72

Irv ing Fisher — Mathematical . investigations

indifference curves, the tangents and normals to which play animportant r61e.

When our individual fixed his whole consumption combination tosuit himself, let us suppose that he spent $25 per year on the twoarticles (a and b) under consideration. We may metaphoricallycompel him, while not altering in the least his purchases of otherarticles and hence having the same $25 to spend on (a) and ( b ), tocontemplate 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 («) and purchase 100 units, or to spend allon (5) and purchase 50 units.

In fig. 18 lay off OA = 100 units and OB = 50 units. Then anypoint on the straight line AB will represent a consumption combina-tion of A and B purchasable for $25* * AB may be called a partialincome line. Our individual is therefore left free only to select hiscombination somewhere on this line. The combination 5 or 5 pre-sent equal inducements hut not as great as 6 or 0 on an arc otgreater 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, whichis evidently at the point I where AB is tangent to an indifferencecurve. f At this point “ he gets the most for his money.”

His selection I is of course just what it was before we began ouranalysis. But we have advanced one step. We have partially anal-yzed this equilibrium, that is we see the equilibrium for A and Bwhile the prices and quantities of other articles remain the same. Itis as if a pendulum free to swing in any vertical plane is found atrfcst 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 mechanicalequilibrium (as of a mill pond or of a suspension bridge) is: thatconfiguration will be assumed which will minimize the potential. Soalso the supreme principle in economic equilibrium is: that arrange-ment will be assumed which will maximize utilityj.

y x

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

25 25

of any point on AB. This becomes y . — + x . = 25 ; that is, x times

OA OB

its price + y times its price equals $25.

f When AB is tangent to two indifference curves that one will be selectedwhich has the greater utility.

t See interesting remarks, Edgeworth : Mathematical Psychics. Also in hisaddress as Pres, section Econ. Sci. and Statistics Brit. Asso., Nature, Sept. 19,1889, p. 496.