26 Irving Fisher — Mathematical i7iv estimations 



face and a uniform thickness of unity so that the vohime of liquid 

 contained is always equal numerically to the area on the face. Hence 

 the amount of liquid in the cistern \n2ij represent commodity and the 

 distance 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. Let 

 us select as a unit of utility the marginal utility of money supposing 

 this to be constant. Thus the cistern is (say) one inch in thickness ; 

 the number of cubic inches of water represents the number of units 

 of the commodit}^ (yards, gallons, or pounds, etc.) consumed by the 

 individual during a given period (say a year) and the ordinate OR 

 (in inches) represents the number of dollars at which the individual 

 prizes 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 a 

 fixed price (^:> dollars). He will therefore consume such an amount 

 of A that its marginal utilitj^ in dollars equals that of the price p, 

 that is, the cistern will be filled till OR =^9. This is evident, for 

 if less should be consumed OR would be greater than ^9, that is, a 

 little more commodity Avould be valued more highly than the dollars 

 exchanged for it and so would be purchased, and \i more should be 

 consumed, reverse considerations hold. 



If the price rises OR will increase and less be consumed but if it 

 falls, more. If the price falls to zero as is the case for water and air 

 the quantity consumed fills the whole cistern up to the horizontal 

 axis. This volume is therefore the quantity of rnaximunx satisfaction. 

 If the price rises to OA the individual will cease consuming. This 

 price is therefore the limiting inaxirninn price at which he will buy. 



The liquid contents of the cistern may be regarded as made up of 

 successive horizontal infinitesimal layers each representing an incre- 

 ment of commodity. Tlie height or distance of each layer from the 

 origin represents the degree of utility of that layer. The last or top 

 layer is on the margin of the whole and its vertical distance from 

 the origin is the degree of utility of that marginal layer or incre- 

 ment of the commodity or briefly its marginal utility. Thus the 

 margin of consumption has in the cistern an actual physical analogue. 



