FOREST TAXATION IN THE UNITED STATES 55 
the number of cents left to the owner. The sum of the two quantities 
is $1. 
The minimum tax ratio resulting from the use of formula 1 is 
wats when n=1. That is, when the forest is on an annual sustained 
Pp 
yield, the tax ratio equals the tax rate divided by the sum of the tax 
and interest rates. In the example already given, for instance, the 
tax ratio is 0.03 £0.01 er £0.01’ or 25 percent. The income cycle is here 1 
year, which means that an equal annual income may be received each 
year. 
The statement that an annual sustained yield forest gives a mini- 
mum tax ratio is true so long as the tacit assumption holds that the 
forest is not overstocked. Of course, if there is more timber on the 
forest than is necessary for sustained ‘yield, it is strictly speaking not 
an annual sustained-yield forest, and in consequence some of the timber 
may be cut off at once, whereupon the extra immediate income will 
reduce the tax ratio to subnormal. This situation is discussed in 
more detail later on in connection with mature forests (p. 63). 
Errect oF CHANGE IN Tax RaTE or INTEREST RATE 
An unexpected increase or decrease in the tax rate or interest rate 
is much more unfavorable or favorable in the case of a deferred-yield 
forest than in the case of an annual sustained-yield forest. The defini- 
tions and assumptions are the same as for formula 3, with the addi- 
tion of the following: (1) The tax rate or interest rate is unexpectedly 
changed by wu at the end of q years, g being less than n, but not nega- 
tive; and (2) the percentage change in value caused by the change in 
rate is called z, the percentage being based, of course, on the value 
before the change. The formula for 2 is 
(Die? at iain) ia ste Date ee UI), 
Cie a feats 2) tear | Date) git 
Formula 4, z=1— 
Wiebe 
Formula 4 may be derived as follows: By definition, g=——— ee, or 
—1 
1 Ot A 
Vea ip tru) 
interest rate, and V, the value 1 year afterwards. V, is discounted 1 year by 
dividing by 1 plus the new interest and tax rates, p+r-+u. 
By analogy from the first part of the proof of formula 3, 
» Vq-1 being the value just before the change in tax or 
y_- PoOEX (lt pty 
ot Garp ae 
Substituting the value of X from formula 3, this becomes 
(Ub sip) ee 
as ey a 
ye a cin Dita) ool 
etre guise (Caest 
if (+p+ne 
ey Mose 
a ee ae 
