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 expenditures,
 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 progressive
 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 ¢,