56 MISC. PUBLICATION 218, U. 8. DEPT. OF AGRICULTURE 
A similar formula holds for V,, though here u must be added to p and r, 
Cope QUE Datta) 
Vom aol Ep ted eee 
Thus 
Gey) Ula EE 
eae (+p+r+u)2—1 : 
(YO) qe tp trty) 
Simplifying this expression—formula 4. 
egal Dat tate) elle palnt) eae 
(Gear) | (pas 
Assume, for instance, that 1 year after the purchase of a forest 
recently cut over, having an income cycle (and rotation) of 50 years, 
the tax rate is unexpectedly increased from 1 to 1% percent, the 
interest rate remaining constant at 3 percent. Or if the tax rate 
remains 1 percent, assume the interest rate to increase to 3% percent. 
In either case, the decrease in value is, according to formula 4, 24 per- 
cent. If, on the other hand, the forest were on annual sustained 
yield, the income cycle being 1 instead of 50, the decrease in value, 
according to formula 4, would be 11 percent, only 46 percent as much 
as in the case of the deferred yield forest. 
Formula 4 applies equally well whether w is negative or positive. 
For negative values of wu, z is negative and measures the percentage 
increase in the value of the forest with an unanticipated decrease in 
the tax rate or interest rate. It will be found that 2 is larger abso- 
lutely the longer the period, n. Deferred-yield forests are thus 
favored more than sustained-yield forests by a drop in tax or interest 
rates. 
— 
Tur GENERAL Tax Ratio FoRMULA 
The effects of thinnings, annual expenses, and regeneration ex- 
penses will now be examined. The possibility of a thinning some- 
time during the income cycle changes the value of a forest property 
and consequently changes the taxes. If a thinning 7, is made at 
the end of m years and every n years thereafter, m being less than n, 
the following formula holds: 
(p Glse ei 
X= V¥—C Tap)" |¥ Ct fd tpn) ei - ee |: 
The tax ratio is defined as = After finding X as above, the result 
must be divided by S, which in this case is Y—C+ T,,(1+p)""”, since 
this quantity is the income with compound interest as of the end of 
the income cycle. Carrying out the division indicated, 
DG | ¥ Ca TE pa) ie eee 
Formula 5, ’9- | (Y=0+T-Fpy "1G +p+rP 1 
Formula 5 may be derived as follows: In the case of no thinnings and no costs 
d+p)2= is 
(+b pthA) ae 
eyes pt asP) awl | : 
Therefore X =(Y of G+p+n*—1_ To this must be added the value of 
of any kind, it was found (formula 3) that the tax ratio, a 1— 
