54 MISC. PUBLICATION 218, U.S. DEPT. OF AGRICULTURE 
Reducing the compound fraction to a simple fraction 
CS er} 
Oa) oa 
Canceling the term r(1+>p)*-! from both numerator and denominator, and 
X=rVo(1+p)— 
substituting Ga peat for its equivalent, Vo, 
ya (Y—C— XN + p+) +p) 
(heel 
Cross multiplying and collecting terms 
Xp) 2 a Se pat) tae op) (YC) (pa) el tp) al nar 
ya (¥=-Ol+p+n"— +p)", 
: Ga pea 
x Cla mDanta lame Olaaa) 2 
Y—C  (1+p+r)"—1 
Dividing the numerator of this fraction by the denominator, 
[poe lp eal 
(Clean) a Gls eset) = Oey) Ole paene a) 
Qa Datah) Saal 
—Use)Part 
there results formula 3, the tax ratio, or 
ss SUL etl ee) dere UN 
Y—C (1+p-+r)7—1 
Suppose that r=1 percent, p=3 percent, and n=50 years. With 
these quantities in formula 3, the tax ratio is 45 percent. If n were 
10 years, p and 7 remaining as above, the tax ratio would be 28.4 
percent; in a previous example, where n was 2 years, it was 25.4 per- 
cent; and for an annual sustained-yield forest, in which n is 1 year, 
the tax ratio has been shown to be 25 percent. In all of these cases, 
the tax ratio under the income tax, according to formula 1, would 
be 25 percent. 
EFFECT OF CHANGE IN INCOME CYCLE 
The tax ratio increases as n increases. In other words, a larger 
portion of the net income before taxes is taken by the Government 
as the income cycle is lengthened. ‘To demonstrate this proposition, 
GSP) iss 
it will be sufficient to prove that (sp decreases as 7 In- 
creases, or what is the same thing, that (1+p-+r)”"—1 increases faster 
than (1+p)"—1. This fact, which is almost self-evident, is capable 
of mathematical verification. 
Resort to the differential calculus (that branch of mathematics which has to 
do with rates of change) will yield the demonstration. Taking the derivative of 
both (1+ p-+r)*—1 and (1+ p)*—1 with respect to n, there results (1+p-+r)” 
log (1+p-+r) and (1+ p)* log (1+ 7). Since both p and r are positive from their 
nature, the first of these derivatives is obviously greater than the second, and 
hence (1+p+r)"—1 increases faster than (1+ p)*—1. 
Cli 
qos represents 
(Laepyeaat 
the number of cents taken by the Government and (ee 
If, in formula 3, Y—C=$1, the term 1— 
