64 Irving Fisher — Mathematical investigations 



Part II. — Utility of one commodity a function of the 



QUANTITIES OF ALL COMMODITIES. 



CHAPTER I. 

 TWO COMMODITIES. 



§1. 



Hitherto it has been assumed that the utility of a commodity is a 

 function of the quantity of that commodity alone. It is true that 

 it depends upon that quantity more than any other and the analysis 

 of Part I is a necessary first approximation. In astronomy the 

 attraction of the sun on the earth is first studied alone to determine 

 the earth's motion; next the moon's influence is admitted, then the 

 occasional "perturbations" due to planets and comets. Absolute 

 accuracy is never attained for the earth's motion is a function of 

 the mass and position of every body in the universe. 



So also the utility of the 100th lb. of butter (100 lbs. per year) 

 depends mostly on that 100 lbs. It would not be perceptibly in- 

 fluenced by a change in the quantity of clothing, but it would be 

 perceptibly reduced if the amount of bread consumed were reduced 

 from 300 loaves to 200, for bread and butter go together. 



It is needful here to distinguish carefully between two ways in 

 which the quantity of one commodity can affect the utility of others. 

 Even under the supposition of Part I, a change in the price of 

 clothes effected a change in tha individual valuation of money and 

 so changed the quantity of bread consumed and so in turn changed 

 the marginal utility and price of bread. But under our new sup- 

 position, a change in the price of butter directly changes the utility 

 of the same quantity of bread. In the first case marginal utility of 

 bread can change only after a change in its quantity. In the second 

 the marginal utility of the same amount of bread changes; the first 

 contemplates a variation in the quantity of water in a cistern, the 

 second contemplates a variation in the cistern wall itself. 



In Part I we assumed: -y^ = F(AJ ; but now we must write: 



