A STATISTICAL METHOD FOR MEASURING “MARGINAL UTILITY” 185 
which, after cancelling S,, may, for mnemonic purposes, best be 
transformed into: 
Ss _ P/ps , Rs/Ry 
Si ¢/é1 ~ Fy/F, 
3) 
where all the “3's” are vertically above corresponding “1’s.” 
Similarly, dividing the lower of equations (2) by the upper 
just as they stand, we get 
WR; WR; 
WF, or WF, 
which, after cancelling W,, may be written mnemonically, 
Ws _ R:/R; 
Wi FF, 
From (3) and (4), by multiplying and cancelling, we obtain 
Eo TRA af RTP 
WSs p2/ P3 
Wis: ¢2/ b1 
(5) 
Formule (3), (4) and (5) afford comparisons between Cases 
1 and 3, both in Oddland; that is, they compare two families in 
exactly the same situation except that their incomes or expendi- 
tures, S; and S;, are different. Formula (3) compares their 
incomes. Formula (4) compares their wants-for-one-more dollar. 
As the want-for-one-more dollar decreases with an increase of 
income, one of these two rations. 
i and Ws 
Ss Ww, 
must be a proper fraction and the other, an improper fraction. 
Their product is given in Formula (5). 
Marginal Want for M oney and the Income Taz 
According to which way this product differs from unity, we 
have a justification for progressive or regressive taxation, while 
if their product is exactly unity, taxation should be neither pro- 
gressive nor regressive, but strictly proportional to income. This 
is all on the assumption that the tax is to be laid according to 
the principle of equal sacrifices to tax payers of different incomes. 
To show these propositions, suppose an income tax, or, to be 
unequivocal, a tax on expenditure, to be levied at the rate of ¢,