186 ECONOMIC ESSAYS IN HONOR OF JOHN BATES CLARK 
per dollar, or per cent, on Si, the income of Case 1, and t3 per 
dollar on Ss, the income of Case 3. The total taxes will be iS; ts 
and S; t3. These are in dollars. 
We assume that the taxes are small so as not appreciably to 
affect the income and the want-for-one-more dollar. The sub- 
jective sacrifices, at W; and Wj; per dollar, will then be 
Wt, S; and Ws t3 Ss. To conform to the principle of equal sacri- 
fices, the above expressions for sacrifices must be equal, z.e., 
Wy t1 Si =W3; ts Ss 
or, otherwise expressed, 
ts _ WiSi = ¢2/ 1 
ft = W 3S; 02/p3 
(6) 
the last part of this continuous equation being equation (5) 
inverted. 
By formula (6) we can now find the theoretically just rate of 
progression (or regression, as the case may be) of an income tax. 
This formula gives, in our hypothetical example, 1.56. Thus, if 
out of S;=%$1000, a tax of 1%, or $10 is paid, then out of 
S;—$1440 a tax of 1.56% or $22.46 should be paid (instead of 
$14.40 as would be the case under proportional taxation). 
Of course these figures are not statistical results, as the reader 
will remember that they are derived from purely hypothetical 
data. But they show how statistical results may be obtained. 
Evidently (assuming the principle of equal sacrifices), a pro- 
gressive income tax is justified if formula (6) gives a result 
greater than unity, a regressive tax if less than unity, and a uni- 
form tax rate, if exactly unity. 
Tt follows, if all our five specified assumptions are correct and 
if we can obtain accurate statistics to which those assumptions 
apply, that it will be possible to turn to practical use this highly 
theoretical study of the most elusive of entities with which eco- 
nomic science is forced to deal, “marginal utility” or the want- 
for-one-more unit of anything. 
As we have seen, Chart I pictures the two families (Cases 1 
and 2); that is, it shows the income ($1440) of Case 3 as con- 
trasted with that ($1000) of Case 1 and the wants-for-more 
dollar, these being respectively .3315 wantabs and .75 wantabs. 
The slope of the line connecting these points determines not only