48 Irving Fisher — Mathematical investigations 



If the price of A rises slightly and of B falls relatively more while 

 the breadth of IIA is less than of IIB, the valuation of money to II 

 will fall. For if not then the ordinates of IIA and IIB must chaugre 

 2?ari 2^cissu with the thickness of the back cisterns. The thickness 

 and ordinate for IIA are, say each increased 10^^ and for IIB 

 reduced 50^r. There is clearly not room m IIA for all the money 

 poured out of IIB. This surplus will spread over all A, B, C, etc., 

 and reduce the ordinates and reduce the money valuation of II. 



These artificially exact cases obviously stand for more general and 

 approximate economic theorems. There are no such delicate adjust- 

 ments in the actual world as here presented, but through ideal cases 

 we study real tendencies. 



4. Depress stopper A. The chief effect will be to lower the price 

 of A. If it is a necessary* a relatively large share of the increase 

 will go to the poor. It will probably occur that while the total 

 money expenditure by the poor for this commodity will increase, 

 that for the rich will decrease.\ The marginal utility of money in 

 general decreases especially for the poor man. 



Most other commodities will rise in price if A decreases in price 

 faster than it increases in quantity. For there will be a saving in 

 the expenditure for A which must be made up elsewhere. But an 

 exceptional commodity may fall in price. Thus if B happens to be 

 extensively used (cisterns deep and broad) only by those who use A 

 slightly, these persons will not save materially in the expenditure 



Hence :rJ.p/^-xkp^ = x^hp^^-x^kp^'K 



Hence x^ : x, y. p^J-p^ : p/'-pJ', 



whicli is the condition required. More generally in order that the valuation of 

 money to an individual shall not change, the cisterns of 11 must be so formed 

 that when the money saved on some articles equals the extra spent on the others 

 the ordinates may all change proportionally with the prices. If the ordinates 

 increase more than this requirement or decrease less, the valuation of money 

 will rise. In the reverse cases it will fall. 



* A necessary may be defined as a commodity whose cistern is relatively deep 

 and narrow, I. e. a very small quantity has a very great utility and a slight 

 addition gives satisfaction very rapidly. A luxury has the reverse properties. 



f "WTien commodity begins to flow into a cistern its money value (the contents 

 of the back cistern) increases in about the same rate as the commodity — it matters 

 little how much the price (thickness) falls. Contrariwise when the cistern is 

 nearly full a fall of price decreases the money value at about the same rate — the 

 increase of commodity matters little. The dividing point is where the commod- 

 ity increases at the same rate as the price decreases. These characters are more 

 plainly shown in the diagrams of Auspitz und Lieben, p. 48, etc. 



