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Irving Fisher—Mathematical investigations

face and a uniform thickness of unity so that the volume of liquidcontained is always equal numerically to the area on the face. Hencethe amount of liquid in the cistern may represent commodity and thedistance of its surface from O, its marginal utility.

ONE COMMODITY (A)—ONE CONSUMER (I).

Let fig. 2 represent the utility cistern for I relative to A. Letus select as a unit of utility the marginal utility of money supposingthis to be constant. Thus the cistern is (say) one inch in thickness ;the number of cubic inches of water represents the number of unitsof the commodity (yards, gallons, or pounds, etc.) consumed by theindividual during a given period (say a year) and the ordinate OR(in inches) represents the number of dollars at which the individualprizes the last yard or gallon (say) of the commodity.

Since the market is large enough to prevent any conscious influ-ence on the price by the individual I, he acts with reference to afixed price (p dollars). He will therefore consume such an amountof A that its marginal utility in dollars equals that of the price p,that is, the cistern will be filled till OR=/>. This is evident, forif less should be consumed OR would be greater than p, that is, alittle more commodity would be valued more highly than the dollarsexchanged for it and so would be purchased, and if more should beconsumed, reverse considerations hold.

If the price rises OR will increase and less be consumed but if itfalls more. If the price falls to zero as is the case for water and airthe quantity consumed fills the whole cistern up to the horizontalaxis. This volume is therefore the quantity of maximum satisfaction.If the price rises to OA the individual will cease consuming. Thisprice is therefore the limiting maximum price at which he will buy.

The liquid contents of the cistern may be regarded as made up ofsuccessive horizontal infinitesimal layers each representing an incre-ment of commodity. The height or distance of each layer from theorigin represents the degree of utility of that layer. The last or toplayer is on the margin of the whole and its vertical distance fromthe origin is the degree of utility of that marginal layer or incre-ment of the commodity or briefly its marginal utility. Thus themargin of consumption has in the cistern an actual physical analogue.