58 MISC. PUBLICATION 218, U. 8. DEPT. OF AGRICULTURE 
Adding this equation to that previously found for taxes where no thinnings 
were involved, and simplifying, 
X=Y—C4Ta(+ p)*—[¥—C4 Tall tpt) a) Gah. 
The tax ratio is then (formula 5) 
ae xX ie i On ee Sy ay ot | 2) 
S  Y—C+T,(1+p)*™ [Y—C+ Tn +p)* lA +p+r)*—]] 
Formula 5 is exactly the same as formula 3, except for the expres- 
Y—C+T,(1+p+r)"™ 
Y— C+ re ae) ae 
integer, and Y—C, T,,, p, and r are also positive, the expression is 
obviously ereater than 1. This means that the subtrahend on the 
right of the tax-ratio formula is greater than if no thinning had 
occurred, and hence the tax ratio is smaller. A thinning, therefore, 
reduces the tax ratio, provided, of course, a net income is received 
from the thinning. A thinning conducted at a loss, on the other 
hand, increases the tax ratio. 
Take, as in the example discussed previously, 7=1 percent, p=3 per- 
cent, and n=50 years, and add to these Y—C=$100, T,,=$10, and 
m=25 years. The tax ratio is then 42 percent, as compared ‘with 
45 percent if no thinning has been made, and 25 percent under the 
income tax. The reduction in tax ratio is logical, since the effect of 
a thinning is to decrease the deferment of income. 
Annual expenses, on the other hand, being in the nature of negative 
intermediate incomes, tend to increase the tax ratio of a deferred- 
yield forest. If, for instance, the annual expenses, e, are 5 cents, and 
the other variables as in the case just mentioned where a thinning 
was made in the twenty-fifth year, the tax ratio becomes 43 percent. 
Under these conditions, as has been shown, the tax ratio under the 
income tax would be 26 percent. Incorporating annual expenses in 
formula 5 gives the following result: 
sion Since n—™m is, by its nature, a positive 
Formula 6, 
X=Y—O+T, (+p) "—[Y¥—C+ Tn +p+n)*-"] cere 
i _| eet). 
aie 
, and the tax ratio 
In this case S= Y—C+ T,,(1+p)*""—e 
is the ratio of X to this quantity. 
The effect of an annual expense on the tax ratio may be deduced 
from the preceding formulas. It has been shown that the tax ratio 
for deferred-yield forests is always greater than the income-tax rate, 
(Ep) iz! 
P 
ae or, in other words, that the numerator of the tax-ratio fraction 
always exceeds the denominator multiplied by Sen It should be 
noted, however, that the effect of the annual expense, e, is to decrease 
