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subject to (a) a higher tax rate and (b) a lower rate of com

mission.

The first limit (a) may be determined by the method given hereunder.

In the example given herein, an income (after deducting commission) of $20,000 (being 20% of the invested capital of $100,000) is the high limit within which the 20% excess-profits tax rate will govern, and the total federal tax payable on that amount is $3,420. The balance of profit after deducting both taxes and commission is therefore $16,580. On $18,000 the commission would be $2,475, being computed at rates decreasing from 20% to 7772%, and on $21,000 it would be $2,675, computed at rates decreasing from 20% to 5%. The problem to solve is to find the amount which after deducting therefrom commission on the above sliding scale will leave $20,000 subject to the federal tax.

It is evident that the commission will be higher than $2,475, for it has to be computed on an amount exceeding $22,475, inasmuch as, otherwise, the balance subject to the federal tax would become less than $20,000. On the other hand, it will be less than $2,625, for, otherwise, it would have to be computed on an amount higher than $23,625, which would bring the taxable income under the second bracket. It follows, therefore, that the lowest rate of commission to be considered in this instance is 5%, applicable to earnings (after deducting taxes) between $18,000 and $21,000.

If we now call the limit beyond which, under the above commission arrangement, the earnings before deducting taxes and commission will be subject to the excess-profits tax under the second bracket x and the commission actually payable y it will be seen

(1) X - $3,420 - y = $16,580.

Deducting from x the amount of $18,000 and from y the commission of $2,475 payable thereon, we obtain the following equation:

(2) (x - $3,420 – $18,000) – (y - $2,475) = $16,580 – $18,000 plus $2,475, or

(3) (x - $21,420) (y - $2,475) = $1,055.

Inasmuch as (y $2,475) represents 5% commission payable on (x - $21,420), we substitute as follows:

95 (4) (x - $21,420)

(x $21,420)

(x - $21,420) = $1,055.

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5

100

100

100

Multiplying both sides of the equation by

we find

95

X

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(5)
- $21,420

$1,110.53, and
(6) x = $22,530.53.
Substituting the value of x in (3) we find
(7) ($22,530.53 — $21,420) – (y – $2,475) = $1,055, or
(8) y = $22,530.53 $21,420 —$1,055 plus $2,475 $2,530.53.

Deducting from $22,530.53 the tax of $3,420, there will be a balance of $19,110.53 on which commission is payable as follows: On the first $18,000, as above...

$2,475.00 On the balance of $1,110.53 @ 5%...

55.53

Total as above

$2,530.53 Deducting this from $22,530.53 leaves a balance of $20,000, on which taxes are payable.

Therefore, if the earnings, before deducting commission on the above sliding scale, are higher than $22,530.53, they will become subject to taxation under both brackets of the excess-profits tax.

Instead of the above algebraic solution, the following general rule may be followed : (a) Take 20% of the invested capital...

$20,000 (b) Compute the tax payable on this amount.

3,420 (c) The balance will represent the net profit after deducting taxes and commission

16,580 (d) Ascertain by inspection the portion of the earnings subject to the higher rate of commission...

18,000 (e) Ascertain the commission payable thereon at these rates.. 2,475 (f) The rate of commission to be considered in subsequent

calculations will be the next lower rate.....
(g) Ascertain the portion of the income remaining after com-

mission at that lower rate has been deducted there-,
from, viz:

(a plus e) (b plus d), or (20,000 plus 2,475)
($3,420 plus 18,000).

1,055 (h) The amount will represent (100 c) % of that portion..

95% (i) The portion of the income to which the lower rate of com

5%

100-C

mission applies is therefore (g) divided by

100

1,110.53

(i)

($1,055 divided by .95).....
The total amount subject to commission will therefore be

(d) plus (i) $18,000 plus $1,110,53.
Add the above amount of the tax (b)...
The sum will represent the limit of profit before deduct-

ing either taxes or commission.

19,110.53 3,420.00

(k) (1)

22,530.53

The limit beyond which the earnings, after deducting taxes, would become subject to a lower commission rate may be ascertained by inspection as already indicated in the particular case discussed in the preceding paragraph. In general, the amount beyond which the earnings will be subject to the higher excess-profits rate ($22,530.53 in the example used herein) should be ascertained first, so that the bracket under which the excess-profits tax is to be considered may be definitely established.

If the given amount of earnings before deducting commission and taxes is, say, $19,000 instead of $70,000, the tax thereon would be $3,140, namely $1,600 for excess-profits and $1,540 for income tax. The commission payable on the remainder of $15,860 would be $2,314.50, as follows: 20% on $5,000..

$1,000.00 15% on $5,000...

750.00 10% on $5,000..

500.00 772% on $ 860..

64.50

$15,860

$2,314.50 It is evident that the taxes actually payable will be less than $3,140 and the commission more than $2,314.50, so that commission will be payable on more than $19,000 — $3,140 = $15,860, and on less than $18,000; the latter because more than $1,000 of taxes will be payable in the given circumstances. Consequently, the lowest rate of commission payable will be 772% applicable to earnings between $18,000 and $15,000.

If the given amount of earnings before deducting commission and taxes should be, say, $23,000, the tax thereon would be $4,800, viz., $1,800 under the first and $1,200 under the second bracket of the excess-profits tax and $1,800 for the income tax. The commission payable on the remainder of $18,200 would be $2,485, viz., $2,475 on the first $18,000 plus 5%, or $10 on the remaining $200. Evidently the tax actually payable will be less than $4,800 and the commission more than $2,485, so that commission will be payable on more than $23,000 - $4,800, or $18,200, and on less than $21,000; the latter obviously because more than $2,000 of tax will be payable in the given circumstances. The lowest rate of commission payable will therefore be 5%, applicable to the earnings between $21,000 and $18,000.

Admittedly, the above method of ascertaining these rates by inspection is not scientific, but it seems hardly necessary to develop an algebraic method that will determine the exact limit beyond which the residuary earnings will become subject to lower commission rates. A close inspection will usually suffice to define a high and a low limit, and even if the wrong rates should be chosen, this can be rectified with less work than a proper determination of the exact limit will require.

PROBLEM V

COMMISSION PAYABLE AFTER DEDUCTING TAXES AND CONSIDERING

THE COMMISSION ITSELF AS A DEDUCTIBLE EXPENSE

In the hypothetical case used throughout in problems I, II and III the consideration of the commission itself as first deductible from income would be equivalent to a reduction of the given percentage of commission (20%) to the actual (16-2/3%), for evidently 16-2/3% of every $100 of income before deducting commission ($16.66 2/3%) is equivalent to 20% of the remainder 83.33 1/3%). The solution of problem I in the circumstances would be as follows:

(a) Preliminary computation of the tax By first method of solution of problem I......

$26,420.00

(b) Preliminary computation of commission Income before deducting commission.... Deduct-preliminary amount of federal tax.

$70,000.00 26,420.00

Amount on which preliminary commission is to be computed....

43,580.00

Preliminary commission of 16-2/3%....

7,263-33

(c) Calculation of correct amount of taxes and commission

Reasoning on the same lines as in the first method of solution of problem I, it will be evident that: (c-4) For every $100 of commission actually payable the tax of

$26,420 will be reduced by $46 and the commission determined ($7,263.33) increased by 16-2/3% of $46, or $7.66 2/3 (instead of 20% of $46, or $9.20), so that

(C-5) The amount of commission ($7,263.33) obtained in the

above preliminary calculation is 92.33-1/3% of the
correct amount, which will therefore be
7,263-33

x 100 = 7,866.43, and

92.33-1/3 (c-6) The amount of the tax will be

46

X 7,866.43 3,618.56, less than $26,420, or $22,801.44.

100

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20% of which equals the amount of commission.

In the above example the solution hinges upon the transformation of the given percentage (20) into the actual percentage (16-2/3) that can be substituted in the solution given for problem I.

To establish a method by which this actual percentage (X) can be directly determined from the given rate (c), it should be observed (a) That the given percentage applies to an amount that is X%

less than the amount of income before deducting com

mission; therefore (b) If the amount of income before deducting commission is 100,

the balance left after deducting commission at the

normal rate (X) would be 100 - X. (c) This amount (100 – X) is the amount to which the given

rate (c) is to be applied, consequently (d) The actual commission payable will be:

с

(100 – X)

100

(e) This amount (d) added to the amount of (100 – X) will,

of course, equal 100, consequently

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