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Dual Income Taxation:
The Choice of the Imputed Rate of Return
by
Paolo M. Panteghini
Università degli Studi di Brescia
Dipartimento di Scienze Economiche,
Via San Faustino 74B, 25122 Brescia, ITALY
E-mail: panteghi@eco.unibs.it
Phone number: 0039-030-2988816
Fax number: 0039-030-2988837
First draft: June 1999
This version: April 2000
ABSTRACT
In this article we study a crucial aspect of a Dual Tax System: the choice of the imputed rate of
return. Under interest rate uncertainty, its optimal value will be shown to depend on the nature of
investment. Following Fane (1987), if investment is reversible, the imputation rate ensuring
neutrality is proportional to the interest rate on default-free bonds. If, instead, investment is
irreversible, the imputation rate must be higher, in order to compensate for the discouraging effects
of irreversibility.
JEL classification: H25; keywords: corporate taxation, tax neutrality, interest rate uncertainty.
The author is grateful to Silvia Giannini, Antonio Guccione, Vesa Kanniainen, and Peter Birch
Sørensen for helpful comments and suggestions. He accepts responsibility for any remaining errors.
1
I. Introduction
In the existing literature on corporate tax neutrality there are two traditional potentially neutral tax
designs. The first one defines true economic profits as its tax base, and is called ‘imputed income
method’ (see Samuelson, 1964). The second one, proposed by Brown (1948), is the cash-flow
method. Unfortunately, both methods are hard to implement. The former is informationally very
demanding, since it requires the knowledge of the rate of return for each firm-specific investment
(see Sandmo, 1979). The latter is an attractive device with some practical disadvantages, such as
difficulties in containing tax evasion and tax avoidance. From the point of view of the taxing
authorities, moreover, the cash-flow tax is hard to implement since it may require tax payments to
expanding firms as well. For these reasons, Boadway and Bruce (1984) proposed ‘a simple and
general result on the design of a neutral and inflation-proof business tax’ [p. 232]. According to this
rule, the business tax base is given by the firm’s current earnings, net of the accounting
depreciation rate (applied to the accounting capital stock) and of the nominal cost of finance. As
argued by Boadway and Bruce (1984), however, each firm may have a different value of the
nominal cost of finance, which must reflect the investment-specific riskiness, and which is not
directly observable.
Fane (1987) takes an important step forward, and finds that the Boadway and Bruce (1984)
general neutrality principle holds even under uncertainty, provided that the tax credit and liabilities
are sure to be redeemed and that the tax rate is known and constant. But, more importantly, he
proves that such a neutrality design could be achieved by simply setting the deductible nominal
cost of finance equal to the risk-free nominal interest rate. ‘Since government bonds provide an
(almost) certain nominal return’, he argues, ‘it is not difficult to estimate the risk-free interest rate’
[p. 99], and the tax design turns to be much less informationally demanding.
Given the above results, in 1991 the IFS Capital Taxes Group proposed the introduction of
the Allowance for Corporate Equity (hereafter ACE), in the UK corporate taxation. According to
this proposal, the corporate tax base should be set equal to the firm's current earnings net of: i) an
arbitrary tax allowance for capital depreciation (not necessarily the cost of economic depreciation)
and ii) the opportunity cost of finance
1
. Moreover, it should entail a symmetric treatment of profits
and losses. Clearly, this proposal is not only based on the theoretical results of Boadway and Bruce
(1984), but also on those of Fane (1987). According to the IFS proposal, in fact, the opportunity
cost of finance should be set equal to the default-free interest rate, thereby making the government
2
‘a sleeping partner in the risky project, sharing in the return, but also sharing some of the risk’
(Devereux and Freeman, 1991, p.8). Recently, Bond and Devereux (1995) have proven that the
ACE system is neutral even when income, capital and bankruptcy risk are introduced. They have
also proven that the rate of relief ensuring neutrality remains the nominal interest rate on default-
free bonds.
The choice of the relevant opportunity cost of finance is an important topic not only on
theoretical grounds but also on tax policy grounds. In the Nineties, in fact, the Nordic countries (see
Sørensen, 1998), and, more recently, Italy (see Bordignon et al., 2000) introduced a dual corporate
tax system
2
. Moreover, a form of ACE taxation was adopted in Croatia in 1994 (see Rose and
Wiswesser, 1998), under the name of Interest Adjusted Income Tax (IAIT). According to these tax
designs, companies' earnings are split into the following two components:
q an imputed return on new investments financed with equity capital, called the ‘ordinary
return’;
q the residual taxable profits, namely profits less the ordinary return.
The ordinary return, approximating the opportunity cost of new equity capital, is calculated
by applying a nominal interest rate to equity capital. According to this dual system, the ordinary
return is taxed at a lower rate than the residual taxable profits
3
.
One of the most controversial aspects of these systems is the choice of the imputation rate.
On the one hand, Denmark sets the imputed rate of return equal to the average market interest rate
on bonds. On the other hand, both the other Nordic countries and Italy prefer to introduce a risk
premium. Finally, under the IAIT the imputed rate of return is set equal to the growth rate of
manufacturing prices plus 300 basis points.
In this article we will show that the choice of the appropriate imputed rate depends on the
characteristics of the dual taxation, on the existence of interest rate uncertainty and, above all, on
the nature of investment. As we know, in fact, investment is, at least partially, irreversible.
Irreversibility may arise from ‘lemon effects’, and from capital specificity (see Dixit and Pindyck,
1994, and Trigeorgis, 1996). Even when brand-new capital can be employed in different
productions, in fact, it may become specific once installed. Furthermore, irreversibility may be
caused by industry comovement as well: when a firm can resell its capital, but the potential buyers
1
For further details on this proposal see IFS Capital Taxes Group (1991), Devereux and Freeman (1991), and Isaac
(1997).
2
Under the Nordic systems, the dual design is applied only to small companies, when labour and capital incomes
accrue jointly. Under the Italian system, instead, this tax design covers all kinds of business activities (except those
activities where simplified accounts are kept). On these grounds, the Italian Dual Income Tax (DIT) is similar to the
ACE system, which proposes the taxation of extra-profits at a higher rate than normal profits, regardless of the
organisational form involved.
3
Of course, the ACE system can be considered as a special case of the dual system where the lower tax rate is null.
3
operating in the same industry are subject to the same market conditions, this comovement obliges
the firm to resort to outsiders. Due to reconversion costs, the firm can sell the capital at a
considerably lower price than an insider would be willing to pay if it did not face the same bad
conditions as the seller.
When investment is reversible, namely when investment can be resold without any
additional cost, the imputed rate of return ensuring neutrality is proportional to the interest rate on
default-free bonds. This result is related to those of Fane (1987), and Bond and Devereux (1995),
who implicitly assume investment reversibility. When, instead, investment is irreversible, the
imputation rate must be higher, in order to compensate for the discouraging effects of investment
irreversibility.
The article is structured as follows. Section II introduces a simple discrete-time model with
a bivariate distribution of interest rates. Section III shows the neutrality results and section IV
discusses the results obtained. In Appendix, we show a generalisation of the results, where
uncertainty does not vanish after one period, but lasts to infinity.
II. The model
In this section we analyse the effects of interest rate uncertainty on the firm's investment decision.
We introduce a simple discrete-time model of a competitive risk-neutral firm. For simplicity, we
omit all the other sources of uncertainty (studied by Bond and Devereux (1995)) and assume that
the firm's lifetime is infinite
4
.
The structure of the model is as follows. Given an investment of 1€, at the end of each
period, the firm will receive a constant cash flow
Π
in perpetuity. The short-term interest rate is a
bivariate stochastic variable. Given its initial value r, at time 1 it will either rise to r
2
with
probability (1-q) or drop to r
1
with probability q. For simplicity, uncertainty vanishes at time 1 and
the interest rate will remain constant forever. Note that with this assumption the model is reduced
to a two-period framework. The first period lasts until time 1 and the second period lasts from time
1 to infinity. This simplifying assumption will be removed in Appendix, where we will assume an
uncertain term structure scenario lasting to infinity.
Let us finally assume the following inequalities
)]1/([)]1/([
1122
rrrr +>Π>+ .
In other words, if the ex-post rate of return of the entrepreneurial investment is less than the rate of
return of a short-term bond, the financial investment is preferred to the entrepreneurial one and vice
4
Note that assuming a finite business lifetime would not change the quality of the results.
4
versa. This implies that an increase (decrease) in the short-term interest rate will represent a bad
(good) news for the firm.
In this article, we compare two different cases, which are both characterised by
irreversibility: in the first case, the representative firm can decide whether investing at time 0 or
never. In the second case, instead, the representative firm can decide whether investing at time 0 or
waiting until time 1, when uncertainty vanishes
5
.
When investment is irreversible and the firm cannot postpone investment, the firm will
invest if the expected net present value at time 0 of its project, net of its investment cost, is positive
[1] 01
1
)1(
1
)1(
)1(
1
1
1
2
1
1
1
0
>−
+
Π
⋅
+
⋅−+
+
⋅=
∑∑
∞
=
−
∞
=
−
r
r
q
r
qNPV
t
t
t
t
,
which can be rewritten as
[1'] 01
0
>−
Π
=
c
r
NPV ,
where
+
⋅−+
+
⋅⋅
+
=
2
2
1
1
1
)1(
1
1
11
r
r
q
r
r
q
rr
c
.
Note that r
c
is the current consol rate of a long-term bond. If there exist perfect capital markets, its
value today (1/r
c
) is equal to its discounted expected value one period from today plus one coupon
payment. As shown in the RHS of the above equality, therefore, its value is given by the weighted
average of the state-contingent prices (discounted one period ahead) of the short-term bond in the
good and bad state, namely
1
1
1
r
r+
and
2
2
1
r
r+
. The weights are represented by the probabilities of
the events, q and (1-q), respectively
6
. The rule expressed by inequality [1'] is straightforward: If the
present discounted value of future cash flows
c
r/Π is greater than the investment cost, then the
firm undertakes investment.
The above case is seldom realistic. When the firm cannot resell the installed capital, in fact,
it is not usually obliged to invest at time 0. Rather, it can decide to observe the realisation of the
future, and wait until time 1, when uncertainty is resolved. In this case the firm will invest at time 0
only if the interest rate is sufficiently low. This implies that the firm is endowed with an option to
delay (see Ingersoll and Ross (1992) and Trigeorgis (1996)). In this case, the positive value of
NPV
0
is no longer a sufficient condition for investing at time 0 to be the optimal strategy. To decide
5
Note that, under reversibility, if inequality r>Π
0
holds, the firm invests and produces at time 0. If, at time 1, the
interest rate decreases, the firm will go on producing. If, instead, the interest rate rises, the firm will stop and resell its
capital without any loss.
6
For further details on the definition of the state-contingent prices see Backus et al. (1998, p.16).
5
when to invest (i.e. when exercising the option), in fact, the firm must compare NPV
0
with its
expected net present value at time 1
[2] .1
1
1
)1(
1
1
1
1
1
1
1
−
Π
⋅
+
=
−Π⋅
+
⋅
+
=
∑
∞
=
−
rr
q
r
r
q
NPV
t
t
As can be noted, NPV
1
does not account for the 'bad news' (i.e. the increase to
2
r ), since a rational
firm would not enter at time 1, after an increase in the interest rate
7
.
Under investment irreversibility, the firm chooses its optimal investment time by comparing
NPV
0
with NPV
1
. If, therefore, inequality NPV
0
> NPV
1
> 0 holds, then immediate investment is
preferred. If, instead, NPV
1
> NPV
0
> 0, postponing investment is more profitable. Equating [1] to
[2] we obtain the trigger value of the short-term interest rate (see Ingersoll and Ross, 1992)
.)1()1(
)1(
2
2
2
2
*
r
r
qq
r
qr
r
Π−
⋅−−Π=−−Π⋅
−+
=
If, therefore, the current interest rate r is less than the trigger value r
*
, then immediate investment is
preferred and vice versa. As can be seen, r
*
depends only on the unfavourable events (namely on
the probability and the seriousness of bad news). Hence, this result is an application of Bernanke's
(1983) Bad News Principle (BNP), which shows that it is the ability to avoid the effects of bad
outcomes that leads the firm to decide whether waiting or not
8
.
For a better understanding of the firm's behaviour, we can rewrite the firm's decision rule by
using equations [1] and [2], thereby obtaining
01
1
1
10
≥
−
Π
⋅
+
+−
=−
m
rr
rq
NPVNPV ,
where
2
2
)1(
)1(
r
qr
qr
r
m
⋅
−+
−
+
= . As can be seen, the firm's decision rule is based on the comparison
between the present discounted value of future cash flows and the investment cost. When the
representative firm holds the option to wait, however, the firm's discount rate is different: easy
computations show that
cm
rr > .
Like r
c
, the rate
m
r has an economic meaning. In fact, r
m
can be obtained by solving the
following equation
[3]
+
⋅−+
+
⋅⋅
+
=
2
2
1
)1(
1
1
11
r
r
q
r
r
q
rr
m
m
m
.
7
For further information on this point see Dixit and Pindyck (1994, pp. 40-41) and Trigeorgis (1996, pp. 197-199).
8
As pointed out by Bernanke (1983), under investment irreversibility "the impact of downside uncertainty on
investment has nothing to do with preferences ... The negative effect of uncertainty is instead closely related to the
search theory result that a greater dispersion of outcomes, by increasing the value of information, lengthens the optimal
search time" [p. 93].
6
The explanation of equation [3] is simple. The LHS is the current price of the long-term consol
bond issued for financing the firm's investment. Similarly to the definition of r
c
, the RHS is the
weighted average of the state-contingent prices (discounted one period ahead) in the good and bad
state. Contrary to the former case (where r
c
was the discount rate), the state-contingent price in the
good state is not
1
1
1
r
r+
but rather
m
m
r
r
+
1
+
>
+
m
m
r
r
r
r
1
1
with
1
1
. To understand this different
evaluation of the good news, let us describe the behaviour of the representative firm. On the one
hand, the firm may decide to delay investment in order to capture the decrease in the interest rate.
On the other hand, the firm is attracted by the immediate payoff. To exploit both these
opportunities, it might, in principle, finance its immediate investment by borrowing in a bond,
which incorporates a prepayment option. Therefore, if the interest rate decreases to r
1
, the firm can
exercise the prepayment option, thereby returning the principle. Then, it issues a new bond at the
current rate r
1
. If capital markets are perfect, therefore, the value of the prepayment option is given
by the difference
+
−
+
m
m
r
r
r
r
1
1
1
1
and the effects of the good news are thus neutralised. If, instead,
the interest rate rises to r
2
, the firm faces a reduction in its NPV and the mortgage rate will be equal
to
m
r =
2
r .
9
Note that the choice of
m
r as the relevant discount rate does not imply that the firm is
obliged to use debt instead of equity to finance its investment. According to the Modigliani-Miller
theorem, if there exist perfect capital markets, equity-financing is equivalent to debt-financing,
namely an equity-debt swap does not alter the value of the firm.
Finally, it is worth noting that neither r
c
nor r
m
are computed by the firm. If capital markets
are perfect, in fact, both of them are given by the existing prices of a default-free long-term
(Treasury) bond and of a callable bond, respectively. Berk (1999) defines this latter bond as a
'mortgage bond', which is equivalent to a portfolio consisting of a long position in a non-callable
consol bond (i.e. Treasury bonds) and a short position in an American call option (the prepayment
option) on the same bond
10
.
9
This implies that the current price of the bond is ./)1(
22
rr+
10
Berk (1999) also argues that such mortgage obligations exist and are listed in the US markets. They are 30 years
bonds, guaranteed by the US government. Thus, their prices do not account for default risk.
7
III. The neutrality results
Let us next introduce taxation. Under a dual tax system, current earnings Π are split into two
components: the ordinary return and, if any, the residual taxable profits. For simplicity, we assume
that profits and losses are treated symmetrically.
To compute the ordinary return we define the imputed rate of return as r
E
, and let
l
τ
and
h
τ
be the tax rate on the ordinary return and that on the residual profits (with τ
l
< τ
h
), respectively. In
order to keep the model as simple as possible, finally, we assume that the project is fully equity-
financed.
Taxes are paid at the end of each period. If the firm cannot postpone investment (namely it
may invest either at time 0 or never), the net present value of the expected tax burden is
[4]
[
]
[
]
c
EhElEhEl
t
t
t
t
r
rr
r
rr
r
q
r
qT
)(
1
)(
)1(
1
)1(
)1(
1
1
1
2
1
1
1
0
−Π+
=
+
−Π+
⋅
+
⋅−+
+
⋅=
∑∑
∞
=
−
∞
=
−
ττττ
.
Clearly, if τ
l
= 0, we turn to the ACE scheme.
Before analysing the effects of the dual tax scheme on investment, let us define neutrality.
As argued by Johansson (1969): 'One way of defining "neutral" corporate taxation is in terms of
identical ranking in a pre-tax and in a post-tax profitability analysis. Corporate income taxation is
neutral, if the concept of income used for taxation purposes is such that identical ranking of
alternative investments is obtained in a pre-tax and post-tax profitability analysis.' [p. 104]. As we
know, the firm's decision depends on the sign of the post-tax net present value of its project at time
0. Namely, if NPV
0
-T
0
is positive, investing is profitable. Following Brown (1948), given a
generic tax rate µ, therefore, a tax system is neutral (namely there exists an identical ranking in a
pre-tax and in a post-tax profitability analysis) if the post-tax net present value is (1-µ) times the
pre-tax net present value
[5] .)1(
000
NPVTNPV ⋅−=−
µ
Using equations [1] and [4], we obtain the post-tax net present value of the project
[6]
−−+
−
Π
−=−
h
c
E
lh
c
h
r
r
r
TNPV
ττττ
)(1)1(
00
.
In order to obtain neutrality, therefore, the second term in the RHS must be null. Therefore, setting
c
lh
h
E
rr ⋅
−
=
ττ
τ
*
equation [6] reduces to the neutrality condition [5] (with
h
τ
µ
= )
−
Π
−=−
1)1(
00
c
h
r
TNPV
τ
.
8
This result is similar to that obtained by Fane (1987), who argues that the benchmark interest rate
must be the nominal interest rate on government bonds since 'tax credits are equivalent bonds, and
the building-up of tax credits by a firm is therefore equivalent to its using equity finance to pay-off
debt' [p. 101]. Since such equity-debt swaps do not alter the value of the firm, the result turns to be
an application of the Modigliani-Miller theorem. Note, however, that, under reversibility, the
imputation rate must be proportional to the current short-term rate. When capital can be resold
without any loss, in fact, production is profitable if the net cash flow is greater than the cost of
capital, namely if
[
]
trrr
tEhEl
∀>−Π+−Π ,)(
τ
τ
. Easy computations show that the neutral
imputation rate must be equal to .0 ,)]/([ ≥∀−= trr
tlhhE
τ
τ
τ
When, instead, investment is
irreversible and the firm cannot postpone it, the appropriate imputation rate must be proportional to
the long-term interest rate on the consol bond r
c.
Under a dual tax system, the neutral imputation rate is higher than the current consol rate r
c
,
due to the different treatment of the ordinary and residual returns
11
. However, the positive
differential (r
E
-r
c
) is independent of any stochastic factor, and cannot be interpreted as a risk
premium
12
. To better understand this point, let us consider two opposite cases. In the first case, we
assume that the firm's return is greater than the ordinary return. Thus, the tax burden on residual
profits is
h
τ
times the residual profits. In the second case, instead, the firm's return is less than the
ordinary return, and the firm receives a subsidy. Despite this negative result, the firm must pay
taxes on the ordinary income in any case. Therefore, the net transfer to the firm is equal to
)(
lh
τ
τ
− times the difference between the effective and the ordinary return, which is negative. To
obtain neutrality, we thus compare the tax burden in the good state with the net transfer in the bad
state. As shown in equation [6], therefore, the present discounted value of the tax benefit due to the
deduction of the investment cost under dual taxation,
c
E
lh
r
r
)( ττ − , is equal to that obtained under a
cash-flow system with rate τ
h
, thereby obtaining
*
E
r .
Let us now turn to the latter case, where investment is irreversible, but the firm can
postpone it. In order to find the value of
E
r ensuring neutrality, we must compute the net present
value of the expected tax burden when the firm invests at time 1
11
Under the ACE system, instead, the appropriate imputation rate is obtained by simply setting τ
l
= 0, thereby
obtaining the r
E
= r
c
(see Bond and Devereux, 1995).
12
Under income uncertainty, the choice of a greater ordinary return than the market rate of interest is due to the
asymmetric treatment of ordinary and residual returns. As explained by Panteghini (1998), however, no risk premium
should be taken into account.
9
[7]
[ ]
−Π+⋅
+
⋅
+
=
∑
∞
=
)(
)1(
1
1
1
1
1 EhEl
t
t
rr
r
r
q
T ττ
.
To check whether neutrality holds, however, the Brown condition must be modified, in order to
account for the firm's option to wait. This modified condition arises from the comparison between
the immediate undertaking of investment and its postponement. This implies that if we have
[8] ))(1()(-)(
101100
NPVNPVTNPVTNPV −−=−−
µ
,
then the trigger value r
*
will be the same as that without taxation. According to condition [8], on
the one hand, an increase in the tax rate reduces the present value of future discounted profits and
induces the firm to delay investment. On the other hand, the increase in the tax rate causes a
decrease in the opportunity cost of investing at time 0, thereby encouraging investment. Thus,
neutrality holds if the net effect of these counteracting factors is null
13
.
Substituting equations [1], [2], [4], and [7] into condition [8] we thus obtain
[9] .)(1)1(
1
1
)(-)(
1100
−−−
Π
−⋅
+
+−
=−−
m
E
lh
m
h
r
r
rr
rq
TNPVTNPV
τττ
Note that if we set
c
lh
h
E
rr ⋅
−
=
ττ
τ
, i.e. the imputed return ensuring neutrality in the former case, a
distortion takes place. Since r
c
< r
m
, in fact, we have an inequality
)-()1(1)1(
1
1
)(-)(
101100
NPVNPV
r
r
rr
rq
TNPVTNPV
hh
m
c
m
h
⋅−<
−−
Π
−⋅
+
+−
=−− τττ
.
As can be seen, the present value of the tax credit, discounted with
c
r , is less than that guaranteed
by the cash-flow taxation, namely the firm underinvests
14
. In order to eliminate this distortion at
time 0, we must set
m
lh
h
E
rr ⋅
−
=
ττ
τ
*
, thereby obtaining the modified Brown condition
).-()1(
1
1
)1()(-)(
101100
NPVNPVI
rr
rq
TNPVTNPV
h
m
h
⋅−=
−
Π
⋅
+
+−
⋅−=−− ττ
The explanation of this result is straightforward: to ensure neutrality, the government must take into
account the same discount rate as that used by the firm, namely r
m
. Thus, the neutrality result is,
13
For further details on the links between corporate income taxation and investment irreversibility, see the survey of
Nieman (1999).
14
A similar result would be obtained if we set the imputation rate proportional to the short-term rate at time 0, namely
rr
lh
h
E
⋅
−
=
ττ
τ
. In this case, underinvestment would be given by the following inequality
)-()1(1)1(
1
1
)(-)(
101100
NPVNPV
r
r
rr
rq
TNPVTNPV
hh
mm
h
⋅−<
−−
Π
−⋅
+
+−
=−− τττ .
10
again, an application of the Modigliani-Miller theorem. However, this theorem now holds on
condition that the alternative bond for the equity-debt swaps incorporates the prepayment option.
To get a clearer picture of this result, let us decompose the appropriate imputation rate as
follows
[10] )(
*
cm
lh
h
c
lh
l
cE
rrrrr −⋅
−
+⋅
−
+=
ττ
τ
ττ
τ
.
The first term of [10] is the interest rate on default-free long-term bonds. The second term is the
compensation for the dual treatment of the firm's earnings. When the firm is endowed with an
option to wait, however, a third term must be added. Since it depends on the differential )(
cm
rr − ,
it represents a risk premium. In particular, the higher the differential (τ
h
-τ
l
) the lower the risk
premium ensuring neutrality.
Setting τ
l
= 0, one obtains the neutral ACE imputation rate
cmE
rrr >=
*
. This rate is higher
than that recommended by Bond and Devereux (1995), since they did not account for the risk
premium due to irreversibility.
Note that
*
E
r
is the neutral imputation rate at time 0. Although the same neutrality rule holds
for t > 0, the result is trivial when uncertainty vanishes. In this case, in fact, the differential between
the short- and long-term interest rates is null (i.e. r
m
= r
c
= r
i
with i =1,2). If, therefore, the firm
found it profitable to wait until uncertainty is resolved (namely at time 1), the neutral imputation
rate is proportional to the current rate. Note, however, that investment is undertaken only if the
interest rate drops to r
1
. Thus, the neutral imputation rate will be
1
*
rr
lh
h
E
⋅
−
=
ττ
τ
.
Finally, it is worth noting that the above results do not depend on the bivariate stochastic
process assumed. In Appendix, we show that the result holds even when the interest rate follows a
generic stochastic process and uncertainty does not vanish after one period. Therefore, the neutral
imputation rate must be proportional to the mortgage rate in order to take into account the firm’s
option to wait. If, otherwise, we set the imputation rate proportional to the current consol rate we
would obtain underinvestment.
IV. Conclusion
The simple model employed in this article deals with the choice of the appropriate imputation rate
under a dual income tax scheme. As we have shown, the rate ensuring neutrality depends on the
nature of the firm's investment. Following Fane (1987), if investment is reversible, the imputation
11
rate ensuring neutrality is proportional to the short-term interest rate on default-free bonds. If,
instead, investment is irreversible, the imputation rate must be higher, in order to compensate for
the discouraging effects of irreversibility. In particular, the imputation rate must be proportional
either to the mortgage or to the consol rate, depending on whether the firm holds the option to wait
or not. It is worth noting that, if capital markets are perfect, the appropriate discount rates (r
m
and
r
c
) are not computed by the firm. Rather, they represent the interest rates of two kinds of bonds
which are priced in the existing capital markets.
The results obtained in Fane (1987) and in this article deal with the benchmark cases of full
reversibility and full irreversibility, respectively. However, they are sufficient to suggest that, in an
intermediate case, the greater the investment irreversibility the higher the imputation rate.
Appendix
In this Appendix we remove the simplifying assumption that the interest rate follows a bivariate
stochastic process. To do so we use Berk's (1999) model. The reader will find in that source careful
statements of the relevant hypotheses and of the results.
When the current consol interest rate r(t) follows a generic stochastic process and
uncertainty does not vanish after one period, the investment rule is as follows: It is optimal to invest
the first time that inequality
[A1] 01
)(
≥
−
Π
tr
m
is satisfied, where )(tr
m
> r(t) is the mortgage rate at time t (see the Proposition at p. 1322 in Berk
(1999)).
Let us now introduce taxation. As we know, neutrality holds if the post-tax ranking in
profitability is the same as the pre-tax one, namely
[A1] 01
)(
)(
]2A[01
)(
≥
−
−Π
⇔≥
−
Π
tr
tT
tr
mm
.
Under a dual tax system, the current tax burden is
[A3]
[
]
))(()()( trtrtT
EhEl
−Π+=
τ
τ
,
where )(tr
E
is the imputation rate at time t. Substituting [A3] into [A2], it is straightforward to see
that neutrality holds if )()( trtr
m
lh
h
E
⋅
−
=
ττ
τ
. In fact, [A2] is simplified as follows
01
)(
)-(1 ]2A[ ≥
−
Π
⋅
tr
m
h
τ ,
12
and, hence, the ranking in profitability is unchanged. If, otherwise, we set the imputation rate equal
to the current consol rate r(t), namely )()( trtr
lh
h
E
⋅
−
=
ττ
τ
, we would obtain
−
Π
⋅>
−
Π
⋅ 1
)(
)-(1
)(
)(
)(
)-(1
trtr
tr
tr
m
h
mm
h
ττ ,
namely the ranking in profitability would be distorted and underinvestment would take place.
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