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by u + Au; then it is obvious that the increment of utility Au belongs to the increment of commodity Ax; and if, for the sake of argument, we suppose the degree of utility uniform over the whole of Ax, which is nearly true, owing to its smallness, we shall find the corresponding degree of utility by dividing Au by A.r.

We find these considerations fully illustrated by the last figure, in which oa represents x, and ab is the degree of utility at the point a. Now, if we increase x by the small quantity , or At, the utility is increased by the small rectangle abb'a', or Au; and since a rectangle is the product of its sides, we find that the length of the line ab, the degree of utility, is represented

Au by the fraction

Ax As already explained, however, the utility of a commodity may be considered to vary with perfect continuity, so that we commit a small error in assuming it to be uniform over the whole increment Ax. To avoid this, we must imagine Ar to be reduced to an infinitely small size, Au decreasing with it. The smaller the quantities are the more nearly we shall have a correct expression for ab, the degree of utility at the point a. Thus the limit Au

du of this fraction , or, as it is commonly expressed, is the Ax

dr degree of utility corresponding to the quantity of commodity I. The degree of utility is, in mathematical language, the differential coefficient of u considered as a function of x, and will itself be another function of c.

We shall seldom need to consider the degree of utility except as regards the last increment which has been consumed, or, which comes to the same thing, the next increment which is about to be consumed. I shall therefore commonly use the expression final degree of utility, as meaning the degree of utility of the last addition, or the next possible addition of a very small, or infinitely small, quantity to the existing stock. In ordinary circumstances, too, the final degree of utility will not be great compared with what it might be. Only in famine or other extreme circumstances do we approach the higher degrees

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of utility. Accordingly we can often treat the lower portions of the curves of variation (pbq) which concern ordinary commercial transactions, while we leave out of sight the portions beyond p or q. It is also evident that we may know the degree of utility at any point while ignorant of the total utility, that is, the area of the whole curve. To be able to estimate the total enjoyment of a person would be an interesting thing, but it would not be really so important as to be able to estimate the additions and subtractions to his enjoyment which circumstances occasion. In the same way a very wealthy person may be quite unable to form any accurate statement of his aggregate wealth, but he may nevertheless have exact accounts of income and expenditure, that is, of additions and subtractions.

Variation of the Final Degree of Utility

The final degree of utility is that function upon which the theory of economics will be found to turn. Economists, generally speaking, have failed to discriminate between this function and the total utility, and from this confusion has arisen much perplexity. Many commodities which are most useful to us are esteemed and desired but little. We cannot live without water, and yet in ordinary circumstances we set no value on it. Why is this? Simply because we usually have so much of it that its final degree of utility is reduced nearly to zero. We enjoy every day the almost infinite utility of water, but then we do not need to consume more than we have. Let the supply run short by drought, and we begin to feel the higher degrees of utility, of which we think but little at other times.

The variation of the function expressing the final degree of utility is the all-important point in economic problems. We may state, as a general law, that the degree of utility varies with the quantity of commodity, and ultimately decreases as that quantity increases. No commodity can be named which we continue to desire with the same force, whatever be the quantity already in use or possession. All our appetites are capable of satisfaction or satiety sooner or later, in fact, both these words mean,

etymologically, that we have had enough, so that more is of no use to us. It does not follow, indeed, that the degree of utility will always sink to zero. This may be the case with some things, especially the simple animal requirements, such as food, water, air, etc. But the more refined and intellectual our needs become, the less are they capable of satiety. To the desire for articles of taste, science, or curiosity, when once excited, there is hardly a limit.

Disutility and Discommodity A few words will suffice to suggest that as utility corresponds to the production of pleasure, or, at least, a favorable alteration in the balance of pleasure and pain, so negative utility will consist in the production of pain, or the unfavorable alteration of the balance. In reality we must be almost as often concerned with the one as with the other; nevertheless, economists have not employed any distinct technical terms to express that production of pain which accompanies so many actions of life. They have fixed their attention on the more agreeable aspect of the matter. It will be allowable, however, to appropriate the good English word discommodity, to signify any substance or action which is the opposite of commodity, that is to say, any. thing which we desire to get rid of, like ashes or sewage. Discommodity is, indeed, properly an abstract form signifying inconvenience, or disadvantage; but as the noun commodities has been used in the English language for four hundred years at least as a concrete term, so we may now convert discommodity into a concrete term, and speak of discommodities as substances or things which possess the quality of causing inconvenience or harm. For the abstract notion, the opposite or negative of utility, we may invent the term disutility, which will mean something different from inutility, or the absence of utility. It is obvious that utility passes through inutility before changing into disutility, these notions being related as +, (), and

Distribution of Commodity in Different Uses The principles of utility may be illustrated by considering the mode in which we distribute a commodity when it is capable of several uses. There are articles which may be employed for many distinct purposes : thus, barley may be used either to make beer, spirits, bread, or to feed cattle ; sugаr may be used to eat, or for producing alcohol ; timber may be used in construction, or as fuel ; iron and other metals may be applied to many different purposes. Imagine, then, a community in the possession of a certain stock of barley ; what principles will regulate their mode of consuming it? Or, as we have not yet reached the subject of exchange, imagine an isolated family, or even an individual, possessing an adequate stock, and using some in one way and some in another. The theory of utility gives, theoretically speaking, a complete solution of the question.

Let 8 be the whole stock of some commodity, and let it be capable of two distinct uses. Then we may represent the two quantities appropriated to these uses by x, and Yı, it being a condition that xi +

The person may be conceived as successively expending small quantities of the commodity ; now it is the inevitable tendency of human nature to choose that course which appears to offer the greatest advantage at the moment. Hence, when the person remains satisfied with the distribution he has made, it follows that no alteration would yield him more pleasure, which amounts to saying that an increment of commodity would yield exactly as much utility in one use as in another. Let Aug, Au, be the increments of utility which might arise respectively from consuming an increment of commodity in the two different ways. When the distribution is completed, we ought to have Au, = Aug; or at the limit we have the equation

du, _ duz
dx

= 8.

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which is true when x, y are respectively equal to rı, y. We must, in other words, have the final degrees of utility in the two uses equal.

The same reasoning which applies to uses of the same commodity will evidently apply to any two uses, and hence to all uses simultaneously, so that we obtain a series of equations less numerous by a unit than the number of ways of using the commodity. The general result is that commodity, if consumed by a perfectly wise being, must be consumed with a maximum production of utility.

We should often find these equations to fail. Even when I is equal to 10% of the stock, its degree of utility might still exceed the utility attaching to the remaining ido part in either of the other uses. This would mean that it was preferable to give the whole commodity to the first use. Such a case might perhaps be said to be not the exception but the rule ; for whenever a commodity is capable of only one use, the circumstance is theoretically represented by saying that the final degree of utility in this employment always exceeds that in any other employment.

Under peculiar circumstances great changes may take place in the consumption of a commodity. In a time of scarcity the utility of barley as food might rise so high as to exceed altogether its utility, even as regards the smallest quantity, in producing alcoholic liquors; its consumption in the latter way would then cease. In a besieged town the employment of articles becomes revolutionized. Things of great utility in other respects are ruthlessly applied to strange purposes.

In Paris a vast stock of horses was eaten, not so much because they were useless in other ways, as because they were needed more strongly as food. A certain stock of horses had, indeed, to be retained as a necessary aid to locomotion, so that the equation of the degrees of utility never wholly failed.

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