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.