68 MISC. PUBLICATION 218, U. S. DEPT. OF AGRICULTURE 
It should be noted that in the above demonstration n measures a 
complete rotation and that consequently V,»—C equals L, the land 
value. If were something less than the rotation, V)>—C would not 
be the land value, and formulas 10 and 11 would not apply. Due to 
the remote possibility of a land tax only, it was not thought worth 
while to derive formulas for cases where n is less than a rotation. 
Such formulas for the combined yield and land taxes are, however, 
derived at a later point. 
DEFERRED TIMBER TAX 
As a second plan for overcoming the inherent disability of the 
property tax when applied to deferred-yield forests, it will be pro- 
posed in part 12 that all of the required tax payments on the timber 
value under the property tax be deferred without interest until 
income from forest products is realized, the loss in revenues being 
made good through payments from a State timber-tax fund. 
The taxes under this plan may be computed from the following 
formulas: 
Formula 12, 
an 
Formula 13, 
es Oey) pee a GSD) he ae 
XT Via Rene + Tet @tnZl| 5 n |. 
When these formulas are applied to uneven-aged forests with regu- 
lar series of age classes (having k income cycles to the rotation of 
kn years) the land value, Z, is the land value of a corresponding 
even-aged forest with yield equal to kY and cost of regeneration 
equal tokC. In symbols, 
kY—kO—X—eC =P} 
laa) a 
To develop formulas 12 and 13 it is necessary first to consider the value sequence 
of a forest having an initial forest value of Vo, with an annual land tax rL and 
annual expense e. From the nature of this. plan the timber tax is only payable 
in case of a yield, and therefore there is no value increment accruing because of 
the payment of annual timber taxes. Assume that there are no intermediate 
yields, in which case T,,=0. The value at the beginning of the first year is 
Vo, the second year, Vo(l+p)+te+rL, the third year, Vo(1+p)?+(e+rL) 
(l+p)+e+rLl, and the nth year V,(1+p)"-!4+ (e+rZ) (1+ p)*?+ (e+rL) 
(1+ p)7-3+...+e+rL. Each year’s timber tax will be the tax rate times the 
difference between the value for the year and the land value. Hence, the timber 
tax sequence is, 
rVo—rLl, rVo(1+p)+rie+trL) —rL, SS Am Sai el ed EEN ot 
, rVo(1-+p)*!-+r(e+rL)(1+p)*2+r(e+rL) (1+p)*3+. . .+r(e+rL)—rL. 
These taxes summed to the end of the income cycle, without interest, by 
methods for summing series as used in connection with preceding formulas, are 
rVo ae ee (e-+r| SFE" =n |=nrl. 
The annual land tax, rl, accumulated to the end of the income cycle, with 
jp aa Ui 1) 
P 
L= lee: 
interest at the p rate, is r 
