Sept. 19, 1889] 



NATURE 



503 



of the " indijfference-curve" employed by the present writer 

 (*' Mathematical Psychics," p. 21). Also the lines /rand/'S are 

 particular cases of the "preference-curve" {ibid. p. 22). If 

 these more general conceptions are employed, the demonstra- 

 tion will not require that we should put the 

 ordinate for money, regarded as a constant 

 measure of utility. The interpretation assigned 

 to the curves OG and OE in our second and 

 third figures may still stand. 



{d) Economic Equilibrium.— By analogy 

 with well-known physical principles, economic 

 equilibrium may be regarded as determined 

 by the condition that the advantage of all 

 parties concerned, the integrated utility of the 

 whole economic system, should be a maximum. 

 This maximum is in general subject, or in 

 technical phrase r^/«/iZ'^, to certain conditions ; 

 in particular what Jevons called the "law of 

 indifference," that in a market all portions of 

 a commodity shall be exchanged at the same 

 rate. But occasionally this condition is sus- 

 pended ; as often as we take what may be 

 called a socialistic or utilitarian view as dis- 

 tinguished from that incommensurability of 

 pleasures appertaining to different persons, 

 which Jevons in a remarkable passage of his 

 "Theory" (p. 15) has postulated. It will be 

 found that this postulate must be abandoned 

 when we consider the gain of trade, as in our 

 note {c), or the theory of combinations, as in 

 note (/), and on other occasions. 



In general, the first condition of a maximum, 

 that the first term of variation should vanish, 

 gives the Jevonian equations of exchange, the 

 demand-curves of other writers. 



The second condition of a maximum, that 

 the second term of variation should be negative, 

 finds its fulfilment in certain well-known pro- 

 positions which involve the conception of a de- 

 creasing rate of increase, viz. the law of dimin- 

 ishing returns, the law, or laws, of diminishing 

 utility and increasing fatigue. 



For some propositions it is proper to take 

 account not only of the sign, but also the 

 magnitude, of the second differential of utility. 

 Thus, when Prof. Walker is contending that 

 in case of "any increase of product resulting 

 from the introduction of any new force into 

 industry," the whole increment will fall to be 

 added to the share of the working class, he 

 argues, quite correctly upon his premises, 

 that if the improvement does not "increase 

 the amount of tools and supplies required in 

 production" — since "there is no greater de- 

 mand for capital in the case supposed— . . , 

 there can be no increase in the rate or amount 

 of interest " {Quarterly Journal of Economics, 

 1887, pp. 283, 284). Analytically we should 

 find that the variation in the rate of interest 

 due to the disturbance of the equilibrium, say 

 At, was indefinitely small as compared with 

 the variation in the rate of wages, say Aw, 

 because the decrease in the rate at which the 

 utility of capital increases is indefinitely great. 

 The argument requires that this second difi"er- 

 ential should be infinite at the position of 

 equilibrium — a postulate which may perhaps 

 give us pause. 



{e) Complex Exchange is the general 

 case of simplex exchange above analyzed. 

 We have now several, instead of two, 

 categories of dealers and commodities. In 

 both cases equilibrium is determined upon 

 the principle that each individual seeks to 

 maximize his own advantage, subject to the 

 conditions (i) that in a market there is only one price for any 

 article, and (2) that all which is bought is sold, and all which 

 is sold is bought. Let there be m dealers and n artitles. 

 And the first article being taken as the measure of value. 



let the prices of the remaining articles be /j, p^, . . . pn. 

 Let the quantities of commodities bought or sold by any 

 individual, say No. r, be Xr\, x^, . . . Xm \ each variable 

 with its sign : plus, if bought, ininus, if sold. Let the 



Fig 4. 



Fig. 5. 



advantage of the individual, regarded as a function of his pur- 

 chases and sales, be »('r(-^n> •^'•j • • . ■x^rn)- There is sought the 

 system of values assigned to the variables for which this function 

 is a maximum, subject (o) to the condition which follows from 



