The leverage–competition nexus with
heterogeneous firms: product
differentiation vs. number of
competitors
Nicolas Gonne ∗
Éric Toulemonde ∗ †
August, 2016
Abstract – The empirical link between leverage and output market
competition is essentially inconclusive. We provide a theoretical explanation based on a heterogeneous firms model augmented with a
financial market where the equilibrium depends upon the competitive
conditions on the output market. Contrary to previous works, which
only consider one-dimensional measures for competition, our framework enables us to disentangle the opposite impacts of product differentiation and the number of firms. Expected profits are higher in
a market with fewer competitors, thus bankruptcy is less likely, debt
is cheaper and leverage, higher. On the contrary, product differentiation brings low-productivity firms into the market, hence increases the
probability of bankruptcy and reduces leverage.
Keywords – corporate financial policy, firm heterogeneity, monopolistic competition, optimal capital structure, product differentiation.
JEL – D43, G32, L11.
∗
†
Department of economics, University of Namur, 8 Rempart de la Vierge, B5000 Namur,
Belgium.
Corresponding author: [email protected] .
1
Introduction
Although both financial and industrial economic theory recognize the interactions between corporate financial policy and the competitive conditions on
the output market, empirical works regarding the correlation between firms
capital structure and competition produce rather inconclusive results. Specifically, studies do find a significant empirical link between financial and real
decisions, but are not consistent in vindicating competing theories regarding
the sign of the correlation.
In this paper we argue that this inconsistency stems from the multidimensional nature of competitive pressures, a fact that empirical exercises typically ignore as they only consider one-dimensional measures of competition. Indeed, for the most part, these studies rest on statistics such as the
Herfindahl-Hirschmann index or Tobin’s q. Strong concentration, indicated
by a high Herfindahl index, and attractive value-adding prospects, implied
by a high Tobin’s q, are both associated with low competition and important
market power. However these aggregate measures integrate different dimensions of competition. In particular, such short-term, cyclical dimension as
the number of competitors in the market intertwines with the longer-run,
structural product differentiation. Now, although more competitors and less
differentiation both correspond to tougher competition, they arguably have
an opposite impact on measures such as the Herfindahl index and Tobin’s
q, because more substitutability contributes to the reallocation of production and profits towards more productive firms. That is, concentration and
profitability may be both a cause and an outcome of competitive pressures.
Therefore it is no surprise that empirical studies on the relationship between
competition and leverage based on such multidimensional aggregates report
now a positive correlation (e.g. Phillips, 1995; Michaelas et al., 1999), now
a negative one (e.g. Chevalier, 1995; Campello, 2003). 1 Some studies find a
1
Note that a large body of work investigates the relationship between profitability and
2
non-linear relationship, with high and low values for Tobin’s q associated with
high debt ratio, whereas leverage is lower for intermediate values (Pandey,
2004; Guney et al., 2011). This suggests that the different dimensions of
competition weigh in preponderantly at different levels of debt.
Two studies hold our attention as they resort to different measures that
arguably isolate a single dimension of competition. Xu (2012) looks at exogenous changes in competition caused by increased import penetration following trade liberalization, and shows that it is associated with a reduction
in firms leverage. This supports the idea that a short-run augmentation of
the number of competitors tends to decrease firms debt ratio. As opposed to
that, Fosu (2013) provides evidence of a positive correlation between leverage and competition as measured by the elasticity of profits with respect to
marginal cost. Because it captures firms ability to pass costs on to consumers,
this suggests that leverage decreases in product differentiation. 2
The purpose of the present paper is to provide theoretical grounds for this
multidimensional impact of output market conditions on firms financial policy. To keep our point simple we restrict the analysis to two generic dimensions of competition which work in opposite directions, namely the number
of firms in the market and product differentiation. Our intuition goes as follows. The cost of leverage increases in the probability of bankruptcy, which
depends on the competitive conditions on the output market. 3 On one hand,
increased competition associated with more firms in the market reduces operating income. Everything else equal, this makes bankruptcy more likely
and decreases leverage. On the other hand, tougher competition stemming
from lower product differentiation implies that low-productivity firms do not
enter the market. Everything else equal, bankruptcy is then less likely in the
2
3
leverage, and looks at competition as a covariate, which raises the issue of the competition/profitability correlation.
About the link between profits elasticity and competition, see the original paper by
Boone (2008).
Valta (2012) provides empirical evidence that the price of bank debt is systematically
higher for firms that operate in competitive output markets.
3
industry, which enhances leverage.
Because we want to focus on the nature of competition rather than on the behaviour of competitors, we work in a monopolistically competitive structure,
that is, an economic environment in which firms enjoy market power but in
which strategic interactions between competitors are absent. Moreover, we
acknowledge that firms are heterogeneous, which implies that competitive
pressures generate intraindustry reallocations. Critically, there exists uncertainty regarding firms idiosyncratic productivity, and it is resolved after
financial decisions are made, so that some firms are not able to repay debt
and go bankrupt.
Accordingly, we build on the standard model of monopolistic competition
with heterogeneous firms developed by Melitz and Ottaviano (2008). In that
model, firms are heterogenous in productivity à la Melitz (2003), that is,
the supply side of the goods market essentially consists of a continuum of
firms that discover their productivity upon the payment of a sunk entry cost.
Firms are monopolists of their own differentiated variety, hence they charge
a markup over marginal cost. Yet they behave competitively in the sense
that they take aggregate variables such as the average price and the number
of competitors as given. On the demand side, the quadratic structure of consumers preferences yields a linear demand system with horizontal product
differentiation, as introduced by Ottaviano et al. (2002). This implies that
there exists a threshold price above which a variety is not consumed. Accordingly, only the firms of which the marginal cost is lower than this threshold
stay in the market and produce, while the others exit. 4
Within this standard setup, we introduce the basic corporate finance decision
whereby firms choose the amount of external financing. It summarizes the
capital structure as the total financing requirement is fixed and amounts to
4
Note that, under the quadratic specification and contrary to the case of constant elasticity of substitution preferences, firm-level performance measures depend on the number
of competitors, which is convenient for our purpose.
4
the entry cost. 5 Specifically, firms can issue bonds on a competitive market
where the equilibrium price is such that risk-neutral agents recover the face
value in expectation. This framework entails a novel outcome from selection
at the firm level. Indeed, for some productivity range, operating income is
positive, although not sufficient to pay off bonds. Somewhat loosely, we label
this situation ‘bankruptcy’, in contrast with the ‘exit’ situation where firms
do not even start producing as their idiosyncratic marginal cost is higher than
the price threshold. In the bankruptcy case, firms stay in the market but
are taken over by bondholders. Therefore there exists a negative relationship
between leverage and the price of bonds, as bankruptcy is more likely for
highly-leveraged firms.
We show that, at a short-run equilibrium where the number of active firms is
fixed, the optimal capital structure depends upon the number of competitors
on the output market in line with the mechanism described above. The
more competitors, the lower expected operating income, hence the higher
the probability of bankruptcy. This implies a relatively low price of bonds,
which penalizes leverage. Next we derive the free-entry equilibrium and show
that, in the long run, a lower degree of product differentiation decreases
the probability of bankruptcy because incumbent firms are on average more
productive. It entails a relatively high price of bonds, which favours leverage.
Note that we force an interior solution to firms financial policy problem by
introducing exogenous transaction costs and tax benefits associated with
leverage. 6
Our paper complements the finance and industrial organization literature by
reconciling the implications of two classes of theoretical models that formal-
5
6
The issue of the impact of investment on the interactions between financial policy and
the output market are beyond the scope of the present paper. On that matter, see the
seminal contribution by Dixit (1980).
These elements entailing a departure from Modigliani and Miller’s (1958) irrelevance
propositions are standard in financial policy models since the seminal theoretical treatment by Kraus and Litzenberger (1973). For recent estimates of bankruptcy costs and
tax benefits, see Elkamhi et al. (2012) and Dwenger and Steiner (2014), respectively.
5
ize the relationship between financial policy and competition on the output
market. One of them rests on Brander and Lewis’ (1986) formulation of the
limited liability effect. They take up the basic idea of Jensen and Meckling
(1976) that, due to limited liability, leverage raises firms incentives to pursue
riskier output market strategies which increase returns in solvent states and
decrease it in case of bankruptcy. Indeed, once debt is issued, shareholders
do not take bankrupt states into account, since captive debt holders become
the residual claimants. Therefore, in an oligopolistic output market, firms
use debt to compete more aggressively, hence leverage is associated with
tough competition. Based on predatory theory, the other class of models
predicts the opposite correlation between the debt level and competition. In
the wake of Fudenberg and Tirole’s (1986) formalization of the long-purse
argument, Bolton and Scharfstein (1990) show how firms in very competitive
markets have an incentive to reduce leverage in order to deter competitors
from preying on them. Against this background, we innovate by abstracting from strategic interactions and focusing on the nature of competition.
Because our model enables us to disentangle the opposite impacts of the
short-run number of firms and the long-run product differentiation, it provides a theoretical justification as to why the correlation can go both ways,
hence why empirical works regarding the impact of competition on leverage
has not been conclusive.
In addition, note that existing works essentially focus on how financial policy
shapes output market outcomes. To the best of our knowledge however, the
reverse causality has received little attention. Still the economic conditions
that firms face arguably condition their external financing opportunities. In
our model the causality explicitly runs from the output market environment
towards financial decision-making at the firm level. Therefore we suggest a
novel determinant of the capital structure. We are not aware of any other
paper that likewise points out competition as a factor determining corporate
financial policy.
The paper is organized as follows. Section 1 describes the output market and
is essentially a reminder of Melitz and Ottaviano’s framework. In section
6
2 we build our model of the financial market and describe the impact of
corporate finance on firm-level selection. Section 3 then looks at the linkage
between competition and financial policy, both in the short run when the
number of competitors matters, and in the longer run when the degree of
product differentiation is decisive. The last section concludes. We relegate
the necessary algebraic details in the appendix.
1
The output market
In this section we sketch out Melitz and Ottaviano’s model of monopolistic
competition between heterogeneous firms. We consider an economy consisting of L agents, each one endowed with one unit of labour.
1.1
Preferences
Agents derive their utility from the consumption of a homogeneous good and
of a continuum of differentiated varieties indexed by j ∈ J. Their preferences
are quasi-linear with a quadratic subutility à la Ottaviano et al. and have
the form
U q0c , qjc = q0c + α
Z

2
Z 2
Z
η
γ

qjc dj −  qjc dj 
qjc dj −
2
2
j∈J
j∈J
(1)
j∈J
where q0c and qjc are the individual consumption of the homogeneous good
and of the differentiated varieties, respectively. The parameters α > 0 and
η > 0 govern the substitution pattern between the differentiated varieties
and the homogeneous good. Characteristically the parameter γ > 0 indexes
the degree of product differentiation between varieties. In the limit case
γ = 0, only the consumption level over all varieties matters as they are
perfect substitutes. As γ rises, varieties become increasingly differentiated
and consumers enjoy a higher utility when spreading consumption across
7
varieties. The degree of product differentiation thus reflects love for variety
and induces a departure from perfect competition.
The maximization of utility under the budget constraint yields consumers
individual inverse demand for any variety j, which we invert to obtain the
linear market demand system
qj = L qjc =
where p̄ ≡
R
j
Lα
L
L ηN
− pj +
p̄
ηN +γ
γ
γ ηN +γ
(2)
pj dj/N is the average price of the differentiated varieties and
N the measure of consumed varieties.
Since the structure of preferences imply that the marginal utility from the
consumption of any variety j is bounded, there exists a choke price above
which the demand is driven to zero. It solves qj (pD ) = 0 in (2) and is given
by
pD ≡
α γ + η N p̄
.
ηN +γ
(3)
Importantly, a lower average price p̄ or a higher number of varieties N reduce
the price choke. Melitz and Ottaviano refer to this as a ‘tougher’ competitive
environment.
We can write the associated indirect utility as
N
(α − p̄)2 N s2p
U (p̄, N ) = I +
+
ηN +γ
2
γ 2
where s2p ≡
R
j
pj − p̄
2
(4)
dj/N is the variance of the prices of the differentiated
varieties and I stands for consumers income. The indirect utility function
decreases in the price index p̄ and exhibits love for variety as it increases in
N.
Finally note that we assume a positive demand for the homogeneous good
R
q0c > 0, which implies I > j pj qjc dj.
8
1.2
Technology
Labour is the only production factor and agents supply it inelastically on a
competitive labour market. Production takes place under constant returns
to scale. We normalize the labour requirement per unit of output in the
homogeneous sector to one and take the homogeneous good as the numeraire.
Accordingly, the wage is unitary.
There is a continuum of firms in the differentiated sector, each producing a
differentiated variety and enjoying market power as a monopolist of its own
variety. Firms are heterogeneous with respect to their marginal production
cost c, equivalent to their labour unit requirement. Therefore we index firms
with their marginal cost from now on.
As in Melitz’s seminal article, there exists an entry barrier as firms only discover their idiosyncratic productivity after making an irreversible investment.
Specifically, they draw their unit labour cost c from a known distribution G(c)
upon the payment of a sunk entry cost fE . For analytical convenience, we
take up the standard assumption that firms productivity 1/c is drawn from
a Pareto distribution of shape parameter κ ≥ 1 and scale parameter 1/cM ,
implying that the unit cost follows the cumulative distribution function
G(c) =
c
cM
κ
(5)
defined on the support [0, cM ]. The shape parameter indexes the dispersion
of firms productivity, with the relative number of high-cost firms increasing
in κ. In the limit case κ = 1, the distribution is uniform, while it degenerates
in cM when κ tends towards infinity. 7
Once firms learn about their productivity, they either select into the market
or leave. To be specific, for the entry cost is sunk and the technology is
7
Combes et al. (2012) show that Melitz and Ottaviano’s core results are robust to not
assuming this specific productivity distribution.
9
linear, firms which are able to cover their marginal cost stay in the industry
and produce, while the others just exit. In line with the classical features
of monopolistic competition, firms ignore their impact on the competitive
environment, as described by the number of varieties N and their average
price p̄, hence precluding strategic interactions. Therefore surviving firms
maximize operating profits taking the residual demand for their variety (2)
as given, and the resulting first-order condition writes
q(c) =
L
p(c) − c .
γ
(6)
Let cD be the idiosyncratic cost level such that p(cD ) = cD , that is, the
marginal production cost which drives demand, hence profits, to zero. From
the demand-side perpective, it corresponds to the choke price pD . Solving
the first order condition for p(c) yields the optimal pricing rule as a function
of the unit cost, which together with (2) gives firms operating profits as
π(c) =
L
(cD − c)2
4γ
(7)
provided it is positive. It is straightforward that only firms with a marginal
cost lower than cD are able to show a positive operating income. Melitz and
Ottaviano refer to this threshold for the idiosyncratic cost level as the cost
cutoff. As expression (3) indicates, it is an inverse index for the number of
firms in the market: the lower cD , the more competitors.
2
The financial market
In this section we model the financial sector in order to introduce the basic
corporate financing choice between debt and equity.
10
2.1
Corporate finance
Although implicit, the conventional understanding of the present type of
model is that firms raise equity capital in order to finance the sunk entry
cost fE . Shareholders are thus the residual claimants, as it gives them the
right to the operating income. The situation is typically thought of as one
where every agent holds an equal share in a mutual fund which owns all the
firms in the industry and which finances entrants as long as it is profitable.
The novel feature of our model consists in introducing an alternative source
of funds for the entry cost. We consider an overly basic corporate financing
framework whereby firms can issue bonds at price pB . In this setting, firms
decide on leverage by choosing the face value of debt 0 ≤ fB ≤ fE /pB . We
assume that the bond market is competitive in the sense that the demand
for bonds is perfectly elastic conditional on fB .
Critically, firms decide on their capital structure given the conditions on the
bond market before they discover their productivity. Subsequently, if they
are efficient enough, they produce and deliver income. Therefore borrowing
induces the risk that operating profits be too low to pay off bonds. In that
case firms file for corporate bankruptcy and are taken over by bondholders.
Moreover, there exist transaction costs associated with bankruptcy and we
assume that they amount to a fraction 0 ≤ δ ≤ 1 of operating profits.
Let cB be the marginal cost draw such that, given its capital structure, the
firm is just able to pay off bondholders. This threshold solves π(cB ) = fB , and
we label it the ‘bankruptcy-limit cost’. Using the expression for idiosyncratic
profits (7) we get
r
cB = cD −
4γ
f B ≤ cD .
L
(8)
In this setting a new outcome emerges for firms which incur the sunk cost.
Like in the standard model, firms with an idiosyncratic cost level above cD
do not start producing. As a new feature however, firms with a marginal cost
comprised between cB and cD show positive operating income, although it is
11
too low to cover debt expenses. Therefore these firms go bankrupt and are
taken over by bond holders, but still produce. In the case where the marginal
cost is below cB , firms are productive enough to pay off bondholders, and
they pay out the residual profits to shareholders.
It is noteworthy that the bankruptcy-limit cost cB is decreasing in the outstanding debt fB , since operating profits are more likely to be used up by
debt expenses when leverage is high. Therefore we often use it as an index
of firms financial structure in what follows.
We sum up the outcome from selection at the firm level in the following
proposition.
Proposition 1 (Financial selection under firm-level heterogeneity). Consider a monopolistically competitive output market and heterogeneous entrants which discover their idiosyncratic productivity level c after they decide
on their capital structure fB . Then
– firms with marginal cost c > cD do not start producing and exit;
– firms with marginal cost cB < c < cD produce but go bankrupt and are
taken over by bondholders;
– firms with marginal cost c < cB produce, pay off bondholders and pay
out the residual profits to shareholders.
The bankruptcy-limit cost cB is given by (8) and decreases in leverage fB .
Note that we assume cM to be high enough for some firms to exit.
2.2
Price of debt
The participation constraint on the bond market governs the formation of
the equilibrium bond price pB . We assume that agents are risk neutral, and
therefore the market price is such that investors recover the face value in expectation. Given the competitive conditions cD , the participation constraint
12
writes
ZcB
pB =
ZcD
1−δ
1 dG(c) +
fB
π(c) dG(c) .
0
(9)
cB
As the expression above indicates, bondholders receive the face value when
firms generate enough income to pay off debt, whereas they only receive their
fair share 1/fB of profits net of transaction costs δ when firms go bankrupt,
and nothing at all in case of exit. Remember that the equilibrium only
supports cB ≤ cD , as fB < 0 otherwise.
Given the definition of profits (7) and the bankruptcy-limit cost (8) it is
straightforward that
∂pB
> 0 and
∂cD
∂pB
<0
∂δ
(10)
which simply comes from the fact that more competitors and higher transaction costs both negatively affect the expected return from holding debt,
hence lower the equilibrium price of bonds. Indeed, everything else equal,
a higher number of incumbent firms makes bankruptcy more likely, while
transaction costs reduce the value of the collateral. Moreover, we show in
appendix A1 that
∂pB
> 0,
∂cB
(11)
that is, our model features a negative relationship between the bond price
and leverage. The intuition is similar: everything else equal, highly leveraged
firms are less likely to generate enough income to meet debt expenses, which
drives down the equilibrium price of bonds. We write these results in the
next proposition.
Proposition 2 (Financial market equilibrium). Consider a monopolistically
competitive output market with heterogeneous firms, and a bond market where
demand is perfectly elastic conditional on leverage and risk neutrality prevails.
13
The equilibrium price of bonds pB
– increases in the cost cutoff cD ;
– increases in the bankruptcy-limit cost cB ;
– decreases in the transaction costs δ.
In the next section we study how price formation on the financial market
interacts with the output market equilibrium.
3
The leverage–competition linkage
This section looks at the relationship between the two markets described
above and contains the crux of our analysis. Specifically, it shows how financial policy depends on the competitive conditions on the output market.
3.1
A short-run equilibrium
We begin by studying the industry equilibrium when the number of firms is
fixed at N̄ , as acceptable in the short run. Like Melitz and Ottaviano, we
assume that the distribution of incumbents productivity Ḡ(c) also follows a
Pareto law on the support [0, c̄M ] during the period under consideration. One
can think of the situation as one where incumbent firms are re-allocated an
κ
idiosyncratic productivity level drawn from c/c̄M without incurring the
sunk cost again. Only firms earning non-negative operating profits produce,
and this determines the cost cutoff cD . Within this time frame we consider
the entry of an incremental firm.
Below we derive this short-run equilibrium, which is fully characterized by
three conditions in pB , cB and cD .
Firms shareholders decide on leverage fB given the equilibrium conditions
on the bond market. Because they are risk-neutral, they choose leverage
with the objective of maximizing the expected equity value V . In our basic
14
framework, this is equivalent to maximizing the net cash flow, that is, the
difference between the sources and uses of funds. Hence the shareholders
program writes
ZcB
max V (fB ) = pB fB − fE + (1 − τ ) π(c) − fB dḠ(c)
fB
(12)
0
subject to the equilibrium price of bonds (9) and where the upper bound of
the integral is the bankruptcy-limit cost (8). Importantly, τ is the rate of
corporate income taxation against which the debt payments fB are written
off the tax base. The examination of the objective function reveals the typical
tradeoff implied by an increment in leverage. One one hand, it increases the
value to shareholders, for it shields more income against corporate taxation.
On the other, it expands the range of idiosyncratic marginal costs over which
profits are too low to meet debt expenses, hence increases the cost of debt. 8
The associated first-order condition suggests another interpretation. Taking
the first derivative of the objective function (12) with respect to fB using
Leibniz’s rule for differentiation under the integral sign and rearranging, we
obtain
1
Ḡ(cB ) =
1−τ
∂pB
pB +
fB
∂fB
!
.
(13)
In this expression the lefthand side indicates the cost of issuing an additional
bond, which is simply the (unitary) face value times the probability of paying
it off. The righthand side corresponds to the benefit, which is equal to the
price of debt discounted at the tax shield rate. Note that the price decreases
in leverage, as stated in proposition 2.
It is worth stressing that our framework does not rely on agency, but well
8
Note that, since shareholders are risk-neutral, the cost of leverage comes from the lower
equilibrium price of bonds and not from the increased bankruptcy risk.
15
on the endogenous equilibrium formation on the bond market. In a standard agency conflict between shareholders and debtholders, the increased
probability of bankruptcy due to leverage is irrelevant to shareholders since
debtholders become the residual claimants. Therefore leverage amounts to
expropriating bondholders. However, this holds true only for a perfectly elastic bonds supply. As opposed to that, in our model, the equilibrium price on
the bonds market decreases in leverage, leaving the expected return from
holding bonds unaffected. Raising leverage essentially increases the cost of
debt to shareholders. 9
For analytical convenience we derive the first-order condition with respect
to the bankruptcy-limit cost cB rather than leverage fB , which is equivalent
given the relationship (8). Remember that cB ≤ cD by construction. We
show in appendix A2 that the only maximum associated with the first-order
condition (12) is
cB =
κδ
cD ,
2τ + κδ
(14)
which, using (8), implies the optimal capital structure
L
fB =
γ
τ
2τ + κδ
2
c2D .
(15)
This expression for the value-maximizing capital structure constitutes the
core of our contribution. Remember that the cost cutoff cD is an inverse
index for the number of firms in the market. Formally, on the demand side,
the choke price (3) determines the number of varieties –equivalently, the
number of surviving firms– as
N=
9
2(1 + κ)γ α − cD
η
cD
(16)
It also increases the variance of the equity value, but by definition it does not affect the
utility of risk-neutral agents. See also footnote 8.
16
where we substituted p̄ with its expression under the Pareto distribution asκ
sumption. Together with the fixed number of incumbents N̄ = N c̄M /cD ,
this zero cutoff profits condition identifies the cost cutoff and the number
of active firms in the short-run. It is straightforward that a lower cutoff is
associated with more competitors in the market. Therefore we can write the
following proposition.
Proposition 3 (Leverage and the number of firms). In the short run, there
exists a negative correlation between leverage and cyclical competition as measured by the number of incumbents in the market.
This essential result comes from the interaction between the output market
and the bond market. Everything else equal, fewer competitors on the output
market imply higher expected operating profits, as firms face higher demand
and are able to charge higher markups. This raises the expected return to
bondholders, which increases the equilibrium price of bonds. Consequently,
the incremental firm resorts to debt to fund the entry cost as it is relatively
cheap. On the contrary, when there are a lot of incumbents, expected operating profits are low, which makes bankruptcy more likely, hence putting an
upwards pressure on the cost of leverage by decreasing the equilibrium price
on the bond market.
Before we proceed to the longer-run analysis, note how the optimal financial
policy corresponds to a corner solution for the firms programme in the absence of taxation or transaction costs. Trivially, firms do not sell debt when
the corporate tax system does not provide any tax shield for debt (fB = 0
for τ = 0), while they only resort to debt when there is no transaction costs
(fB = fE when δ = 0) . Note that Modigliani and Miller’s irrelevance result
is restored when both transaction costs and taxation are absent (δ = τ = 0):
when leverage bears no cost nor benefit, financial policy is irrelevant.
17
3.2
The long-run equilibrium
The short-run equilibrium we derive above is characterized by the zero-profits
cutoff (3), the participation constraint on the bond market (9) and the optimal leverage (13).
However, this short-term equilibrium concept implies that firm-level performance variables, such as operating profits π(c), are not constant as the entry
of an incremental firm affect the residual demand q(c) which incumbents face.
Put differently, firms cannot form perfect expectations about all industry aggregates, notably the average price p̄. An important corollary is that, in the
short-run, the expected return on debt is not equal to the expected return
on equity because firms make pure profits at the aggregate level.
In what follows we consider the longer term, where the unrestricted entry
of firms eliminates pure profits, implying that the competitive conditions on
the output market cD get to their long-run level and the return on funds is
equalized across markets. For this purpose we introduce a free entry condition, which constitutes a fourth equilibrium equation and determines the
long-run equilibrium number of firms N .
In a free-entry equilibrium, the expected profits from entering the market is
nil, which determines the cost cutoff cD . The free entry condition writes
(1 − τ )
ZcB
π(c) − fB dG(c) = f − pB fB ,
(17)
0
and simply states that expected operating profits net of debt expenses and
taxes just cover the equity investment.
In appendix A3 we show that, together with the equilibrium price of bond
(9) and the optimal leverage rule (14), the free entry condition renders the
18
long-run cost cutoff as a function of parameters only as
cD =
2γ
(2τ + κδ)1+κ (1 + κ)(2 + κ)cκM fE
L (1 − δ)(2τ + κδ)1+κ + δ(κδ)κ (2τ + κδ + κτ )
1
! 2+κ
.
(18)
Together with the zero cutoff profits condition (16), it closes the model and
determines the long-run number of firms N .
Note that this expression is isomorphic to the one in Melitz and Ottaviano
when either δ = 0 or τ = 0. In particular, the long-run cutoff increases in
product differentiation γ, as it shelters low-productivity firms from cannibalization by more productive competitors.
We can now turn to the capital structure. Remember that, in the short-run,
optimal leverage depends on the number of incumbents. At the long-run
equilibrium number of firms described by the free-entry cost cutoff (18), the
optimal leverage rule (15) gives the equilibrium leverage of the marginal
entrant as
2
2+κ

κ 2
2+κ


τ
2κ(1 + κ)(2 + κ)cκM fE
L

fB =
1+κ


γ
2τ + κδ
(1 − δ)κ+(2τ + κδ + κτ ) 2τκδ
+κδ
(19)
and our last proposition follows straightforwardly.
Proposition 4 (Leverage and product differentiation). In the long run, there
exists a positive correlation between leverage and structural competition as
inversely measured by the degree of product differentiation.
Observe how this result comes from the probability of bankruptcy. First note
that, under the Pareto-distributed productivity assumption, the density of
marginal costs is strictly increasing and convex on [0, cM ], implying that
the probability of bankruptcy monotonically increases in cD − cB . Next
19
rearranging expression (8) as
r
cD − cB =
4γ
fB
L
(20)
indicates that bankruptcy is more likely when product differentiation is high.
The intuition is the following. When varieties are very differentiated, the
industry accommodates low-productivity competitors, hence average profits
are low. 10 As we stress above, this reduces the price of bonds and penalizes
leverage.
Conclusion
This paper has integrated a basic corporate finance framework in an otherwise
standard model of monopolistic competition with heterogeneous firms. In
that setting we have investigated the interactions between corporate financial
policy and the competitive conditions on the output market.
We have argued competition is not one-dimensional, and that the direction
of the correlation between leverage and competition depends on the nature
of the latter. Our model has enabled us to disentangle a cyclical dimension of
competition, related to the number of incumbents in the market, and a structural dimension, determined by the degree of product differentiation. In the
short-run, more competitors imply less leverage, whereas in the longer-run,
less differentiation means more leverage. We have also described the mechanism at work, which goes through the equilibrium price on the corporate
bond market. Because it makes bankruptcy more likely, more competitors
and more differentiation both increase the cost of debt, hence decrease firms
optimal leverage.
10
The negative correlation between product differentiation and average industry profits is typical of monopolistically competitive heterogeneous firms models, both in the
standard Melitz model and in the present framework. Intuitively, more differentiation
entails more firms in the market for a constant revenue/expenditure.
20
Our contribution has shed new light on the seemingly inconsistent results
reported by the vast body of empirical work that looks at the competitionleverage relationship. Moreover, from a theoretical perspective, we have put
forward new grounds for understanding the interactions between financial
and real economic decisions. In particular, contrary to existing work in financial and industrial economics, we have emphasized how the causality can
go from to output market to the financial sphere. In doing so, we have suggested that the competitive conditions on the output market in themselves
may be a determinant of the capital structure.
We do not claim that this model is elaborate enough to describe the complex interactions between competition and financial decisions. Nevertheless,
we believe that it might help analyze some topical economic policy issues.
In particular, we think of the public finance aspect of the tax treatment of
debt. Another intuitive direction towards which the model could be extended
is related with regional economics and trade. In particular, capital tax competition could be analyzed taking into account tax planning based on the
difference between debt and equity income. The present analysis suggests
that competition can either alleviate or in the contrary exacerbate the tax
advantage of debt shifting.
More generally our analysis suggests a promising way to read again an important body of research at the crossroads of corporate finance finance and
industrial organization.
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23
Appendices
Below are the developments necessary to obtain the results in the text. The
first appendix provides comparative statics regarding the equilibrium on the
financial market, the next one deals with the shareholders program, and the
last one tackles the long-run equilibrium.
A1
Comparative statics for the price of debt
We show that the equilibrium price of bonds decreases in leverage. To do
so, we substitute the expression for profits (7) and the definition of the
bankruptcy-limit cost (8) in the bond market equilibrium equation to obtain
ZcB
ZcD 1 dḠ(c) + (1 − δ)
pB =
0
cD − c
cD − cB
2
dḠ(c) ,
(A1)
cB
where the argument in the first integral is bigger than the one in the second
integral by construction. Since the definition of the bankruptcy-limit cost
implies ∂cB/∂fB < 0, an increment in leverage decreases the range of the first
integral and increases the range of the second, which gives the result (11).
A2
Firms financial program
We maximize the objective function (12) with respect to the debt-limiting
cost cB rather than leverage fB . This is equivalent except for the sign given
that the relationship between the two variables is monotonically decreasing,
as expression (8) indicates. We start by substituting the price equilibrium
24
on the financial market (9) in the objective function (12) as
V fB (cB ), cB
ZcD
ZcB
= π(c) dG(c) + τ fB 1 dG(c)
0
0
ZcB
ZcB
− τ π(c) dG(c) − δ π(c) dG(c) − fE .
0
(A2)
cD
Next using definition (8) under the form fB = L(cD − cB )2 /4γ and solving
under Pareto renders the maximand
V (cB ) =
1
L
4γ (1 + κ)(2 + κ)cκM
τ (1 + κ)(2 + κ)(cD − cB )2 cκB
+(1 + κ)(2 + κ)c2D (1 − δ)ckD − (τ − δ)cκB
1+κ
−2cD κ(2 + κ) (1 − δ)c1+κ
D − (τ − δ)cB
2+κ
+κ(1 + κ) (1 − δ)c2+κ
−
(τ
−
δ)c
D
B
4γ
− (1 + κ)(2 + κ)cκM fE
L
(A3)
of which we take the first derivative with respect to cB to obtain the firstorder condition
(2 + κ)(1 + κ) cκ−1
B (cD − cB ) κδ(cD − cB ) − 2τ cB = 0.
(A4)
This first-order condition for firms financial programme cancels out for three
values of the debt-limiting cost. Either (i) cB = 0, which does not satisfy the
associated second-order condition κδ(cD − cB ) − 2τ cB < 0. Or (ii) cB = cD ,
which does not satisfy the associated second-order condition κδ(cD − cB ) −
2τ cB > 0 either. Or finally (iii) κδ(cD −cB )−2τ cB = 0, and the second-order
condition −κδ − 2τ < 0 is always satisfied provided τ > 0 or δ > 0. This
gives the result (14).
Further, observe that when τ = 0, then cB = cD , and fB = 0 from the
25
definition (8). Note this is an interior solution: the first-order condition
2
writes (2 + κ)(1 + κ)c1+κ
B (cD − cB ) κδ = 0 and the second-order condition
(cB )k−1 2 (cD − cB ) (−1) < 0 is satisfied.
Likewise, when δ = 0, then cB = 0, and fB = c2D L/4γ. However firms
cannot borrow more than fE < c2D L/4γ by construction. Note this is an
interior solution as well: the fist-order condition is (2 + κ)(1 + κ)c1+κ
B (cD −
cB )(−2τ cB ) = 0 and the second-order one c1+κ
B (cD −cB )(−2τ ) < 0 is satisfied.
Finally, for τ = δ = 0, the first-order condition is indeterminate. This is
reminiscent of the irrelevance result: V is independent of cB .
A3
Free-entry equilibrium
Free entry simply implies that V = 0. Evaluating the maximand (A3) at the
optimal capital structure rule (14) yields the long-run cost cutoff (18).
26