12 Irving Fisher — MathematiGal investigations 



indifference curves, the tangents and normals to which play an 

 important role. 



When our individual fixed his whole consumption combination to 

 suit himself, let us suppose that he spent $25 per year on the two 

 articles {a and h) under consideration. We may metaphorically 

 compel him, while not altering in the least his purchases of other 

 articles and hence having the same |25 to spend on («) and (h), to 

 contemplate spending it in a different way. If the price of (a) is 

 $0.25 and of (b) is $0.50, the two simplest methods of spending his 

 $25 is to spend it all on [a) and purchase 100 units, or to spend all 

 on ih) and purchase 50 units. 



In fig. 18 lay off OAz=: 100 units and OB = 50 units. Then any 

 point on the straight line AB loill represent a consumption combina- 

 tion of A and B purchasable for $25.^ AB may be called 2^ partial 

 income line. Our individual is therefore left free only to select his 

 combination somewhere on this line. The combination 5 or 5 pre- 

 sent equal inducements but not as great as 6 or 6 on an arc of 

 greater utility, nor there as much as at I. He will select his combina- 

 tion in such a manner as to obtain the maximum total utility, which 

 is evidently at the point I where AB is tangent to an indifference 

 curve.\ At this point " he gets the most for his money." 



His selection I is of course just what it was before we began our 

 analysis. But we have advanced one step. We have partially anal- 

 yzed this equilibrium, that is we see the equilibrium for A and B 

 while the prices and quantities of other articles remain the same. It 

 is as if a pendulum free to swing in any vertical plane is found at^ 

 rest and a scientist attempts to analyze its equilibrium. He forth- 

 with confines its motion to a single plane and discusses its equilib- 

 rium there. The analogy suggested may be extended. The prin- 

 ciple underlying the equilibrium of a pendulum or any mechanical 

 equilibrium (as of a mill pond or of a suspension bridge) is: that 

 configuration will be assumed which will minimize the potential. So 

 also the supreme principle in economic equilibrium is; that arrange- 

 ment will be assumed which will maximize utility^. 



* Proof : Equation of AB is — - + -— = 1 where x and y are the co-ordinates 



OA OB 



25 25 



of any point on AB. This becomes ]/ . -— + x . pr^ = 25 ; that is, x times 



its price + \j times its price equals $25. 



f When AB is tangent to two indifference curves that one will be selected 

 which has the greater utility. 



X See interesting remarks, Edgeworth : Mathematical Psychics. ^Iso in his 

 address as Pres. section Econ. Sci. and Statistics Brit. Asso., Nature, Sept. 19, 

 1889, p. 496. 



