteem for a thing. A silver ornament is a beautiful object apart from all ideas of traffic; it may thus be valued or esteemed simply because it suits the taste and fancy of its owner, and is the only one possessed. Even Robinson Crusoe must have looked upon each of his possessions with varying esteem and desire for more, although he was incapable of exchanging with any other person. Now, in this sense value seems to be identical with the final degree of utility of a commodity, as defined in a previous page (p. 49); it is measured by the intensity of the pleasure or benefit which would be obtained from a new increment of the same commodity. No doubt there is a close connection between value in this meaning, and value as ratio of exchange. Nothing can have a high purchasing power unless it be highly esteemed in itself; but it may be highly esteemed apart from all comparison with other things; and, though highly esteemed, it may have a low purchasing power. because those things against which it is measured are still more esteemed.

Thus I come to the conclusion that, in the use of the word value, three distinct meanings are habitually confused together, and require to be thus distinguished—

(1) Value in use = total utility;
(2) Esteem = final degree of utility;
(3) Purchasing power = ratio of exchange.

It is not to be expected that we could profitably discuss such matters as economical doctrines, while the fundamental ideas of the subject are thus jumbled up together in one ambiguous word. The only thorough remedy consists in substituting for the dangerous name value that one of three stated meanings which is intended in each case. In this work, therefore, I shall discontinue the use of the word value altogether, and when, as will be most often the case in the remainder of the book, I need to refer to the third meaning, often called by economists exchange or exchangeable value, I shall substitute the wholly unequivocal expression Ratio of exchange, specifying at the same time what are the two articles exchanged. When we speak of the ratio of exchange of pig-iron and gold, there can be no possible doubt that we intend to refer to the ratio of the number of units of the one commodity to the number of units of the other commodity for which it exchanges, the units being arbitrary concrete magnitudes, but the ratio an abstract number.

When I proposed, in the first edition of this book, to use Ratio of Exchange instead of the word value, the expression had been so little, if at all, employed by English economists, that it amounted to an innovation. J. S. Mill, indeed, in his chapters on Value, speaks once and again of things exchanging for each other "in the ratio of their cost of production;" but he always omits to say distinctly that exchange value is itself a matter of ratio. As to Ricardo, Malthus, Adam Smith, and other great English economists, although they usually discourse at some length upon the meanings of the word value, I am not aware that they ever explicitly apply the name ratio to exchange or exchangeable value. Yet ratio is unquestionably the correct scientific term, and the only term which is strictly and entirely correct.

It is interesting, therefore, to find that, although overlooked by English economists, the expression had been used by two or more of the truly scientific French economists, namely, Le Trosne and Condillac. Le Trosne carefully defines value in the following terms:*68 "La valeur consiste dans le rapport d'échange qui se trouve entre telle chose et telle autre, entre telle mesure d'une production et telle mesure des autres." Condillac apparently adopts the words of Le Trosne, saying*69 of value: "Qu'elle consiste dans le rapport d'échange entre telle chose et telle autre." Such economical works as those of Baudeau, Le Trosne, and Condillac were almost wholly unknown to English readers until attention was drawn to them by Mr. H. D. Mácleod and Professor Adamson; but I shall endeavour for the future to make proper use of them.

Dimension of Value.

There is no difficulty in seeing that, when we use the word Value in the sense of ratio of exchange, its dimension will be simply zero. Value will be expressed, like angular magnitude and other ratios in general, by abstract number. Angular magnitude is measured by the ratio of a line to a line, the ratio of the are subtended by the angle to the radius of the circle. So value in this sense is a ratio of the quantity of one commodity to the quantity of some other commodity exchanged for it. If we compare the commodities simply as physical quantities, we have the dimensions M divided by M, or MM^-1, or M⁰. Exactly the same result would be obtained if, instead of taking the mere physical quantities, we were to compare their utilities, for we should then have MU divided by MU or M⁰U⁰, which, as it really means unity, is identical in meaning with M⁰.

When we use the word value in the sense of esteem, or urgency of desire, the feeling with which Oliver Twist must have regarded a few more mouthfuls when he "asked for more," the meaning of the word, as already explained, is identical with degree of utility, of which the dimension is U. Lastly, the value in use of Adam Smith, or the total utility, is the integral of U d M, and has the dimensions MU. We may thus tabulate our results concerning the ambiguous uses of the word value—

Popular Expression of Meaning.	Scientific Expression.	Dimensions.
(1) Value in use	Total Utility	MU.
(2) Esteem, or Urgency of Desire for more	Final Degree of Utility	U.
(3) Purchasing Power	Ratio of Exchange	M0.

Definition of Market.

Before proceeding to the Theory of Exchange, it will be desirable to place beyond doubt the meanings of two other terms which I shall frequently employ.

By a Market I shall mean much what commercial men use it to express. Originally a market was a public place in a town where provisions and other objects were exposed for sale; but the word has been generalised, so as to mean any body of persons who are in intimate business relations and carry on extensive transactions in any commodity. A great city may contain as many markets as there are important branches of trade, and these markets may or may not be localised. The central point of a market is the public exchange,—mart or auction rooms, where the traders agree to meet and transact business. In London, the Stock Market, the Corn Market, the Coal Market, the Sugar Market, and many others, are distinctly localised; in Manchester, the Cotton Market, the Cotton Waste Market, and others. But this distinction of locality is not necessary. The traders may be spread over a whole town, or region of country, and yet make a market, if they are, by means of fairs, meetings, published price lists, the post office, or otherwise, in close communication with each other. Thus, the common expression Money Market denotes no locality: it is applied to the aggregate of those bankers, capitalists, and other traders who lend or borrow money, and who constantly exchange information concerning the course of business.*70

In Economics we may usefully adopt this term with a clear and well-defined meaning. By a market I shall mean two or more persons dealing in two or more commodities, whose stocks of those commodities and intentions of exchanging are known to all. It is also essential that the ratio of exchange between any two persons should be known to all the others. It is only so far as this community of knowledge extends that the market extends. Any persons who are not acquainted at the moment with the prevailing ratio of exchange, or whose stocks are not available for want of communication, must not be considered part of the market. Secret or unknown stocks of a commodity must also be considered beyond reach of a market so long as they remain secret and unknown. Every individual must be considered as exchanging from a pure regard to his own requirements or private interests, and there must be perfectly free competition, so that any one will exchange with any one else for the slightest apparent advantage. There must be no conspiracies for absorbing and holding supplies to produce unnatural ratios of exchange. Were a conspiracy of farmers to withhold all corn from market, the consumers might be driven, by starvation, to pay prices bearing no proper relation to the existing supplies, and the ordinary conditions of the market would be thus overthrown.

The theoretical conception of a perfect market is more or less completely carried out in practice. It is the work of brokers in any extensive market to organise exchange, so that every purchase shall be made with the most thorough acquaintance with the conditions of the trade. Each broker strives to gain the best knowledge of the conditions of supply and demand, and the earliest intimation of any change. He is in communication with as many other traders as possible, in order to have the widest range of information, and the greatest chance of making suitable exchanges. It is only thus that a definite market price can be ascertained at every moment, and varied according to the frequent news capable of affecting buyers and sellers. By the mediation of a body of brokers a complete consensus is established, and the stock of every seller or the demand of every buyer brought into the market. It is of the very essence of trade to have wide and constant information. A market, then, is theoretically perfect only when all traders have perfect knowledge of the conditions of supply and demand, and the consequent ratio of exchange; and in such a market, as we shall now see, there can only be one ratio of exchange of one uniform commodity at any moment.

So essential is a knowledge of the real state of supply and demand to the smooth procedure of trade and the real good of the community, that I conceive it would be quite legitimate to compel the publication of any requisite statistics. Secrecy can only conduce to the profit of speculators who gain from great fluctuations of prices. Speculation is advantageous to the public only so far as it tends to equalise prices; and it is, therefore, against the public good to allow speculators to foster artificially the inequalities of prices by which they profit. The welfare of millions, both of consumers and producers, depends upon an accurate knowledge of the stocks of cotton and corn; and it would, therefore, be no unwarrantable interference with the liberty of the subject to require any information as to the stocks in hand. In Billingsgate fish market there was long ago a regulation to the effect that salesmen shall fix up in a conspicuous place every morning a statement of the kind and amount of their stock.*71 The same principle has long been recognised in the Acts of Parliament concerning the collection of statistics of the quantities and prices of corn sold in English market towns. More recently similar legislation has taken place as regards the cotton trade, in the Cotton Statistics Act of 1868. Publicity, whenever it can thus be enforced on markets by public authority, tends almost always to the advantage of everybody except perhaps a few speculators and financiers.

Definition of Trading Body.

I find it necessary to adopt some expression for any number of people whose aggregate influence in a market, either in the way of supply or demand, we have to consider. By a trading body I mean, in the most general manner, any body either of buyers or sellers. The trading body may be a single individual in one case; it may be the whole inhabitants of a continent in another; it may be the individuals of a trade diffused through a country in a third. England and North America will be trading bodies if we are considering the corn we receive from America in exchange for iron and other goods. The continent of Europe is a trading body as purchasing coal from England. The farmers of England are a trading body when they sell corn to the millers, and the millers both when they buy corn from the farmers and sell flour to the bakers.

We must use the expression with this wide meaning, because the principles of exchange are the same in nature, however wide or narrow may be the market considered. Every trading body is either an individual or an aggregate of individuals, and the law, in the case of the aggregate, must depend upon the fulfilment of law in the individuals. We cannot usually observe any precise and continuous variation in the wants and deeds of an individual, because the action of extraneous motives, or what would seem to be caprice, overwhelms minute tendencies. As I have already remarked (p. 15), a single individual does not vary his consumption of sugar, butter, or eggs from week to week by infinitesimal amounts, according to each small change in the price. He probably continues his ordinary consumption until accident directs his attention to a rise in price, and he then, perhaps, discontinues the use of the articles altogether for a time. But the aggregate, or what is the same, the average consumption, of a large community will be found to vary continuously or nearly so. The most minute tendencies make themselves apparent in a wide average. Thus, our laws of Economics will be theoretically true in the case of individuals, and practically true in the case of large aggregates; but the general principles will be the same, whatever the extent of the trading body considered. We shall be justified, then, in using the expression with the utmost generality.

It should be remarked, however, that the economical laws representing the conduct of large aggregates of individuals will never represent exactly the conduct of any one individual. If we could imagine that there were a thousand individuals all exactly alike in regard to their demand for commodities, and their capabilities of supplying them, then the average laws of supply and demand deduced from the conduct of such individuals would agree with the conduct of any one individual. But a community is composed of persons differing widely in their powers, wants, habits, and possessions. In such circumstances the average laws applying to them will come under what I have elsewhere*72 called the "Fictitious Mean," that is to say, they are numerical results which do not pretend to represent the character of any existing thing. But average laws would not on this account be less useful, if we could obtain them; for the movements of trade and industry depend upon averages and aggregates, not upon the whims of individuals.

The Law of Indifference.

When a commodity is perfectly uniform or homogeneous in quality, any portion may be indifferently used in place of an equal portion: hence, in the same market, and at the same moment, all portions must be exchanged at the same ratio. There can be no reason why a person should treat exactly similar things differently, and the slightest excess in what is demanded for one over the other will cause him to take the latter instead of the former. In nicely-balanced exchanges it is a very minute scruple which turns the scale and governs the choice. A minute difference of quality in a commodity may thus give rise to preference, and cause the ratio of exchange to differ. But where no difference exists at all, or where no difference is known to exist, there can be no ground for preference whatever. If, in selling a quantity of perfectly equal and uniform barrels of flour, a merchant arbitrarily fixed different prices on them, a purchaser would of course select the cheaper ones; and where there was absolutely no difference in the thing purchased, even an excess of a penny in the price of a thing worth a thousand pounds would be a valid ground of choice. Hence follows what is undoubtedly true, with proper explanations, that in the same open market, at any one moment, there cannot be two prices for the same kind of article. Such differences as may practically occur arise from extraneous circumstances, such as the defective credit of the purchasers, their imperfect knowledge of the market, and so on.

The principle above expressed is a general law of the utmost importance in Economics, and I propose to call it The Law of Indifference, meaning that, when two objects or commodities are subject to no important difference as regards the purpose in view, they will either of them be taken instead of the other with perfect indifference by a purchaser. Every such act of indifferent choice gives rise to an equation of degrees of utility, so that in this principle of indifference we have one of the central pivots of the theory.

Though the price of the same commodity must be uniform at any one moment, it may vary from moment to moment, and must be conceived as in a state of continual change. Theoretically speaking, it would not usually be possible to buy two portions of the same commodity successively at the same ratio of exchange, because, no sooner would the first portion have been bought than the conditions of utility would be altered. When exchanges are made on a large scale, this result will be verified in practice.*73 If a wealthy person invested £100,000 in the funds in the morning, it is hardly likely that the operation could be repeated in the afternoon at the same price. In any market, if a person goes on buying largely, he will ultimately raise the price against himself. Thus it is apparent that extensive purchases would best be made gradually, so as to secure the advantage of a lower price upon the earlier portions. In theory this effect of exchange upon the ratio of exchange must be conceived to exist in some degree, however small may be the purchases made. Strictly speaking, the ratio of exchange at any moment is that of dy to dx, of an infinitely small quantity of one commodity to the infinitely small quantity of another which is given for it. The ratio of exchange is really a differential coefficient. The quantity of any article purchased is a function of the price at which it is purchased, and the ratio of exchange expresses the rate at which the quantity of the article increases compared with what is given for it.

We must carefully distinguish, at the same time, between the Statics and Dynamics of this subject. The real condition of industry is one of perpetual motion and change. Commodities are being continually manufactured and exchanged and consumed. If we wished to have a complete solution of the problem in all its natural complexity, we should have to treat it as a problem of motion—a problem of dynamics. But it would surely be absurd to attempt the more difficult question when the more easy one is yet so imperfectly within our power. It is only as a purely statical problem that I can venture to treat the action of exchange. Holders of commodities will be regarded not as continuously passing on these commodities in streams of trade, but as possessing certain fixed amounts which they exchange until they come to equilibrium.

It is much more easy to determine the point at which a pendulum will come to rest than to calculate the velocity at which it will move when displaced from that point of rest. Just so, it is a far more easy task to lay down the conditions under which trade is completed and interchange ceases, than to attempt to ascertain at what rate trade will go on when equilibrium is not attained.

The difference will present itself in this form: dynamically we could not treat the ratio of exchange otherwise than as the ratio of dy and dx, infinitesimal quantities of commodity. Our equations would then be regarded as differential equations, which would have to be integrated. But in the statical view of the question we can substitute the ratio of the finite quantities y and x. Thus, from the self-evident principle, stated on pp. 91, 92, that there cannot, in the same market, at the same moment, be two different prices for the same uniform commodity, it follows that the last increments in an act of exchange must be exchanged in the same ratio as the whole quantities exchanged. Suppose that two commodities are bartered in the ratio of x for y; then every m^th part of x is given for the m^th part of y, and it does not matter for which of the m^th parts. No part of the commodity can be treated differently to any other part. We may carry this division to an indefinite extent by imagining m to be constantly increased, so that, at the limit, even an infinitely small part of x must be exchanged for an infinitely small part of y, in the same ratio as the whole quantities. This result we may express by stating that the increments concerned in the process of exchange must obey the equation

The use which we shall make of this equation will be seen in the next section.

The Theory of Exchange.

The keystone of the whole Theory of Exchange, and of the principal problems of Economics, lies in this proposition—The ratio of exchange of any two commodities will be the reciprocal of the ratio of the final degrees of utility of the quantities of commodity available for consumption after the exchange is completed. When the reader has reflected a little upon the meaning of this proposition, he will see, I think, that it is necessarily true, if the principles of human nature have been correctly represented in previous pages.

Imagine that there is one trading body possessing only corn, and another possessing only beef. It is certain that, under these circumstances, a portion of the corn may be given in exchange for a portion of the beef with a considerable increase of utility. How are we to determine at what point the exchange will cease to be beneficial? This question must involve both the ratio of exchange and the degrees of utility. Suppose, for a moment, that the ratio of exchange is approximately that of ten pounds of corn for one pound of beef: then if, to the trading body which possesses corn, ten pounds of corn are less useful than one of beef, that body will desire to carry the exchange further. Should the other body possessing beef find one pound less useful than ten pounds of corn, this body will also be desirous to continue the exchange. Exchange will thus go on until each party has obtained all the benefit that is possible, and loss of utility would result if more were exchanged. Both parties, then, rest in satisfaction and equilibrium, and the degrees of utility have come to their level, as it were.

This point of equilibrium will be known by the criterion, that an infinitely small amount of commodity exchanged in addition, at the same rate, will bring neither gain nor loss of utility. In other words, if increments of commodities be exchanged at the established ratio, their utilities will be equal for both parties. Thus, if ten pounds of corn were of exactly the same utility as one pound of beef, there would be neither harm nor good in further exchange at this ratio.

It is hardly possible to represent this theory completely by means of a diagram, but the accompanying figure may, perhaps, render it clearer. Suppose the line pqr to be a small portion of the curve of utility of one commodity, while the broken line p'qr' is the like curve of another commodity which has been reversed and superposed on the other. Owing to this reversal, the quantities of the first commodity are measured along the base line from a towards b, whereas those of the second must be measured in the opposite direction. Let units of both commodities be represented by equal lengths: then the little line áa indicates an increase of the first commodity, and a decrease of the second. Assume the ratio of exchange to be that of unit for unit, or 1 to 1: then, by receiving the commodity áa the person will gain the utility ad, and lose the utility ác; or he will make a net gain of the utility corresponding to the mixtilinear figure cd. He will, therefore, wish to extend the exchange. If he were to go up to the point b', and were still proceeding, he would, by the next small exchange, receive the utility be, and part with b'f; or he would have a net loss of ef. He would, therefore, have gone too far; and it is pretty obvious that the point of intersection, q, defines the place where he would stop with the greatest advantage. It is there that a net gain is converted into a net loss, or rather where, for an infinitely small quantity, there is neither gain nor loss. To represent an infinitely small quantity, or even an exceedingly small quantity, on a diagram is, of course, impossible; but on either side of the line mq I have represented the utilities of a small quantity of commodity more or less, and it is apparent that the net gain or loss upon the exchange of these quantities would be trifling.

Symbolic Statement of the Theory.

To represent this process of reasoning in symbols, let Dx denote a small increment of corn, and Dy a small increment of beef exchanged for it. Now our Law of Indifference comes into play. As both the corn and the beef are homogeneous commodities, no parts can be exchanged at a different ratio from other parts in the same market: hence, if x be the whole quantity of corn given for y, the whole quantity of beef received, Dy must have the same ratio to Dx as y to x: we have then,

   or   

In a state of equilibrium, the utilities of these increments must be equal in the case of each party, in order that neither more nor less exchange would be desirable. Now the increment of beef, Dy, is y/x times as great as the increment of corn, Dx, so that, in order that their utilities shall be equal, the degree of utility of beef must be x/y times as great as the degree of utility of corn. Thus we arrive at the principle that the degrees of utility of commodities exchanged will be in the inverse proportion of the magnitudes of the increments exchanged.

Let us now suppose that the first body, A, originally possessed the quantity a of corn, and that the second body, B, possessed the quantity b of beef. As the exchange consists in giving x of corn for y of beef, the state of things after exchange will be as follows:—

A holds a - x of corn, and y of beef.
B holds x of corn, and b - y of beef.

Let f₁ (a - x) denote the final degree of utility of corn to A, and f₂ x the corresponding function for B. Also let y₁ y denote A's final degree of utility for beef, and y₂ (b - y) B's similar function. Then, as explained on p. 96, A will not be satisfied unless the following equation holds true:—

f₁ (a - x). dx = y₁ y. dy;

or

Hence, substituting for the second member by the equation given on p. 95, we have

What holds true of A will also hold true of B, mutatis mutandis. He must also derive exactly equal utility from the final increments, otherwise it will be for his interest to exchange either more or less, and he will disturb the conditions of exchange. Accordingly the following equation must hold true:—

y₂ (b - y). dy = f₂ x. dx:

or, substituting as before,

We arrive, then, at the conclusion, that whenever two commodities are exchanged for each other, and more or less can be given or received in infinitely small quantities, the quantities exchanged satisfy two equations, which may be thus stated in a concise form—

The two equations are sufficient to determine the results of exchange; for there are only two unknown quantities concerned, namely, x and y, the quantities given and received.

A vague notion has existed in the minds of economical writers, that the conditions of exchange may be expressed in the form of an equation. Thus, J. S. Mill has said:*74 "The idea of a ratio, as between demand and supply, is out of place, and has no concern in the matter: the proper mathematical analogy is that of an equation. Demand and supply, the quantity demanded and the quantity supplied, will be made equal." Mill here speaks of an equation as only a proper mathematical analogy. But if Economics is to be a real science at all, it must not deal merely with analogies; it must reason by real equations, like all the other sciences which have reached at all a systematic character. Mill's equation, indeed, is not explicitly the same as any at which we have arrived above. His equation states that the quantity of a commodity given by A is equal to the quantity received by B. This seems at first sight to be a mere truism, for this equality must necessarily exist if any exchange takes place at all. The theory of value, as expounded by Mill, fails to reach the root of the matter, and show how the amount of demand or supply is caused to vary. And Mill does not perceive that, as there must be two parties and two quantities to every exchange, there must be two equations.

Nevertheless, our theory is perfectly consistent with the laws of supply and demand; and if we had the functions of utility determined, it would be possible to throw them into a form clearly expressing the equivalence of supply and demand. We may regard x as the quantity demanded on one side and supplied on the other; similarly, y is the quantity supplied on the one side and demanded on the other. Now, when we hold the two equations to be simultaneously true, we assume that the x and y of one equation equal those of the other. The laws of supply and demand are thus a result of what seems to me the true theory of value or exchange.

Analogy to the Theory of the Lever.

I have heard objections made to the general character of the equations employed in this book. It is remarked that the equations in question continually involve infinitesimal quantities, and yet they are not treated as differential equations usually are, that is integrated. There is, indeed, no reason why the process of integration should not be applied when it is required, and I will here show that the equations employed do not differ in general character from those which are really treated in many branches of physical science. Whenever, in fact, we deal with continuously varying quantities, the ultimate equations must lie between infinitesimals. The process of integration, if I understand the matter aright, only ascertains other equations, the truth of which follows from the fundamental differential equation.

The mode in which mechanies is usually treated in elementary work tends to disguise the real foundation of the science which is to be found in the so-called theory of virtual velocities. Let us take the description of the lever of the first order as it is given in some of the best modern elementary works, as, for instance, in Mr. Magnus's Lessons in Elementary Mechanics, p. 128. We here read as follows:—

"Let AB be a lever turning freely about C, the fulcrum, and let P be the force applied at A, and W the force exerted, or resistance overcome, or weight raised at B. Suppose the lever turned through the angle ACA', then the work done by P equals P × are AA', and work done by W equals W × arc BB', if P and W act perpendicularly to the arm. Therefore, by the law of energy,

P × AA' = W × BB', and since we have

P × AC = W × BC,

or, P × its arm = W × its arm."

Now, in such a statement as this, we seem to be dealing with plain finite quantities, and there is no apparent difficulty in the matter. In reality the difficulty is only disguised by assuming that P and W act perpendicularly to the arm through finite arcs. This condition is, indeed, carried out with approximate exactness in the problem of the wheel and axle,*75 which may be regarded as combining together an infinite series of straight levers, coming successively into operation. In this machine, therefore, the weights, roughly speaking, always act perpendicularly to arms of invariable length. But, in the generality of cases of the lever, the theory is only true for infinitely small displacements, and no sooner has the lever begun to move through any finite arc AA', than it ceases to be exactly true that the work done by P equals P × arc AA'. Nevertheless, the theory is quite correct as applied to the lever considered statically, that is, as in a state of rest and equilibrium, because the finite arcs of displacement, when it really is displaced, are exactly proportional to the infinitely small arcs, known as virtual velocities, through which it would be displaced, if instead of being at rest, it suffered an infinitely small displacement.

It is curious, moreover, that, when we take the theory of the lever treated according to the principle of virtual velocities, we get equations exactly similar in form to those of the theory of value as established above. The general principle of virtual velocities is to the effect that, if any number of forces be in equilibrium at one or more points of a rigid body, and if this body receive an infinitely small displacement, the algebraic sum of the products of each force into its displacement is equal to zero. In the case of a lever of the first order, this amounts to saying that one force multiplied into its displacement will be neutralised by the other force multiplied into its negative displacement. But inasmuch as the displacements are exactly proportional to the lengths of the arms of the lever, we obtain as a derivative equation, that the forces multiplied each by its own arm are equal to each other. No doubt in the quotation given above, P × AC = W × BC is an equation between finite quantities; but the real equation derived immediately from the principle of virtual velocities, is P × AA' = W × BB', in which P and W are finite, but AA' and BB' are in strictness infinitely small displacements. Let us write this equation in the form

then as we also have

we can substitute; hence

I dwell upon this matter at some length because we here have exactly the forms of the equations of exchange. As we have seen, the original equation is of the general form

where fx and yy represent finite expressions for the degrees of utility of the commodities Y and X, as regards some individual, and dy and dx are infinitesimal quantities of those commodities exchanged. But these infinitesimals may in this case at least be eliminated, because, in virtue of the Law of Indifference, they are exactly proportional to the whole finite quantities exchanged. Hence for dy/dx we substitute y/x. We may write the equations one below the other, so as to make the analogy visible—thus

To put this analogy of the theories of exchange and of the lever in the clearest possible light, I give below a diagram, in which the several economic qualities are represented by the parts of the diagram to which they correspond or are proportional.

Now in statical problems no such process as integration is applicable. The equation lies actually between imaginary infinitesimal quantities, and there is no effect to be summed up. Yet there is no statical problem which is not subject to the principle of virtual velocities, and Poisson, in his Traité de Mécanique, which commences with statical theorems, asserts explicitly,*76 "Dans cet ouvrage, j'emploierai exclusivement la méthode des infiniment petits."

Impediments to Exchange.

We have hitherto treated the theory of exchange as if the action of exchange could be carried on without trouble or cost. In reality, the cost of conveyance is almost always of importance, and it is sometimes the principal element in the question. To the cost of mere transport must be added a variety of charges of brokers, agents, packers, dock, harbour, light dues, etc., together with any customs duties imposed either on the importation or exportation of commodities. All these charges, whether necessary or arbitrary, are so many impediments to commerce, and tend to reduce its advantages. The effect of any one such charge, or of the aggregate of the costs of exchange, can be represented in our formulæ in a very simple manner.

In whatever mode the charges are payable, they may be conceived as paid by the surrender on importation of a certain fraction of the commodity received; for the amount of the charges will usually be proportional to the quantity of goods, and, if expressed in money, can be considered as turned into commodity.

Thus, if A gives x in exchange, this is not the quantity received by B; a part of x is previously subtracted, so that B receives say mx, which is less than x, and the terms of exchange must be adjusted on his part so as to agree with this condition. Hence the second equation will be

Again, A, though giving x, will not receive the whole of y; but say ny, so that his equation similarly will be

The result is, that there is not one ratio of exchange, but two ratios; and the more these differ, the less advantage will there be in exchange. It is obvious that A has either to remain satisfied with less of the second commodity than before, or has to give more of his own in purchasing it. By an obvious transfer of the factors m and n we may state the equations of impeded exchange in the concise form—

Illustrations of the Theory of Exchange.

As stated above, the Theory of Exchange may seem to be of a somewhat abstract and perplexing character; but it is not difficult to find practical illustrations which will show how it is verified in the actual working of a great market. The ordinary laws of supply and demand, when properly stated, are the practical manifestation of the theory. Considerable discussion has taken place concerning these laws, in consequence of Mr. W. T. Thornton's writings upon the subject in the Fortnightly Review, and in his work on the Claims of Labour. Mill, although he had previously declared the Theory of Value to be complete and perfect (see p. 76), was led by Mr. Thornton's arguments to allow that modification was required.

For my own part, I think that most of Mr. Thornton's arguments are beside the question. He suggests that there are no regular laws of supply and demand, because he adduces certain cases in which no regular variation can take place. Those cases might be indefinitely multiplied, and yet the laws in question would not be touched. Of course, laws which assume a continuity of variation are inapplicable where continuous variation is impossible. Economists can never be free from difficulties unless they will distinguish between a theory and the application of a theory. Because, in retail trade, in English or Dutch auction, or other particular modes of traffic, we cannot at once observe the operation of the laws of supply and demand, it is not in the least to be supposed that those laws are false. In fact, Mr. Thornton seems to allow that, if prospective demand and supply are taken into account, they become substantially true. But, in the actual working of any market, the influence of future events should never be neglected, neither by a merchant nor an economist.

Though Mr. Thornton's objections are mostly beside the question, his remarks have served to show that the action of the laws of supply and demand was inadequately explained by previous economists. What constitutes the demand and the supply was not carefully enough investigated. As Mr. Thornton points out, there may be a number of persons willing to buy; but if their highest offer is ever so little short of the lowest price which the seller is willing to take, their influence is nil. If in an auction there are ten people willing to buy a horse at £20, but not higher, their demand instantly ceases when any one person offers £21. I am inclined not only to accept such a view, but to carry it further. Any change in the price of an article will be determined not with regard to the large numbers who might or might not buy it at other prices, but by the few who will or will not buy it according as a change is made close to the existing price.

The theory consists in carrying out this view to the point of asserting that it is only comparatively insignificant quantities of supply and demand which are at any moment operative on the ratio of exchange. This is practically verified by what takes place in any very large market—say that of the Consolidated Three Per Cent Annuities. As the whole amount of the English funds is nearly eight hundred millions sterling, the quantity bought or sold by any ordinary purchaser is inconsiderably small in comparison. Even £1000 worth of stock may be taken as an infinitesimally small increment, because it does not appreciably affect the total existing supply. Now the theory consists in asserting that the market price of the funds is affected from hour to hour not by the enormous amounts which might be bought or sold at extreme prices, but by the comparatively insignificant amounts which are being sold or bought at the existing prices. A change of price is always occasioned by the overbalancing of the inclinations of those who will or will not sell just about the point at which prices stand. When Consols are at 93½, and business is in a tranquil state, it matters not how many buyers there are at 93, or sellers at 94. They are really off the market. Those only are operative who may be made to buy or sell by a rise or fall of an eighth. The question is, whether the price shall remain at 93½, or rise to 93 5/8, or fall to 93 3/8. This is determined by the sale or purchase of comparatively very small amounts. It is the purchasers who find a little stock more profitable to them than the corresponding sum of money who make the price rise by 1/8. When the price of the funds is very steady and the market quiescent, it means that the stocks are distributed among holders in such a way that the exchange of more or less at the prevailing price is a matter of indifference.

In practice, no market ever long fulfils the theoretical conditions of equilibrium, because, from the various accidents of life and business, there are sure to be people every day compelled to sell, or having sudden inducements to buy. There is nearly always, again, the influence of prospective supply or demand, depending upon the political intelligence of the moment. Speculation complicates the action of the laws of supply and demand in a high degree, but does not in the least degree arrest their action or alter their nature. We shall never have a Science of Economics unless we learn to discern the operation of law even among the most perplexing complications and apparent interruptions.

Problems in the Theory of Exchange.

We have hitherto considered only one simple case of the Theory of Exchange. In all other cases where the commodities are capable of indefinite subdivision, the principles will be exactly the same, but the particular conditions may be subject to variation.

We may, firstly, express the conditions of a great market where vast quantities of some stock are available, so that any one small trader will not appreciably affect the ratio of exchange. This ratio is, then, approximately a fixed number, and each trader exchanges at that ratio just so much as suits him. These circumstances may be represented by supposing A to be a trading body possessing two very large stocks of commodities, a and b. Let C be a person who possesses a comparatively small quantity c of the second commodity, and gives a portion of it, y, which is very small compared with b, in exchange for a portion x of a, which is very small compared with a. Then, after exchange, we shall find A in possession of the quantities a - x and b + y, and C in possession of x and c - y. The equations will become

Since a - x and b + y, by supposition, do not appreciably differ from a and b, we may substitute the latter quantities, and we have, for the first equation, approximately,

The ratio of exchange being an approximately fixed ratio determined by the conditions of the trading body A, there is, in reality, only one undetermined quantity, x, the quantity of commodity which C finds it advantageous to purchase by expending part of c. This will now be determined by the equation

This equation will represent the condition in regard to any one distinct commodity of a very small country trading with a much larger one. It might represent, to some extent, the circumstances of trade between the Channel Islands and the great markets of England, though, of course, it is never absolutely verified, because the smallest purchasers do affect the market in some degree. The equation still more accurately represents the position of an individual consumer with regard to the aggregate trade of a large community, since he must buy at the current prices, which he cannot in an appreciable degree affect.

A still simpler formula, however, is needed to represent the conditions of a large part of our purchases. In many cases we want so little of a commodity, that an individual need not give more than a very small fraction of his possessions to obtain it. We may suppose, then, that y in the last problem is a very small part of c, so that y₂ (c - y) does not differ appreciably from y₂c. Taking m as before to be the existing ratio of exchange, we have only one equation—

This means that C will buy of the commodity until its degree of utility falls below that of the commodity he gives. A person's expenditure on salt is in this country an inconsiderable item of expense; what he thus spends does not make him appreciably poorer; yet, if the established price or ratio is one penny for each pound of salt, he buys in any time, say one year, so many pounds of salt that an additional pound would not have so much utility to him as a penny. In the above equation m. y₂c represents the utility to him of a penny, which being an inconsiderable fraction of his possessions, is approximately invariable in utility, and he buys salt until f₂x, which is approximately the utility of the next pound, is equal to, or it may be somewhat less than that of the penny. But this case must not be confused with that of purchases which appreciably affect the possessions of the purchaser. Thus, if a poor family purchase much butchers'-meat, they will probably have to go without something else. The more they buy, the lower the final degree of utility of the meat, and the higher the final degree of utility of something else; and thus these purchases will be the more narrowly limited.

Complex Cases of the Theory.

We have hitherto considered the Theory of Exchange as applying only to two trading bodies possessing and dealing in two commodities. Exactly the same principles hold true, however numerous and complicated may be the conditions. The main point to be remembered in tracing out the results of the theory is, that the same pair of commodities in the same market can have only one ratio of exchange, which must therefore prevail between each body and each other, the costs of conveyance being considered as nil. The equations become rapidly more numerous as additional bodies or commodities are considered; but we may exhibit them as they apply to the case of three trading bodies and three commodities.

Thus, suppose that

A possesses the stock a of cotton, and gives x₁ of it to B, x₂ to C.
B possesses the stock b of silk, and gives y₁ to A, y₂ to C.
C possesses the stock c of wool, and gives z₁ to A, z₂ to B.

We have here altogether six unknown quantities—x₁, x₂, y₁, y₂, z₁, z₂; but we have also sufficient means of determining them. They are exchanged as follows—

A gives x₁ for y₁, and x₂ for z₁.
B gives 'y₁ for x₁, and y₂ for z₂.
C gives z₁ for x₂, and z₂ for y₂.

These may be treated as independent exchanges; each body must be satisfied in regard to each of its exchanges, and we must therefore take into account the functions of utility or the final degrees of utility of each commodity in respect of each body. Let us express these functions as follows—

f1, y1, c1 are the respective functions of utility for	A.
f2, y2, c2 are the respective functions of utility for	B.
f3, y3, c3 are the respective functions of utility for	C.

Now A, after the exchange, will hold a - x₁ - x₂ of cotton and y₁ of silk; and B will hold x₁ of cotton and b - y₁ - y₂ of silk: their ratio of exchange, y₁ for x₁, will therefore be governed by the following pair of equations:—

The exchange of A with C will be similarly determined by the ratio of the degrees of utility of wool and cotton on each side subsequent to the exchange; hence we have

There will also be interchange between B and C which will be independently regulated on similar principles, so that we have another pair of equations to complete the conditions, namely—

We might proceed in the same way to lay down the conditions of exchange between more numerous bodies, but the principles would be exactly the same. For every quantity of commodity which is given in exchange something must be received; and if portions of the same kind of commodity be received from several distinct parties, then we may conceive the quantity which is given for that commodity to be broken up into as many distinct portions. The exchanges in the most complicated case may thus always be decomposed into simple exchanges, and every exchange will give rise to two equations sufficient to determine the quantities involved. The same can also be done when there are two or more commodities in the possession of each trading body.

Competition in Exchange.

One case of the Theory of Exchange is of considerable importance, and arises when two parties compete together in supplying a third party with a certain commodity. Thus, suppose that A, with the quantity of one commodity denoted by a, purchases another kind of commodity both from B and from C, who respectively possess b and c of it. All the quantities concerned are as follows—

A gives x₁ of a to B and x₂ to C,
B gives y₁ of b to A,
C gives y₂ of c to A.

As each commodity may be supposed to be perfectly homogeneous, the ratio of exchange must be the same in one case as in the other, so that we have one equation thus furnished—

(1)

Now, provided that A gets the right commodity in the proper quantity, he does not care whence it comes, so that we need not, in his equation, distinguish the source or destination of the quantities; he simply gives x₁ + x₂, and receives in exchange y₁ + y₂. Observing, then, that by (1)

we have the usual equation of exchange—

(2)

But B and C must both be separately satisfied with their shares in the transaction. Thus

(3) (4)

There are altogether four unknown quantities—x₁, x₂, y₁, y₂; and we have four equations by which to determine them. Various suppositions might be made as to the comparative magnitudes of the quantities b and c, or the character of the functions concerned; and conclusions could then be drawn as to the effect upon the trade. The general result would be, that the smaller holder must more or less conform to the prices of the larger holder.

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.*77 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:—

u1 = the	utility of	A's book to A,
u2 = the	utility of	A's book to B,
v1 = the	utility of	B's book to A,
v2 = the	utility of	B's book to B.

Then the conditions of exchange are simply

v₁> u₁,
u₂> v₂.

We might indeed theoretically contemplate the case where the utilities were exactly equal on one side; thus

v₁> u₁,
u₂ = v₂;

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."*78

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, p₁ q₂, p₂ q₃, etc., represent the total utilities of successive bottles of ink; while the equal spaces o r₁, p₁ r₂, 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 p₃ q₃ q₄ p₄ exceed in area the rectangle p₃ r₃ r₄ p₄.

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.

Negative and Zero Value.

Only a few economists, notably Mr. H. D. Macleod in several of his publications, have noticed the fact that there may be such a thing as negative value. Yet there cannot be the least doubt that people often labour, or pay money to other labourers, in order to get rid of things, and they would not do this unless such things were hurtful, that is, had the opposite quality to utility—disutility. Water, when it gets into a mine, is a costly thing to get out again, and many people have been ruined by wet mines. Quarries and mines usually produce great quantities of valueless rock or earth, variously called duff, spoil, waste, rubbish, and no inconsiderable part of the cost of working arises from the need of raising and carrying this profitless mass of matter and then finding land on which to deposit it. Every furnace yields cinders, dross, or slag, which can seldom be sold for any money, and every household is at the expense of getting rid, in one way or another, of sewage, ashes, swill, and other rejectanea. Reflection soon shows, in short, that no inconsiderable part of the values with which we deal in practical economics must be negative values.

It will hardly be needful to show at full length that this negative value may be regarded as varying continuously in the same way as positive value. If after a long drought rain begins to fall heavily, it is at first hailed as a great benefit; the rain-water may be so valuable as to produce a crop, when otherwise successful agriculture would have been impossible. Rain may thus avert famine; but after the rain has fallen for a certain length of time, the farmer begins to think he has had enough of it; more rain will retard his operations, or injure the growing plants. As the rain continues to fall he fears further injury; water begins to flood his land, and there is even danger of the soil and crops being all washed away together. But the rain unfortunately pours down more and more heavily, until at length perhaps the crops, soil, house, stock,—nay, the farmer himself, are all swept bodily away. That same water, then, which in moderate quantity would have been of the greatest possible benefit, has only to be supplied in greater and greater quantities to become injurious, until it ends with occasioning the ruin, and even the death, of the individual. Those acquainted with the floods and droughts of Australia know that this is no fancy sketch.*79

In many other cases it might be shown similarly that matter, we can hardly call it commodity, acquires a higher and higher degree of disutility the greater the quantity which has to be disposed of. Such is the case with the sewage of great towns, the foul or poisoned water from mines, dye-works, etc. Any obstacle, however, may be regarded as so much discommodity, whether it be a mountain which has to be bored through to make a railway, or a hollow which has to be filled up with an expensive embankment. If a building site requires a certain expenditure in levelling and draining before it can be made use of, the cost of this work is, of course, subtracted from the value which the land would otherwise possess. As every advantage in property gives rise to value, so every disadvantage must be set against that value.

We now come to the question how negative value is to be represented in our equations. Let us suppose a person possessing a of some commodity to find it insufficient: then it has positive degree of utility for him, that is to say f(a) is positive. Suppose x to be added to a and gradually increased: f(a + x) will gradually decrease. Let us assume that for a certain value of x it becomes zero; then, if the further increase of x turns utility into disutility, f(a + x) will become a negative quantity. How will this negative sign affect the validity of the equations which we have been employing in preceding pages, and in which each member has appeared to be both formally and intrinsically positive? It is plain that we cannot equate a positive to a negative quantity; but it will be found that if, at the same time that we introduce negative utility, we also assign to each increment of commodity the positive or negative sign, according as it is added to or subtracted from the exchanger's possessions, that is to say, received or given in exchange, no such difficulty arises.

Suppose A and B respectively to hold a and b, and to exchange dx and dy of the commodities X and Y. Then it will be apparent from the general character of the argument on pp. 98-100, that the fundamental equation there adopted will be included in the more general form—

f(a±x).dx + y (b±y). dy = 0.

In this equation either factor of either term may be intrinsically negative, while the alternative signs before x and y allow for every possible case of giving and receiving in exchange.

Four possible cases will arise. In the first case, both commodities have utility for each person, that is to say, f and y are both positive functions; but A gives some of X in return for some of Y. This means that dx is negative, and dy positive, while the quantities in possession after exchange are a - x, and b+y. Thus the equation becomes

- f (a - x). dx + y (b+y). dy = 0.

We should have merely to transpose the negative term to the other side of the equation, and to assume b = 0, to obtain the equation on p. 99.

As the second case, suppose that Y possesses disutility for A, so that the function y becomes for him negative; in order to get rid of y, he must also pay x with it, and both these quantities as well as dy and dx receive the negative sign. Then the equation takes the shape

f(a-x) × (-dx) - y (b-y) × (-dy) = 0,

or

-f(a-x).dx + y(b-y). dy = 0.

The third case is the counterpart of the last, and represents B's position, who receives both x and y, on the ground that one of these quantities is discommodity to him. But putting the matter as the case of A, we may assume f to be positive, y negative, and giving the positive sign to all of x, y, dx, and dy, we obtain the equation—

f(a + x). dx - y (b+y). dx = 0.

It is possible to conceive yet a fourth case in which people should be exchanging two discommodities; that is to say, getting rid of one hurtful substance by accepting in place of it what is felt to be less hurtful, though still possessing disutility. In this case we have both f and y negative, as well as one of the quantities exchanged; taking x and dx as positive, and y and dy as negative, the equation assumes the form

- f (a + x). dx - y (b-y). (-dy) = 0,

or

- f(a+x). dx + y(b-y). dy = 0.

It might be difficult to discover any distinct cases of this last kind of exchange. Generally speaking, when a person receives assistance in getting rid of some inconvenient possession, he pays in money or other commodity for the service of him who helps to remove the burden. It must naturally be a very rare case that the remover has some burden which it would suit the other party to receive in exchange. Yet the contingency may, and no doubt does, sometimes occur. Two adjacent landowners, for instance. might reasonably agree that, if A allows B to throw the spoil of his mine on A's land, then A shall be allowed to drain his mine into B's mine. It might happen that B was comparatively more embarrassed by the great quantity of his spoil than by water, and that A had room for the spoil, but could not get rid of the water in other ways without great difficulty. An exchange of inconveniences would then be plainly beneficial.

Looking at the equations obtained in the four cases as stated above, it is apparent that the general equation of exchange consists in equating to zero the sum of one positive and one negative term, so that the signs, both of the utility functions and of the increments, may be disregarded. Thus the fundamental equation may be written in the general form

We may express the result of this theory in general terms by saying that the algebraic sum of the utility or disutility received or parted with, as regards the last increments concerned in an act of traffic, will always be zero. It also follows that, without regard to sign, the increments are in magnitude inversely as their degrees of utility or disutility. The reader will not fail to notice the remarkable analogy between this theory and that of the equilibrium of two forces regarded according to the principle of virtual velocities. A rigid lever will remain in equilibrium under the action of two forces, provided that the algebraic sum of the forces, each multiplied by its infinitely small displacement, be zero. Substitute for force degree of utility, positive or negative, and for infinitely small displacements infinitely small quantities of commodity exchanged, and the principles are identical.

It still remains to consider the imaginary case in which substances possess or are supposed to possess neither utility nor disutility, and are yet exchanged in finite quantities. Substituting the ratio of y and x for that of dy and dx, the general equation

will give the value

both the functions of utility being zero. This means that the quantities exchanged will be indeterminate so far as the theory of utility goes. If one substance possesses utility, and the other does not, the ratio of exchange becomes either y/0 or 0/y, infinity or zero, indicating that there can be no comparison in our theory between things which do and those which do not possess utility. Practically speaking, such cases do not occur except in an approximate manner. Such things as cinders, shavings, night soil, etc., have either low degrees of utility or disutility. If the dustman takes them away for nothing, they must have utility for him sufficient to pay the cost of removal. When the dust is riddled, one part is usually found to have utility just sufficient to balance the disutility of the remainder, giving us an instance of the second or third form of the equation of exchange according as we regard the matter from the householder's or the dustman's point of view.

Equivalence of Commodities.

Much confusion is thrown into the statistical investigation of questions of supply and demand by the circumstance that one commodity can often replace another, and serve the same purposes more or less perfectly. The same, or nearly the same, substance is often obtained from two or three sources. The constituents of wheat, barley, oats, and rye are closely similar, if not identical. Vegetable structures are composed mainly of the same chemical compound in nearly all cases. Animal meat, again, is of nearly the same composition from whatever animal derived. There are endless differences of flavour and quality, but these are often insufficient to prevent one kind from serving in place of another.

Whenever different commodities are thus applicable to the same purposes, their conditions of demand and exchange are not independent. Their mutual ratio of exchange cannot vary much, for it will be closely defined by the ratio of their utilities. Beef and mutton, for instance, differ so slightly, that people eat them almost indifferently. But the whole-sale price of mutton, on an average, exceeds that of beef in the ratio of 9 to 8, and we must therefore conclude that people generally esteem mutton more than beef in this proportion, otherwise they would not buy the dearer meat. It follows that the final degrees of utility of these meats are in this ratio, or that if fx be the degree of utility of mutton, and yy that of beef, we have

8. fx = 9. yy.

This equation would doubtless not hold true in extreme circumstances; if mutton became comparatively scarce, there would probably be some persons willing to pay a higher price, merely because it would then be considered a delicacy. But this is certain, that, so long as the equation of utilities holds true, the ratio of exchange between mutton and beef will not diverge from that of 8 to 9. If the supply of beef falls off to a small extent, people will not pay a higher price for it, but will eat more mutton; and if the supply of mutton falls off, they will eat more beef. The conditions of supply will have no effect upon the ratio of exchange; we must, in fact, treat beef and mutton as one commodity of two different strengths, just as gold at eighteen and gold at twenty carats are hardly considered as two but rather as one commodity, of which twenty parts of one are equivalent to eighteen of the other.

It is upon this principle that we must explain, in harmony with Cairnes' views, the extraordinary permanence of the ratio of exchange of gold and silver, which from the commencement of the eighteenth century up to recent years never diverged much from 15 to 1. That this fixedness of ratio did not depend entirely upon the amount or cost of production is proved by the very slight effect of the Australian and Californian gold discoveries, which never raised the gold price of silver more than about 4 2/3 per cent, and failed to have a permanent effect of more than 1½ per cent. This permanence of relative values may have been partially due to the fact, that gold and silver can be employed for exactly the same purposes, but that the superior brilliancy of gold occasions it to be preferred, unless it be about 15 or 15½ times as costly as silver. Much more probably, however, the explanation of the fact is to be found in the fixed ratio of 15½ to 1, according to which these metals are exchanged in the currency of France and some other continental countries. The French Currency Law of the Year XI. established an artificial equation—

Utility of gold = 15½ × Utility of silver;

and it is probably not without some reason that Wolowski and other recent French economists attributed to this law of replacement an important effect in preventing disturbance in the relations of gold and silver.

Since the first edition of this work was published, the views of Wolowski have received striking verification in the unprecedented fall in the value of silver which has occurred in the last three or four years. T