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.
In attempting to represent these conditions of labour with accuracy, we shall find that there are no less than four quantities concerned; let us denote them as follows:—
t = time, or duration of labour.
l = amount of labour, as meaning the aggregate balance of pain accompanying it, irrespective of the produce.
x = amount of commodity produced.
v = total utility of that commodity.
The amount of commodity produced will be very different in different cases. In any one case the rate of production will be determined by dividing the whole quantity produced by the time of production, provided that the rate of production has been uniform; it will then be . But if the rate of production be variable, it can only be determined at any moment by comparing a small quantity of produce with the small portion of time occupied in its production. More strictly speaking, we must ascertain the ratio of an infinitely small quantity of produce to the corresponding infinitely small portion of time. Thus the rate of production is properly denoted by or at the limit by .
Again, the degree of painfulness of labour would be if it remained invariable; but as it is highly variable, we must again compare small increments, and or, at the limit, correctly represents the degree of painfulness of labour. But we must also take into account the fact that the utility of commodity is not constant. If a man works regularly twelve hours a day, he will produce more commodity than in ten hours; therefore the final degree of utility of his commodity, whether he consume it himself or whether he exchange it, will not be quite so high as when he produced less. This degree of utility is denoted, as before, by the ratio of the increment of utility to the increment of commodity.
The amount of reward of labour can now be expressed; for it is
that is to say, it is the product of the ratio of the commodity produced to the time, multiplied by the ratio of the utility to the amount of produce. Thus, the last two hours of work in the day generally gives less reward, both because less produce is then created in proportion to the time spent, and because that produce is less necessary and useful to one who makes enough to support himself in the other ten hours.
We can now ascertain the length of time which should be selected as the most advantageous term of labour. A free labourer endures the irksomeness of work because the pleasure he expects to receive, or the pain he expects to ward off, by means of the produce, exceeds the pain of exertion. When labour itself is a worse evil than that which it saves him from, there can be no motive for further exertion, and he ceases. Therefore he will cease to labour just at the point when the pain becomes equal to the corresponding pleasure gained; and we thus have t defined by the equation
In this, as in the other questions of Economics, all depends upon the final increments, and we have expressed in the above formula the final equivalence of labour and utility. A man must be regarded as earning all through his hours of labour an excess of utility; what he produces must be considered not merely the exact equivalent of the labour he gives for it, for it would be, in that case, a matter of indifference whether he laboured or not. As long as he gains, he labours, and when he ceases to gain, he ceases to labour.
In some cases, as in some kinds of machine labour, the rate of production is uniform, or nearly so, and by choice of suitable units may be made equal to unity; the result may then be put more simply in this way. Labour may be considered as expended in successive small quantities, Dl, each lasting, for instance, for a quarter of an hour; the corresponding benefit derived from the labour will then be denoted by Du. Now, so long as Du exceeds in amount of pleasure the negative quantity or pain of Dl, the difference of sign being disregarded, there will be gain inducing to continued labour. Were Du to fall below Dl, there would be more harm than good in labouring; therefore, the boundary between labour and inactivity will be defined by the equality of Du and Dl, and at the limit we have the equation