A STATISTICAL METHOD FOR MEASURING ‘MARGINAL UTILITY” 187
whether progressive or regressive taxation is indicated but the
exact degree of progressiveness or regressiveness.
The most satisfactory way to picture this mathematically is to
plot the two points S;, W; and S3;, W3 on “doubly logarithmic”
paper, join these two points by a straight line, and measure
the slope of that line. If the slope is 45°, then S; W,=8; W3
and the tax should be at a uniform rate; if it slopes downward
more steeply than 45°, the tax should be progressive; if less
steeply, regressive. The slope itself tells us at what percentage
rate the want for a dollar decreases for each 1 per cent increase in
income.
This figure for the slope can, of course, be attained arith-
metically without plotting." This slope is what Marshall, in a
different application, called “elasticity.”
Extension of the Theory
All the essentials of the method have now been stated. But it
may be well to point out that, by successive applications, its
range can be extended indefinitely or as far as the budgetary
statistics are available.
That is, we may continue to choose identical families con-
formably to the same prescription that for every family in Odd-
land there will exist in Evenland another family provided with
an income such as will lead it to choose the same, and equally
desirable, food ration; whereas for every such family chosen in
Evenland there must be another in Oddland that will have an
income such as will lead it to choose the same, and equally desir-
able, housing accommodation. We have hitherto supposed only
Cases 1, 2,3. We now add Cases 4, 5, 6, 7, etc., all the odd figures
referring to Cases in Oddland and all the even figures to Cases
in Evenland, as shown in Chart II which is merely a schedule of
Cases 1, 2, 3, 4, 5, etc., with a chasm or ocean between Oddland
and Evenland. Our calculations evidently constitute a sort of
triangulation by which we pass back and forth from Case 1 via
Case 2 to Case 3, thence, via Case 4 to Case 5 and so on. The
Chart shows schematically what I mean by “triangulation.”
* We need merely equate the logarithms of the two sides of equation
(3) and likewise of equation (4) and then divide one of these new equations
by the other and calculate out the right hand side on the basis of the
statistical figures it contains.