FOREST TAXATION IN THE UNITED STATES 71 
Subtracting these two equations, 
O=Vel(1t ptr’)»— 14+ [+ ptr!" 1-Y+6, 
or 
y—c- SF a +ptr/)9—-1 
on CFE ST 
which becomes 
SL CiN Wane (r—r')L+e 
ee 
Cpr nae 
Substituting this value for Vo in the second equation for X above, 
Y—C (r—r’)L+e Gepm)esi 
2 er ae SE lat 9-114 Y-c-6 ; 
This reduces to 
a _ (Gsrpp@)e=1 _er'—p(r—r')L  (+p)*—1 
x=(y-0)|1 Cel et ee 
which is the same as formula 15. 
The tax ratio will be the ratio of either expression for X to 
i nm] 
OS ee. 
Theoretically it is possible to construct a formula showing the rela- 
tionship of the various factors to give the precise reduction factor 
which, if applied to timber only, would result in the same tax ratio as 
an income tax or tax on net yield. 
Formula 17, below, is an expression of this relationship: 
[ pC) pp Y—C-ne) +L +p )"—r [tp tr’) 
+| Y-0-5 (Y¥—C—ne)—(e+pL) CPPS ate te ye 
tic a0 te ay (l+p)"—1], 
-| @—O0+p)"— 5h P—C—ne)— (e+ ph) |, 
r 
ae is Mis silts ai (nD) eae 
SE SON NIE) ie re OUD Se 
Formula 17 may be derived in the following manner: As previously stated, the 
income tax formula is X = a (Y—C—ne). 
According to the above assumption the tax under the differential timber tax plan 
must equal this income tax. Hence 
‘ (1+ p)*—1 SN Cth hc ta (ap eeel 
p+r (a arr) el Patni p 
Simplifying and collecting the terms containing the unknown, r’, to the left 
member of the equation the formula reduces to formula 17 above. 
It may be noted that the unknown, 7’, in formula 17 occurs in the 
three terms of the left member in the following forms (1+p-+7’)”, 
r’(1+p-+r’)”, and r’. While these are somewhat complicated 
relationships, it is possible to determine values for 7’, when all other 
factors are known, which will give a value equal to the right member 
of the equation to as great a degree of accuracy as desired. Then 
(Y—C—ne) =(Y—C) E = 
