Front Page Titles (by Subject) Symbolic Statement of the Theory. - The Theory of Political Economy
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Symbolic Statement of the Theory. - William Stanley Jevons, The Theory of Political Economy 
The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.
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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,
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 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 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 f2x the corresponding function for B. Also let y1y 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:—
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 = f2x.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:1 "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.
[]Principles of Political Economy, book iii., chap. ii. sec. 4.