686 



EEPOET 1889. 



contending that in case of ' any increase of product resulting from the introduction 

 of any new force into industry,' the whole increment wiU fall to be added to the 

 share of the working class ; he argues, quite correctly upon his premisses, that if 

 the improvement does not ' increase the amount of tools and supplies required in 

 production ' — since ' there is no greater demand for capital in the case supposed — 

 .... there can be no increase in the rate or amount of interest ' (' Quarterlj' 

 Journal of Economics,' 1887, pp. 283, 284). Analytically we should find that the 

 variation in the rate of interest due to the disturbance of equilibrium, say At, was 

 indefinitely small as compared with the variation in the rate of wages, say A&j, 

 because the decrease in the rate at which the utility of capital increases is in- 

 definitely great. The argument requires that this second differential should be 

 infinite at the position of equilibrium. 



(e) Complex Exchange is the general case of Simplex Exchange above ana- 

 lysed. 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 



Fig. 5. 



to maximise his own advantage, subject to tlie conditions (1) 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 articles. And the first 

 article being taken as the measure of value, let the prices of the remaining articles 

 be p.i, Ps, . . . p„. Let the quantities of commodities bought or sold by any indivi- 

 dual, say No. r, be a.v„ Xr., .... a?™; each variable with its sign : plus, if bought, 

 minus, if sold. Let the advantage of the individual, regarded as a function of liis 

 purchases and sales, be ^r(.Xr\, x,-^ .... x,-,,). There is sought the system of values 

 assigned to the variables for which tliis function is a maximum, subject (a) to the condi- 

 tion which follows from the first assumption above made : x^, +p„x,.2 + 4'C. +PnXrv = o. 

 In order to determine the maximum of tf/,. subject to this condition, we obtain ()3) by 

 the Calculus of Variations (»- 1) equations of the form — 



\dXr, / p\.da:,., / 



p,\dXm / 



