FOREST TAXATION IN THE UNITED STATES 57 
the taxes to be imposed on the material entering into the thinnings 7’, occurring 
at the end of m years and every n years thereafter. The value of these thinning 
taxes as of the time the principal yield, Y, is received will be called X ,. 
The discounted worth of all future thinnings directly after one thinning has 
been made is This is the ordinary formula for the present worth 
ee 
sea) eel 
of a perpetuity wherever the tax rate r is levied on value in exactly the same way 
as is the interest rate p. At the beginning of the next income cycle, this dis- 
counted worth of all future thinnings must be revalued to allow for the decreased 
period of waiting before the next thinning. By analogy from the proof of for- 
mula 1, it is obvious that the revaluation factor is (1+p+r)"-™. At the beginning 
of the Y income cycle, therefore, the value of future expected thinnings is 
ct Dake cn : Sol LAA Urs ne ale : 
(oes Next year the value is Ciscerarery samen and so on until the 
aie Dn Lats Bite ee 
beginning of the mth year, when the value becomes Cau moa 
» but thereafter it increases again until it 
One year 
dh 
1 Bi SARS AME 
later the value drops to G=ptn 1 
n—m—1 
reaches ce the year prior to the final yield. The thinning taxes 
are levied at the rate r on these values, and with compound interest to the end 
of the rotation, they amount to X,. Hence 
(here 
(eee Pa n—m (1+ p) t+ (i+ptr) 1+ p) + 
a Cera ter Lat pm) mantle 7) Mantas teat) List) tam tae ataerine lie 
she vont) ameme Fe 
The bracket, which will be called B, may be divided into two power series, 
the point of division occurring between the terms (1l+p-+r)*"1(1+p)*-™ and 
(1+ p)-""!. This point marks the end of the mth year, when the thinning 
is made. Factoring out (1l-++-p+r)"-"(1+p)”~-! from the first part of B ani 
(1+ p)"-*—! from the second part, there is 
2 
o — Ao ne a Te | 
QP) (1+ p) 1 CLINT 1+p ag a (1+ p)=1 
pel fa a en = |- 
(1+p) 1 ee te Ar (1 p)s-m1 
Combining these terms in each of the brackets according to the power series 
law, 
(heat Dey OP 
a nm petal eta) rac penetrate ae 
arya la eyo 
Simplifying the complex fractions and canceling terms occurring in both 
numerator and denominator, 
patetr) Gam ia) ene CR ahi atats) orn Uist) Mints Utes Data Nee ot) ve 
Tr 
Rearranging terms 
pe lint Pita te Plata ved! Noes Ol Sie iat eed siege) Baa L) 
r 
But it was found above that X “Gin ct 
Therefore, 
geet) camel Use Dicta hi) ier ln aioe eal te Peal) 
UE ide eel 
