-12-

If the deductibility of interest is eliminated, (22) becomes

= (Rei) (1-k) - 6
Py rT +6 (23)

This formula is neutral between debt and equity financing if there is no
investment tax credit,

E. A Neutral Investment Tax Credit with Respect _to the Longevity

of Capital, The investment tax credit may be understood as a substitute
for a capital levy, If the Federal Government were to confiscate half
of all capital and sell it to establish a trust fund, we could maintain
the level of public services while greatly lowering existing taxes and
their distorting effects. As long as no one thought there would be another
round of confiscation, there would be an increase in productivity, If
instead of confiscating the capital, the Government were to stamp all
existing capital "old," and enact a special tax on all old capital, the
effect would be nearly the same, At a further level of refinement, the
Government could say that all capital will be subject to tax, but anyone
who accumulates capital from now on will be given a check for some fraction
of the present value of the taxes that will be paid on it. The effect
would again be the same. This is the principle by which the investment
tax credit works. New capital is subsidized to lower the effective level
of taxes on it, and old capital, though it is not confiscated, falls in
value because it must compete with new, subsidized capital.

An investment tax credit is neutral with respect to the longevity

of capital only if the price of capital services, less depreciation, is


-3-
pay its debt and equity investors the market rates of return. In the
long run, competition will force excess profit to zero. The discount
rate of R reflects the fact that an investment is a joint venture of
debt and equity, even 4f legal control is in the hands of equity.

Define P, as the expected rental price of time t of a machine
that might be purchased at time 0. The present value of the anticipated

future rental payments for the asset when it is new, Po» is

eo Rt
Poe g Pre dt (2)
Define US as the present value of the corporate taxes on the rental
Payments on a new asset, and k as the investment tax credit that the

government would provide, In market equilibrium,
Poo Uy zsl-k (3)

Corporate taxes are levied at a nominal rate of u, but only on
realized income, net of depreciation allowances, that goes to equity.
Additional income for capital, exempt from the corporation income tax,
accrues in the form of interest on debt and in the form of changes in
capital value for equity, of just the right amount in market equilibrium
to insure that current income plus the capital gain or loss will provide
the market rate of return. When a loss is involved, it is called de~
preciation, and the tax rules seek to exempt it, though they do not
succeed perfectly, But there is no counterpart (negative depreciation)
for capital gains, The deductibility of interest and the imperfections in

tax depreciation imply that the true tax rate on capital is very complex.


-16-
REFERENCES

Hall, R.E. and Jorgenson, D.W. 1967 "Tax Policy and Investment Behavior"
American Economic Review 57 (June 1967), pp. 391-414.

Hall, R.E. and Jorgenson, D.W. 1971 "Application of the Theory of
Optimal Capital Accumulation" in G. Fromm, ed., Tax Incentives and

Capital Spending (The Brookings Institution, Washington) pp. 9-60.


-8-

III. TAX RULES FOR NEUTRALITY

A. Equalization of Rates of Return. The true before-tax rate of

return of an asset is not the p that Hall and Jorgenson use, but
rather the price of its services less depreciation. The price paid is
the gross productivity of an asset when users are in competitive
equilibrium; depreciation is the measure of diminution of real resources
associated with its use. Any remainder is a real rate of return.
Efficient allocation requires that this real productivity be the same
for all assets. Thus a sensible goal for tax rules is to insure that
uniformity of real returns is achieved. One way of seeking to do this

would be to define a target real return, e*, take the equation

= Po 55> (17)

substitute from (7) for Po» and then solve for k. The result is
k= 1 - u(Z 41.) - (0% 85) ¥G-u) (18)

An investment tax credit of this complex form would achieve equalization
of returns. But it is simpler to introduce other changes that obviate
the need for most of the complexity of the investment tax credit.

B, Price of Capital Services with Exponential Depreciation and Inflation.

For simplicity, take the case of exponential decay in the services of an asset
and assume that tax authorities achieve their stated goal of permitting tax de:

preciation that exactly equals economic depreciation, in the absence of



-5-

ces at the rate 6, the rental value at time t in the absence

inflation would be

= - st
Py TM Poe (8)

ential growth in prices at a rate of i implies that the nominal

al price will be

a a i ae (10)

resent value of interest payments will be

(1-k) fr

©
T= f° (1-k)frje( St1-Rtae =
078 a R-1i+6

uting from (10) and (11) into (7),

(R-it 6) (1-k-uZ_) - ufr, (1-k) :
Po 2. 4 (12) /

° l-u

arison, the Hall-Jorgenson formula for the price of capital

s (1971, p. 16], with slight changes to make the terminology

» is

[(l-u)p +6] (1-k-uz )

> — (13)

is "the before tax rate of return [adjusted for] deductions

est allowed for tax purposes" [1971, p. 16]. The two differences



-4-
Tf 2, is the present value of depreciation allowances on a new
asset and Ty is the present value of interest deductions, then the

present value of the taxes on a new asset is given by
U, = u(P,-Z5-1,) (4)

Substituting from (4) into (3) and solving for Poe

— l-k- uZ, - ul, (5)
° l-u

To transform (5) into a statement about rental payments, the economic
life of the asset must be introduced. Define Y as the present equiva-
lent of the years of service that will be delivered by the asset. That

is, if P, is the rental price of a new asset,
wa Pot eR at ep / (6)
Oo Po oP

Therefore

l-k-uZ - ul
‘°

oO
Po ® YG-u) (7)

This formula for the rental price of capital can be applied to an
asset with any pattern of depreciation, as long as Py #0, which
would invalidate (6).

Further interesting results can be obtained by applying the formula
to the special case of exponential decay in the services of an asset,

with exponential growth in the price level. With exponential decay in


-7-
which is the same as (14), since what Hall and Jorgenson appear to

mean by p is

R - ufrg (16)
p= T-u

This correct result for a special case may have caused them to overlook
the fact that the formula does not work generally. To show that the
two formulas do not always yield the same results, consider the case
where tax depreciation is a full write-off on the fifth anniversary of
purchase, Then, with no tax credit and no inflation, the two formulas
would yield the same result if and only if (Rt 6)e7 OR were equal to

-5(1-u)pt 6

[(1-u)et 6 Je » and this is generally not true,


Office of Tax Analysis
U.S, Treasury Department
Washington, D.C, 20220
Issued: September 1976

MEASURING THE COST OF CAPITAL SERVICES

T, Nicolaus Tideman
Virginia Polytechnic Institute

OTA Paper 4 April 1975


-10-
so that (19) becomes

(Reit 6) (1-k) - w - ufr

La leu

(Red) (1-k) - ké - ufty (1-k)

= Toa +6 (21)

2. The inflation component of interest must not be deductible.
Then the interest deduction for any year is £(ry-t) and (21) becomes

(Rot) lnk) = 15 ~ uf (4-4) (1-k)

PO = Toa +6 (22)

Now the price of capital services has a form such that equal increases
in interest rates and inflation will have no effect on it. To keep
interest rates from rising by more than inflation, two more changes are
needed,

3. The inflation component of interest received by individuals
must not be taxed. If interest income is not adjusted for inflation,
the income of capital will be taxed more heavily when there is inflation,
affecting saving and real interest rates.

4. The inflation component of capital gains must not be taxed.
This insures that the relative advantages of capital gains and ordinary
income will not be affected by inflation.

D. Tax Neutrality between Debt and Equity Financing. The deduc-

tibility of interest causes firms to benefit from debt finance even when

their nominal capital costs are raised in the process. This distortion


-9-
inflation. With exponential decay, the present value of economic
depreciation will equal é/(R+é). Substituting this into (12), the

price of capital services reduces to

5 ué
. (R-it 6) (1-k - Rs

° leu

y= ufr ,(1-k)

P (19)

C. Tax Neutrality with Respect to Inflation. The first sensible

adjustment would be to eliminate the effects of inflation on the price

of capital services. One way of achieving this would be to require all
financial records to be indexed for changes in the value of dollars, so
that when dollars of different periods are added and subtracted, inflation
would cause no distortions. If the sweeping change is not enacted,

four separate changes are needed.

1. Depreciation allowances must be indexed. That is, the deprecia-
tion that is appropriate when there is no inflation must be multiplied
by the ratio of the price index at the time of the allowance to the
price index at the time the asset was purchased. Some people would
Propose that indexation be linked to changes in the price of the indi-
vidual asset, but this would improve neutrality only if all asset price
changes were included as income, whether realized as capital gains and
losses or not. The indexation of depreciation allowances converts their

present value to

Z 2 —— (20)


-6-
between the formulas are: 1) Inflation appears in (12) and not in (13).
A real discount rate is applied to rental payment and future interest
cost, while a nominal discount rate is applied to depreciation allowances,
in (12). 2) Interest appears as an explicit deduction in (12), while in
(13) an adjustment is made in p to allow for the deductibility of
interest.

If there is no inflation and no tax credit, and if tax depreciation
equals economic depreciation, then the two formulas yield the same result.
With exponential decay, the present value of economic depreciation is
6/(R+6) at a discount rate of R, and é/[(l-u)p +6] at a discount
rate of (l-u)p. Substituting 6/(R+ 6) for Z, in (12), and setting

i=0 and k= 0,

ud ufrg
(Rt 8) (1 - Ras - BED

RF l-u

Rt 6 - WJ - ufrg
l-u

R- utr,
"joa ts 4)

When 6/[(l-u)p +6] is substituted for Z, in (13) along with k=0,

the result is

py = Gowe +8 =

l-u

=p +s (15)


-~2-
II, THE PRICE OF CAPITAL SERVICES

To avoid having to deal with labor, consider a firm that only buys
capital equipment and rents it out. To avoid the need for a scale
Parameter, assume that each investment under consideration is the pur-
chase of an asset that costs $1. Each investment is financed by a
combination of debt and equity in the proportions f of debt and (1-£)
of equity. The firm issues and retires debt and equity as necessary
so as to keep the debt/equity ratio at f/(1-£), which it has chosen to
minimize costs, Debt must be paid a market return (before individual
taxes) of Ta, while equity receives dividends and anticipated capital
gains that the market values at XQ: These returns can be measured either
as nominal returns, or as real returns, where the real return is the
nominal return minus the inflation rate. Some of the steps in the suc-
ceeding analysis will vary with the choice of nominal or real measures,
but equivalent real results can be obtained from the two approaches.
Nominal analysis has been chosen for this paper because fixed nominal tax
rates make it slightly simpler.

The nominal return, R, that the firm must earn after corporate

taxes is a weighted average of Tyg and ry:
R= fr, + (l-f) ry. (1)

By making only investments that yield a non-negative profit (after taxes)

when discounted at the rate R, the firm insures that it will be able to


~13-
independent of the rate of depreciation, Taking (23) as the price of
capital services, this means
(Rei) (1-k) = 15 * 24
Too = (24)

or, solving for k,

k= Roi- p*(1-u) (25)

R-it6é

This implies that after other distortions are eliminated, an investment
tax credit that is neutral with respect to longevity must be of the

relatively simple form
k= RTT (26)

where x is any positive number. If .05 is a reasonable estimate of

R- i, then a value of .02 for x Produces a neutral investment tax
credit with approximately the same average value as the present credit,
which is 10.0% for assets with lives over 6 years, 6.67% for lives of 5 or
6 years, 3.33% for lives of 3 or 4 years, and nothing for shorter lives,
Taking the rate of depreciation as 2/T, where T is economic life for
tax purposes (as if double declining balance depreciation in perpetuity
were exactly right), the comparison of the Present credit with a neutral
credit is shown in Table 1,

F. Neutrality toward Appreciating Assets. At the beginning of the

discussion of neutrality, it was assumed that tax depreciation equals

economic depreciation. For appreciating assets, such as trees, wine, and


-15-
land development, this requires that negative depreciation be included
as income, in other words, that capital gains be taxed as they accrue,
To defer the taxes on capital gains is to lower their present value,

therefore lowering the equilibrium rate of return of such investments.


I, INTRODUCTION

Hall and Jorgenson developed a formula for the price of capital
services [1967, 1971] that is widely used. Their formula, however,
fails to distinguish between the real discount rate for future services
of capital and the nominal discount rate for depreciation. It also
adjusts the discount rate to allow for the deductibility of interest,
when the proper procedure is to include interest as an explicit deduc~
tion, like depreciation, The Hall-Jorgenson procedure yields the correct
answer in the special case of exponential decay and tax depreciation
equal to economic depreciation, but not in general,

This paper slowly develops a formula for the cost of capital services,
treating each complication as it arises. The formula is then applied
in a discussion of the tax rules that would be neutral with respect to
inflation, forms of financing, the longevity of capital, and appreciating

assets,


I.
Il.

IIl.

TABLE OF CONTENTS

Introduction... ee ee ee eee eee

The Price of Capital Services ........

Tax Rules for Neutrality ....... eee

Equalization of Rates of Return .....
Tax Neutrality in the Absence of Inflation

Tax Neutrality with Respect to Inflation .

Tax Neutrality between Debt and Equity Financing

A Neutral Investment Tax Credit with Respect to the

Longevity of Capital... 1. eee ee eee

Neutrality toward Appreciating Assets ..

Page

10

12

13


-ll-
increases the risk of bankruptcy and favors investments with lower risk.
One way of eliminating this distortion would be to abolish the corporate
profits tax, attribute all corporate income to shareholders, and make up
any revenue loss by higher taxes on all non-labor income. But that is
not likely to happen because voters tend to favor the idea that corpora~
tions as entities pay taxes. It may be more politically feasible to
convert the corporation income tax to a withholding tax, by letting share-
holders take tax credits for their portion of corporation income taxes
withheld, again attributing all corporate income, whether distributed or
not, to shareholders. This approach may be more politically feasible
simply because it preserves the appearance of a corporate tax.

Another way of eliminating the bias toward debt financing would be
to make debt as well as equity subject to the corporate tax, and lower
the tax rate to keep revenue constant. This would preserve the discrimina-
tion against corporate activity compared to unincorporated enterprise,
while eliminating the bias against equity financing.

If voters really want discrimination against corporate activity,
because of an evil associated with bigness, it would make more sense
to put a more continuous degree of progression into the tax, so that firms
would have an incentive to break themselves up. It might also be
desirable to base the tax on assets rather than income, as assets are
probably more highly correlated than income with the excess power of

corporations that people perceive,


-16-

TABLE 1

Tax Present Neutral Tax Present Neutral

Life Credit Credit Life Credit Credit
2 0% 1.90% 11 10.00% 8.63%
3 3.33 2.79 12 10.00 9,23
4 3.33 3.64 15 10.00 10.91
5 6.67 4.44 20 10.00 13.33
6 6.67 5.22 25 10.00 15.38
7 10.00 5.96 30 10.00 17.14
8 10.00 6.67 40 10.00 20.00
9 10.00 7.35 50 10.00 22,22

10 10.00 8.00 ° 10.00 40.00