The Online Library of Liberty

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• Preface to the First Edition (1871)
• Preface to the Second Edition (1879)
• Preface to the Third Edition By Harriet Jevons
• Errata
• Chapter I: Introduction
• Mathematical Character of the Science.
• Confusion Between Mathematical and Exact Sciences.
• Capability of Exact Measurement.
• Measurement of Feeling and Motives.
• Logical Method of Economics.
• Relation of Economics to Ethics.
• Chapter II: Theory of Pleasure and Pain
• Pleasure and Pain As Quantities.
• Pain the Negative of Pleasure.
• Anticipated Feeling.
• Uncertainty of Future Events.
• Chapter III: Theory of Utility
• Definition of Terms.
• The Laws of Human Want.
• Utility Is Not an Intrinsic Quality.
• Law of the Variation of Utility.
• Total Utility and Degree of Utility.
• Variation of the Final Degree of Utility.
• Disutility and Discommodity.
• Distribution of Commodity In Different Uses.
• Theory of Dimensions of Economic Quantities.
• Actual, Prospective, and Potential Utility.
• Distribution of a Commodity In Time.
• Chapter IV: Theory of Exchange
• Importance of Exchange In Economics.
• Ambiguity of the Term Value.
• Value Expresses Ratio of Exchange.
• Popular Use of the Term Value.
• Dimension of Value.
• Definition of Market.
• Definition of Trading Body.
• The Law of Indifference.
• The Theory of Exchange.
• Symbolic Statement of the Theory.
• Analogy to the Theory of the Lever.
• Impediments to Exchange.
• Illustrations of the Theory of Exchange.
• Problems In the Theory of Exchange.
• Complex Cases of the Theory.
• Competition In Exchange.
• Failure of the Equations of Exchange.
• Negative and Zero Value.
• Equivalence of Commodities.
• Acquired Utility of Commodities.
• The Gain By Exchange.
• Numerical Determination of the Laws of Utility.
• Opinions As to the Variation of Price.
• Variation of the Price of Corn.
• The Origin of Value.
• Chapter V: Theory of Labour
• Definition of Labour.
• Quantitative Notions of Labour.
• Symbolic Statement of the Theory.
• Dimensions of Labour.
• Balance Between Need and Labour.
• Distribution of Labour.
• Relation of the Theories of Labour and Exchange.
• Relations of Economic Quantities.
• Various Cases of the Theory.
• Joint Production.
• Over-production.
• Limits to the Intensity of Labour.
• Chapter VI: Theory of Rent
• Accepted Opinions Concerning Rent.
• Symbolic Statement of the Theory.
• Illustrations of the Theory.
• Chapter VII: Theory of Capital
• The Function of Capital.
• Capital Is Concerned With Time.
• Quantitative Notions Concerning Capital.
• Expression For Amount of Investment.
• Dimensions of Capital, Credit and Debit.
• Effect of the Duration of Work.
• Illustrations of the Investment of Capital.
• Fixed and Circulating Capital.
• Free and Invested Capital.
• Uniformity of the Rate of Interest.
• General Expression For the Rate of Interest.
• Dimension of Interest.
• Peacock On the Dimensions of Interest.
• Tendency of Profits to a Minimum.
• Advantage of Capital to Industry.
• Are Articles In the Consumers' Hands Capital?
• Chapter VIII: Concluding Remarks
• The Doctrine of Population.
• Relation of Wages and Profit.
• Professor Hearn's Views.
• The Noxious Influence of Authority.
• Appendix I
• Appendix Ii

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 x 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 D 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 Dx means a small portion of x, and x + Dx is therefore a quantity a little greater than x. Now, when x is a quantity of commodity, the utility of x + Dx will be more than that of x as a general rule. Let the whole utility of x + Dx be denoted by u + Du; then it is obvious that the increment of utility Du belongs to the increment of commodity Dx; and if, for the sake of argument, we suppose the degree of utility uniform over the whole of Dx, which is nearly true owing to its smallness, we shall find the corresponding degree of utility by dividing Du by Dx.

We find these considerations fully illustrated by Fig. IV., 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 aá, or Dx, the utility is increased by the small rectangle abb'a', or Du; 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 by the fraction .

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 Dx. To avoid this we must imagine Dx to be reduced to an infinitely small size, Du 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 of this fraction or, as it is commonly expressed, , is the degree of utility corresponding to the quantity of commodity x.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 x.

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 of utility. Accordingly, we can often treat the lower portions of the curves of variation (pbq, Fig. IV.) 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.