46 Irving Fisher — Mathematical investigations 



decrease. He is benefited not injured by the increase of income of 

 his neighbor I. 



3. Press stopper I and raise III. I, II, III now represent a wealthy 

 middle class and poor man respectively. We observe first that this 

 change causes the poor man to relinquish entirely some things 

 (luxuries) as C while decreasing his necessaries slightly; second that 

 the rich man increases his luxuries enormously and his necessaries 

 slightly, and thirdly that slight modifications will appear in the 

 prices and hence in the middle-class consumption. 



The nature of the effect on prices depends on the character of the 

 cisterns of I and III, and on the magnitude of the changes in their 

 incomes. In order that prices may not change, one condition (neces- 

 sary but not sufficient) is that the amount of money income added 

 to I must equal that taken from III, for if the amounts of com- 

 modities are not to change, nor their prices, their total values cannot. 

 If all prices rise it proves a net increase of money income in the 

 whole system. 



If the increase of income of I equals the decrease of that of III, 

 so that the total money value in the market is unchanged, and if 

 furthermore all the cisterns of I and III have straight walls on the 

 right and have their breadths* proportional, there will be no change 

 in price. For if the cistern breadths of the III row are each, (say) 

 half the corresponding ones in the I row, equilibrium will clearly be 

 satisfied by shortening each ordinate of the I row by a uniform per- 

 centage (say 10^), and lengthening those of the III row by just 

 twice the amount of shortening in the corresponding I ordinates. 

 This will evidently cause the lengthening of the III ordinates to be 

 uniform (say 15^). The ratio of marginal utilities has thus been 

 preserved and hence the prices. Obviously the contents added to 

 lA equals that taken from IIIA and equilibrium is reestablished by a 

 simple transfer from III to I. In this case there is no effect on II 

 or any individual save I and III. 



* The breadth of a cistern is evidently the differential of its area divided by 

 the differential of the ordinate that is the fluxion of commodity in reference to its 

 'marginal utility. It is a magnitude important in the discussion of distribution 

 of commodities. Involving as it does the second differential of utility it has no 

 perfectly distinct recognition in popular language. A narrow cistern means that 

 a slight reduction of its contents causes its ordinate to increase much, i. e. causes 

 it to be greatly desired. The individual is very sensitive to a change in that 

 commodity. He misses a little less of it and appreciates a little more. Eeversely 

 a broad cistern signifies that it is hard to satisfy the man by increase and hard 

 to annoy him by decrease. These two sorts of cisterns may be called " sensitive " 

 and " callous" (see Appendix I). 



