of the machine, and an obsolescence cost, which results from the development of an im-
proved model. For example, at the end of the fourth year the deterioration cost is $16,500
- $12,000 = $4500, and the obsolescence cost is $600 x 3 = $1800. If the quality of the
product declines as the machine ages, the resulting loss of income can be added to the de-
terioration cost.
ECONOMY OF REPLACEMENT
ON AFTER-TAX BASIS
A machine was purchased 3 years ago at a cost of $45,000. It had a life expectancy of 7
years and anticipated salvage value of $3000. It has been depreciated by the sum-of-digits
method. The net resale value of the machine is $13,000 at present and is expected to be
$9000 a year hence. The operating cost during the coming year will be $2600. A newly
developed machine can be substituted for the existing one. According to estimates, this
machine will have an optimal life of 5 years with an annual cost of $4800 on an after-tax
basis. The tax rate is 45 percent for ordinary income and 30 percent for long-term capital
gains. The desired investment rate on an after-tax basis is 8 percent. Determine whether
the existing machine should be replaced at present.
Calculation Procedure:- Compute the depreciation charges for the first 4 years
Refer to an earlier calculation procedure for sum-of-digits depreciation. The charges are
Z) 1 = $10,500; D 2 = $9000; D 3 = $7500; D 4 = $6000. - Compute the book value at the end of the third and fourth years
Let BR = book value at end ofRth year. Then B 3 = $45,000 - ($10,500 + $9,000 + $7,500)
- $18,000 and B 4 = $18,000 - $6000 - $12,000.
- Compute the cost of retaining the machine through the
fourth year
Income that accrues from normal business operations is called ordinary income; other
forms of income are called capital gains. The difference between the net income that ac-
crues from selling an asset and its book value at the date of sale is a capital gain (or loss).
If the asset was held for a certain minimum amount of time, this capital gain (or loss) is
subject to a tax rate different from that for ordinary income.
Let R = age of asset, years. The after-tax cost of retaining the asset through the (R +
l)st year, as evaluated at the end of that year, is [LR(\ - tc) + BRtc](\ + ia) + cR+l(\ - t 0 ) -
LR+l(\ - tc) - BR+ltc - DR+lt 09 where ia = after-tax investment rate; t 0 and tc = tax rate on or-
dinary income and long-term capital gains, respectively; L = true salvage value; c = annu-
al operating cost; and the subscript refers to the age of the asset. The expression in brack-
ets is the income that would be earned if the asset were sold at the end of the Rth year; if
the asset is retained, this income is forfeited and becomes part of the cost of retention.
With ia9 = 8 percent, t 0 = 45 percent, tc = 30 percent, L 3 = $13,000, L 4 = $9000, and C 4 =
$2600, the cost of retaining the machine through the fourth year is $13,000(0.70) +
$18,000(0.30) + $2600(0.55) - $9000(0.70) - $12,000(0.30) - $6000(0.45) =
$4490.