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.
The accepted Theory of Rent, as given above, needs little or no alteration to adapt it to expression in mathematical symbols. For doses or increments of capital I shall substitute increments of labour, partly because the functions of capital remain to be considered in the next chapter, and partly because James Mill, J. S. Mill, and MacCulloch hold the application of capital to be synonymous with the application of labour. This assumption is implied in James Mill's statement (p. 13); it is expressly stated in J. S. Mill's First Fundamental Proposition concerning the Nature of Capital;1 and MacCulloch adds a footnote2 to make it clear, that as all capital was originally produced by labour, the application of additional capital is the application of additional labour. "Either the one phrase or the other may be used indiscriminately." This doctrine is in itself altogether erroneous, but it will not be erroneous to assume as a mode of simplifying the problem that the increments of labour applied are equally assisted by capital. It is a separate and subsequent problem to determine how rent or interest arises when the same labour is assisted by different quantities of capital.
I shall suppose that a certain labourer, or, what comes to exactly the same thing, a body of labourers, expend labour on several different pieces of ground. On what principle will they distribute their labour between the several pieces? Let us imagine that a certain amount has been spent upon each, and that another small portion, Dl, is going to be applied. Let there be two pieces of land, and let Dx1, Dx2, be the increments of produce to be expected from the pieces respectively. They will naturally apply the labour to the land which yields the greatest result. So long as there is any advantage in one use of labour over another, the most advantageous will certainly be adopted. Therefore, when they are perfectly satisfied with the distribution made, the increment of produce to the same labour will be equal in each case; or we have
To attain scientific accuracy, we must decrease the increments infinitely, and then we obtain the equation—
Now represents the ratio of produce, or the productiveness of labour, as regards the last increment of labour applied. We may say, then, that whenever a labourer or body of labourers distribute their labour over pieces of land with perfect economy, the final ratios of produce to labour will be equal.
We may now take into account the general law, that when more and more labour is applied to the same piece of land, the produce ultimately does not increase proportionately to the labour. This means that the function dx/dl diminishes without limit towards zero after x has passed a certain quantity. The whole produce of a piece of land is x, the whole labour spent upon it is l; and x varies in some way as l varies, never decreasing when l increases. We may say, then, that x is a function of l; let us call it Pl. When a little more labour is expended, the increment of produce dx is dPl, and is the final rate of production, the same as was previously denoted by .
In the Theory of Labour it was shown that no increment of labour would be expended unless there was sufficient recompense in the produce, but that labour would be expended up to the point at which the increment of utility exactly equals the increment of pain incurred in acquiring it. Here we find an exact definition of the amount of labour which will be profitably applied.
It was also shown that the last increment of labour is the most painful, so that if a person is recompensed for the last increment of labour which he applies to land by the rate of production , it follows that all the labour he applies might be recompensed sufficiently at the same rate. The whole labour is l, so that if the recompense were equal over the whole, the result would be . Consequently, he obtains more than the necessary return to labour by the amount
or, as we may write it,
Pl-l · P'l,
in which P'l is the differential coefficient of Pl, or the final rate of production. This expression represents the advantage he derives from the possession of land in affording him more profit than other methods of employing his labour. It is therefore the rent which he would ask before yielding it up to another person, or equally the rent which he would be able and willing to pay if hiring it from another.
The same considerations apply to every piece of land cultivated. When the same person or body of labourers cultivates several pieces, P'l will be of the same magnitude in each case, but the quantities of labour, and possibly the functions of labour, will be different. Thus with two pieces of land the rent may be represented as
P1l1 + P2l2 - (l1 + l2) P1'l1;
or, speaking generally of any number of pieces, it is the sum of the quantities of the form Pl, minus the sum of the quantities of the form l.P'l.
[]Book i., chap. v. sec. I.
[]Wealth of Nations, p. 445.