Discussion Papers In Economics
And Business

Optimal capital structure with earnings above a floor

Michi NISHIHARA, Takashi SHIBATA

Discussion Paper 23-09

June 2023

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANOptimal capital structure with earnings above a ﬂoor ∗Michi NISHIHARA†,

Takashi SHIBATA‡

Abstract
This paper derives the optimal capital structure of a ﬁrm whose earnings follow a geometric
Brownian motion with a lower reﬂecting barrier. The barrier can be interpreted as a market intervention threshold (e.g., a price ﬂoor) by the government or an exit threshold of weak competitors
in the market. Unlike in the standard model with no barrier, the ﬁrm is able to issue riskless debt
to a certain capacity determined by the barrier. The higher the barrier, the larger the riskless debt
capacity, and the ﬁrm prefers riskless capital structure rather than risky capital structure. Notably,with intermediate barrier levels, the ﬁrm can choose riskless capital structure with lower leveragethan the level with no barrier. This mechanism can help explain debt conservatism observed inpractice. The paper also entails several implications of public intervention by examining the lowest
barrier (i.e., the weakest intervention) to achieve riskless capital structure.
JEL Classiﬁcation Codes: G13; G28; G32.
Keywords: Capital structure; Real options; Regulated market; Price ﬂoor; Competitive advantage.

1

Introduction

This paper analyzes an optimal capital structure model of a ﬁrm that receives stochastic ﬂows ofearnings above a ﬂoor. The ﬂoor can be interpreted as the government’s intervention to protect
particular companies or industries (e.g., transportation, utility, agricultural, or ﬁnancial industries)
against downside risks. Apart from the regulated markets, the ﬂoor can be interpreted as an exit
threshold of relatively weak competitors. Then, the model can approximate a ﬁrm with a certain
competitive advantage or protection against downside risks. In the model, we reveal that a ﬁrm∗

This version was written on June 1, 2023. This work was supported by the JSPS KAKENHI (Grant numbers

JP20K01769, JP21H00730). The author thanks Artur Rodrigues for helpful comments. The author also thanks theparticipants at FMA 2022 in Kyoto and Real Options Workshop 2022 (online) for helpful feedback.
†
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka
560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277‡
Graduate School of Management, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397,
Japan, E-mail: tshibata@tmu.ac.jp, Phone: 81-42-677-2310, Fax: 81-42-677-2298

1

can determine capital structure by a mechanism diﬀerent from standard trade-oﬀ theory; indeed,
a ﬁrm can optimally choose capital structure with no risky debt.
The baseline model builds on the standard real options models of optimal capital structure(e.g., Leland (1994), Goldstein, Ju, and Leland (2001), and Shibata and Nishihara (2012)). As
in the standard literature, we assume that a ﬁrm has an option to issue consol debt at an initialtime and that shareholders of the ﬁrm have an option to default debt in place. The ﬁrm’s earnings
are modeled by a geometric Brownian motion (GBM) with a lower reﬂecting barrier (i.e., a ﬂoor),
which is a diﬀerence from the standard models. We also extend the baseline model to a model witha debt ﬁnancing constraint. In the models with a barrier, we analytically derive the equity, debt,
ﬁrm values, leverage, and credit spreads, as well as their sensitivities to barrier levels. The results
are explained below.
A most notable diﬀerence from the standard results with no barrier (e.g., Leland (1994), Goldstein, Ju, and Leland (2001), and Shibata and Nishihara (2012)) is that a barrier generates acapacity of riskless debt ﬁnancing. Naturally, the higher the barrier, the larger the riskless debt
capacity. Compared to risky debt, riskless debt has an advantage of no bankruptcy cost but adisadvantage of the debt level being limited by the capacity. If the barrier is lower than a criticallevel (i.e., the riskless debt capacity is insuﬃcient), then the ﬁrm prefers risky capital structure. In
this case, the presence of a barrier hardly aﬀects the equity, debt, ﬁrm values, leverage, and credit
spreads because the ﬁrm chooses leverage by the standard trade-oﬀ between the tax beneﬁts andbankruptcy costs of debt.
If the barrier is higher than the critical level (i.e., the riskless debt capacity is suﬃcient), thenthe ﬁrm prefers riskless capital structure. In the no-default case, the barrier level greatly aﬀects
all the values because the riskless debt capacity (depending on the barrier level) rather than thestandard trade-oﬀ is a key determinant of capital structure. Notably, the barrier close to thecritical level leads to lower debt and leverage levels than the optimal levels with no barrier. Thatis, in contrast to the straightforward intuition that the ﬁrm increases debt with a ﬂoor, the ﬁrm
voluntarily reduces debt to take advantage of having no bankruptcy risk. This result can help
explain empirical observations of debt conservatism (e.g., Graham (2000), Strebulaev and Yang
(2013), and El Ghoul, Guedhami, Kwok, and Zheng (2018)); some ﬁrms have quite low leverageand bankruptcy risk compared to the optimal levels predicted by standard trade-oﬀ theory. Indeed,
our model suggests that ﬁrms do not choose risky capital structure based on the standard trade-oﬀ
but optimally choose riskless capital structure with low leverage if they have certain degrees ofcompetitive advantage or protection against downside risks.
We also examine the comparative statics with respect to key parameters. With a given barrier
level, a higher volatility, bankruptcy cost, lower growth rate, corporate tax rate, and stronger debt
issuance constraint tend to lead to the no-default case. The switch to the no-default case can cause
the comparative static results to diﬀer from those of the standard trade-oﬀ models. For instance,in the no-default case, a higher growth rate increases equity value more than riskless debt capacity,
decreasing leverage contrary to the standard result. This mechanism can explain empirical evidence2

of a negative relation between leverage and proﬁtability, which is well known as an inconsistency
between trade-oﬀ theory and practice (e.g., Frank and Goyal (2015) and Demarzo (2019)).
This paper focuses on the lowest barrier to achieve the no-default case, because it can beinterpreted as the weakest intervention by the government that prevents the ﬁrm from bankruptcy.
We show that the critical level is lower than the level necessary to save the ﬁrm from bankruptcy ex
post. This result emphasizes the importance of the ex ante information disclosure of the intervention
policy. The appropriate commitment by the government leads a ﬁrm to adopt riskless capital
structure with low leverage rather than leading the ﬁrm to take moral hazard behavior of increasing
debt. A higher volatility and lower growth rate decrease the critical barrier level but increase thefrequency of hitting the barrier. A higher bankruptcy cost, lower corporate tax rate, and stronger
debt issuance constraint decrease advantages of risky debt, decreasing the critical barrier level andthe frequency of hitting the barrier. These results suggest that the government can weaken andreduce market interventions with a more stringent bankruptcy law (i.e., a higher bankruptcy cost),
lower corporate tax rate, and stronger leverage regulation.
Last, we will brieﬂy explain technically related literature. Dixit and Pindyck (1994) solve theinvestment and exit timing models with a price ceiling and ﬂoor and entail many implicationsof market competition and regulation. By extending the models, Dobbs (2004) shows that the
optimal price ceiling delays investment in a monopoly, whereas Roques and Savva (2009) showthat it accelerates investment in an oligopoly. Evans and Guthrie (2012) study a ﬁrm’s production
capacity adjustments under a price ceiling and quantity ﬂoor and show that with economies of scale,the ﬁrm invests in smaller, more frequent, increments than the social planner. Adkins, Paxson,
Pereira, and Rodrigues (2019) examine the optimal duration of regulation in the investment timing
model with a ﬁnite/retractable price ceiling and ﬂoor. Unlike this paper, the above papers assume
all-equity ﬁrms and do not examine any capital structure problem.
Sarkar (2016) develops a Leland-type capital structure model with a price ceiling and shows thatthe price ceiling signiﬁcantly increases leverage. He also shows that the price ceiling can counterintuitively decrease consumer welfare. Rodrigues (2022) is closest to this paper. He investigates theinvestment timing and capital structure model with a revenue ceiling and ﬂoor. His model, which
includes the investment timing and price ceiling, is more generalized than our baseline model, but
due to the model complexity, most of the results are shown numerically. Unlike Rodrigues (2022),
this paper shows the explicit mechanism of how the price ﬂoor generates riskless debt capacity andleads to riskless capital structure in the simple model. In contrast to Sarkar (2016), this paper
shows that leverage can either increase or decrease (i.e., it can be nonmonotonic) with ﬂoor levels.
The paper is organized as follows. Section 2 explains the model setup. In Section 3.1, we
explain the solutions in the benchmark model with no barrier, and in Section 3.2, we derive theexplicit solutions in the baseline model with a barrier. We also analytically derive the sensitivities
to barrier levels. In Section 3.3, we derive the explicit solutions in the extended model with a debt
issuance constraint. Section 4 numerically examines the sensitivities to the key parameters, andSection 5 concludes.

32Model setup

The baseline model builds on the standard capital structure model based on trade-oﬀ theory (e.g.,
Leland (1994) and Goldstein, Ju, and Leland (2001)). Consider a ﬁrm that receives continuous
streams of earnings before interest and taxes (EBIT) X(t) until bankruptcy. EBIT X(t) follows a
GBM
dX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = xwith lower reﬂecting barrier xB (> 0), where B(t) denotes the standard Brownian motion deﬁned
in a ﬁltered probability space (Ω, F, P, {Ft }) and growth rate µ, volatility σ(> 0), and initial value
X(0) = x(≥ xB ) are constants. For convergence, r > µ is assumed, where a positive constant r
denotes the risk-free interest rate, and X(0) = x is assumed to be suﬃciently high level so that theﬁrm is not bankrupt at time 0.
At time 0, the ﬁrm issues consol debt to maximize the ﬁrm value, where the tax beneﬁts andbankruptcy costs of debt will be clariﬁed in the next section. For debt in place, shareholders canstop coupon payments (i.e., declare default) to maximize the equity value. In the default case,shareholders receive nothing, and debt holders receive the post-bankruptcy ﬁrm value, which isequal to the unlevered ﬁrm value multiplied by (1 − α). This means that a fraction α ∈ (0, 1) of theunlevered ﬁrm value is lost to the deadweight costs of bankruptcy. Equity, debt, and ﬁrm values
are fairly priced based on the rational expectation of ex post shareholders’ default behavior.
The presence of lower reﬂecting barrier xB is a diﬀerence from the standard capital structure
model.1 Intuitively, lower reﬂecting barrier xB means that X(t) is pulled back to xB and moves
again from xB immediately after X(t) falls below xB . It is diﬀerent from the assumption that
X(t) equals a GBM S(t) (i.e., the shadow process) for S(t) ≥ xB but remains at xB for S(t) < xB .
Although the shadow process model requires the value function in regions S(t) < xB and S(t) ≥ xB ,the reﬂecting barrier model requires only one region, X(t) ≥ xB .2 By this technical simplicity, we
can solve the model explicitly in the next section.
The model with barrier xB can approximate the following two situations. First, governments
might try to intervene in markets to protect speciﬁc ﬁrms or industries (e.g., transportation, utility,
agricultural, or ﬁnancial industries) against downside risks. For instance, European Union countries’ governments purchase particular agricultural products to prevent their prices from dropping
to unsustainably low levels. Although such market interventions require direct and indirect costs,
1

Reﬂecting barriers frequently appear in dynamic ﬁnancing and payout models. For instance, in Bolton, Chen, andWang (2011), the state variable moves between a lower reﬂecting barrier (i.e., the ﬁnancing threshold) and an upper
reﬂecting barrier (i.e., the payout threshold), although unlike in our model, the barrier levels are determined endogenously
in their model.
2
Most of the literature studies the shadow process models (e.g., Sarkar (2016), Adkins, Paxson, Pereira, and Rodrigues(2019), and Rodrigues (2022)). An exception is Chapter 8 of Dixit and Pindyck (1994), where the reﬂecting barrier models
are examined. We do not think that the technical diﬀerence matters in terms of economic implications. Indeed, the same
results on ﬂoors as in our results are observed in Rodrigues (2022), who studies a more complicated model based on the
shadow process. For instance, a higher ﬂoor also leads to riskless capital structure in Rodrigues (2022).

4

governments can adopt the market measures if the bankruptcy costs of these ﬁrms, including indirect costs, such as threats to national security, are higher than the intervention costs. In thiscase, xB is interpreted as the intervention threshold.3 Chapter 9 of Dixit and Pindyck (1994),
Adkins, Paxson, Pereira, and Rodrigues (2019), and Rodrigues (2022) also examine real options
models with ﬂoors (and ceilings) in terms of public intervention. Sections 3.3 and 4.6 study theeﬀects of leverage regulation in addition to the public intervention by incorporating a debt issuanceconstraint into the baseline model.
Second, the baseline model may capture ﬁrms with strong competitive advantage against downside risks. For instance, consider oil prices. Relatively weak shale oil producers tend to exit themarkets when oil prices fall to unsustainably low levels for them. After the exit of shale oil producers, oil prices are likely to rebound. Thus, the biggest oil companies, which have suﬃcientcompetitive advantage to survive downturns, could receive cash ﬂows above certain levels. Moregenerally, the cash ﬂow dynamics of resilient ﬁrms may have such a trend. In this case, xB is
regarded as the exit threshold of weak competitors. Chapter 8 of Dixit and Pindyck (1994) also
examine entry and exit timing models with lower and upper reﬂecting barriers in terms of thecompetitive market.

3

Model solutions

3.1

EBIT with no barrierThis subsection explains the benchmark model with no barrier (i.e., xB = 0). The following results
are well known in previous literature (e.g., Goldstein, Ju, and Leland (2001), Shibata and Nishihara(2012), and Sundaresan, Wang, and Yang (2015)), and hence, the details of derivation are omitted.
First, suppose that the ﬁrm issues consol debt with coupon C. For given coupon C, the equity,debt, and ﬁrm values are expressed asγ (1 − τ )Cx(1 − τ )C
E0 (x; C) = πx −+− πx0 (C)rx0 (C)
r

γ 

C
x
C
D0 (x; C) =−− (1 − α)πx0 (C)r
x0 (C)
r

γ 

τC
x
τC
F0 (x; C) = πx +−
απx0 (C) +rx0 (C)
r

(1)
(2)
(3)

for x ≥ x0 (C), where τ denotes a corporate tax rate, π = (1−τ )/(r−µ) denotes the unlevered ﬁrm’s
q2
coeﬃcient, γ = 0.5 − µ/σ 2 − (µ/σ 2 − 0.5) + 2r/σ 2 denotes a negative characteristic root, andx0 (C) denotes the default threshold. Throughout the paper, subscript 0 stands for the benchmark
model with no barrier. The ﬁrst, second, and last terms in equity value (1) correspond to theunlevered ﬁrm value, perpetual coupon payments, and the value of the default option, respectively.
The ﬁrst and second terms in debt value (2) are the perpetual coupon receipts (i.e., the riskless3

When the product price follows a GBM with a ﬂoor, EBIT also follows a GBM with a ﬂoor in the standard setups(e.g., Dixit and Pindyck (1994)). Then, for simplicity, this paper directly assumes EBIT with a ﬂoor.

5

debt value) and loss due to default risk, respectively. The ﬁrst, second, and last terms in ﬁrm value(3) are the unlevered ﬁrm value, perpetual tax beneﬁts of debt, and bankruptcy costs, respectively.
Note that shareholders determine x0 (C) to maximize its own value E0 (x; C) for debt in place.
By solving arg maxx0 (C)≥0 E0 (x; C), we obtain default threshold
x0 (C) = C/δ,where δ is a constant given byδ=(4)(γ − 1)r(> 1).
γ(r − µ)Now, consider the optimal capital structure. The ﬁrm chooses coupon C to maximize ﬁrm value
F0 (x; C) based on the trade-oﬀ between the tax beneﬁts and bankruptcy costs of debt. By solving
arg maxC≥0 F0 (x; C), we obtain optimal coupon
C0 (x) = δx/h,(5)

where h is a constant given byh
α i− γ1h= 1−γ 1−α+(> 1).
τThe optimally levered ﬁrm value isF0 (x; C0 (x)) = ψπx,(6)where ψ is a constant given by
ψ =1+

τ
(> 1)
(1 − τ )hand is interpreted as the leverage eﬀect. Indeed, the levered ﬁrm value F0 (x; C0 (x)) is the unleverd
ﬁrm value πx multiplied by ψ(> 1). The ﬁrm’s leverage is
LV0 (x) =γ − 1 (1 − τ )(1 − ξ)
D0 (x; C0 (x))=,
F0 (x; C0 (x))
γ
hψ

(7)

and credit spreads are
CS0 (x) =C0 (x)rξ
−r =,D0 (x; C0 (x))
1−ξ

(8)

where ξ is a constant deﬁned byγξ = 1 − (1 − α)(1 − τ )hγ (∈ (0, 1)).
γ−1

3.2

EBIT with a lower reﬂecting barrierThis subsection solves the baseline model with lower reﬂecting barrier xB > 0. First, suppose thatthe ﬁrm issues debt with coupon C. For given C, shareholders choose whether they default. Then,equity value E(x; C) is expressed asE(x; C) = max{Ed (x; C), En (x; C)},6(9)

where Ed (x; C) and En (x; C) represent the equity values in the default-possible and no-default
cases, which will be deﬁned later. The next proposition shows the equity, debt, and ﬁrm values,
denoted by E(x; C), D(x; C), and F (x; C), respectively, for given coupon C. For the proof, see
Appendix A.
Proposition 1 For C > δxB , the ﬁrm goes bankrupt at default threshold x0 (C) = C/δ (i.e., thedefault-possible case). The equity, debt, and ﬁrm values are given byE(x; C) = Ed (x; C) = E0 (x; C),γ   γx
Cx
C(1 − α)πxB
D(x; C) = Dd (x; C) =−− (1 − α)πx0 (C) −,rx0 (C)
r
xB
γ
|
{z
}
τC
F (x; C) = Fd (x; C) = πx +
−
r
|



=D0 (x;C)γ xx0 (C){zτC
απx0 (C) +
r




−

}

x
xB

γ

(1 − α)πxB
.
γ

(10)
(11)

(12)

=F0 (x;C)

Otherwise, the ﬁrm never goes bankrupt (i.e., the no-default case). The equity, debt, and ﬁrm
values are given by γ
x(1 − τ )C
πxB
E(x; C) = En (x; C) = πx −
−
,
r
xB
γ
C
D(x; C) = Dn (x; C) = ,
r
 γ
τC
x
πxB
F (x; C) = Fn (x; C) = πx +
−
.
r
xB
γ

(13)
(14)
(15)

Note that C > δxB is equivalent to x0 (C) > xB . First, we explain the default-possible case,in which x0 (C) is higher than xB . Equity value Ed (x; C) does not depend on xB because X(t)does not hit xB before bankruptcy. Then, Ed (x; C) is the same as the benchmark value E0 (x; C)(see (10)). However, debt and ﬁrm values, Dd (x; C) and Fd (x; C), respectively, beneﬁt by xB . The
ﬁrst and second terms in (11) coincide with D0 (x; C), and the last term is the additional value
by xB (note that γ < 0). The additional value arises from the fact that X(t) can hit xB after
default. That is, the post-default value, which debt holders obtain, increases with higher xB .4
Firm value Fd (x; C) has the same beneﬁt from xB (see the last terms in (11) and (12)) because of
Fd (x; C) = E0 (x; C) + Dd (x; C).
Now, we explain the no-default case. By C ≤ δxB , En (xB ; C) ≥ 0 holds in (13). Then, for
any X(t) ≥ xB , shareholders are better oﬀ continuing operation with coupon payments rather thandeclaring default. Shareholders beneﬁt by xB because X(t) can hit xB . The ﬁrst, second, and thirdterms in (13) represent the unlevered ﬁrm value, perpetual coupon payments, and additional value
by xB (note that γ < 0). Debt holders also beneﬁt by xB because xB removes the default risk.
4

As in Leland (1994) and Goldstein, Ju, and Leland (2001), our model does not specify either liquidation or reorgani-zation bankruptcy but assumes that the post-default ﬁrm value is the unlevered value discounted by bankruptcy costs.
The unlevered value increases in xB .

7

Then, Dn (x; C) agrees with the riskless debt value in (14). In (15), ﬁrm value Fn (x; C) consistsof the unlevered ﬁrm value, perpetual tax beneﬁts, and additional value by xB . Unlike F0 (x; C),Fn (x; C) does not include any term representing bankruptcy costs.
Proposition 1 implies that Dn (x; δxB ) = δxB /r is the capacity of riskless debt. Of course,
for C ≤ xB , the ﬁrm always receives nonnegative cash ﬂows X(t) − C, and hence, shareholders
continue operation perpetually. Note that δ > 1. Considering the possibility that X(t) goesbeyond xB due to volatility σ, shareholders prefer to operate perpetually for C ≤ δxB . Indeed,
the expected cash ﬂows of perpetual operation are nonnegative (i.e., En (x; C) ≥ 0) for C ≤ δxB .
This is how the presence of barrier xB creates the riskless debt capacity δxB /r. Proposition 1 nests
the benchmark case with no barrier as the limiting case of xB → 0. Indeed, limxB →0 E(x; C) =E0 (x; C), limxB →0 D(x; C) = D0 (x; C), and limxB →0 F (x; C) = F0 (x; C) hold.
Next, consider the optimal capital structure. We need to solve max C≥0 F (x; C). By (5) and(12), we have arg maxC≥0 Fd (x; C) = arg maxC≥0 F0 (x; C) = C0 (x), which reﬂects the standard
trade-oﬀ between the tax beneﬁts and bankruptcy costs of debt. By (15), Fn (x; C) increases
linearly in C, implying arg maxC∈[0,δxB ] Fn (x; C) = δxB . This reﬂects the fact that a higher debt
level increases ﬁrm value via greater tax beneﬁts in the no-default case. Comparing (12) and(15), we have Fd (x; C) < Fn (x; C) for any (x, C) because Fd (x; C), unlike Fn (x; C), includes theterm of bankruptcy costs. Therefore, maxC≥0 F (x; C) = max{Fd (x; C0 (x)), Fn (x; δxB )} holds. Forgraphical images of F (x; C), see Figure 1. Deﬁne the diﬀerence asG(x; xB ) = Fn (x; δxB ) − Fd (x; C0 (x))(γ − 1)τ xB
τx−−=γ(r − µ)(r − µ)h



x
xB

γ

απxB
,
γ

(16)

where we used (6), (12), and (15) to obtain (16). The next proposition shows the equity, debt,
ﬁrm values, coupon, leverage, and credit spreads, denoted by E(x), D(x), F (x), C(x), LV (x), andCS(x) respectively, under optimal capital structure. For the proof, see Appendix B.
Proposition 2 There exists a unique solution x∗B ∈ (0, γx/(γ−1)h) to G(x; x∗B ) = 0, and xB < x∗Bis equivalent to G(x; xB ) < 0.
For xB < x∗B , the ﬁrm issues risky debt with coupon C0 (x) and goes bankrupt at default threshold x0 (C0 (x)) = x/h (i.e., the default-possible case).

The equity, debt, ﬁrm values, coupon,leverage, and credit spreads are given by E(x) = E0 (x; C0 (x)), D(x) = Dd (x; C0 (x)), F (x) =Fd (x; C0 (x)), C(x) = C0 (x), LV (x) = Dd (x; C0 (x))/Fd (x; C0 (x)), and CS(x) = C0 (x)/Dd (x; C0 (x))−
r, respectively.
Otherwise, the ﬁrm issues riskless debt with coupon δxB and never goes bankrupt (i.e., theno-default case). The equity, debt, ﬁrm values, coupon, leverage, and credit spreads are given by
E(x) = En (x; δxB ), D(x) = δxB /r, F (x) = Fn (x; δxB ), C(x) = δxB , LV (x) = δxB /rFn (x; δxB ),and CS(x) = 0, respectively.
First, we explain the default-possible case (i.e., xB < x∗B ). The ﬁrm prefers to issue risky debt
due to the insuﬃcient riskless debt capacity. In this case, the optimal coupon and default timing

8

are the same as those of the benchmark model with no barrier. The other values are obtained bysubstituting coupon C0 (x) into the default-possible case of Proposition 1. Note that C0 (x) > δxB
holds by Fn (x; δxB ) < Fd (x; C0 (x)) < Fn (x; C0 (x)).
Next, we focus on the no-default case (i.e., xB ≥ x∗B ). The ﬁrm is better oﬀ issuing risklessdebt δxB /r because of the suﬃcient riskless debt capacity. Notably, δxB is not necessarily higher
than C0 (x). Both δxB < C0 (x) and Fn (x; δxB ) ≥ Fd (x; C0 (x)) can be satisﬁed because Fn (x; δxB ),unlike Fd (x; C0 (x)), includes no term of bankruptcy costs (e.g., see the top panel of Figure 1). In
other words, the ﬁrm chooses riskless capital structure if the gain from having no bankruptcy risk
dominates the ineﬃciency from the upper limit of riskless debt.
In Proposition 2, x∗B stands for the lowest barrier to achieve the no-default case.5 The criticallevel x∗B is lower than the benchmark default threshold x0 (C0 (x)) = x/h by x∗B < γx/(γ − 1)h <x/h. This result has the following implication for public intervention. Suppose that the government
attempts to prevent the ﬁrm from bankruptcy. Without the government’s ex ante commitment of
an intervention threshold, the ﬁrm issues debt with coupon C0 (x) (as in the benchmark case withno barrier). In this case, by Proposition 1 with C = C0 (x), the government needs intervention
threshold xB = C0 (x)/δ = x/h to prevent the ﬁrm from bankruptcy. However, the ex ante
commitment of intervention threshold x∗B (< x/h) prevents the ﬁrm from bankruptcy. This isbecause by considering intervention threshold x∗B , the ﬁrm strategically reduces debt (i.e., δx∗B <C0 (x)) and chooses riskless capital structure. That is, the government’s credible commitment does
not cause the ﬁrm’s moral hazard (i.e., increasing debt) but instead improves the eﬃciency of the
public intervention policy. As we will check numerically in Section 4, the ex ante required level x∗B
is much lower than the ex post required level x/h, which highlights the importance of the credible
commitment of the public intervention policy.
We can analytically prove the comparative statics with respect to barrier xB because Proposition
2 derives all the values explicitly. For the proof, see Appendix C.
Proposition 3 For xB < x∗B (i.e., the default-possible case), D(x), F (x), and LV (x) increase inxB , CS(x) decreases in xB , and C(x) = C0 (x), x0 (C0 (x)) = x/h, and E(x) = E0 (x; C0 (x)) areconstant.
At xB = x∗B (i.e., the switching point), E(x) jumps upward, D(x), C(x), LV (x), and CS(x)jump downward, and F (x) is continuous.
For xB ≥ x∗B (i.e., the no-default case), D(x), F (x), C(x) = δxB , and LV (x) increase in xB ,E(x) decreases in xB , and CS(x) is 0.
Note that the values approach the benchmark values with no barrier for xB → 0. At xB = x∗B ,the values, except ﬁrm value F (x), jump because the ﬁrm switches coupon C(x) from C0 (x) (i.e.,the default-possible case) to δx∗B (i.e., the no-default case). The switch results from maximization
5

By (16), x∗B depends on initial EBIT X(0) = x. Deﬁne yB = xB /x and G(x; xB ) = xg(yB ). Then, there is a unique∗∗∗solution yB∈ (0, γ/(γ − 1)h) to g(yB
) = 0, and yB
is the threshold between the default-possible and no-default cases.

In other words, the standardized barrier level xB /x matters to the ﬁrm.

9

of F (x), and hence, the switch does not cause a jump in F (x). Interestingly, E(x), D(x), C(x),and LV (x) are nonmonotonic with respect to xB because of the switch. Section 4.1 will show thequantitative eﬀects of xB on the results in numerical examples.
The no-default case with xB close to x∗B is most notable. In this region, D(x) and LV (x) arelower than D0 (x) and LV0 (x) due to δx∗B < C0 (x). As explained previously, this result implies
that by the credible commitment of the market intervention threshold, the government can decrease the ﬁrm’s debt and remove its bankruptcy risk. Furthermore, this result can help explain
debt conservatism. It is well known as debt conservatism that some ﬁrms have quite low leverageand bankruptcy risk compared to the optimal level predicted by trade-oﬀ theory (e.g., Graham(2000), Strebulaev and Yang (2013), and El Ghoul, Guedhami, Kwok, and Zheng (2018)). Debt
conservatism is often explained by theories of dynamic (and infrequent) leverage adjustment andﬁnancial slack for future investments and downside risks, but our model adds an alternative mechanism. Indeed, a ﬁrm can optimally choose riskless capital structure with low leverage if it has acertain degree of competitive advantage or protection against downside risks.
Following the standard models (e.g., Leland (1994) and Goldstein, Ju, and Leland (2001)), ourmodel assumes the post-bankruptcy value as the discounted value of the unlevered value. However,some papers, including Mella-Barral and Perraudin (1997), Lambrecht and Myers (2008), andShibata and Nishihara (2018), assume a constant component of the post-bankruptcy value (e.g.,constant scrap value). The presence of constant liquidation value generates the possibility of risklessdebt ﬁnancing, but its mechanism is diﬀerent from that of this paper. In these models, shareholders
can retire the principal of debt by a part of the constant liquidation value and obtain the residual
value. In such a situation, the ﬁrm exits the market, but debt becomes riskless. In contrast, in ourmodel, protection against downside risks can lead the ﬁrm to operate perpetually in the market,
which makes debt riskless.

3.3

Debt issuance constraint

This subsection interprets xB as the public intervention threshold. In such regulated markets (e.g.,transportation, utility, agricultural, and ﬁnancial industries), the government might not only saveﬁrms from ﬁnancial distress but also regulate excessive uses of debt to remove bankruptcy risk.
To explore the eﬀects of such a leverage regulation on the outcome, this subsection extends thebaseline model to a model with an upper limit of debt issuance. The extended model assumes that
coupon C must satisfy C ≤ C̄ for a given upper limit C̄(> 0). This is interpreted as a constraint
on the book value of debt (i.e., C/r ≤ C̄/r). However, as we will see in Section 4.5, D(x) andLV (x) monotonically increases in C̄. Hence, the results will remain unchanged even if we assume
a constraint on the market value of debt or leverage. Assume that C̄ < C0 (x) because the ﬁrm isunconstrained for C̄ ≥ C0 (x).
For C̄ ≤ δxB , the ﬁrm optimally chooses the maximum coupon C̄ and obtain ﬁrm value Fn (x; C̄)
because there is no possibility of bankruptcy (see Proposition 1). For C̄ ∈ (δxB , C0 (x)), the ﬁrm
solves max{Fd (x; C̄), Fn (x; δxB )} because Fd (x; C) increases in C ≤ C0 (x). As in the previous

10subsection, deﬁne the diﬀerence asḠ(x; xB ) = Fn (x; δxB ) − Fd (x; C̄)
 γ

γ 

τ C̄
τ (δxB − C̄)
x
απxB
x
=
−
+
απx0 (C̄) +
,
r
xB
γ
r
x0 (C̄)(17)

where we used (12) and (15) to obtain (17). The next proposition shows the equity, debt, ﬁrm¯ (x), and CS(x),¯values, coupon, leverage, and credit spreads, denoted by Ē(x), D̄(x), F̄ (x), C̄(x), LV
respectively, under the debt issuance constraint. For the proof, see Appendix D.
Proposition 4 There exists a unique solution x¯B ∈ (0, min{x∗B , C̄/δ}) to Ḡ(x; x¯B ) = 0. The
solution x¯B increases in C̄.
For xB < x¯B , the ﬁrm issues risky debt with coupon C̄ and goes bankrupt at default threshold
x0 (C̄) (i.e., the default-possible case). The equity, debt, ﬁrm values, coupon, leverage, and credit¯ (x) =
spreads are given by Ē(x) = E0 (x; C̄), D̄(x) = Dd (x; C̄), F̄ (x) = Fd (x; C̄), C̄(x) = C̄, LV¯
Dd (x; C̄)/Fd (x; C̄), and CS(x)= C̄/Dd (x; C̄) − r, respectively.
For xB ∈ [x¯B , C̄/δ], the ﬁrm riskless debt with coupon δxB and never goes bankrupt (i.e., theno-default case). The equity, debt, ﬁrm values, coupon, leverage, and credit spreads are given by¯ (x) = δxB /rFn (x; δxB ),Ē(x) = En (x; δxB ), D̄(x) = δxB /r, F̄ (x) = Fn (x; δxB ), C̄(x) = δxB , LV¯
and CS(x)= 0, respectively.
For xB > C̄/δ, the ﬁrm riskless debt with coupon C̄ and never goes bankrupt (i.e., the nodefault case). The equity, debt, ﬁrm values, coupon, leverage, and credit spreads are given by¯ (x) = C̄/rFn (x; C̄), and CS(x)¯Ē(x) = En (x; C̄), D̄(x) = C̄/r, F̄ (x) = Fn (x; C̄), C̄(x) = C̄, LV=
0, respectively.
Proposition 4 can be interpreted in the same way as Proposition 2. Indeed, x¯B represents thelowest barrier to achieve the no-default case. For xB < x¯B (i.e., the default-possible case), risklessdebt capacity δxB /r is not suﬃcient, and hence, the ﬁrm issues risky debt with the maximum
coupon C̄. For xB ∈ [x¯B , C̄/δ] (i.e., the no-default case), riskless debt capacity δxB /r is large
enough to lead the ﬁrm to choose riskless debt. For xB > C̄/δ (i.e., the no-default case), debt with
any C ≤ C̄ becomes riskless, and hence, the ﬁrm issues riskless debt with the maximum couponC̄.6 Proposition 4 nests Proposition 2 as the limiting case of C̄ → C0 (x) because Ḡ(x; xB ) and x¯Bagree with G(x; xB ) and x∗B in the limiting case.
Proposition 4 shows that lower C̄ decreases x¯B and ﬁrm value Fn (x; δxB ). That is, with astronger leverage regulation, the government can weaken market intervention, but the ﬁrm value
lowers. In reality, the government’s regulation and intervention require direct and indirect costs,
causing spillover eﬀects on other ﬁrms and industries. The combination of strong regulation (i.e.,
low C̄) and weak intervention (i.e, low x¯B ) will decrease the intervention cost but increase theregulation cost. The government chooses one from the set {(C̄, x¯B ) | 0 ≤ C̄ ≤ C0 (x)} so that it
6

Although the debt issuance constraint is imposed even for riskless debt in this paper, it may be imposed only forrisky debt to reduce default risk (see Nishihara, Shibata, and Zhang (2023)). In that setup, the ﬁrm issues riskless debt
δxB /r for any xB ≥ x¯B , and it does not matter whether x¯B is higher than C̄/r.

11

can minimize the total costs based on the trade-oﬀ. It is beyond the scope of this paper to model
the total social costs and derive the optimal choice. Note that x¯B is lower than x0 (C̄) = C̄/δ. As
in Proposition Proposition 2, this implies that by the ex ante commitment, the government can
improve the eﬃciency of the market intervention policy.
Although this paper interprets C̄ as a leverage regulation, it can be be interpreted as a ﬁnancing constraint imposed by debt holders. In this context, some papers investigate the eﬀects
of a borrowing constraint on the investment and ﬁnancing timing problems with no barrier (e.g.,
Shibata and Nishihara (2012), Shibata and Nishihara (2015), and Shibata and Nishihara (2018)).
In particular, Shibata and Nishihara (2018) and Nishihara, Shibata, and Zhang (2023) show that
under very hard borrowing constraints, the ﬁrm tends to issue riskless debt in the models withconstant liquidation value.7 The previous results align with our result that lower C̄ decreases x¯B .
As in Proposition 3, we can analytically prove the comparative statics with respect to barrier
xB . For the proof, see Appendix E.
¯ (x) increase in
Proposition 5 For xB < x¯B (i.e., the default-possible case), D̄(x), F̄ (x), and LV¯
xB , CS(x)decreases in xB , and C̄(x) = C̄, x0 (C̄) = C̄/δ, and Ē(x) = E0 (x; C̄) are constant.
¯ (x), and CS(x)¯
At xB = x¯B (i.e., the switching point), Ē(x) jumps upward, D̄(x), C̄(x), LV
jump downward, and F̄ (x) is continuous.
¯ (x) increase in xB ,For xB ∈ [x¯B , C̄/δ] (i.e., the no-default case), D̄(x), F̄ (x), C̄(x), and LV¯
Ē(x) decreases in xB , and CS(x)is 0.
¯ (x) decrease in
For xB > C̄/δ (i.e., the no-default case), Ē(x) and F̄ (x) increase in xB , LV¯
xB , and D̄(x) = C̄/r, C̄(x) = C̄, CS(x)= 0 are constant.
Proposition 5 shows that the comparative statics with respect to barrier xB are mostly unchanged from Proposition 3, even if the model includes the debt issuance constraint. At xB = x¯B ,the values, except ﬁrm value F (x), jump because coupon C̄(x) jumps from C̄ (i.e., the defaultpossible case) to δx∗B (i.e., the no-default case). All the values are continuous at xB = C̄/δ because¯ (x) becomeC̄(x) is continuous. Due to the switching point xB = x¯B , Ē(x), D̄(x), C̄(x), and LV
nonmonotonic with respect to xB .

4

Numerical analysis and implications

4.1

Baseline results

This section conducts numerical analyses, including comparative statics with respect to barrier xB ,volatility σ, growth rate µ, bankruptcy cost α, and upper limit C̄. The baseline parameter values
are set as in Table 1, where the values of r, µ, σ, τ , and α are standard in dynamic corporate ﬁnance7

In these previous models, a ﬁrm optimally chooses risky debt ﬁnancing in the ﬁrst-best case with no ﬁnancing

constraint. This diﬀers from this paper’s result (cf. Proposition 2). Regarding this issue, Shibata and Nishihara (2023)
show that with a high degree of information asymmetry between managers and shareholders. the ﬁrm can use risklessdebt ﬁnancing.

12

literature and reﬂect a typical S&P ﬁrm (e.g., Morellec (2001), Arnold (2014), and Nishihara,
Shibata, and Zhang (2023)). The initial EBIT value is normalized as x = X(0) = 1. For theseparameter values, the lowest barrier to achieve the no-default case becomes x∗B = 0.153, whichamounts to 15.3% of the initial EBIT. In the baseline case, we set xB = 0.2 (i.e., the no-defaultcase), which is close to x∗B = 0.153, so that the outcome will switch between the no-default anddefault-possible cases with varying levels of other parameters (cf. Sections 4.3, 4.4, and 4.5).
Sections 4.1–4.5 focus on the baseline model, and Section 4.6 examines the model with a debt
issuance constraint.
Figure 1 depicts ﬁrm value F (x; C) for varying levels of C and xB . As shown by (15) in
Proposition 1, F (x; C) increases linearly in C up to C = δxB = 0.435, 0.333, and 0.217 in thetop, center, and bottom panels, respectively. In each panel, F (x; C) jumps downward after pointx = δxB because Fn (x; δxB ) > Fd (x; δxB ) holds in (12) and (15). All the results are shown byline graphs in Section 4, and lines that look vertical stand for jumps. For the baseline parametervalues, we have C0 (x) = 0.623. For C > δxB , F (x; C) = Fd (x; C) takes its maximum value
at C = C0 (x) = 0.623, whereas for C ≤ δxB , F (x; C) = Fn (x; C) takes its maximum value at
C = δxB = 0.435, 0.333, and 0.217 in the panels. In the center panel (i.e., xB = x∗B = 0.153),
Fn (x; δxB ) agrees with Fd (x; C0 (x)), and hence, the ﬁrm is indiﬀerent to the choice between coupon
δxB = 0.333 or C0 (x) = 0.623. In the baseline case (i.e., xB = 0.2; see the top panel), Fn (x; δxB ) is
higher than Fd (x; C0 (x)), and the ﬁrm chooses C(x) = δxB = 0.435 (i.e., riskless capital structure).
In the bottom panel (i.e., xB = 0.1), Fn (x; δxB ) is lower than Fd (x; C0 (x)), and the ﬁrm chooses
C(x) = C0 (x) = 0.623 (i.e., risky capital structure).
Tables 2 and 3 show the baseline results (i.e., xB = 0.2) and benchmark results with no barrier,respectively. As mentioned above, in the baseline model, the ﬁrm prefers riskless capital structure(i.e., the no-default case). The ﬁrm issues debt up to the riskless debt capacity (i.e., D(x) =δxB /r = 8.7) to obtain the maximum tax beneﬁts. Leverage becomes LV (x) = 0.38, but credit
spreads are CS(x) = 0 because of riskless debt. In the benchmark model with no barrier, the ﬁrm
chooses coupon C0 (x) = 0.623 based on the trade-oﬀ between the tax beneﬁts and bankruptcy costsof debt. The ﬁrm will go bankrupt when X(t) falls to x0 (C0 (x)) = 0.287. Hence, D0 (x) = 10.84
is discounted from C0 (x)/r = 12.46 due to default risk, and credit spreads are positive (i.e.,
CS0 (x) = 0.00751).
In Tables 2 and 3, it holds that C(x) < C0 (x), E(x) > E0 (x), D(x) < D0 (x), F (x) > F0 (x), LV (x) <LV0 (x), and CS(x) < CS0 (x). It is straightforward that F (x) > F0 (x) and CS(x) < CS0 (x), andE(x) > E0 (x) readily follows from C(x) < C0 (x). Inequalities C(x) < C0 (x), D(x) < D0 (x),and LV (x) < LV0 (x) are notable. As discussed after Proposition 3, these inequalities imply thatbarrier xB = 0.2 leads the ﬁrm to strategically reduce debt to take advantage of riskless capital structure rather than to increase debt. In particular, LV (x) = 0.3798 is much lower than
LV0 (x) = 0.4854. As explained after Proposition 3, the model can help explain ﬁrms with lowleverage and no bankruptcy risk observed in the real world.

13

4.2

Eﬀects of barrier xB

Figure 2 depicts C(x), x0 (C(x)), E(x), D(x), F (x), LV (x), and CS(x) for varying levels of barrier
xB .8 The other parameter values are set as in Table 1. Region xB < x∗B = 0.153 is the defaultpossible case, whereas region xB ≥ x∗B = 0.153 is the no-default case. Default threshold x0 (C(x))
is depicted only in the default-possible case. For comparison, Figure 2 also depicts the benchmark
results with no barrier by dashed lines. The benchmark results do not depend on xB .
Although Proposition 3 has already shown the comparative static results analytically, Figure
2 shows them more closely and quantitatively. For instance, we ﬁnd that the eﬀects of xB on
D(x), F (x), LV (x), and CS(x) are very weak in the default-possible region (i.e., xB < x∗B = 0.153).
This is because C(x) = C0 (x) and E(x) = E0 (x) do not depend on xB and the third term in(11) is very small. Thus, barrier xB does not largely change D(x), F (x), LV (x), and CS(x) fromthe benchmark values D0 (x), F0 (x), LV0 (x), and CS0 (x). However, in the no-default region (i.e.,xB ≥ x∗B = 0.153), the eﬀects of xB on E(x), D(x), F (x), and LV (x) are strong because C(x) = δxB
increases linearly in xB . As discussed after Proposition 3, C(x), D(x), and LV (x) are lower thanC0 (x), D0 (x), and LV0 (x) for xB close to x∗B = 0.153, whereas C(x), D(x), and LV (x) are higher
than C0 (x), D0 (x), and LV0 (x) for xB > 0.3.
These results entail several implications. First, we interpret xB as the degree of competitive
advantage. Then, the model shows that leverage can be nonmonotonic with respect to the degreeof competitive advantage. Indeed, ﬁrms with intermediate levels of competitive advantage can takelow leverage with only riskless debt.
Next, we interpret xB as the strength of public intervention. The critical intervention thresholdx∗B = 0.153 (i.e., 15.3% of the initial EBIT) is not very high. In absence of the ex ante commitment of the intervention threshold, as explained after Proposition 2, the government would
need intervention threshold xB = x0 (C0 (x)) = 0.287 to prevent the ﬁrm from bankruptcy ex post.
With a credible commitment, the government can prevent the ﬁrm from bankruptcy by almosta half intervention threshold (i.e., x∗B = 0.153). Of course, in the real world including uncertainty and diversity of ﬁrm parameter values, it may be diﬃcult for the government to match
xB = x∗B perfectly. A low market intervention threshold (i.e., xB < x∗B = 0.153) hardly inﬂuences capital structure, bankruptcy probability, and ﬁrm value, whereas a high market intervention
threshold (say, xB > 0.3) prevents bankruptcy but leads to the ﬁrm’s moral hazard (i.e., increasing
leverage to gain tax beneﬁts). It is important to set an appropriate intervention threshold (i.e.,xB ≈ x∗B = 0.153) to prevent bankruptcy and reduce leverage eﬀectively.

8

Rodrigues (2022) also studies the comparative statics with respect to ﬂoor levels in numerical examples. Although

his model is more complicated than our model, the eﬀects of ﬂoor levels on capital structure are qualitatively unchanged
from our results. Indeed, Rodrigues (2022) also shows that a higher ﬂoor leads to riskless capital structure.

14

4.3

Eﬀects of volatility σ

Figure 3 depicts C(x), x0 (C(x)), E(x), D(x), F (x), LV (x), CS(x), x∗B , and (x/x∗B )γ for varying levels of volatility σ.9 The other parameter values are set as in Table 1. Region σ < 0.17 is thedefault-possible case, whereas region σ ≥ 0.17 is the no-default case. Default threshold x0 (C(x))
is depicted only in the default-possible case. For comparison, Figure 3 also depicts the benchmark
results with no barrier by dashed lines.
In the default-possible region (i.e., σ < 0.17), each value moves in the same way as in thebenchmark case with no barrier. In fact, higher σ decreases C(x), x0 (C(x)), D(x), F (x), and LV (x)and increases E(x). These results can be intuitively interpreted as follows. Higher σ increases
bankruptcy risk, and the ﬁrm reduces leverage to mitigate bankruptcy risk. However, decreased
leverage does not fully oﬀset increased bankruptcy risk with higher σ, and hence, credit spreads
increase in σ. Firm value decreases in σ due to the lower leverage eﬀect, although equity value
increases due to decreased coupon payments. These results align with the standard results inprevious literature (e.g., Leland (1994)). As in Figure 2, Figure 3 also shows that the presence ofxB hardly aﬀects the values in the default-possible region.
At σ = 0.17, the result switches from the default-possible case to the no-default case. Then,
C(x), E(x), D(x), LV (x), and CS(x) jump at this point. In the no-default region (i.e., σ ≥ 0.17),
C(x), E(x), D(x), F (x), and LV (x) move contrary to the benchmark case with no barrier. The
comparative statics are explained by the sensitivity of riskless debt capacity δxB /r to σ. Note that
δxB /r increases in σ by ∂δ/∂σ > 0. Then, C(x) = δxB , D(x) = δxB /r, F (x), and LV (x) increasein σ, whereas E(x) decreases in σ due to increased C(x). The results imply that the eﬀects ofvolatility on ﬁrm value and capital structure for ﬁrms with suﬃcient competitive advantage orprotection against downside risks can greatly diﬀer from those for ordinary ﬁrms.
Note that the above results are based on the assumption of constant barrier level xB = 0.2.
Barrier xB = 0.2 is more eﬀective with higher σ because the probability of X(t) hitting xB = 0.2
increases with higher σ. The bottom-right panel of Figure 3 shows that the critical level x∗B and thestate price (x/x∗B )γ 10 decrease and increase, respectively, in σ. The comparative statics of x∗B are
explained by the decrease in Fd (x) and increase in Fn (x) with higher σ (see F (x) of Figure 3). By
these two eﬀects, x∗B , which is the unique solution to (16), decreases in σ. Despite the decrease inσ, (x/x∗B )γ increases in σ due to ∂γ/∂σ > 0. In other words, higher σ makes X(t) more volatile andincreases the probability of X(t) hitting x∗B . In terms of public intervention, these results suggestthat the government needs a lower market intervention threshold but more frequent interventions
to prevent a more volatile ﬁrm from bankruptcy.

9Rodrigues (2022) also studies the comparative statics with respect to volatility in numerical examples. The eﬀects ofvolatility on capital structure in his shadow process model are qualitatively the same as in our reﬂecting barrier model.
10
The state price denotes the present values of $1 contingent on X(t) hitting x∗B .

15

4.4

Eﬀects of growth rate µ

Figure 4 depicts C(x), x0 (C(x)), E(x), D(x), F (x), LV (x), CS(x), x∗B , and (x/x∗B )γ for varying levels of growth rate µ.11 The other parameter values are set as in Table 1. Region µ ≤ 0.0252 is the
no-default case, whereas region µ > 0.0252 is the default-possible case. Default threshold x0 (C(x))
is depicted only in the default-possible case. For comparison, Figure 4 also depicts the benchmark
results with no barrier by dashed lines.
As in Figures 2 and 3, Figure 4 shows that all the values in the baseline case are almost thesame as those in the benchmark case in the default-possible region (i.e., µ > 0.0252). One reason
is that the ﬁrm choose the same coupon C(x) = C0 (x), and the other reason is that the state pricecontingent on X(t) hitting xB (i.e., (x/xB )γ ) is very low. We omit explaining the details of thecomparative statics in the default-possible case because they are the same as those in the standard
model with no barrier (e.g., Leland (1994)).
At the switching point µ = 0.0252, C(x), E(x), D(x), LV (x), and CS(x) jump. Even in the
no-default region (i.e., µ ≤ 0.0252), C(x), E(x), D(x), and F (x) change with µ in the same way
as in the benchmark values. More notably, LV (x) decreases in µ, contrary to LV0 (x). The reason
is as follows. Riskless debt capacity D(x) = δxB /r increases in µ by ∂δ/∂µ > 0, and equity
value E(x) = E(x; δxB ) also increases in µ by ∂π/∂µ > 0 in (13). The latter eﬀect dominates
the former eﬀect, and hence LV (x) decreases in µ. This sensitivity is novel and contrasted withthe standard result. In fact, the standard trade-oﬀ models (e.g., Leland (1994)) predict a positive
relation between leverage and growth rate (see LV0 (x) in Figure 4), but empirical studies (e.g.,
Titman and Wessels (1988) and Frank and Goyal (2015)) show a negative relation. This is wellknown as a deﬁcit of the standard trade-oﬀ models (e.g., Demarzo (2019)). Our model may help
resolve the problem. Indeed, the model predicts a negative relation between leverage and cash
ﬂows for ﬁrms with suﬃcient competitive advantage or protection against downside risks. 12 This isbecause such a ﬁrm can set debt level by riskless debt capacity rather than the trade-oﬀ between
the tax beneﬁts and bankruptcy costs.
The bottom-right panel of Figure 4 shows that x∗B and (x/x∗B )γ increase and decrease, respectively, in µ. The former result is caused by Fd (x) increasing in µ more than Fn (x) does. Despite
the increase in σ, (x/x∗B )γ decreases in µ due to ∂γ/∂µ < 0 (i.e., higher µ decreases the probabilityof X(t) hitting x∗B ). These results entail a policy implication that the government needs a higher
market intervention threshold but less frequent market interventions to prevent a high-growth ﬁrmfrom bankruptcy.

11

Rodrigues (2022) also studies the comparative statics with respect to growth rate in numerical examples. The eﬀects

of growth rate on capital structure in his shadow process model are qualitatively the same as in our reﬂecting barriermodel.
12
The relation between X(0) = x and LV (x) is also negative in the no-default region because as explained in footnote
5, the sensitivities to x is inverse to the sensitivities to xB (see Figure 2).

16

4.5

Eﬀects of bankruptcy cost α

Figure 5 depicts C(x), x0 (C(x)), E(x), D(x), F (x), LV (x), CS(x), x∗B , and (x/x∗B )γ for varying levels of bankruptcy cost α. The other parameter values are set as in Table 1. Region α < 0.199 is the
no-default case, whereas region α ≥ 0.199 is the default-possible case. Default threshold x0 (C(x))
is depicted only in the default-possible case. For comparison, Figure 5 also depicts the benchmark
results with no barrier by dashed lines.
In the no-default region (i.e., α ≥ 0.199), neither value depends on α because the ﬁrm will
never go bankrupt. In the default-possible region (i.e., α < 0.199), all the values change with α inthe same way as in the benchmark values with no barrier. In this region, higher α increases the
disadvantages of debt and hence decreases C(x), D(x), and LV (x). Firm value F (x) and CS(x)also decrease in α due to the decreased leverage eﬀect, whereas E(x) increases in α due to decreased
coupon payments.
By (16) and ∂h/∂α > 0, we can easily prove that x∗B decreases in α. The bottom-right panelof Figure 3 numerically veriﬁes the sensitivity of x∗B to α. Note that (x/x∗B )γ changes in the same
way as x∗B because γ does not depend on α. This result is intuitively explained as follows. Higher α
increases the disadvantages of risky debt and decreases the leverage eﬀect. Then, the ﬁrm is more
likely to be better oﬀ using riskless debt rather than relying on risky debt. For the same reason,
lower corporate tax rate τ decreases x∗B and (x/x∗B )γ . although we omit depicting a ﬁgure withvarying levels of τ . Indeed, lower τ decreases the tax advantages of debt, which decreases the ﬁrm’s
motive to use risky debt. The comparative static results have the following implications of public
intervention. The government can prevent the ﬁrm from bankruptcy by weaker and fewer market
interventions if it imposes a lower corporate tax rate and a more stringent bankruptcy law withhigher bankruptcy penalty. This is because with such public policies, the ﬁrm has fewer advantages
from issuing risky debt and is more likely to choose riskless capital structure.

4.6

Eﬀects of upper limit C̄So far, we have examined the eﬀects of the key parameters on the results in the baseline model.
This subsection studies the eﬀects of upper limit C̄ in the extended model of Section 3.3. For thebaseline parameter values (i.e., Table 1), the no-default case holds by x∗B = 0.1529 < xB = 0.2
in absence of C̄. By Proposition 4, we have x¯B < x∗B = 0.153 < xB = 0.2 for any C̄, andhence, the no-default case holds for any C̄. We reset xB = 0.1 to depict both the no-default
and default-possible cases.

The other parameter values are set as in Table 1.

Figure 6 de-¯ (x), CS(x),¯picts C̄(x), x0 (C̄(x)), Ē(x), D̄(x), F̄ (x), LV
x¯B , and (x/x¯B )γ for varying levels of C̄(≤C0 (x) = 0.623). Note that C̄ does not bind the ﬁrm for C̄ ≥ C0 (x) = 0.623. Region C̄ ≤ 0.261 is
the no-default region, whereas region C̄ > 0.261 is the default-possible region. Default threshold
x0 (C̄(x)) is depicted only in the default-possible case. The no-default region is classiﬁed into regionC̄ ∈ [0.218, 0.261], where C̄(x) = δxB = 0.218 does not depend on C̄, and region C̄ < 0.218, whereC̄(x) = C̄ (see Proposition 4). For comparison, Figure 5 also depicts the benchmark results with17

no barrier under upper limit C̄ by dashed lines. In this benchmark case, the ﬁrm chooses couponC̄ because ﬁrm value F0 (x; C) (see (3)) monotonically increases in C up to C = C0 (x) = 0.623.
In the default-possible region (i.e., C̄ > 0.261) of Figure 6, the presence of xB hardly aﬀects
each value. The main reason is that the ﬁrm chooses the maximum coupon C̄ regardless of xB .
All the comparative static results are straightforward and the same as the benchmark results with
¯ (x). The increased leverage eﬀects increase F̄ (x),no barrier. Higher C̄ increases D̄(x) and LV¯
although the increased coupon payments decrease Ē(x) and increase CS(x).
Note that each value
agrees with that of the unconstrained baseline model for C̄ = C0 (x) = 0.623 (i.e., the right end ofeach panel of Figure 6).
The no-default region C̄ ∈ [0.218, 0.261] is most notable. In this region, the ﬁrm chooses riskless
capital structure because x¯B ≤ xB = 0.1 (see the bottom-right panel of Figure 6). Riskless debt capacity δxB = 0.218 rather than debt issuance limit C̄ binds the ﬁrm due to δxB = 0.218 ≤ C̄. Then,coupon C̄(x) = δxB = 0.218 is constant in this region. This also implies that Ē(x), D̄(x), F̄ (x),¯ (x) are constant in this region. These results are contrasted with the benchmark resultsand LV
with no barrier.
Last, we turn to the no-default region C̄ < 0.218. In this region, debt issuance limit C̄ rather
than riskless debt capacity δxB = 0.218 binds the ﬁrm due to C̄ < δxB = 0.218. Then, the ﬁrm
chooses the maximum coupon C̄ as in the benchmark case with no barrier. The comparative static¯
results other than CS(x)= 0 are the same with the standard results with no barrier. Note thateach value converges to that of the all-equity ﬁrm for C̄ → 0 (i.e., the left end of each panel of
Figure 6).
As shown by Proposition 4, the bottom-right panel of Figure 6 shows that the critical levelx¯B increases in C̄. State price (x/x¯B )γ similarly increases in C̄ because γ does not depend on C̄.
These results show that by regulating leverage, the government can reduce the market intervention
threshold and frequency to prevent the ﬁrm from bankruptcy. As discussed after Proposition 4, theoptimal policy would lie in {(C̄, x¯B ) | 0 ≤ C̄ ≤ C0 (x)}, but it may be diﬃcult for the government toﬁnd a perfectly optimal pair (C̄, x¯B ). In fact, the government tends to impose a uniform regulationand protection policy over ﬁrms within the same industry, although cash ﬂows are aﬀected by ﬁrmspeciﬁc factors and risks. That is, (C̄, x¯B ) diﬀers over ﬁrms in the industry, but the government
must choose one policy for all the ﬁrms. Regulation that is too weak cannot prevent bankruptcy(cf. the region C̄ > 0.261), whereas regulation that is too strong decreases ﬁrm value ineﬃciently(cf. the region C̄ < 0.218). Even if the government cannot ﬁnd a perfect solution for all the ﬁrms,
it can choose a policy within the plausible region (cf. the region C̄ ∈ [0.218, 0.261]).

5

Conclusion

This paper investigates the capital structure model with earnings above a reﬂecting barrier. The
model can approximate a ﬁrm with competitive advantage or public protection against downside
risks. In the former, the barrier represents an exit threshold of competitors, whereas in the latter,18

it represents a public intervention threshold. This paper explicitly derives the equity, debt, ﬁrm
values, leverage, and credit spreads and shows their comparative statics with respect to barrier
level. The main results are summarized below.
First, and most notably, the barrier generates the riskless debt capacity, and the ﬁrm chooses
either riskless or risky capital structure by comparing the values with the maximum riskless debt
and with risky debt. The higher the barrier, the larger the riskless debt capacity, and the ﬁrm
tends to prefer riskless capital structure. With intermediate barrier levels, the ﬁrm chooses lower
leverage than the level with no barrier to take advantage of riskless debt.
This result can help explain debt conservatism observed in the real world. Indeed, the model
predicts that ﬁrms with certain degrees of competitive advantage or public protection can issue
lower levels of riskless debt rather than adjusting risky debt levels based on the trade-oﬀ between
the tax beneﬁts and bankruptcy costs of debt. In the no-default case, contrary to the results instandard trade-oﬀ theory, leverage increases with higher volatility and lower growth rate. The latter
result can account for empirical ﬁndings of a negative relation between leverage and proﬁtability.
The model also entails several implications of public intervention to protect speciﬁc ﬁrms orindustries from ﬁnancial distress. Using the ex ante commitment of an appropriate intervention
threshold, the government can eﬃciently lead ﬁrms to adopt riskless capital structure with lowleverage. With a more stringent bankruptcy law (i.e., higher bankruptcy cost), lower corporate tax
rate, and stronger leverage regulation, the government needs weaker and fewer market interventions
to prevent the ﬁrms from bankruptcy.

A

Proof of Proposition 1

First, derive the equity value of the ﬁrm that operates perpetually, i.e., En (x; C). The derivation
process is the same as in the reﬂecting barrier models in Chapter 8 of Dixit and Pindyck (1994).
Equity value En (x; C) satisﬁes the diﬀerential equationµ∂En (x; C)∂ 2 En (x; C)+ 0.5σ 2+ (1 − τ )(x − C) = rEn (x; C)
∂x
∂2x

(18)

for x > xB with the boundary conditions∂En (xB ; C)
= 0,∂x
En (x; C)
lim
< ∞.
x→∞
π(x)

(19)
(20)

Note that (19) means that the derivative of En (x; C) must be 0 at reﬂecting barrier xB because
X(t) surely increases from X(0) = xB , while (20) stems from the fact the probability of X(t)hitting xB approaches 0 for X(0) → ∞. By (18) and (20), En (x; C) is expressed as
En (x; C) = πx −(1 − τ )C+ Axγ ,
r

where A is a constant. By (19), we can derive A as
A=−1−γ
(1 − α)πxB.
γ19

Then, En (x; C) is expressed as (13). Note that En (xB ; C) ≥ 0 holds if and only if C ≤ δxB .
Accordingly, for C ≤ δxB , En (x; C) ≥ 0 holds for all x ≥ xB , which implies that shareholders
do not prefer to receive default value 0 by declaring default. Then, equity value E(x; C) becomesEn (x; C) in this case. Debt is riskless, and hence Dd (x; C) = C/r holds. By summing this andEn (x; C), we have Fn (x; C) as (15).
On the other hand, for C > δxB , En (xB ; C) < 0 holds, which implies that shareholders preferto declare default at a suﬃciently low threshold xd (≥ xB ). Note that C > δxB is equivalent tox0 (C) ≥ xB . As in the standard literature (e.g., Goldstein, Ju, and Leland (2001), Shibata andNishihara (2012), and Sundaresan, Wang, and Yang (2015)), the equity value of the ﬁrm that
defaults at the optimal timing, i.e., Ed (x; C), is expressed as γ 
x(1 − τ )C(1 − τ )C+− πxd
Ed (x; C) = sup πx −rxdrxd ≥xBγ 
(1 − τ )Cx(1 − τ )C= πx −+− πx0 (C)rx0 (C)r= E0 (x; C).
Hence, equity value E(x; C) becomes Ed (x; C) = E0 (x; C) in this case. It should be noted thatE0 (x; C) > En (x; C) holds for x ≥ max{xB , x0 (C)} if and only if C > δxB .
Debt value is derived as


γ 

C
x
C
Dd (x; C) =−− (1 − α)Fn (x0 (C); 0)rx0 (C)rγ   γx
Cx
C(1 − α)πxB=−− (1 − α)πx0 (C) −.
rx0 (C)
rxBγ

By summing this and E0 (x; C), we also obtain Fd (x; C) as (12).

B

Proof of Proposition 2

By (16) and γ < 0, G(x; xB ) is continuously increases in xB ∈ [0, x]. By (16), we also haveτx<0
(r − µ)h γxαπ x˜B
G(x; x˜B ) = −
> 0,
x˜B
γ
G(x; 0) = −

(21)
(22)

where x˜B = γx/(γ −1)h. Therefore, a unique solution x∗B ∈ (0, γx/(γ −1)h) exists to G(x; x∗B ) = 0.
For xB < x∗B , G(x; xB ) < 0 holds, which leads to maxC≥0 F (x; C) = max{Fd (x; C0 (x)), Fn (x; δxB )} =Fd (x; C0 (x)). Hence, the ﬁrm chooses coupon C(x) = C0 (x) at time 0. Note that C0 (x) = δx/h >δxB follows from xB < x∗B < γx/(γ − 1)h. Then, the equity, debt, ﬁrm values, coupon, default threshold, leverage, and credit spreads are equal to those of the default-possible case with
C = C0 (x) in Proposition 1.
For xB ≥ x∗B , G(x; xB ) ≥ 0 holds, which leads to maxC≥0 F (x; C) = max{Fd (x; C0 (x)), Fn (x; δxB )} =Fn (x; δxB ). Hence, the ﬁrm chooses coupon C(x) = δxB at time 0. Then, the equity, debt, ﬁrm
values, coupon, default threshold, leverage, and credit spreads are equal to those of the no-default
case with C = δxB in Proposition 1.

20

C

Proof of Proposition 3

By Propositions 1 and 2, for xB < x∗B , D(x) = Dd (x; C0 (x)) increases in xB , while C(x) = C0 (x),x0 (C0 (x)) = x/h, and E(x) = E0 (x; C0 (x)) are constant. Then, F (x) and LV (x) increase in xB ,CS(x) decreases in xB .
At xB = x∗B , coupon C(x) changes from C0 (x) = δx/h to δx∗B . It follows from x∗B < γx/(γ −1)hthat C0 (x) = δx/h > δx∗B (i.e., a downward jump). At xB = x∗B , E(x) changes from E0 (x; C0 (x))
to(1 − τ )δx∗B
En (x; δx∗B ) = πx −
−



r

x
x∗B

γ

πx∗B
γ

= E0 (x; δx∗B )> E0 (x; C0 (x))(i.e., an upward jump), where we obtained the last inequality by C0 (x) > δx∗B . By deﬁnition ofx∗B (i.e., G(x; x∗B ) = 0), Fd (x; C0 (x)) continuously changes to Fn (x; δx∗B )) at xB = x∗B . By thecontinuity of F (x) and the upward jump of E(x), D(x) must jump downward at xB = x∗B . Then,LV (x) = D(x)/F (x) jumps downward at xB = x∗B , and CS(x) also jumps downward to 0 (i.e.,
riskless debt).
By Propositions 1 and 2, for xB ≥ x∗B , D(x) = δxB /r and F (x) = Fn (x; δxB ), C(x) = δxB
increase in xB , while CS(x) is 0. Deﬁne
H(xB ) = En (x; δxB ) = πx −(1 − τ )δxB−
r

and compute the derivativedH(xB )
(γ − 1)π
=−
dxB
γ




1−

x
xB



x
xB

γ

πxB
γ

γ 
< 0,

where the last inequality follows from x > xB and γ < 0. Hence, E(x) = En (x; δxB ) decreases in
xB . By the decrease of E(x) and increase of D(x), LV (x) increases in xB .

D

Proof of Proposition 4

By (17), and γ < 0, Ḡ(x; xB ) is continuously increases in xB ∈ [0, x]. By (17), we can show that
γ 


τ C̄
C̄
x
απx0 (C̄) +Ḡ(x; 0) = − +rrx0 (C̄)= πx − F0 (x; C̄) < 0,(23)Ḡ(x; x∗B ) = Fn (x; δx∗B ) − Fd (x; C̄)= Fd (x; C0 (x)) − Fd (x; C̄) > 0,
γ 
 γ


τ C̄
δx
απ C̄
x
απx0 (C̄) +Ḡ(x; C̄/δ) = −
+
> 0,
γδ
r
C̄
x0 (C̄)(24)

where we used πx = F0 (x; 0) < F0 (x; C̄) in (23), and we used Fn (x; δx∗B ) = Fd (x; C0 (x)) and the
optimality of C0 (x) in (24). Hence, a unique solution x¯B ∈ (0, min{x∗B , C̄/δ}) exists to Ḡ(x; x¯B ) =0.

21

For xB < x¯B , Ḡ(x; xB ) < 0 holds, which leads to maxC∈[0,C̄] F (x; C) = max{Fd (x; C̄), Fn (x; δxB )} =Fd (x; C̄). Then, the ﬁrm chooses coupon C(x) = C̄ at time 0. The results follow from the defaultpossible case with C = C̄ in Proposition 1.
For xB ∈ [x¯B , C̄/δ], Ḡ(x; xB ) ≥ 0 holds, which leads to maxC∈[0,C̄] F (x; C) = max{Fd (x; C̄), Fn (x; δxB )} =Fn (x; δxB ). Then, the ﬁrm chooses coupon C(x) = δxB at time 0. The results follow from theno-default case with C = δxB in Proposition 1.
For xB > C̄/δ, debt with any coupon C(≤ C̄) becomes riskless, which leads to maxC∈[0,C̄] F (x; C) =maxC∈[0,C̄] Fn (x; C̄) = Fn (x; C̄). Then, the ﬁrm chooses coupon C(x) = C̄ at time 0. The results
follow from the no-default case with C = C̄ in Proposition 1.

E

Proof of Proposition 5

By Propositions 1 and 4, for xB < x¯B , D̄(x) = Dd (x; C̄) increases in xB , while C̄(x) = C̄,¯ (x) increase in xB , while
x0 (C̄) = C̄/δ, and Ē(x) = E0 (x; C̄) are constant. Then, F̄ (x) and LV¯
CS(x)decreases in xB .
At xB = x¯B , coupon C̄(x) changes from C̄ to δ x¯B By Proposition 4, C̄ > δ x¯B holds. At
xB = x¯B , Ē(x) changes from E0 (x; C̄) to
En (x; δ x¯B ) = πx −(1 − τ )δ x¯B
−
r



x
x¯B

γ

π x¯B
γ

= E0 (x; δ x¯B )> E0 (x; C̄)(i.e., an upward jump), where we obtained the last inequality by C̄ > δ x¯B . By deﬁnition of x¯B(i.e., Ḡ(x; x¯B ) = 0), Fd (x; C̄) continuously changes to Fn (x; δ x¯B ) at xB = x¯B . By the continuity of¯ (x) jumpsF̄ (x) and the upward jump of Ē(x), D̄(x) must jump downward at xB = x¯B . Then, LV¯
downward at xB = x¯B , and CS(x)also jumps downward to 0.
For xB ∈ [x¯B , C̄/δ], the results follow from the proof of Proposition 3 (see the third paragraphof Appendix C).
By Propositions 1 and 4, for xB > C̄/δ, Ē(x) = En (x; C̄) increases in xB , while D̄(x) =¯¯ (x) decreases inC̄/r, C̄(x) = C̄, and CS(x)= 0 are constant. Then, F̄ (x) increases in xB , and LV
xB .

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24

Table 1: Baseline parameter values.
r

µ

σ

τ

α

xB

x = X(0)

0.05

0.01

0.2

0.15

0.4

0.2

1

Table 2: Baseline results.
C(x)

E(x)

D(x)

F (x)

LV (x)

CS(x)

0.435

14.21

8.7

22.91

0.38

0

Table 3: Benchmark results with no barrier.
C0 (x)x0 (C0 (x))

E0 (x)

D0 (x)F0 (x)

LV0 (x)CS0 (x)

0.623

0.287

11.49

10.84

22.32

0.485

0.00751

25

23
22.5
22
21.5
21
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

23
22.5
22
21.5
21

23
22.5
22
21.5
21

Figure 1: Firm value F (x; C) for varying levels of C. The top, center, and bottom panels show F (x; C)
for xB = 0.2 (Baseline), xB = x∗B = 0.153, and xB = 0.1, respectively, where F (x; C) = Fn (x; C) forC ≤ δxB = 0.435, 0.333, and 0.217, respectively, and F (x; C) = Fd (x; C) for C > δxB = 0.435, 0.333,
and 0.217, respectively. In all the panels, Fd (x; C) takes the maximum at C = C0 (x) = 0.623.

26

1

1

0.8

0.5

0.6

0

0.4

-0.5
0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.3

0.4

0.5

20

14
12

15

10
10

8
6
0

0.1

0.2

0.3

0.4

0.5

0.7
26
0.6
0.5

24

0.4
0.3

22
0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

10 -3

8
6
4
2
0
0

Figure 2: Comparative statics with respect to reﬂecting barrier xB . The ﬁgure depicts coupon C(x),default threshold x0 (C(x)), equity value E(x), debt value D(x), ﬁrm value F (x), leverage LV (x), andcredit spread CS(x) in the baseline model by solid lines. Region xB < x∗B = 0.153 is the defaultpossible case, whereas region xB ≥ x∗B = 0.153 is the no-default case. The dashed lines represent thebenchmark results with no barrier.
27

0.7

0.4

0.6

0.3

0.5
0.2
0.4
0.1

0.2

0.3

0.4

0.1

14

0.2

0.3

0.4

0.2

0.3

0.4

0.2

0.3

0.4

0.2

0.3

0.4

14

12

12

10

10
8

8
0.1

0.2

0.3

0.4

0.1

0.6
26
0.5
24
0.4
22
0.1

0.2

0.3

0.4

0.1

0.4
0.02

0.3

0.015
0.2
0.01
0.1

0.005
0
0.1

0.2

0.3

0
0.1

0.4

Figure 3: Comparative statics with respect to volatility σ. The ﬁgure depicts coupon C(x), default
threshold x0 (C(x)), equity value E(x), debt value D(x), ﬁrm value F (x), leverage LV (x), credit
spread CS(x), critical barrier x∗B , and state price (x/x∗B )γ in the baseline model by solid lines. Region
σ < 0.17 is the default-possible case, while region σ ≥ 0.17 is the no-default case. The dashed lines
represent the benchmark results with no barrier.
28

0.4
2
0.35
1

0.3

0
0

0.01

0.02

0.03

0.04

0

0.01

0.02

0.03

0.04

0

0.01

0.02

0.03

0.04

0

0.01

0.02

0.03

0.04

0

0.01

0.02

0.03

0.04

40
30
20

20

10

0
0

0.01

0.02

0.03

0.04

0.6

100

0.55
0.5
50
0.45
0.4
0
0

0.01

0.02

0.03

0.04

0.01

0.25
0.2

0.005

0.15
0.1

0
0

0.01

0.02

0.03

0.04

Figure 4: Comparative statics with respect to growth rate µ. The ﬁgure depicts coupon C(x), default
threshold x0 (C(x)), equity value E(x), debt value D(x), ﬁrm value F (x), leverage LV (x), credit
spread CS(x), critical barrier x∗B , and state price (x/x∗B )γ in the baseline model by solid lines. Regionµ ≤ 0.0252 is the default-possible case, whereas region µ > 0.0252 is the no-default case. The dashedlines represent the benchmark results with no barrier.
29

0.9

0.4

0.8

0.35

0.7

0.3

0.6
0.25
0.5
0.1

0.2

0.3

0.4

0.5

0.2
0.1

0.6

0.2

0.3

0.4

0.5

0.6

0.2

0.3

0.4

0.5

0.6

0.2

0.3

0.4

0.5

0.6

0.2

0.3

0.4

0.5

0.6

14
14
12
12
10
10
8
0.1

0.2

0.3

0.4

0.5

0.6

0.1

23
0.6
0.5

22.5

0.4
22
0.1

0.2

0.3

0.4

0.5

0.6

0.1

0.01
0.2
0.15

0.005

0.1
0
0.1

0.2

0.3

0.4

0.5

0.6

0.1

Figure 5: Comparative statics with respect to bankruptcy cost α. The ﬁgure depicts coupon C(x),default threshold x0 (C(x)), equity value E(x), debt value D(x), ﬁrm value F (x), leverage LV (x),credit spread CS(x), critical barrier x∗B , and state price (x/x∗B )γ in the baseline model by solid lines.
Region α < 0.199 is the default-possible case, whereas region α ≥ 0.199 is the no-default case. The
dashed lines represent the benchmark results with no barrier.
30

0.6
0.2

0.4

0.1

0.2
0

0
0

0.2

0.4

0.6

20

10

15

5

10

0

0.2

0.4

0.6

0

0.2

0.4

0.6

0

0.2

0.4

0.6

0

0.2

0.4

0.6

0
0

0.2

0.4

0.6

0.4

22

0.3
0.2

21.5

0.1
21

0
0

0.2

0.4

0.6

10 -3
0.15
6
0.1
4
0.05

2
0

0
0

0.2

0.4

0.6

Figure 6: Comparative statics with respect to upper limit C̄. The ﬁgure depicts coupon C̄(x), default¯ (x), credit spreadthreshold x0 (C̄(x)), equity value Ē(x), debt value D̄(x), ﬁrm value F̄ (x), leverage LV¯
CS(x),
critical barrier x¯B , and state price (x/x¯B )γ in the extended model with upper limit C̄ by solidlines. Region C̄ ≤ 0.261 is the no-default case, whereas region C̄ > 0.261 is the default-possible case.
The dashed lines represent the benchmark results with no barrier under upper limit C̄.
31