Taxation and the Optimal Constraint on Corporate
Debt Finance: Why a Comprehensive Business
Income Tax is Suboptimal
Peter B. Sorensen
October 2015
TAXATION AND THE OPTIMAL CONSTRAINT
ON CORPORATE DEBT FINANCE:
WHY A COMPREHENSIVE
BUSINESS INCOME TAX IS SUBOPTIMAL
Peter Birch Sørensen
Abstract: The tax bias in favour of debt finance under the corporate income tax means that
corporate debt ratios exceed the socially optimal level. This creates a rationale for a general
thin-capitalization rule limiting the amount of debt that qualifies for interest deductibility.
This paper sets up a model of corporate finance and investment in a small open economy to
identify the optimal constraint on tax-favoured debt finance, assuming that a given amount
of revenue has to be raised from the corporate income tax. For plausible parameter values
the socially optimal debt-asset ratio is 2-3 percentage points below the average corporate debt
level currently observed. Driving the actual debt ratio down to this level through limitations
on interest deductibility would generate a total welfare gain of about 5 percent of corporate
tax revenue. The welfare gain would arise mainly from a fall in the social risks associated
with corporate investment, but also from the cut in the corporate tax rate made possible by a
broader corporate tax base.
Address for correspondence:
Peter Birch Sørensen
University of Copenhagen, Department of Economics
Øster Farimagsgade 5, 1353 Copenhagen K, Denmark.
E-mail: [email protected]
TAXATION AND THE OPTIMAL CONSTRAINT
ON CORPORATE DEBT FINANCE:
WHY A COMPREHENSIVE
BUSINESS INCOME TAX IS SUBOPTIMAL
Peter Birch Sørensen1
1. The problem: Addressing the debt bias of the corporate income tax
A conventional corporate income tax allows deductibility of interest but does not grant an
allowance for the cost of equity finance. This tax bias in favour of debt is causing concern
among policy makers, for two reasons. First, there is mounting evidence that the shifting
of debt and interest deductions within a multinational group is a major tax planning
instrument whereby multinational companies reallocate taxable profits towards low-tax
jurisdictions (see, e.g., Desai, Foley and Hines (2004); Huizinga, Laeven and Nicodème
(2008), Gordon (2010), and the survey by de Mooij (2011)). Second, in the wake of the
recent financial crisis, there is a growing awareness that excessive use of debt finance
makes companies more vulnerable to business cycle downturns and to credit crunches
caused by financial instability, just as excessive gearing makes financial institutions more
unstable (Keen and De Mooij (2012)).
During the last two decades the international tax policy debate has focused on two
alternative designs for eliminating the debt bias under the corporation tax. One option for
reform is the Allowance for Corporate Equity (ACE) originally proposed by the Capital
Taxes Group of the Institute for Fiscal Studies (1991) and recently recommended by the
Mirrlees Review (Mirrlees et al. (2011)). The ACE system allows companies to deduct an
imputed return on equity as well as interest on debt, essentially turning the corporation
tax into a tax on rents. The ACE is a logical policy implication of the theoretical insight
1
I wish to thank Guttorm Schjelderup for comments on an earlier version of this paper. The paper is a
further development of work originally carried out for the Norwegian government tax reform committee,
presented in Sørensen (2014). I thank members of the committee and its secretariat for fruitful discussions
of many of the issues involved. Any remaining shortcomings are my own responsibility.
2
that it is inoptimal to levy a source-based tax on the normal return to capital in a
small open economy (Griffith, Hines and Sørensen (2010)). In recent years countries
like Belgium and Italy have in fact experimented with versions of the ACE system (see
Zangari (2014)), but generally policy makers have been reluctant to embrace the ACE,
mainly due to the revenue loss it would imply.2
An alternative design for neutral tax treatment of debt and equity is the Comprehensive Business Income Tax (CBIT) originally described by the US Department of the
Treasury (1992) and recently proposed (in a modified form) by the Swedish Corporate
Tax Reform Committee (2014). In its clean version, the CBIT fully eliminates interest
deductibility and turns the corporate income tax into a source-based tax on the full return to capital, regardless of the mode of finance. However, policy makers have generally
shyed away from full elimination of interest deductibility for fear that it might generate
capital flight and might cause severe transition problems for heavily indebted companies.
The CBIT also raises difficult issues of corporate-personal tax integration and creates a
need for special tax rules for deposit-taking financial institutions.
Instead of pursuing ambitious reforms like the ACE or the CBIT, most OECD countries have tried to tackle the debt bias and the problem of international debt shifting
in more pragmatic ways. Early policy responses to debt shifting took the form of rules
against thin capitalization. Such rules typically stipulate that companies can only deduct
interest on debt up to a certain percentage of total assets. Unfortunately thin capitalization rules are potentially vulnerable to manipulation of asset values through clever
accounting practices or to manipulation of interest rates on intra-company loans. Recently several countries have therefore supplemented their thin capitalization rules by
direct limitations on the total amount of interest a corporate entity is allowed to deduct.
Typically such a cap means that the total deduction for interest expenses cannot exceed a certain percentage of the company’s EBIT (Earnings Before Interest and Tax) or
EBITDA (Earnings Before Interest, Tax, Depreciation, and Amortization).
Thin capitalization rules and caps on interest deductibility usually only apply to entities within a corporate group and they have mainly aimed at curbing international
2
The likely revenue loss has often been overstated in the debate on the ACE. According to the
estimates by de Mooij (2012), an ACE system would involve a budgetary cost of around 15 per cent of
current corporate tax revenue, on average for a selection of advanced economies.
3
profit-shifting. This paper shows that the tax distortion in favour of debt finance provides a rationale for applying such rules to all companies, as recently recommended by
the Norwegian tax reform committee (2014). To drive home this point in the clearest
possible manner, I will simplify the exposition by abstracting from multinational group
structures.3 I present a method of identifying and quantifying the second-best optimal
level of corporate debt and the welfare gain from constraining the use of debt finance. I
show that the deadweight loss from the non-neutral tax treatment of debt and equity is
intimately linked to the rise in risk premiums generated by the tax bias in favour of debt.
In this context, the ‘risk premiums’ do not only include compensation for uncertainty;
they also compensate for the costs of financial distress and the agency costs incurred
by investors as a consequence of imperfect and asymmetric information. Thus the risk
premiums reflect genuine welfare and resource costs.
Debt finance and equity finance involve different types of risk and agency costs. According to the conventional trade-off theory of corporate finance companies weigh the
various costs of equity and debt finance against each other to arrive at a privately optimal debt-equity ratio. Since the deductibility of interest allows firms to shift some of
the cost of debt finance onto other taxpayers, the privately optimal level of debt exceeds
the level companies would have chosen in the absence of tax. As we shall see, this excess borrowing generates a welfare-reducing rise in the social risk premium, defined as
the weighted average of the pre-tax risk premiums on debt and equity. However, the
deductibility of interest also serves as a shield against the corporate income tax, thereby
alleviating the tax distortion to the level of real investment.
To identify the socially optimal constraint on the use of debt finance, policy makers
must therefore account for the following mechanisms: i) A lower corporate debt ratio
will reduce the social risk premium included in the cost of corporate finance. This is
welfare-improving since the initial social risk premium is inoptimally high due to excessive (tax-induced) use of debt finance. ii) Restricting the use of debt will broaden the
corporate tax base, thereby allowing a cut in the corporate tax rate. This will raise
welfare by stimulating investment which is initially too low due to taxation. iii) Forcing
3
Adding the possibility of internal debt shifting within multinational groups would only stregthen the
case for constraints on corporate debt finance. See Egger et al. (2010) for an analysis of internal debt
shifting and Møen et al. (2012) for a paper that studies internal as well as external debt shifting.
4
the corporate debt ratio below the privately optimal level will ceteris paribus increase the
private cost of corporate finance. This will generate a welfare loss by reducing corporate
investment.
If there is no constraint on debt finance in the initial equilibrium, the welfare loss from
the effect in iii) will be negligible, since a small reduction in debt will have no first-order
effect on the private cost of capital when companies have initially optimized their use of
debt. At the same time the mechanisms in i) and ii) will generate first-order welfare gains
from a small cut in the debt ratio. Hence it will always be optimal to introduce some
constraint on the use of debt. The optimal constraint on the amount of tax-favoured
debt is found where the marginal welfare gains from the effects in i) and ii) are just offset
by the marginal welfare loss from the mechanism in iii). The quantitative analysis in
this paper suggests that, for empirically plausible parameter values, the socially optimal
amount of debt qualifying for interest deductibility is far above zero. In other words, it
will not be optimal to introduce a Comprehensive Business Income Tax that completely
eliminates interest deductibility, even though the resulting broadening of the tax base will
allow a lower corporate tax rate. According to the analysis below the preservation of some
amount of interest deductibility is a more effective way of reducing the cost of capital
than cutting the corporate tax rate all the way down to the level allowed by a CBIT.
This is an example of second-best tax policy: the existing corporate tax system distorts
the level of corporate investment as well as the pattern of corporate finance, but rather
than eliminating the latter distortion altogether, it is better to accept some distortion
to financing decisions if this is the most effective way of alleviating the distortion to
investment decisions.
For convenience the exposition below pretends that the policy maker can directly control the corporate debt ratio. In practice the government can only disallow deductibility
of interest on debt exceeding a certain debt ratio (call it β max ). If companies are willing
to forego interest deductions on their marginal debt, they may thus choose a debt level
exceeding β max . However, the analysis in this paper indicates that the socially optimal
value of β max is lower than the debt ratio which the representative company would prefer
under unlimited interest deductibility but higher than the debt ratio it would choose if
debt and equity were granted equal tax treatment at the margin. Hence we may expect
5
that the representative profit-maximizing company will voluntarily choose the debt ratio
β max when this maximum debt ratio qualifying for interest deductibility is set optimally.4
The rest of the paper is structured as follows. Section 2 presents a model of corporate
finance and investment to derive the cost of capital and the privately optimal pattern of
finance for companies faced with asymmetric tax treatment of debt and equity. Section 3
illustrates the efficiency loss from the corporate income tax, including the deadweight loss
from the tax bias in favour of debt finance and the loss from lower domestic investment.
Section 4 derives the second-best optimal debt-to-asset ratio of the corporate sector,
assuming that the government has to raise a given amount of revenue from the corporate
income tax. Section 5 calibrates the model and offers numerical estimates of the optimal
level of debt and the welfare gain from the optimal constraint on debt finance. In section
6 I summarize the main findings of the paper and discuss some limitations of the analysis.
2. A model of corporate finance and investment in a small open
economy
This section sets up a simple model of corporate finance and investment. I focus on a
small open economy facing an exogenous risk-free world interest rate and risk premiums
on corporate debt and equity that depend on the debt ratio chosen by the representative domestic company. Production requires the input of perfectly mobile capital and a
domestic fixed factor which is immobile across borders.
The model uses the following notation:
ρ = user cost of capital
δ = real rate of economic depreciation
c = real private cost of capital
cs = real social cost of capital
r = risk-free real interest rate
π = rate of inflation
pd = risk premium in the interest rate on corporate debt
pe = risk premium in the required return on equity
4
See section 5.2.2 for an elaboration of this point.
6
q = real private cost of corporate finance
β = debt-asset ratio
τ = corporate income tax rate
K = capital stock invested in the domestic economy
L = domestic fixed factor of production
w = price of domestic fixed factor
Π = total after-tax profit
R = corporate income tax revenue
The total real output of the representative company is given by the concave production
function F (K, L). As a benchmark, I assume that the tax code allows companies to
deduct the true economic depreciation of their assets from the corporate income tax base.
The real after-tax profit of the representative competitive firm may then be written as
Π = (1 − τ ) [F (K, L) − wL − δK] − qK,
FK > 0,
FKK < 0,
FL > 0,
FLL < 0,
(2.1)
FKL > 0,
where the subscripts denote partial derivatives. A key feature of the model is the relationship between the individual company’s debt ratio and its cost of finance. Following Boadway (1987) and numerous other writers, I assume that the risk premiums in
the required returns on a company’s debt and equity depend on its debt-to-asset ratio.
Specifically, the company’s real cost of finance is
Cost of equity finance
q = (1 − β)
[r + pe (β)]
Cost of debt finance
+ β {r + pd (β) − τ [r + pd (β) + π]}.
(2.2)
According to (2.2) the cost of finance is a weighted average of the cost of equity finance
and the cost of debt finance, with weights determined by the debt-asset ratio. The cost
of debt finance is reduced by the fact that the company may deduct its nominal interest
payments from the corporate tax base. It is reasonable to assume that a company starting
out with zero debt will face a zero risk premium on debt initially, but as it starts to borrow,
the risk premium will gradually become positive and rise at an increasing rate as the debt
ratio increases, reflecting the growing risk that a more indebted firm will not be able to
service (all of) its debt. I therefore assume that the risk premium on debt, pd (β), has
7
the following properties:
pd (0) = p′d (0) = 0,
p′d (β) > 0 for β > 0,
p′′d (β) > 0.
(2.3)
The required risk premium on equity - which must also compensate shareholders for the
agency costs of controlling the firm and its management - is given by the function pe (β).
The shareholders’ agency costs of monitoring the firm may be reduced if the task of
monitoring can be shared with the debtholders, as emphasized by Jensen (1986). Up to a
certain point the required risk premium on equity may therefore decline as the company
increases its debt-asset ratio. However, as the debt ratio grows, the risk of bankruptcy
becomes a growing concern. Sooner or later this will generate conflicts of interest between
shareholders and debtholders, as argued by Jensen and Meckling (1976). Beyond a certain
debt ratio the required premium on equity will therefore start to increase at a growing
rate as shareholders face accelerating risks and costs of controlling the firm. Formally,
these mechanisms mean that p′e (β) will be negative at low levels of β, but positive at
high values of β, and that p′′e (β) > 0. Consequently, even if debt were not favoured by
the tax system, a company seeking to minimize its cost of finance would want to choose
a positive debt ratio between zero and one.
The cost of finance in (2.2) may be rewritten as
q = r + p (β) − βτ (r + π) ,
(2.4)
where p (β) is the total private (i.e., after-tax) risk premium defined as
p (β) ≡ (1 − β) pe (β) + β (1 − τ ) pd (β) .
(2.5)
In order to derive an explicit analytical solution for the company’s optimal debt ratio,
I will work with a second-order Taylor approximation of the expression for p (β), where
the Taylor expansion is made around the cost-minimizing debt ratio β that the company
would choose in the absence of tax. In section 1 in the appendix I show that such an
approximation yields
p (β) ≈ p β − τ a β − β +
b
β−β
2
2
,
p β ≡ 1 − β pe β + β (1 − τ ) pd β ,
a ≡ pd β + βp′d β > 0,
8
b ≡ p′′ β ,
(2.6)
where a is the marginal risk premium on debt at the debt level β. A necessary condition
for profit maximization is that the company minimizes its cost of finance. Using (2.4)
and (2.6), one can show that the first-order condition for minimization of q with respect
to β implies
β=β+
τ (r + π + a)
.
b
(2.7)
From (2.4) and (2.6) I find the second-order condition for a cost minimum to be d2 q/ (dβ)2 =
b > 0. Since a is also positive (see (2.6)), it follows from (2.7) that the (marginal) tax
shield provided by debt finance - captured by the term τ (r + π + a) - induces the company to choose a higher debt ratio than the ratio β it would have preferred in the absence
of tax. Three features of the solution in (2.7) should be noted. First, we see from the
definition of a that the marginal value of the tax shield includes the tax saving on the
intra-marginal debt occurring when the risk premium increases due to a rise in the company’s debt. Second, β is not only the debt ratio that would prevail in a hypothetical
world without taxes; in the present model it is also the rate of indebtedness that would be
chosen under a CBIT or an ACE with neutral tax treatment of debt and equity. Third,
β is the marginal as well as the average debt ratio.
Equations (2.4), (2.5) and (2.7) determine the company’s cost of finance, given that
it chooses the privately optimal combination of equity and debt. The firm then adjusts
its capital stock so as to maximize its profit given by (2.1). Using (2.4), (2.5) and the
first-order condition ∂Π/∂K = 0, we obtain the following expression for the company’s
cost of capital, defined as the required real pre-tax return on the marginal investment:
c ≡ FK − δ =
q
(1 − β) [τ r + pe (β)] − βτ π
= r + βpd (β) +
.
1−τ
1−τ
(2.8)
Profit maximization also requires that the marginal product of the domestic fixed factor
be equal to its price, i.e., FL = w.
3. The deadweight loss from the corporate income tax
It is clear from (2.8) that the corporate income tax distorts the cost of capital. The
corporate tax system also creates a deadweight loss by distorting corporate financing
decisions. More precisely, the tax system increases the risk premiums that companies
9
must pay. These risk premiums include real resource costs in the form of agency and
bankruptcy costs. The deadweight loss from tax distortions to financing decisions may
therefore be measured by the increase in the risk premiums caused by the tax system. I
will now demonstrate this proposition.
3.1. The risk premium and the deadweight loss from the tax bias against
equity
The impact of a change in the debt ratio β on social welfare is captured by its impact
on the total rent to society generated by the investment undertaken by firms. The total
rent (W ) is
W = F (K, L) − [r + ps (β) + δ] K,
(3.1)
where ps (β) is the social (i.e., pre-tax) risk premium included in the cost of finance,
defined as
ps (β) ≡ (1 − β) pe (β) + βpd (β) .
(3.2)
Using the definitions in (2.1), (2.2), and (3.2), we can write the total rent in (3.1) as the
sum of after-tax profits, the income accruing to the domestic fixed factor (wL), and total
corporate tax revenue (R):
W = Π + wL + R.
(3.3)
The revenue from the corporate income tax is
R = τ {F (K, L) − δK − wL − β [r + pd (β) + π] K} .
(3.4)
When the debt ratio β increases by a small amount, the after-tax profits and the income
of the fixed factor will be unaffected. The reason is that, when firms have optimized their
debt ratios, a small increase in β will have no first-order effect on the cost of capital and
the amount of investment, since the cost increase caused by higher risk premiums will be
just offset by the tax savings from higher interest deductions. Hence it follows from (3.3)
that the effect of a small increase in the debt ratio β on total rents may be measured by
the resulting change in tax revenue. When β increases by one unit, the per-period loss
of corporate tax revenue (∆R) stemming from larger interest deductions will be
∆R = τ [r + π + pd (β) + βp′d (β)] K.
10
(3.5)
The term [r + π + pd (β)] K in (3.5) is the interest payment on the additional debt, while
the term βp′d (β) K captures the increase in the interest payments on the pre-existing
debt caused by the rise in the risk premium on debt induced by the higher debt ratio.
Since ∆R reflects the fall in total rents, it is a measure of the marginal deadweight
loss (M DW L) from the increase in β. When the firm has optimized its initial debt ratio,
it follows from (2.6) and (2.7) that τ (r + π) = b β − β − τ a = p′ (β). Inserting this
into (3.5), we find that
M DW L (β) = {p′ (β) + τ [pd (β) + βp′d (β)]} K.
(3.6)
According to (3.6), the marginal deadweight loss from a rise in the debt ratio (measured per unit of capital) equals the sum of the increase in the private after-tax risk
premium, p′ (β), and the loss of corporate tax revenue following from the higher risk
premium on debt, τ [pd (β) + βp′d (β)]. From (2.5) and (3.2) it follows that p′ (β) +
τ [pd (β) + βp′d (β)] = dps (β) /dβ, so from (3.6) we get
M DW L (β) =
dps (β)
K.
dβ
(3.7)
In other words, the marginal deadweight loss per unit of capital is simply equal to the
rise in the social risk premium defined in (3.2). Recalling that β ∗ is the cost-minimizing
debt ratio in the absence of tax while β is the corresponding ratio in the presence of tax,
we obtain the following first-order approximation to the total deadweight loss (T DW L)
caused by the tax distortion to financing decisions:
β
T DW L (β) =
β
dps (β)
Kdβ = ps (β) − ps β
dβ
MDW L (β) dβ =
β
K.
(3.8)
β
According to (3.8) the total increase in the social risk premium caused by the tax system
provides a measure of the total efficiency loss from the tax distortion to corporate financing decisions. This result is intuitive, since the rise in the social risk premium reflects the
welfare loss from a distortion to the allocation of risk and higher agency and bankruptcy
costs.
3.2. Illustrating the deadweight loss from the corporate income tax
Drawing on the above analysis, figure 1 illustrates the distortions caused by the taxation
of the representative firm. The horizontal axis measures the total stock of capital invested
11
in the firm, and the vertical axis measures its cost of capital. The downward-sloping curve
K (c + δ) indicates the firm’s demand for capital which is a decreasing function of the
user cost of capital, c + δ. The capital demand curve (which is not necessarily linear)
reflects the (declining) marginal productivity of capital, so the total area under the curve
K (c + δ) measures the total output of the firm.
Figure 1 includes three measures of the cost of capital. The first one, c, is the private
cost of capital given by eq. (2.8). Faced with this ‘hurdle’ rate of return, the firm will
install the capital stock K0 .
The second measure, c∗ , is the cost of capital that would prevail in the absence of
taxation. It is found from (2.4) and (2.8) by setting β = β and τ = 0, yielding
c∗ = r + ps β ,
(3.9)
since it follows from (2.6) and (3.2) that p β = ps β for τ = 0. In a hypothetical
no-tax world where the cost of capital would be given by (3.9), the firm would install the
capital stock K ∗ indicated in figure 1.
Finally, we have the measure cs which is the social cost of capital in the presence of
taxation, defined as
cs = r + ps (β) .
(3.10)
This is the marginal cost of capital to society, consisting of the international risk-free
interest rate plus the social risk premium ps (β) implied by the actual debt ratio β chosen
by the firm. Note that the social risk premium ps (β) includes the added agency and
bankruptcy costs caused by the additional debt β − β induced by the tax system.
From (3.8), (3.9) and (3.10) we see that the area B in figure 1 is
Area B = (cs − c∗ ) K0 = ps (β) − ps β
K0 = T DW L (β) .
(3.11)
Thus area B measures the total deadweight loss caused by the tax bias against equity
finance.
In addition, the corporate income tax drives up the private cost of capital c for any
given value of the social risk premium. On the vertical axis in figure 1 this increase in
the required return on investment is measured by the distance c − cs which represents
the effective marginal corporate tax wedge. It generates an additional deadweight loss by
12
discouraging investment, thereby reducing total real income. If production takes place
under constant returns to scale, the average return to capital equals the marginal return.
The marginal effective tax rate will then coicide with the average effective tax rate, and
the total corporate tax revenue collected from the firm will equal the area C in figure 1.
Figure 1. The deadweight loss from the corporate income tax in a small open economy
The efficiency loss from the fall in investment caused by the excess risk premium
cs − c∗ and the marginal corporate tax wedge c − cs is given by the familiar Harberger
triangle A in figure 1. Thus the total deadweight loss generated by the corporate income
tax equals the sum of the areas A and B. Restricting the use of debt finance will tend to
reduce area B by lowering the social risk premium (and hence cs ), but driving β below
the privately optimal level will also tend to increase the private cost of capital c, thereby
expanding the area of the Harberger triangle A. Finding the optimal value of β involves
a trade-off between these offsetting welfare effects. I will now derive a formula that will
allow us to characterize and quantify the socially optimal corporate debt ratio.
13
4. The socially optimal corporate debt ratio
The optimal debt ratio is the value of β that will maximize the total rent to society
given by (3.1). Suppose the government must raise a fixed amount of revenue from the
corporate income tax. A cut in the debt ratio β will then allow a cut in the corporate tax
rate τ . Defining the user cost of capital as ρ ≡ c + δ and applying the implicit function
theorem to the revenue constraint (3.4), one can show that
τ r + π + pd (β) + βp′d (β) +
dτ
=
dβ
A 1 − τ η ∂c
ρ
η≡−
dK ρ
,
dρ K
ηA
ρ
∂c
∂β
,
(4.1)
∂τ
A ≡ c − β [r + π + pd (β)] ,
where η is the numerical user cost elasticity of capital demand. When choosing the
optimal value of β, the policy maker must account for the relationship between β and τ
given by (4.1). From (3.1) the first-order condition dW/dβ = 0 for the maximization of
total rent with respect to the debt ratio may then be written as
Effect i)
p′s (β) + η
Effect ii)
c − cs
c+δ
Effect iii)
∂c dτ
+η
∂τ dβ
c − cs
c+δ
∂c
= 0,
∂β
(4.2)
where we recall from (3.10) that cs = r + ps (β) is the social cost of capital. In the normal
case we have ∂c/∂τ > 0, and the firm’s second-order condition for minimization of c with
respect to β implies that ∂c/∂β < 0 when β is below its privately optimal level. Given
the plausible parameter values presented in the next section, we also have dτ /dβ > 0, as
one would expect.
With this in mind, we can interpret the optimum condition (4.2). The first term on
the LHS is the welfare gain from the fall in the social risk premium induced by a marginal
fall in β. The second term captures the welfare gain from the rise in investment generated
by the cut in the corporate tax rate made possible as the lower debt ratio broadens the
corporate tax base. Higher investment raises social welfare because the marginal pre-tax
rate of return to investment (c) exceeds the social cost of capital (cs ), due to the corporate
tax wedge between these two rates of return. The third term on the LHS of (4.2) reflects
the welfare loss from the fall in investment occurring as the debt ratio is driven below
the level that would minimize the private cost of finance to companies.
14
We see that the terms i), ii) and iii) appearing on the LHS of (4.2) correspond to the
three effects of a lower debt ratio explained in the introductory section. As already noted,
the effect iii) will be zero when the economy starts out from an equilibrium without any
constraint on debt finance, since profit maximization will then imply that ∂c/∂β = 0.
Hence it will always be optimal to impose some (small) constraint on the use of debt
finance since the first two terms in (4.2) are positive. But as the debt ratio is forced
below the value of β satisfying the private optimum condition ∂c/∂β = 0, ∂c/∂β will
turn negative and will become numerically larger as β is futher reduced. At the same
time the efficiency gains from further drops in the risk premium and in the corporate tax
rate will diminish as β and τ are lowered. The socially optimal value of β is found where
the sum of the positive marginal welfare effects i) and ii) is just offset by the negative
marginal welfare effect iii). If this point were not reached until β = 0, a Comprehensive
Business Income Tax allowing no interest deductibility at all would be optimal. But as
the quantitative analysis in the next section suggests, the socially optimal amount of debt
qualifying for interest deductibility will most likely be far above zero.
5. Quantitative analysis
5.1. Calibration
I will now illustrate how the model may be calibrated for the purpose of quantitative
analysis. To estimate the effects of a change in the debt ratio on private and social risk
premiums we must quantify the parameters a, b and β appearing in (2.6). To this end
I will use the following second-order approximation for the risk premium on debt which
satisfies the plausible assumptions made in (2.3):
pd (β) ≈
k 2
β ,
2
k > 0.
(5.1)
Suppose the observed corporate debt ratio in the economy’s initial equilibrium with no
restriction on interest deductibility is β 0 and that the associated initial risk premium on
corporate debt is pd0 . From (5.1) we can then find the value of the parameter k that
generates a realistic initial risk premium on debt, given the observed initial debt ratio:
pd0 =
k 2
β
2 0
=⇒
15
k=
2pd0
.
β 20
(5.2)
From the definition of a given in (2.6) and the approximation in (5.1) it follows that
a ≡ pd β + βp′d β = 3pd β
2
=⇒
2
a = 1.5kβ .
(5.3)
The debt ratio β that would prevail in the absence of tax (or with neutral tax treatment
of debt and equity) is not directly observable, so it is calibrated to ensure that the initial
model equilibrium reproduces the observed initial debt ratio β 0 , given a realistic value of
the semi-elasticity of the debt-asset ratio with respect to the corporate tax rate. Using
(2.7), we find the semi-elasticity at the initial debt ratio to be:
ε≡
∂β 1
r+π+a
.
=
∂τ β 0
β 0b
(5.4)
Inserting (5.4) in the condition (2.7) for the firm’s optimal choice of capital structure, we
get
β = β 0 (1 − τ 0 ε) .
(5.5)
The parameter ε has been estimated in numerous empirical studies. Inserting an estimate
for ε and the observed initial debt ratio β 0 as well as the initial corporate tax rate τ 0
into (5.5), one obtains an estimate for β. This may be inserted in (5.3) to provide an
estimate for a. Given the resulting value of a, the value of b may then be derived from
(5.4), assuming plausible rates of interest and inflation.
To apply the formula (4.2) determining the socially optimal value of β, we also need
an estimate of the social risk premium ps (β) and its derivative. In section 1 of the
appendix I show that our second-order approximations (2.6) and (5.1) combined with
the assumption of optimal financing decisions imply that the social risk premium is given
by
ps (β) ≈ ps β +
bs
β −β
2
2
,
bs = b + 3τ kβ.
(5.6)
The estimates of b, k and β may thus be used to derive an estimate of the parameter bs
in (5.6). From (5.6) and (3.11) we may then estimate the size of the welfare loss caused
by the asymmetric tax treatment of debt and equity. Measured as a share of the total
corporate tax revenue specified in (3.4), the welfare loss in the initial equilibrium is
ps (β 0 ) − ps β K0
T DW L (β 0 )
=
.
R
τ 0 [F (K0 , L) − δK0 − wL − β (r + pd0 + π) K0 ]
(5.7)
I assume that the production function takes the Cobb-Douglas form F (K, L) = K α L1−α ,
0 < α < 1, and without loss of generality I normalize the input of the domestic fixed factor
16
at L = 1. The firm’s first-order condition FL = w then implies that F (K0 , L) − wL =
αK0α , and the first-order condition (2.8) yields FK = αK0α−1 = c0 + δ. Using these results
along with (5.6) and dividing through by K0 in (5.7), I get
2
bs
β0 − β
T DW L (β 0 )
2
=
.
R
τ 0 [c0 − β 0 (r + pd0 + π)]
(5.8)
Furthermore, by taking total differentials in (3.1) around the initial equilibrium, exploiting the firm’s first-order condition FK = c0 + δ, and using the notation ps0 ≡ ps (β 0 ),
I obtain the following approximation to the social welfare gain from a reduction in the
corporate debt ratio from β 0 to β, measured relative to corporate tax revenue,
Welfare gain from fall
Welfare gain from fall
in cost of capital
in social risk premium
dW
=
R
(c0 − r − ps0 )
η (c0 − c)
c0 + δ
,
τ 0 [c0 − β 0 (r + pd0 + π)]
ps (β 0 ) − ps (β)
+
(5.9)
where c0 − r − ps0 is the rent created by an additional unit of capital, starting from the
initial equilibrium. Recalling that ρ ≡ c + δ, the first-order condition FK = αK α−1 = ρ
implies that the user cost elasticity of capital demand is given by
η≡−
1
dK ρ
=
.
dρ K
1−α
(5.10)
The expressions for the partials ∂c/∂τ and ∂c/∂β appearing in the key formula (4.2) are
derived in the appendix. The other important derivative dτ /dβ appearing in (4.2) may
then be found from (4.1).
Table 1 provides the complete list of parameters needed to derive quantitative estimates from the model. The parameters in the first column are chosen exogenously to
reflect empirically plausible values (see below). Once these parameter values are given,
the calibrated parameters in the second column may be found by following the calibration
methods explained above. This procedure ensures that the calibrated values of a, b and
β ∗ actually generate (via the firm’s first-order condition (2.7)) the observed value of the
initial debt ratio β 0 as an endogenous outcome in the initial equilibrium.
17
Table 1. Benchmark calibration
Exogenous parameters Calibrated parameters
r
0.025
k
0.12
π
0.02
a
0.038
pe0
0.05
b
0.5534
pd0
0.015
bs
0.598
τ0
0.27
η
1.5
β0
0.5
β
0.4595
ε
0.3
p0
0.0305
δ
0.06
ps0
0.0325
α
0.333
ps β
0.032
Note: β 0 , pe0 , pd0 , p0 and ps0 are the initial debt ratio and the initial risk premiums, respectively.
A key parameter is the semi-elasticity of debt with respect to the corporate tax rate.
According to the survey by de Mooij and Ederveen (2008), a plausible value for this
semi-elasticity is ε = 0.3. In an updated meta-analysis of a large number of empirical
studies, de Mooij (2011) finds an average value of ε closer to 0.2, but with a tendency for
studies based on more recent data to find higher elasticities. Against this background, I
have chosen to set ε = 0.3 in the benchmark calibration of the model.
The rates of interest and inflation stated in table 1 are ‘normal’ values for most
small Western European economies. The initial 1.5 percent risk premium on corporate
debt (pd0 ) assumed in the table is a plausible average value for corporate bonds with
intermediate maturity and good ratings.5 The assumption of an initial 5 percent risk
premium on equity (pe0 ) is based on the empirical survey by Dimson et al. (2008) of
historical long run equity risk premiums in Western countries. These values for pe0 and
pd0 are introduced for the purpose of calibrating the parameters b and k so that the model
endogenously generates realistic risk premiums in the initial equilibrium.
5
For example, according to table 1 in Chen et al. (2007), the difference between the average yield
on U.S. corporate bonds with an AA-rating and medium maturity (7-15 years) and the average yield
on comparable maturity treasury bonds from 1995 to 2003 was 146.27 basis points. For AAA-rated
corporate bonds the yield spread was 82.44 basis points, and for A-rated bonds it was 177.68 basis
points.
18
The 27 percent corporate tax rate assumed in table 1 corresponds to the GDPweighted average corporate tax rate in the EU in 2014 (see Pomerleau (2014, table 4)).
According to estimates by the European Central Bank (2012), the ratio of debt to total
assets of non-financial corporations in the euro zone was 46 percent in the second quarter
of 2009 and 43 percent in the first quarter of 2011. However, since the debt-asset ratio of
financial corporations is considerably higher, table 1 assumes an initial debt-asset ratio of
50 percent. The assumed annual real depreciation rate δ = 0.06 is a rough average across
different types of real capital. When a depreciation rate of this magnitude is plugged into
a standard Solow growth model (and combined with plausible values of the other parameters of the Solow model), the model predicts a realistic speed of income convergence
across countries (see Sørensen and Whitta-Jacobsen (2010, ch. 5). Finally, given our
model assumptions the capital income share of total output will equal the elasticity of
output with respect to capital input (α). A capital income share around 1/3 fits well with
the historical experience of Western countries, so table 1 assumes α = 0.333. According
to (5.10), this implies a user cost elasticity of capital demand (η) equal to 1.5.6
5.2. Results
5.2.1. The deadweight loss from the tax bias against equity
With the benchmark calibration described above, we can offer an estimate of the total
deadweight loss caused by the non-neutral tax treatment of debt and equity, measured as
a fraction of total corporate tax revenue (T DW L/R). From (5.8) we find T DW L/R =
0.0482. In other words, the deadweight loss amounts to almost 5 percent of corporate tax
revenue. From (5.6), (5.7) and (5.8) it follows that the numerator in the latter formula
measures the deadweight loss relative to the initial corporate capital stock (T DW L/K0 ).
With our benchmark calibration this number is 0.049 percent. This is very close to the
estimates provided by Weichenrieder and Klautke (2008) who found (using a different
method) that the total deadweight loss from the tax distortion to corporate financing
decisions varies between 0.05 and 0.15 percent of the corporate capital stock.
6
This value of η is higher than the user cost elasticity found in most of the empirical studies surveyed
by Hassett and Hubbard (2002), but as we shall see in the next section, the quantitative results from
our model are not very sensitive to the value of η.
19
In recent years the revenue from the corporate income tax has fluctuated around 3
percent of GDP in most Western European countries. Given this revenue share of GDP,
our estimate of the deadweight loss from the financial distortion amounts to about 0.14
percent of GDP.
5.2.2. Effects of the optimal thin-capitalization rule
The benchmark calibration of the model also delivers the results reported in the first
column of table 2.7 The optimal limit on debt finance reduces the debt-asset ratio of the
corporate sector by 2.6 percentage points. The second-best optimal debt ratio β = 0.474
is seen to be higher than the debt ratio β = 0.46 which would be chosen if debt and
equity were treated equally by the tax code. We may therefore assume that profitmaximizing companies will not choose a debt ratio exceeding β = 0.474 if debt beyond
that level is treated on par with equity for tax purposes. At the same time, since the
debt limit β = 0.473 is lower than the debt ratio β 0 = 0.5 that would maximize profits if
all interest payments were fully deductible, an optimizing company will want to exploit
the opportunity for interest deductibility to the maximum possible extent by incurring a
debt ratio equal to β = 0.473. Hence we may expect that, by denying interest deductions
for debt beyond a certain level in the interval between β 0 and β, the government can
also induce the representative company to choose that level of debt, as argued in the
introductory section.
A model simulation reveals that, at the optimal corporate debt ratio implied by the
benchmark calibration, total corporate interest expenses make up 31.9 percent of the total
nominal Earnings Before Interest and Tax (EBIT ) generated by the corporate sector.
For comparison, total interest payments amount to 34.2 percent of EBIT in the initial
unconstrained equilibrium. The optimal constraint on interest deductibility is thus fairly
modest.
The broadening of the corporate tax base allows a 1.5 percentage point cut in the
corporate tax rate. Since the forced reduction in the debt ratio and the resulting increase
7
To derive the optimal constraint on debt finance from formula (4.2) and the resulting welfare gain
from formula (5.9), I use an iterative solution algorithm implemented in an Excel spreadsheet available
on request.
20
in the private cost of corporate finance is moderate, the fall in the corporate tax rate
is sufficient to generate a small drop in the cost of capital, thereby inducing firms to
increase their capital stock by 0.8 percent.8
According to the second row from the bottom of table 2, the optimal thin capitalization rule generates a total welfare gain of 5 percent of the revenue from the corporate
income tax in the benchmark calibration. Some of this gain is due to the increase in the
capital stock, but from the third row from the bottom of the table we see that most of the
gain (4.2 percent of tax revenue) arises from the fall in the risk premium induced by the
smaller distortion to the choice between debt and equity. Thus the optimal constraint on
debt finance eliminates a very large part of the total deadweight loss from the financial
distortion (which amounts to 4.8 percent of tax revenue, as shown in the bottom row of
table 1) even though it only eliminates a little more than half of the initial ‘excess’ debt
β 0 − β. This result reflects that the welfare loss from the financial distortion increases
with the square of the difference between β and β (see the numerator of (5.8)).
As mentioned earlier, some empirical studies have found a lower semi-elasticity of corporate debt with respect to the tax rate than the value of 0.3 assumed in the benchmark
calibration. If the semi-elasticity is only 0.2, and if the parameters β, a, b and k are
recalibrated so that the model still reproduces the initial debt ratio and risk premiums
stated in table 1, we obtain the results shown in the second column in table 2. Not
surprisingly, we see that the welfare gain from the optimal thin capitalization rule is now
smaller, because a lower tax elasticity of debt implies lower initial distortions to corporate
financing decisions (i.e., a smaller difference between the average initial debt ratio and
the average debt ratio that would be optimal in the absence of tax).
The third column of table 2 reports the simulation results obtained when the capital
income share is 0.2 rather than 0.333. According to (5.10) the user cost elasticity of capital
demand will then be 1.25 rather than the elasticity of 1.5 assumed in the benchmark
calibration. As one would expect, the increase in the capital stock is now more modest,
because capital demand is less sensitive to changes in the cost of capital. However, the
overall effects of the optimal thin capitalization rule do not change very much. The reason
8
Recall from (2.8) that the relationship between the cost of finance (q) and the cost of capital (c) is
c = q/ (1 − τ ).
21
is that the net change in the cost of capital for the corporate sector as a whole is quite
modest (due to the offsetting effects of the cuts in β and τ ) which makes the sensitivity
of capital demand to the cost of capital less important.
Table 2. Effects of the optimal thin capitalization rule
Benchmark Low debt
Low user
Low initial Low initial High initial
calibration
elasticity
cost elasticity
tax rate
debt ratio
debt ratio
(β 0 =0.5)
(ε=0.2)
(η=1.25)
(τ 0 =0.2)
(β 0 =0.4)
(β 0 =0.6)
β
0.474
0.483
0.473
0.481
0.379
0.569
β∗
0.46
0.473
0.46
0.47
0.368
0.551
∆β
-0.026
-0.017
-0.027
-0.019
-0.021
-0.031
∆τ
-0.015
-0.011
-0.016
-0.009
-0.01
-0.025
∆K/K
0.008
0.006
0.007
0.004
0.006
0.011
W G (∆ps )1
0.042
0.028
0.043
0.034
0.025
0.076
W G/R2
0.05
0.033
0.05
0.038
0.031
0.087
T DW L/R3
0.048
0.032
0.048
0.039
0.029
0.088
1. Welfare gain from fall in social risk premium (as a fraction of corporate tax revenue). 2.
Total welfare gain (as a fraction of corporate tax revenue). 3. Total deadweight loss from initial
distortion to debt-equity choice (as a fraction of corporate tax revenue).
A ∆-prescript denotes a change in the relevant variable.
The fourth column of table 2 shows that the welfare gain from the optimal constraint
on debt finance will be smaller if the initial corporate tax rate is only 20 percent rather
than the 27 percent assumed in the benchmark scenario. This is intuitive, since a lower
initial tax rate implies lower initial distortions to financing and investment decisions.
The last two columns indicate that the welfare gain from the optimal thin capitalization rule is fairly sensitive to the initial debt level. With an initial debt ratio of 40
percent, the welfare gain is only 3.2 percent of corporate tax revenue, whereas it is 8.7
percent of tax revenue if the initial debt ratio is 60 percent. Again, this is intuitive since
the tax discrimination in favour of debt creates smaller distortions when debt is a less
important source of finance.
22
Apart from illustrating the importance of the initial degree of indebtedness, table 2
leaves the impression that the optimal thin capitalization rule is not particularly sensitive
to uncertainty about parameter values.
6. Conclusions and discussion
According to the analysis in this paper the current tax bias in favour of debt finance
makes it second-best optimal to impose a limit on the debt-asset ratio of companies even
if they do not have the opportunity to engage in international profit-shifting. Implicitly,
the analysis also provides a rationale for a general cap on deductible interest expenses.
For plausible parameter values derived from international empirical studies, the analysis
indicates that the socially optimal debt ratio is about 2-3 percentage points below the
current average corporate debt ratio observed in Europe. The analysis also suggests
that the socially optimal debt ratio may be implemented through the voluntary financial
choices of companies by denying deductibility of interest on debt beyond the socially
optimal level. At the same time the analysis clearly suggests that complete elimination
of interest deductibility - as a Comprehensive Business Income Tax would imply - is not
an optimal second-best tax policy.
Reducing the average debt ratio to the socially optimal level would allow a cut in
the corporate tax rate of 1-2 percentage points and would generate a net welfare gain
of about 3-5 percent of corporate tax revenue, according to the model estimates. The
bulk of the welfare gain would arise from a fall in the risk premium included in the cost
of corporate finance. Indeed, the drop in the risk premium induced by the optimal thin
capitalization rule would eliminate around four fifths of the total deadweight loss from
the current tax distortion to the choice between debt and equity. The remaining part of
the welfare gain from the optimal constraint on debt finance would stem from the fall in
the corporate tax rate made possible through the broadening of the corporate tax base.
My modelling of corporate financing decisions relied on the so-called trade-off theory
of capital structure. According to this theory companies trade off the marginal tax benefit
from additional debt finance against the resulting increase in the costs of financial distress.
The model allowed for the possibility that companies will want to issue a certain amount
23
of debt even in the absence of interest deductibility because the need to service debt
may help to discipline corporate managers, thereby reducing the shareholders’ agency
costs of monitoring the firm. Importantly, the model implies that companies will make a
socially optimal trade-off between the marginal costs and benefits of debt in the absence
of taxation. However, once interest deductibility is allowed, corporate debt ratios will
rise above the efficient level.
An alternative theory of corporate finance is the so-called pecking-order theory originally developed by Myers and Majluf (1984) and Myers (1984). These authors stressed
that asymmetric information between managers and outside investors give rise to an adverse selection problem where investors may interpret new issues of shares or debt as a
sign that the company is in trouble and in need of liquidity. A company’s use of external
finance may thus lead to a fall in its market value. To the extent possible, companies will
therefore prefer to finance investment out of retained earnings. If they need additional
capital, companies will generally prefer debt over new share issues, because share prices
will react more negatively when firms use new equity rather than debt finance, since
then investors do not only have to worry about the risk of default, but also about the
entire distribution of returns to investment.9 The pecking-order theory only attaches a
second-order importance to the tax benefits of debt finance. Further, as noted by Gordon
(2010), this theory implies that corporate debt ratios may be inefficiently low due to the
adverse selection problem that may lead stock markets to punish the issue of debt.
As Brealy, Myers and Allen (2009) explain, the trade-off theory and the peckingorder theory of corporate finance both have strengths and weaknesses when it comes to
explaining the details of corporate financing patterns and their variations across firms
and industries. In the present paper I have chosen to work with a version of the tradeoff theory because the pecking-order theory does not lead to a well-defined target for a
company’s debt-equity mix. However, in so far as adverse selection problems in capital
markets would lead to an inoptimally low use of debt finance in the absence of tax, the
analysis in the present paper may overstate the benefits of constraints on corporate debt
finance and/or interest deductibility.
9
Another way of explaining the firm’s preference for debt over new equity is that a manager who
believes that the stock market undervalues the company’s shares will want to finance new profitable
investment by debt rather than new shares to avoid “giving away a free gift” to new investors.
24
The analysis in this paper does not allow for the fact that the tax-favored status of
debt under the corporate income tax may be (partly) offset by tax favours for equity
income under the personal income tax. In Sørensen (2014) I extend the model presented
above by introducing a distinction between ‘large’ companies financing investment via the
international capital market and ‘small’ companies with controlling shareholders subject
to domestic personal income tax. This allows an analysis of the effects of introducing
domestic tax relief for dividends and capital gains at the personal shareholder level. For
plausible parameter values it turns out that such domestic shareholder tax relief does not
affect the size of the welfare gain from the optimal thin capitalization rule very much.
This result reflects that the bulk of corporate finance is supplied by international investors
and tax-exempt institutional investors.
However, the heterogeneity of firms and industries means that the socially optimal
debt ratios of individual firms are likely to differ substantially. For example, within the
framework of the trade-off theory of finance, firms that rely more heavily on tangible and
relatively safe assets will have higher optimal debt ratios, because their marginal costs
of financial distress at any given debt ratio will be lower. The extended model presented
in Sørensen (2014) does allow for some heterogeneity across large and small companies,
but it does not capture all of the real-world diversity in the capacity for debt finance
across firms. This limitation of the model suggests that a general thin capitalization rule
applied to all firms (or a general cap on interest deductibility) should be more lenient
than the optimal debt ratio derived from the quantitative version of the model presented
in this paper.
25
TECHNICAL APPENDIX
1. Approximations to risk premiums
The private after-tax risk premium included in the cost of corporate finance is
p (β) ≡ (1 − β) pe (β) + β (1 − τ ) pd (β) .
(A.1)
A second-order Taylor approximation of this expression around β = β ∗ yields
dp β
dβ
p (β) ≈ p β +
β −β +
1 d2 p β
2 (dβ)2
β −β
2
,
(A.2)
where
dp β
= (1 − τ ) pd β − pe β + 1 − β p′e β + β (1 − τ ) p′d β ,
dβ
dp2 β
(dβ)2
= 2 (1 − τ ) p′d β − p′e β
+ 1 − β p′′e β + β (1 − τ ) p′′d β .
(A.3)
(A.4)
The social risk premium is
ps (β) ≡ (1 − β) pe (β) + βpd (β) .
(A.5)
In the absence of tax (τ = 0), private and social risk premiums would coincide, and firms
would minimize their cost of finance by minimizing the expression in (A.5), implying the
first-order condition
dps β /dβ ≡ 0
=⇒
pd β − pe β + 1 − β p′e β + βp′d β = 0.
(A.6)
Inserting (A.6) into (A.3), we get
dp β
≡ −τ a,
dβ
a ≡ pd β + βp′d β .
(A.7)
Moreover, defining
b≡
d2 p β
(dβ)2
,
(A.8)
and inserting (A.7) and (A.8) into (A.2), we obtain
p (β) ≈ p β − τ a β − β +
26
b
β−β
2
2
,
(A.9)
as stated in (2.6) in section 2. Further, by using (A.6), we can write the second-order
Taylor approximation to the social risk premium (A.5) around β = β as
ps (β) ≈ ps β +
1 d2 ps β
2 (dβ)2
β−β
2
,
(A.10)
where
d2 ps β
2
= 2 p′d β − p′e β
(dβ)
From (A.4), (A.8), and (A.11) it follows that
d2 ps β
(dβ)2
+ 1 − β p′′e β + βp′′d β .
= b + τ 2p′d β + βp′′d β
.
(A.11)
(A.12)
In section 5.1 we introduced the second-order approximation
k 2
β .
2
pd (β) ≈
(A.13)
Using (A.12) and (A.13), we may therefore write (A.10) as
ps (β) ≈ ps β +
bs
β −β
2
2
,
bs ≡ b + 3τ kβ.
(A.14)
Eq. (A.14) is seen to be identical to eq. (5.6) in the main text. Note from (A.1), (A.5)
and (A.13) that
ps β
= p β + τ βpd β
k 3
= p β +τ β .
2
(A.15)
When calibrating the model, I use (A.15) and the specification of bs stated in (A.14) to
ensure consistency between the approximations made in (A.9), (A.13) and (A.14).
2. The cost of capital and its derivatives
From (2.4), (2.6), (2.8) and (A.13) one finds that

c=
1
1−τ
≡p(β )

r − τ β (r + π) + 1 − β pe β + β (1 − τ ) pd β

−τ a β − β +
b
β −β
2
2
,
(A.16)
b β − β − τ (r + π + a)
∂c
=
,
∂β
1−τ
(A.17)
3
c − β (r + π + a) + aβ − 0.5kβ
∂c
=
.
∂τ
1−τ
27
(A.18)
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