FOREST TAXATION IN THE UNITED STATES 63 
Formula 7 may be derived as follows: It can be shown by analogy from the 
proof of formula 3 that X= V’o[(1+p+r)*—(1+p)*], and since V’ = any 
DS (O% 2.0) eeaaarce= ae “|. Multiplying both sides by (1+ p)* and com- 
bing the X’s, X|(l-+p)F-+-(1--p-+-7)?—(1+ p) *)]— Vid + p+r)*—C-+ p)*l, or 
US R sei) Fei Xe ee ane 
$= Cemaene This reduces to formula 7, =! Cee 
In other words, formula 7 is identical with formula 3 except that 
the —1 in both numerator and denominator is lacking, thus denoting 
its noncyclical character. If land is not abandoned following cut- 
ting, but is used to produce a second crop, formula 3 applies rather 
than formula 7. 
Suppose that p=3 percent, r=1 percent, and k=20 years. The 
tax ratio is then 17.6 percent. If k=40 years, however, the tax 
ratio is 32.1 percent; if k=50 years, it is 38.3 percent. The longer 
the period of waiting, the greater the tax ratio. Over any given 
period of waiting, however, the tax ratio is somewhat less than that 
for a deferred-yield second-growth forest. In the case of a 50-year 
period of waiting, for instance, the tax ratios are 38.3 and 44.6 percent, 
respectively. 
Formulas corresponding to formulas 4 and 5 could also be derived, 
but they are not necessary to prove the main point here presented— 
that financially immature old-growth forests, although they partake 
of much the same nature as deferred-yield forests in regard to the 
tax ratio, are at less disadvantage under the property tax. 
OLD-GROWTH FORESTS BEING LIQUIDATED 
If cutting can be commenced immediately in an old-growth forest, 
to be carried on for, say, k years, at the end of which period either all 
of the timber will have been removed and the bare land will be without 
appreciable value, or the residual growing stock will be reduced to a 
sustained-yield basis, the tax ratio will be small. Such an old-growth 
forest is in whole or in part in the nature of a mine. The value is 
depleted from year to year and taxes take a lower percentage of in- 
come than in the case of other wealth. This fact may be proved, 
assuming that the net income each year before taxes is d, and the 
value which the forest would have had if there had been no taxes is 
W,, all other variables being as in formula 1. The tax ratio in this 
case is the ratio of the difference in values of a forest property before 
and after the imposition of a property tax to the value before such 
imposition. If the present value is V’, (to distinguish it from the 
sa yee 
initial forest value, V,), the tax ratio is, in symbols, Yo and is 
0 
equal to the following: 
Wise Wa Fis (Crem PUD) a) 
Wo Gor? te pan) LOGE pas 1 
Formula 8, 
