Front Page Titles (by Subject) Law of the Variation of Utility. - The Theory of Political Economy
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Law of the Variation of Utility. - William Stanley Jevons, The Theory of Political Economy 
The Theory of Political Economy (London: Macmillan, 1888) 3rd ed.
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Law of the Variation of Utility.
Let us now investigate this subject a little more closely. Utility must be considered as measured by, or even as actually identical with, the addition made to a person's happiness. It is a convenient name for the aggregate of the favourable balance of feeling produced—the sum of the pleasure created and the pain prevented. We must now carefully discriminate between the total utility arising from any commodity and the utility attaching to any particular portion of it. Thus the total utility of the food we eat consists in maintaining life, and may be considered as infinitely great; but if we were to subtract a tenth part from what we eat daily, our loss would be but slight. We should certainly not lose a tenth part of the whole utility of food to us. It might be doubtful whether we should suffer any harm at all.
Let us imagine the whole quantity of food which a person consumes on an average during twenty-four hours to be divided into ten equal parts. If his food be reduced by the last part, he will suffer but little; if a second tenth part be deficient, he will feel the want distinctly; the subtraction of the third tenth part will be decidedly injurious; with every subsequent subtraction of a tenth part his sufferings will be more and more serious, until at length he will be upon the verge of starvation. Now, if we call each of the tenth parts an increment, the meaning of these facts is, that each increment of food is less necessary, or possesses less utility, than the previous one. To explain this variation of utility we may make use of space-representations, which I have found convenient in illustrating the laws of Economics in my College lectures during fifteen years past.
Let the line ox be used as a measure of the quantity of food, and let it be divided into ten equal parts to correspond to the ten portions of food mentioned above. Upon these equal lines are constructed rectangles, and the area of each rectangle may be assumed to represent the utility of the increment of food corresponding to its base. Thus the utility of the last increment is small, being proportional to the small rectangle on x. As we approach towards o, each increment bears a larger rectangle, that standing upon III being the largest complete rectangle. The utility of the next increment, II, is undefined, as also that of I, since these portions of food would be indispensable to life, and their utility, therefore, infinitely great.
We can now form a clear notion of the utility of the whole food, or of any part of it, for we have only to add together the proper rectangles. The utility of the first half of the food will be the sum of the rectangles standing on the line oa; that of the second half will be represented by the sum of the smaller rectangles between a and b. The total utility of the food will be the whole sum of the rectangles, and will be infinitely great.
The comparative utility of the several portions is, however, the most important point. Utility may be treated1 as a quantity of two dimensions, one dimension consisting in the quantity of the commodity, and another in the intensity of the effect produced upon the consumer. Now, the quantity of the commodity is measured on the horizontal line ox, and the intensity of utility will be measured by the length of the upright lines, or ordinates. The intensity of utility of the third increment is measured either by q, or and its utility is the product of the units in pp multiplied by those in pq.
But the division of the food into ten equal parts is an arbitrary supposition. If we had taken twenty or a hundred or more equal parts, the same general principle would hold true, namely, that each small portion would be less useful and necessary than the last. The law may be considered to hold true theoretically, however small the increments are made; and in this way we shall at last reach a figure which is undistinguishable from a continuous curve. The notion of infinitely small quantities of food may seem absurd as regards the consumption of one individual; but, when we consider the consumption of a nation as a whole, the consumption may well be conceived to increase or diminish by quantities which are, practically speaking, infinitely small compared with the whole consumption. The laws which we are about to trace out are to be conceived as theoretically true of the individual; they can only be practically verified as regards the aggregate transactions, productions, and consumptions of a large body of people. But the laws of the aggregate depend of course upon the laws applying to individual cases.
The law of the variation of the degree of utility of food may thus be represented by a continuous curve pbq (Fig. IV.), and the perpendicular height of each point of the curve above the line ox, represents the degree of utility of the commodity when a certain amount has been consumed.
Thus, when the quantity oa has been consumed, the degree of utility corresponds to the length of the line ab; for if we take a very little more food, aá, its utility will be the product of aá and ab very nearly, and more nearly the less is the magnitude of aá. The degree of utility is thus properly measured by the height of a very narrow rectangle corresponding
to a very small quantity of food, which theoretically ought to be infinitely small.
[]The theory of dimensions of utility is fully stated in a subsequent section.