48

Irving Fisher—Mathematical investigations

If the price of A rises slightly and of B falls relatively more whilethe breadth of IIA is less than of IIB, the valuation of money to IIwill fall. For if not then the ordinates of IIA. and IIB must changepari passu with the thickness of the back cisterns. The thicknessand ordinate for IIA are, say each increased 10$ and for IIBreduced 50$. There is clearly not room in IIA for all the moneypoured 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 andapproximate economic theorems. There are no such delicate adjust-ments in the actual world as here presented, but through ideal caseswe study real tendencies.

4. Depress stopper A. The chief effect will be to lower the priceof A. If it is a necessary* * a relatively large share of the increasewill go to the poor. It will probably occur that while the totalmoney expenditure by the poor for this commodity will increase,that for the rich will decrease .f The marginal utility of money ingeneral decreases especially for the poor man.

Most other commodities will rise in price if A decreases in pricefaster than if Increases in quantity For there will be a saving inthe expenditure for A which must be made up elsewhere. But anexceptional commodity may fall in price. Thus if B happens to beextensively used (cisterns deep and broad) only by those who use Aslightly, these persons will not save materially in the expenditure

Hence xkp'^—xkp^ = xjcp^ — xjvp™

Hence x a : x h :: P^-P^ : P^-pJ,

which is the condition required. More generally in order that the valuation ofmoney to an individual shall not change, the cisterns of II must be so formedthat when the money saved on some articles equals the extra spent on the othersthe ordinates may all change proportionally with the prices. If the ordinatesincrease more than this requirement or decrease less, the valuation of moneywill rise. In the reverse cases it will fall.

* A necessary may be defined as a commodity whose cistern is relatively deepand narrow. I. e. a very small quantity has a. very great utility and a slightaddition gives satisfaction very rapidly. A luxury has the reverse properties.

f When commodity begins to flow into a cistern its money value (the contentsof the back cistern) increases in about the same rate as the commodity—it matterslittle how much the price (thickness) falls. Contrariwise when the cistern isnearly full a fall of price decreases the money value at about the same rate—theincrease of commodity matters little. The dividing point is where the commod-ity increases at the same rate as the price decreases. These characters are moreplainly shown in the diagrams of Auspitz und Lieben, p. 48, etc.