172 ECONOMIC ESSAYS IN HONOR OF JOHN BATES CLARK 
rests, in order to make sure whether those hypotheses are true or 
reasonable. 
The hypotheses have already been stated, but the equations 
help us to see more clearly what they involve, and one cannot see 
too clearly when trying to peer into a region supposed to be dark, 
and filled with elusive will-o-the-wisps of thought. 
If the equations for food, rely Soa = Ss and W, F1— 
Ws F,, are correct (and also the corTesrending fale for rent), all 
the rest follows indubitably. Any critic, in order to discredit the 
method, must discredit one or another of these four equations. 
Certainly no criticism of the first equation is possible, except 
as to the statistical accuracy of the numerical data. The equa- 
tion merely describes the two families enjoying the same, or 
equivalent, food rations. That is, if we locate in the statistical 
tables of budgets two groups of families, one in Oddland having 
a total budget of S;, or $1000, and the other in Evenland having 
a total budget of S., or $600, and if it be true, as the statistical 
tables are here assumed to state, that in Oddland a $1000 family 
averages ¢1, or 40%, (of his income and expenditure) on food, 
making S; ¢;, or $400, while a $600 Evenland family averages 
¢2, or 50%, on food, making $300; and if, furthermore, the rela- 
tive food prices in the two countries (for food of the same 
quality) are as F;—F,, or as 1.3313--1.00, then these two 
families certainly do have the relationship 
Boy 
Sits _ Sut (;, 1000 X 40% _ 600 X 0%) 
Fone 7% Eas 331 1.00 
For convenience we have described this relationship by saying 
that the two families are selected to have the same food rations. 
But if we prefer meticulous exactitude, we should say, instead, 
that they are selected such that their food expenditures are pro- 
portional to the food price indexes. This is evident if the first 
equation hh = hh is written oh = > . This states that the 
expenditures for food in Cases 1 and 2 are proportional to their 
prices. It is only in this specific sense that the food can be 
said to be “the same” in the two Cases. 
We need, therefore, no longer picture this sameness as same- 
ness in “pounds,” nor need we longer conceive of the index number