110

A TREATISE ON MONEY

BK. II

ing limits within which the true comparison must lie,as follows :

Let P and Q be the composite commodities repre-sentative of expenditure in the first and second posi-tions respectively.

Let the amount of P which can be purchased for£l in the first position be chosen as our unit of P, andthe amount of Q which can be purchased for £l in thesecond position be chosen as our unit of Q ; and let pbe the price of a unit of P in the second position, and

1

the price of a unit of Q in the first position. Let similar

persons having a real-income E buy n x units of P in thefirst position and n 2 units of Q in the second position.

Then, since the money-incomes of similar personsare £n t in the first position and £n 2 in the second posi-tion, the index-number comparing purchasing powers

in the two positions = —. It can be shown that this

must lie between p and q.

For since the consumer has a choice in the firstposition between buying n x units of P or n x q units ofQ and prefers the former, and since by hypothesis thesatisfaction of the former purchase is equal to that ofbuying n 2 units of Q, it follows that n 2 >n 1 q ; andsimilarly, since in the second position he has a choicebetween buying n 2 units of Q, equal in satisfaction by

n 9

hypothesis to n x units of P, or buying — units of P, and

prefers the former, it follows that n x >

n 2 _p ’

hence

n 2

n,

is greater than q and less than p.

Thus if q is greater than 1, the value of money hascertainly fallen ; and if p is less than 1, its value hascertainly risen ; and in any case the measure of thechange in the value of money lies between p and q.

This conclusion is not unfamiliar—though the above