310 



BCIENGE, 



[Vol. VIII, No. 191 



relation to value. This may be indicated with 

 sufficient precision in a brief space. One of the 

 first points noticed by economists in the theory of 

 value is that the exchange values of different com- 

 modities are not at all proportioned to their util- 

 ities. The theory advanced by Jevons — and 

 Walras's is substantially identical with it — points 

 out, that while it is true that the aggregate utility 

 of the whole amount of a given kind of commod- 

 ity has no relation to its exchange value, yet in a 

 certain sense commodities do exchange in the 

 ratio of their utilities. The total utility of differ- 

 ent amounts of the same commodity is not pro- 

 portional Lo the amount : as successive equal in- 

 crements are added to the existing quantity, they 

 add less and less to the aggregate utility. Now, 

 what the theory asserts is, that the exchange value 

 of any commodity is determined by the utility 

 which would result from the addition of a small 

 quantity of it to the amount already possessed. 

 Thus commodities do not, indeed, exchange in the 

 ratio of their total utility, but they do exchange 

 in the ratio of their jinal utility ; that is, of the 

 utility of the last small portion produced, or, what 

 is the same thing, of the next small portion that 

 might be produced. The total utility, u, of the 

 whole quantity, x, of a given commodity, is, then, 

 given by an equation, 



u T=. f (x), 



which may be called the utility-equation ; and the 

 exchange value of the commodity is proportional 

 to the derivative of xi with respect to x. We 

 might conceivably obtain the form of the utility- 

 equation of any article from a study of its com- 

 mercial statistics ; but this has not been done for 

 any commodity, and it may be doubted whether 

 it ever can be done — with even the lowest toler- 

 able degree of accuracy — unless, possibly, in some 

 very j)eculiar cases. We do know, however, in 

 practically every case, that f {x) increases with a?, 

 but increases at a diminishing rate ; that it is 

 when a; = 0, and reaches a maximum for some 

 value of X. This last point might at first sight be 

 doubted, for it is equivalent to saying that for 

 eveiy commodity there is a point beyond which 

 the quantity on hand cannot be increased without 

 its becoming a nuisance ; but it is plain that such 

 a point does in general exist, though it may be 

 very far beyond the quantity actually possessed. 



What Launhardt has added to the work of his 

 predecessors is chiefly the discussion of a large 

 number of applications of the general theory, — a 

 discussion which was in most instances made pos- 

 sible only by a special and arbitrary assumption 

 concerning the form of the utility -equation. Since 

 the function ax — hx^ (where a and 6 are positive 



constants) is a very simple function, possessing the 

 properties above mentioned as belonging to the 

 utility-function, — viz.. it is when x is 0, then 

 increases but at a diminishing rate, and reaches a 

 maximum at a certain point, — Launhardt adopts 

 it, stating at tlje outset that he would employ it 

 for purposes of illustration, but insensibly falling 

 into the way of deducing from the assumption of 

 its sufficiency the greater part of his theorems. 

 That the form is not sufficiently general for even 

 the roughest approximation, despite the fact that 

 the choice of different coefficients, a and h, gives 

 a wide range for the different characters of differ- 

 ent commodities, one may easily convince himself. 

 The derivative of ax — hx^ is a — 2&a7: accord- 

 ingly, the exchange value of a unit of any com- 

 modity would be a linear function of the entire 

 quantity of that commodity available ; so that, if 

 we consider any three quantities, x^, x^, x^, such 

 that x.^ is the arithmetical mean of a?i and x.^, the 

 exchange value of the article when the quantity 

 is Xg would necessarily be a mean between its 

 values when the quantity is x^ and x^. This is 

 certainly not even approximately true for com- 

 modities in general ; and this consideration alone 

 would be sufficient to justify us in not accepting 

 the form ax — bx" as sufficiently general for pur- 

 poses of investigation. Indeed, as ah-eady stated, 

 the author seems to have had no deliberate inten- 

 tion of so using it. 



We have dwelt at soma length on this point, 

 because the most striking conclusions in the first 

 section of the book — that devoted to exchange — 

 are dependent upon it. One or two theorems of 

 this kind may be quoted, and they will also serve 

 to indicate the nature of the questions discussed 

 by the author. The theorems are printed in 

 italics, as embodying the net outcome of the 

 mathematical investigations which pi-ecede them. 



" When the merchant is so jjlaced that he can 

 fix his rate of profit at the point most advanta- 

 geous to him, he obtains two-thirds of the entire 

 economic gain accomplished by the exchange, or 

 twice as much as the producer and consumer to- 

 gether. 



"The most advantageous duty is therefore equal 

 to one-third the difference between the price 

 which the domestic goods would bring if there 

 were no importation, and the price at which the 

 foreign goods could be sold with no profit to the 

 producer." 



The simplicity of these results is equalled by 

 their unreliability. It is not very surprising that 

 a simple result should be reached from a mathe- 

 matical hypothesis so much simpler than the facts 

 warrant, even for the purposes of the purest 

 theory ; but, in spite of the small value of the re- 



