42 MISC. PUBLICATION 218, U. 8. DEPT. OF AGRICULTURE 
years, the value of the capital may be obtained by considering that 
the perpetual annual income of $674, less the tax at 1 percent, must 
be 3 percent of the value of the capital, just as in the case of the first 
brother. In equation form, 674—0.01X=0.03X, where X stands for 
the value of the capital. Solving this equation gives $16,850 as the 
value of the capital, and the property tax on this value at 1 percent 
will be $168.50, or the same as the income tax of 25 percent. The 
present worth of this value, assuming a property tax of 1 percent, is 
that amount which, with 3 percent for interest and 1 percent for 
taxes, will equal $16, 850 in 17% years. In other words, the present 
worth (the value of the capital after taxes) is $16,850 discounted at 
4 percent for the period in question. Since money doubles at 4 
percent in approximately 17% years, the value of the capital after 
taxes is one-half the above amount, or $8,425. The present worth of 
the property taxes is the difference between this amount and $13,333, 
the value of the capital before taxes. This difference is $4,908, 36. 8 
percent of $13,333. The corresponding reduction in the tax-free 
value of the capital under an income tax has just been shown to be 
$3,333, 25 percent of $13,333. 
The third brother receives an annual income of $3,333 for a little 
under 5 years, thereby diminishing and finally exhausting entirely his 
capital. An income tax, at 25 percent, would be $833 each of the first 
4 years, $274 the fifth year, and nothing thereafter. The present 
worth of these amounts is $3,333, which is, as in the case of the other 
two brothers, exactly 25 percent of the capital before taxes. <A 
1-percent property tax, on the other hand, takes just enough each year 
to make the capital for the succeeding year, before deduction of the 
_ $3,333, greater than that for the preceding by interest (8 percent) 
and taxes (1 percent). In mathematical terms, taking V,, as the value 
at the end of year n, V,ii:=V,(1-+0.03-+-0.01)—$3,333. Where 
n=4, however, the equation is: V;=0=V,(1--0. 03-40. 01)—$1,094, 
since the i income for the fifth year is only $1,094. The solution of this 
last equation is V,=$1,052. From this the value of V; can be found, 
by use of the first equation, to be $4,216, and so on to Vo, which is 
$12,998, the value of the capital after property taxes. Since the value 
of the capital before property taxes is $13,333, property taxes have 
reduced this value by $335, or 2.5 percent. 
From these three typical examples, it may readily be concluded 
that, when measured by the effect on the present worth of the capital, 
a net-income tax under the given assumptions would treat with 
equality all forms of investment. A permanent net income tax at 
any given rate has the same effect as taking once and for all that 
fraction of the original capital represented by the tax rate. The 25- 
percent income tax is the same as taking at the beginning 25 percent 
of the capital of each of the brothers in the example. Each could 
have ‘‘compounded”’ his taxes forever by setting aside a permanent 
fund of one-fourth of his capital before taxes, $3,333. 
On the contrary, the property tax deals unequally with the several 
forms of investment, in terms of its effect on the present worth of the 
capital. Only in the case of an investment producing a regular 
annual income equal to the interest on the capital does the property 
tax (at an appropriate rate) produce the same effect as the net-income 
tax. This is illustrated by the investment of the first brother, whose 
