52

Irving Fisher—Mathematical investigations

ates in a uniform percentage without loss or gain of commodity. Inthis case the price of A will decrease while that of all other articleswill increase exactly alike.* The valuation of money will be re-duced since the ordinate of a B cistern (say) has not changed whileits back thickness has increased. The changes just considered maybe brought about if A suddenly goes out of fashion.

Perfectly analogous changes occur if a cistern or cisterns becomenarrower. The individual is then more keenly “ sensitive ” tochanges of quantities. This change may occur through a discoveryby which a little of the commodity is made to “go farther” thanbefore.

Reverse changes occur if cisterns are broadened or deepened.

It is impossible to combine all the A cisterns into a single demandcistern for A as was done in Ch. II or to combine all the I cisternsinto an income cistern as in Ch. Ill, for we can no longer overlookthe influence of other commodities and other individuals. Theanalysis therefore which treats of but one commodity at a time andconstructs a demand curve for it is a superficial one for it does notreach all the independent variables.

§10. Analytical.

Suppose there are n individuals and m commodities in our givenisolated market during the given period and suppose the amountsof the commodities A, B, C, etc., are given K„. K„ K e , etc., and thegiven incomes of I, II, III, etc. are K i; K a , K t , etc. Then the con-dition that the commodity-sums are given is:

+ A„=K, )

+B„= K b

-A, -f A a 4- A„ -f

B, + B a + B a +

C, + C a + C a +

rn equations.

mn unknowns.

+ M„=K,

M 1 + M, + M,+

* For their mutual ratios cannot change since the ordinates to which they areproportional do not.