FOREST TAXATION IN THE UNITED STATES 53 
Assume now a deferred-yield forest having an income cycle of 2 
years and a value at the beginning of the cycle of $20,000. The in- 
terest and taxes at the end of the first year amount to $800, but since 
no income is actually realized this interest and tax charge serves to 
augment the principal which becomes $20,800. Taxes on this sum 
for the second year amount to $208, while interest is $624, a total of 
$832. <A total net income before taxes of $1,632 must, therefore, be 
received every 2 years in order to justify the initial value of $20,000. 
Taxes have amounted to $408 without interest. When 1 year’s in- 
terest is added to the first year’s taxes of $200, the total tax cost as 
of the end of the second year js $414. The tax ratio is ate awa? OF 20.4 
1,632 
percent. The general formula for the tax ratio is as follows: 
Formula 3, vena) (aL aE 
Formula 3 may be derived as follows: The initial forest value at the beginning 
of an income cycle is, by definition, the sum of all future net incomes discounted 
at the interest rate, p. In symbols: Vo Gast This is the ordinary type 
of formula for finding the present worth of a perpetuity. 
The value at the beginning of the second year, V;, equals Vo plus 1 year’s 
interest and taxes, or Vo(1+p-+r). Ther in this expression capitalizes the taxes, 
that is, V; would differ from Vy by a less amount if there were no taxes. Capi- 
talization of taxes reduces the value in the early stages of a cycle in comparison 
with the value at later stages. 
Similarly, the value at the beginning of the third year equals V, plus 1 year’s 
interest and taxes, or 
V.=Vi(1+p+r)=V.(1+p+7r)?, ete. 
Since the tax equals the tax rate, r, times the value, the first year’s tax=rVo» 
the second year’s tax=rV,(1+p-+r), etc. And the first year’s tax payable at 
the end of the year, must be compounded at the interest rate, p, through each 
of the remaining years of the income cycle after the first, or in all n—1 years, to 
obtain its worth at the end of the income cycle. At this time, then, the worth 
amounts to rV)(1+>p)"—!. The second year’s tax likewise must be compounded 
in all for n—2 years, and it equals, at the end of income cycle, rVo(1-+p+r) 
(1+ p)"-2. The third year’s tax=rV)(1+p+7r)2(1+p)"-, and so on. The sum 
of all these quantities is 
X=rVill +p)" +0 +ptHd+p)?+d+p+n2l+p)3+.. + +ptr) 
Factoring out the quantity (1+ p)”—! from the bracketed part of the expression 
results in 
i AL slarianeo Clieiarie ime. eee] 
The above expression is that of a geometric series of the general form, 
1— 
UE irr 
Cee ; and n is the number of terms in the series. 
a where Q, the desired summation, is X; g is rVo(1+p)""; q is 
“I+p- 
Therefore 
per)" 
pl oe aia 
A =rV,(1+p:) : _itp+r 
l+p 
