be regulated. For our purposes wants are simply motives of varying power which universally exist, and the laws of which we propose to investigate. We have to deal with them merely as forces, without any other estimate of their characters than the intensity with which they are felt by the persons who experience them. Nor are we any more concerned to appreciate the character of the means of enjoyment than we are to appreciate the character of the want. It is enough that the want is felt, and that it can be satisfied.

2. The Theory of Utility1

Utility is not an Intrinsic Quality

My principal work now lies in tracing out the exact nature and conditions of utility. It seems strange indeed that economists have not bestowed more minute attention on a subject which doubtless furnishes the true key to the problem of

economics.

In the first place, utility, though a quality of things, is no inherent quality. It is better described as a circumstance of things arising out of their relation to man's requirements. As Senior most accurately says, "Utility denotes no intrinsic quality in the things which we call useful; it merely expresses their relations to the pains and pleasures of mankind." We can never, therefore, say absolutely that some objects have utility and others have not. The ore lying in the mine, the diamond escaping the eye of the searcher, the wheat lying unreaped, the fruit ungathered for want of consumers, have no utility at all. The most wholesome and necessary kinds of food are useless unless there are hands to collect and mouths to eat them sooner or later. Nor, when we consider the matter closely, can we say that all portions of the same commodity possess equal utility. Water, for instance, may be roughly described as the most useful of all substances. A quart of water per day has the high utility of saving a person from dying in a most distressing manner. Several

1 By W. S. Jevons. Reprinted from Jevons's Theory of Political Economy, third edition [London, 1888].

gallons a day may possess much utility for such purposes as cooking and washing; but after an adequate supply is secured for these uses, any additional quantity is a matter of comparative indifference. All that we can say, then, is that water, up to a certain quantity, is indispensable; that further quantities will have various degrees of utility; but that beyond a certain quantity the utility sinks gradually to zero; it may even become negative, that is to say, further supplies of the same substance may become inconvenient and hurtful.

Exactly the same considerations apply more or less clearly to every other article. A pound of bread per day supplied to a person saves him from starvation, and has the highest conceiv able utility. A second pound per day has also no slight utility; it keeps him in a state of comparative plenty, though it be not altogether indispensable. A third pound would begin to be superfluous. It is clear, then, that utility is not proportional to commodity: the very same articles vary in utility according as we already possess more or less of the same article. The like may be said of other things. One suit of clothes per annum is necessary, a second convenient, a third desirable, a fourth not unacceptable, but we sooner or later reach a point at which further supplies are not desired with any perceptible force unless it be for subsequent use.

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 favorable 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.

q'

Let the line or 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 repre

b

04

III IV v a vi VII VIII IX X

X

sent 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 1, 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. Utility may be treated 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 pq, or p'q', 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 consump

y

b b'

m

a a'

q

n

X

tions 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, and the perpendicular height of each point at the curve above the line or 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, aa', its utility will be the product of aa' and ab very nearly, and more nearly the less is the magnitude of aa'. 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.

Total Utility and Degree of Utility

We are now in a position to appreciate perfectly the difference between the total utility of any commodity and the degree of utility of the commodity at any point. These are, in fact, quantities of altogether different kinds, the first being represented by an area, and the second by a line. We must consider how we may express these notions in appropriate mathematical language.

Let a signify, as is usual in mathematical books, the quantity which varies independently, in this case the quantity of commodity. Let u denote the whole utility proceeding from the consumption of x. Then u will be, as mathematicians say, a function of x; that is, it will vary in some continuous and regular, but probably unknown, manner, when x is made to vary. Our great object at present, however, is to express the degree of utility.

Mathematicians employ the sign ▲ prefixed to a sign of quantity, such as x, to signify that a quantity of the same nature as x, but small in proportion to x, is taken into consideration. Thus ▲x means a small portion of x, and x + ▲x is therefore a quantity a little greater than x. Now when x is a quantity of commodity, the utility of x + Ax will be more than that of x as a general rule. Let the whole utility of x + Ax be denoted