Debt-Ridden Borrowers and Productivity Slowdown(Incomplete and preliminary)Keiichiro Kobayashi∗ and Daichi Shirai†
May 2012 (First version: March, 2012)

Abstract
Many authors argue that ﬁnancial constraints have been tightened since the greatrecession in 2007–2009. To explain this, we construct a model in which borrowing
constraints for ﬁrms are tightened as a result of mass default due to a bubble collapse.
In Jermann and Quadrini (2012)’s model, a defaulted ﬁrm either goes back to a normal
ﬁrm by (partial) repayment of the debt or is liquidated. We assume that there is anintermediate status: a “debt-ridden” ﬁrm, which is deﬁned as a ﬁrm whose lender
retains the right to liquidate them. The lender allows the debt-ridden ﬁrm to continue
if it pays continuation fee. In our model debt forgiveness is infeasible: once a ﬁrm
defaults on the debt, it is either liquidated or kept as a debt-ridden ﬁrm. The defaulter
can never go back to a normal ﬁrm unless it repays all the unpaid debt. Prohibitionof debt forgiveness can be justiﬁed as a collective choice of the society to expand theborrowing limit of normal ﬁrms.
It is shown that borrowing constraints are tighter for debt-ridden ﬁrms than for
normal ﬁrms. This implies that emergence of a large mass of debt-ridden borrowers
may be a cause of the “ﬁnancial shocks” in the recent macroeconomic literature.
Tightened borrowing constraints due to emergence of debt-ridden ﬁrms lowers theaggregate productivity. The negative eﬀect on productivity can be permanent. In a
version of the model with endogenous growth the growth rate of aggregate productivity
becomes zero if the number of debt-ridden ﬁrms exceeds a certain threshold.
Keywords: Borrowing constraint, working capital, ﬁnancial shocks, productivity
slowdown.
JEL Classiﬁcation numbers: E30, G01, O40.

∗

Institute of Economic Research, Hitotsubashi University; CIGS; and RIETI. Email: kcrkbys@ier.hit-

u.ac.jp†

The Canon Institute for Global Studies. Email: shirai.daichi@gmail.com

1

1

Introduction

The decade after a ﬁnancial crisis tends to be associated with low economic growth (Reinhart and Rogoﬀ 2009, Reinhart and Reinhart 2010). It is also known that the growthof total factor productivity slows down or even becomes negative for a decade (Kehoeand Prescott 2007). Why productivity growth slows down persistently after a ﬁnancial
crisis? Many authors argue that ﬁnancial constraints have been tightened during/after
the great recession in 2007–2009. Why can ﬁnancial constraints be tightened due to aﬁnancial crisis or a collapse of asset-price bubble? Can tightening of ﬁnancial constraints
cause a persistent slowdown of economic growth? These are the question we consider
in this paper. In this paper we propose a theoretical model in which the emergence ofdebt-ridden borrowers lowers the aggregate productivity persistently through tightening
ﬁnancial constraints.
We construct a general equilibrium model, in which an exogenous shock makes substantial number of ﬁrms default on their debt. The model is based on Jermann andQuadrini’s (2006, 2012) models, which explicitly consider the bargaining after default toderive a borrowing constraint à la Kiyotaki and Moore (1997). In Jermann and Quadrini,ﬁrms’ ability to borrow is bounded by the limited enforceability of debt contract. The
borrowing ﬁrm can default on the debt obligation and renegotiate on the repayment. The
borrowing is limited such that the amount borrowed is renegotiation-proof. In the imaginary renegotiation in Jermann and Quadrini, the defaulted ﬁrm can go back to a normal
ﬁrm if the lender and the ﬁrm agree on repayment, or the ﬁrm is liquidated if the renegotiation breaks down. Thus there are two status for a ﬁrm: being a normal ﬁrm or being
liquidated.
A novel feature of our model is that we assume that there is an intermediate status ofa ﬁrm: being a “debt-ridden” ﬁrm. We deﬁne a debt-ridden ﬁrm as a ﬁrm whose lender
retains the right to liquidate it. The lender allows the debt-ridden ﬁrm to continue its
operation if she agrees the amount of continuation fee to be paid by the ﬁrm. We assume
that once a ﬁrm defaults on its debt, it can never go back to normal unless it pays all theoriginal debt. The defaulted ﬁrm is either liquidated or kept as a debt-ridden ﬁrm.
We analyze the borrowing constraint for the debt-ridden ﬁrms and show that they facetighter borrowing constraint when they borrow working capital loans than normal ﬁrms
do. This result seems counterintuitive since debt-ridden ﬁrms are under the control of
their lenders, while the normal ﬁrms are not. The reason for this counterintuitive result
is that a normal ﬁrm loses more when it defaults on its debt than a debt-ridden ﬁrm do.
In our model, if a normal ﬁrm defaults on its debt, it inevitably becomes a debt-ridden
ﬁrm unless it repays all the original debt. If a normal ﬁrm defaults it loses the status ofa normal ﬁrm, while if a debt-ridden ﬁrm defaults it continues as a debt-ridden ﬁrm after2

renegotiation. Thus a normal ﬁrm loses more by defaulting than a debt-ridden ﬁrm do.
Because of this fact the incentive-compatibility condition (or no-default condition) implies
that a normal ﬁrm can borrow more than a debt-ridden ﬁrm can.
Tighter borrowing constraint for working capital loans of debt-ridden ﬁrms makes theirproduction activity ineﬃcient. If a substantial number of ﬁrms become debt-ridden, boththe aggregate borrowing capacity and the aggregate productivity decline. This implies thatemergence of debt-ridden borrowers may be a cause of the “ﬁnancial shocks” in the recent
macroeconomic literature. After the great recession in 2007–2009, many authors argue thata shock in the ﬁnancial sector can cause a severe recession (e.g., a risk shock in Christiano,
Motto, and Rostagno 2009, and a ﬁnancial shock in Jermann and Quadrini 2012). In
our model, emergence of a substantial number of debt-ridden ﬁrms manifests itself astightening of aggregate borrowing constraint, which can be interpreted as a ﬁnancial shock.
We also show that emergence of debt-ridden ﬁrms has a persistent negative eﬀect on
productivity. The higher ineﬃciency due to tighter borrowing constraints lowers the valueof new entry to the market for a potential entrant, and discourages the R&D activity. The
decrease in R&D activity leads the economy into a steady state with low productivity. We
consider a modiﬁed version of the model where the economy grows endogenously, and show
in a numerical example that the growth rate of aggregate productivity may become zero
permanently if the number of debt-ridden ﬁrms exceeds a certain threshold. This result
seems consistent with the facts that we observe in persistent recessions after ﬁnancial crises(see Section 2).
Related literature:

Our theory is related to the literature on debt overhang. Myers(1977) pointed out the suboptimality of debt in the corporate ﬁnance literature and Lamont (1995) applied the notion of debt overhang in macroeconomics.1 Debt overhang
problem is typically that a ﬁrm cannot borrow new money for a productive projects when
it has a too large amount of existing debt. The debt overhang arises if the existing debt
holder is diﬀerent from the potential lender who would lend new money. In this paper wetake a small step forward by proposing a new theory that an ineﬃciency can arise even
if the lender of new money is the existing debt holder. This paper is also close to Caballero, Hoshi, and Kashyap (2008). They analyze the “zombie lending” which is deﬁned
as a de facto subsidy to unproductive ﬁrms from the banks. They argue that congestingzombie ﬁrms hinder entry of highly productive ﬁrms and lower the aggregate productivity.
In this paper, we make a complementary point to their argument: even an intrinsically
productive ﬁrm can become unproductive when it becomes debt-ridden. This is because
a debt-ridden ﬁrm is subject to tighter borrowing constraint for working capital loans.
1

See also Krugman (1988) on debt overhang in the international ﬁnance. See Moyen (2007) and Chen

and Manso (2010), for example.

3

This paper is organized as follows. In the next section we review the facts on persistentrecessions after ﬁnancial crises. Section 3 proposes and analyzes the model. We modify
the model to an endogenous growth model in Section 3.4. Section 4 concludes.

2

Facts on persistent recessions after ﬁnancial crises

There observed numerous examples of productivity slowdown after a ﬁnancial crisis. The
most notable is the Great Depression in the 1930s in the US and the similar depressions
in that period in the major nations. Ohanian (2001) shows that there was a large productivity decline during the US Great Depression, which is unexplained by capital utilizationor labor hoarding. Kehoe and Prescott (2007) drew our attention to the fact that many
countries experienced a decade-long severe recessions, which they call the “great depressions” in the twentieth century. Papers in their book unanimously emphasize that the
declines in the growth rate of total factor productivity was the primal cause of the great
depressions.
Another example of a decade-long recession after a ﬁnancial crisis is the 1990s in Japan.
The growth rates of the gross domestic product (GDP) and the total factor productivity(TFP) in the 1990s are both lower than in the 1980s. Figure 1 shows the GDP along withthe potential capacity, which has an apparent kink at the beginning of the 1990s, whenhuge asset-price bubbles burst in the stock market and the real estate market. See Figure 2
for asset prices in Japan in the 1990s. Table 1 shows various estimates of TFP growth ratein Japan. Hayashi and Prescott (2002) emphasize that the growth of TFP slowed downin the 1990s in Japan.2 One notable feature in the 1990s is signiﬁcant decrease in entriesof new ﬁrms and increase in exits. See Figure 3 for comparison of entry and exit of ﬁrms
between Japan and the US. In the literature, the procyclicality of net entry is well known(Bilbiie, Ghironi, and Melitz 2007). Net entry also contributes signiﬁcantly to TFP growthfor US manufacturing establishments (Bartelsman and Doms 2000). Nishimura, Nakajima
and Kiyota (2005) argue that malfunctioning of entry and exit contributes substantially toa fall in Japan’s TFP in the late 1990s. Another characteristic in the 1990s in Japan was
the persistently lingering nonperforming loans (NPLs) in the banking sector. The NPLs
were the excess debt for nonﬁnancial ﬁrms in mainly real-estate, wholesale, retail, andconstruction sectors. Figure 4 shows development in the amount of the NPLs in Japanin the 1990s and the 2000s. The delayed disposal of the huge NPLs was seen as de facto
subsidy to nonviable ﬁrms (“zombie lending”). The zombie lending has been named as

2

There are substantial debate on whether the TFP slowdown in Japan is truly a slowdown of technicalprogress or just a measurement error (see Kawamoto 2005, Fukao and Miyagawa 2008). Tentative conclusion on this issue in the literature is that there was a slowdown in technical progress in Japan, though it
may not be so severe as Hayashi and Prescott originally measure.

4

the cause of Japan’s persistent recession (Peek and Rosengren 2005, and Caballero, Hoshi,
and Kashyap 2008).

3

Model

We consider a closed economy in which the ﬁnal good is produced competitively fromlabor input and varieties of intermediate goods. The ﬁrms are monopolistic competitors
and they produce respective varieties of intermediate goods from material input, which is
the ﬁnal good, and capital input. In our model, when a ﬁrm defaults on the debt, thelender cannot forgive debt. Instead, the lender can choose whether to liquidate the ﬁrm
or to allow it to continue operation as a “debt-ridden ﬁrm.” In this paper a debt-ridden
ﬁrm is a ﬁrm whose lender retains a unilateral discretion to liquidate the ﬁrm. Later
we clarify the diﬀerence between normal ﬁrms and debt-ridden ﬁrms by formally deﬁning
respective optimization problems they solve. The model is a version of expanding variety
model in which new entry of ﬁrms increases the aggregate productivity. The expanding
variety model is proposed by Rivera-Batiz and Romer (1991) and simpliﬁed by Acemoglu(2009). We basically follow Acemoglu’s exposition. In Sections 3.1–3.3, we analyze themodel in which the economy converges to a steady state. In Section 3.4, we introduce a
positive externality from the variety of goods to the aggregate productivity, which enables
endogenous growth. We show in Section 3.4 that the growth rate falls to zero if there
emerge suﬃciently many debt-ridden ﬁrms.

3.1

Basic setup

There are ﬁxed supplies of capital K and labor L. The labor is used for production of the
ﬁnal good, and the capital is used for production of the intermediate goods. There is amass of ﬁrms indexed by i ∈ [0, Nt ], which produce intermediate goods, where Nt is themeasure of varieties of the intermediate goods in period t. The ﬁrms are either normal
ﬁrms or debt-ridden ﬁrms, while the status of being normal and debt-ridden are clariﬁed
later. A representative household owns these ﬁrms and solve the following program:
[∞
]
∑
max
E0
β t ln Ct ,Ct ,It ,bt+1

subject to

t=0

bt+1
Ct +
+ It ≤ wt L +1 + rt∫ Nt
πit di + ξt bt + dt ,0

Nt+1 = (1 − δ)Nt + χIt ,It ≥ 0.
where β is the subjective discount factor, Ct is consumption, L is the ﬁxed amount oflabor supply, It is the R&D investment, wt is wage rate, rt is the market interest rate, bt
5

is bond issued by the normal ﬁrms, dt is payment by the debt-ridden ﬁrms, and πit is the
proﬁt from ﬁrm i, where i ∈ [0, Nt ]. The bond is risky debt and ξt is the recovery rateof the bond, where ξt = 1 if no ﬁrm defaults and ξt < 1 if some ﬁrms default. The costof R&D investment is measured in the unit of the ﬁnal good and one unit of the R&D
investment creates χ units of new variety of the intermediate goods. The δ fraction of thevarieties depreciates every period. The ﬁrst-order conditions (FOCs) with respect to It
and Nt+1 imply that the R&D investment takes place if χVnt = 1 and it does not takeplace if χVnt < 1, where Vnt is the value of a normal ﬁrm. The cost of the R&D investment
is unity and the expected gain from the R&D is χVnt . The amount of the R&D investment
is determined such that the cost and the gain are equialized: χVnt = 1.
The ﬁnal good is produced competitively from the intermediate goods xit , i ∈ [0, Nt ]and the labor by the following production function:(∫ N t)1
η
xit di L1−η ,
Yt =
η
0
∫N
where 0 < η < 1. Since the ﬁnal good producer maximizes Yt − 0 t pit xit di − wt Lt , where
pit is the real price of the intermediate good i, perfect competition in the ﬁnal good market
implies thatη−1
pit = p(xit ) = L1−η xit,

wt =1 − η Yt.
η L

Firm i produces the intermediate good i from capital input kit and the material input
mit , where mit is the ﬁnal good, by the following production function:α 1−αxit = Ait kit
mit ,where Ait is the productivity parameter. For simplicity of the analysis, we assume thatthe productivity is identical across ﬁrms and constant over time:
Ait = Afor all i and t.

Firm i needs to buy kit at the price of qt−1 in period t − 1 and buys mit at the price ofunity in period t. We assume for simplicity that on one hand there is no ﬁnancial friction
in buying and selling the physical capital kit in the market. Thus ﬁrm i can pay any
amount of qt−1 kit . On the other hand, ﬁrm i needs to borrow working capital mit fromthe representative household to buy the material input and this debt is subject to theﬁnancial constraint that we specify below. Firm i borrows both inter-temporal debt b′it
and intra-temporal debt (working capital) mit . The inter-temporal debt contract is suchb′that ﬁrm i borrows 1+ritt−1 at the end of period t − 1 where rt−1 is the market rate. The

6

sole shock in this economy is the idiosyncratic redistribution shock ∆it that changes theamount of the inter-temporal debt to be repaid in period t. We deﬁne bit bybit ≡ b′it + ∆it ,which is the amount that ﬁrm i must repay in period t to the lender of the inter-temporal
debt. All agents know the probability distribution function of ∆it and they take theexpectation over the distribution of ∆it when they decide their actions.
Timing of actions of ﬁrm i is as follows. At the end of period t − 1 ﬁrm i borrowsb′it
1+rt−1and purchases capital stock kit at price qt−1 . At the beginning of period t theredistribution shock ∆it is revealed and the inter-temporal debt becomes bit = b′it + ∆it ,then the ﬁrm borrows working capital mit , purchases the material input, and produces
xit . After selling xit it repays debt mit + bit and sells kit at price qt . Note that there is
no stochastic shock during the short span when the ﬁrm borrows the working capital. So
the gross interest rate for the intra-period working capital is 1.
Now we specify the borrowing constraint for ﬁrm i. In what follows we omit thesubscript i unless there is a risk of confusion. It will be shown that the ﬁrm’s debt issubject to the borrowing constraint:
mt + bt ≤ ϕqt kt + Vnt − Vzt ,(1)

where Vnt is the value of a normal ﬁrm and Vzt is the value of a debt-ridden ﬁrm, while later
we will specify Vnt by (2) and Vzt by (9). It will be shown that Vnt > Vzt in equilibrium.
The borrowing limit is derived from the limited enforceability of debt contracts as ﬁrms
can default on their debt obligations. The basic logic is the same as Jermann and Quadrini(2012). The decision of default arises after the realization of revenues but before repaying
the inter-period and intra-period loans. At this stage the total liabilities are mt + bt , thatis, the intra-period loan plus the inter-period debt due in period t. As in the JermannQuadrini model, the lender and the ﬁrm renegotiate on repayment if the ﬁrm defaults, andif the renegotiation breaks down and the ﬁrm is liquidated at this point in time the lenderobtains (ϕ + ψ)qt kt , by conﬁscating a part of capital stock (kt ), where 0 < ϕ + ψ < 1. A
departure from the Jermann-Quadrini model is the following assumption:
Assumption 1. The institutional environment of this economy is such that a lendercannot forgive debt of the borrowing ﬁrm if it defaults on the debt. Instead the lendereither liquidates the defaulted ﬁrm immediately or allows it to continue operation as adebt-ridden ﬁrm. The debt-ridden ﬁrm is a ﬁrm whose lender retains the right to liquidate
it and the lender allows it to continue operation if she agrees on the continuation fee thatit pays to the lender. The lender and the debt-ridden ﬁrm negotiate at most twice in eachperiod. The ﬁrst negotiation arises if the ﬁrm defaults on the debt obligation in the middleof the period. In this negotiation the lender and the ﬁrm negotiate on the repayment f ,7

while the lender obtains (ϕ + ψ)qt kt if she liquidates the ﬁrm at this stage. The second
negotiation, which arises at the end of the period, is over the continuation fee dt+1 thatthe ﬁrm should pay in the next period. If the lender liquidates the ﬁrm in the second
negotiation, she obtains only ψqt kt .
The diﬀerence of the liquidation value of the ﬁrm in the ﬁrst and the second negotiation
can be justiﬁed as follows: the ﬁrst negotiation takes place in the middle of the period
and the second negotiation takes place at the end of the period; and the ﬁrm can conceal
or sell oﬀ a part of capital (ϕkt ) during the period between the two negotiations. Given
this institutional environment, a normal ﬁrm inevitably becomes a debt-ridden ﬁrm once it
defaults on the debt. The defaulted ﬁrm loses Vnt −Vzt inevitably. Appendix A shows thatthe defaulted ﬁrm and the lender renegotiate on repayment and agree that the defaulted
ﬁrm repays ϕqt kt in period t. The incentive-compatibility constraint for a normal ﬁrm
not to default is that the original debt (mt + bt ) is no greater than the value it loses bydefaulting (ϕqt kt + Vnt − Vzt ). Thus a normal ﬁrm solves the following Bellman equation:[}]
λt+1 {b′− qt k + Et max β
p(x)x − m − b̃ + qt+1 k + Ṽt+1 ,(2)
Vnt = maxmk,b′ 1 + rt
λt
subject to

x = At+1 k α m1−α ,(3)b = b′ + ∆,(4)m ≤ max{0, ϕqt+1 k + Vnt+1 − Vzt+1 − b},(5)b̃ = b and Ṽt+1 = Vnt+1

if b ≤ ϕqt+1 k + Vnt+1 − Vzt+1 ,b̃ = ϕqt kt and Ṽt+1 = Vzt+1

if b > ϕqt+1 k + Vnt+1 − Vzt+1 ,

(6)
(7)

where λt is the Lagrange multiplier of the budget constraint in the household’s problem
and b̃ is the realized repayment of the inter-temporal debt b. Constraints (5) says that
if the realization of ∆ is such that ϕqt+1 k + Vnt+1 − Vzt+1 − b < 0, the ﬁrm sets m = 0.
Since Appendix A shows that the repayment after default is ϕqt kt , it is easily shownthat the payoﬀ of no default (qt kt + Vnt+1 − b) is strictly less than that of defaulting((1 − ϕ)qt kt + Vzt+1 ). Therefore, the ﬁrm defaults on b if ϕqt+1 k + Vnt+1 − Vzt+1 − b < 0.
To exclude the equilibrium in which all ﬁrms intentionally defaults after borrowing too
much bt , we assume that the values of the parameters are chosen such that∀t,

Vnt − Vzt ≥ ψqt kt ,(8)

in the equilibrium path.33Note that the lender is willing to lend as long as mt + bt ≤ (ϕ + ψ)qt kt , because she can obtain(ϕ + ψ)qt kt by liquidating the ﬁrm if it defaults on the debt. So if Vnt − Vzt < ψqt kt , all ﬁrms set thehighest possible bt at the end of period t − 1 such that ϕqt kt + Vnt − Vzt < mt + bt ≤ (ϕ + ψ)qt kt , andall of them intentionally default in period t. In this case, all ﬁrms become debt-ridden ﬁrms from period t
on. This equilibrium path is self-consistent but does not seem to be relevant to the reality. Condition (8)excludes the possibility of emergence of this “all default” equilibrium.

8Why debt forgiveness is not allowed? Our institutional setting that debt forgivenessis not allowed is justiﬁed as a collective choice of the society to expand the borrowing limit.
Given that the parameter values satisfy (8), the borrowing limit for a normal ﬁrm in theeconomy where debt forgiveness is feasible is (ϕ + ψ)qt kt as we show in Appendix D,
while the limit is ϕqt kt + Vnt − Vzt in the economy where debt forgiveness is prohibited.
(8) implies that the borrowing limit is higher in the economy where debt forgiveness is
prohibited than in the economy it is not. Thus the prohibition of debt forgiveness can bejustiﬁed as a social choice to expand the borrowing limit.
Now we consider the behavior of a debt-ridden ﬁrm. We assume the following for the
institutional setting surrounding the debt-ridden ﬁrm.
Assumption 2. At the end of period t the lender and the debt-ridden ﬁrm bargain overthe continuation fee, dt+1 , that the debt-ridden ﬁrm should pay to the lender in periodt + 1. If they agree on dt+1 , the lender allows the ﬁrm to continue operation in periodt + 1 and the ﬁrm maximize its net proﬁt after paying dt+1 . The debt-ridden ﬁrm cannot
accumulate ﬁnancial assets inter-temporally.
The last sentence of the above assumption implies that the debt-ridden ﬁrm cannot
make savings. As we show in Appendix C, this assumption is not necessary for our resultsin the case of the deterministic equilibrium.
The bargaining over dt+1 takes place at the end of period t and we assume that if thebargaining breaks down and the ﬁrm is liquidated at this point, the lender obtains only
ψqt kt by conﬁscating a part of capital stock. We assume that the lender cannot conﬁscate(ϕ + ψ)qt kt because ϕqt kt has been concealed from the lender at the end of the period.4[{}]
Therefore, if the debt-ridden ﬁrm continues the lender can obtain Dt ≡ βEt λt+1 d˜t+1 + Dt+1 ,λtwhere d˜t ≤ dt is the realized payment, while the promised amount is dt , and Dt is thepresent value of the ﬂow of payments from the ﬁrm, while the lender can obtain ψqt kt if
she liquidates the ﬁrm immediately. As we see in Appendix B, the bargaining outcome
dt+1 is determined by[
Dt = βEt}]
λt+1 { ˜
dt+1 + Dt+1
= ψqt kt .
λt

The lender and the ﬁrm determine dt+1 , taking {kt+j }∞j=1 as given. See Appendix B for
detailed description of the values of dt+1 and d˜t+1 . After dt+1 is agreed at the end of
period t, the debt-ridden ﬁrm is allowed to operate in period t + 1. Actions of the debtridden ﬁrm are as follows. At the end of period t, after dt+1 is agreed it sells kt and buys
4

The debt-ridden ﬁrm and the lender can determine the value of dt+1 in the ﬁrst negotiation in the

middle of period t. But since they have the second chance of the renegotiation at the end of period t,
the value of dt+1 must be identical to the value which which is determined in the second renegotiation.
Otherwise dt+1 is revised in the second renegotiation at the end of period t.

9kt+1 in the market. At the beginning of period t + 1 after realization of At+1 , it borrows
working capital mt+1 to purchase the material input (the ﬁnal good). It produces the
α m1−α and sells it in the monopolistically competitive market.
intermediate good At+1 kt+1
zt+1

After it receives the proceeds of sales, it repays the debt mzt+1 + dt+1 .
The debt-ridden ﬁrm’s ability to borrow the working capital mzt+1 is bounded by thelimited enforceability of debt contracts. After realization of revenues and before repaying
mzt+1 , the debt-ridden ﬁrm has a chance to default on its debt (mzt+1 + dt+1 ). If the ﬁrm
defaults, the ﬁrm and the lender renegotiate on the repayment f as described in Appendix
B. If they agree on f , the lender obtains f + Dt+1 , while she obtains (ϕ + ψ)qt+1 kt+1 byliquidating the ﬁrm if the bargaining breaks down. Therefore, f + Dt+1 = (ϕ + ψ)qt+1 kt+1
and the no-default condition (m ≤ max{0, f − dt+1 }) implies that
mzt+1 ≤ max{0, (ϕ + ψ)qt+1 kt+1 − dt+1 − Dt+1 }.
Since it is shown that Dt+1 = ψqt+1 kt+1 in Appendix B, the borrowing constraint for mzt+1
is mzt+1 ≤ max{0, ϕqt+1 kt+1 − dt+1 }. The debt-ridden ﬁrm maximizes its own value Vzt ,which is deﬁned by the following Bellman equation. Given {dt+j }∞j=1 , the debt-ridden ﬁrm
solves][λt+1
Vzt = max −qt kt+1 + Et max β{p(x)x − mt+1 − dt+1 + qt+1 kt+1 + Vzt+1 } ,
mt+1
kt+1
λt
subject to(9)αx = At+1 kt+1
m1−α
t+1 ,

(10)

mt+1 ≤ max{0, ϕqt+1 kt+1 − dt+1 }.

(11)

Note that dt+1 does not depend on the ﬁrm’s choice of kt+1 , since the lender and the ﬁrm
agree on dt+1 at the end of t, taking the expected value of kt+1 as given.
One may suspect that prohibition of corporate savings by the debt-ridden ﬁrms in
Assumption 2 is crucial in deriving a persistent ineﬃciency due to the binding borrowing
constraint (11). If the ﬁrm can accumulate ﬁnancial assets, it could be possible to relax
the borrowing constraint eventually. But we show in Appendix C that at least in thedeterministic equilibrium the borrowing constraint is identical to (11) even if the debtridden ﬁrm can make savings, under the assumption that the lender can conﬁscate thesavings when she liquidates the ﬁrm. Thus there is no incentive for the debt-ridden ﬁrm
to make savings in the deterministic case.

3.2

Equilibrium without debt-ridden ﬁrms

In this subsection we characterize the deterministic equilibrium in which ∆it = 0 withprobability one. We also assume that δ = 0, that is, a variety of good is not depleted once
it is created.

10We consider the economy in which all ﬁrms are normal ﬁrms and there is no debt-ridden
ﬁrm. The resource constraints of the economy are∫ Nt
Ct + It +mit = Yt ,
0
∫ Nt
K=
kit di.
0

The economy converges to a stationary equilibrium, in which the number of ﬁrms Nt = N
is time-invariant and determined by
Vnt =

1
.
χ

In the symmetric stationary equilibrium, there is no R&D investment (It = 0) and theresource constraints are C = Y − N m and K = N k, where k and m are the capital andmaterial inputs for one ﬁrm. We focus on the parameter values that makes the borrowing
constraint (5) non-binding in the stationary equilibrium. In this case m = m̂, where
α m1−α )Aα1−α − m. That the borrowing constraint ism̂ = arg maxm p(Ait+1 kt+1
it+1 kt+1 m

non-binding implies thatm̂ ≤ ϕqt+1 kt+1 + Vnt+1 − Vzt+1 .
In this case, bt+1 is indeterminate. But we assume that there is inﬁnitesimally small tax
beneﬁt for issuing intertemporal debt such that ﬁrms are willing to borrow intertemporaldebt up to the borrowing limit. In the case of a deterministic equilibrium, the amountof bt+1 is determined by bt+1 = ϕqt+1 kt+1 − m̂ + Vnt+1 − Vzt+1 . Note that Vnt does notdepend on bt+1 , because the tax beneﬁt is inﬁnitesimal, and the loan rate and the market
rate are equal and satisfy
[
]
bt+1
λt+1
= βEt
bt+1 .
1 + rtλt

3.3Equilibrium with debt-ridden ﬁrms

We assume that the economy is initially in the symmetric stationary equilibrium where allﬁrms are normal, It = 0, and Nt = N , where N is determined so that Vn = χ1 . We assume
that in period τ the economy is hit by redistribution shocks {∆iτ }i∈[0,Z] , which makesbiτ > ϕqτ kτ + Vnτ − Vzτ for i ∈ [0, Z] and biτ ≤ ϕqτ kτ + Vnτ − Vzτ for i ∈ (Z, N ]. Thusﬁrm i ∈ [0, Z] choose to default on biτ and pay ϕqt kt , and ﬁrm i ∈ (Z, N ] do not default
on their debt. At the end of period τ , the ﬁrms in [0, Z] and their lenders negotiate on theamount of the continuation fee dτ +1 . We conjecture and verify later that Vnt < χ1 alongwith the equilibrium path. Since Vnt < χ1 , there takes place no R&D investment and Nt
does not change from N . Since N , K, and L are all constant over time, the equilibriumis the steady state. We focus on the equilibrium where the borrowing constraint (5) for11

normal ﬁrms is not binding and (11) for debt-ridden ﬁrms does bind. Denoting variables
for debt-ridden ﬁrms and normal ﬁrms by those with subscript z and n, respectively, theequilibrium is described by the following system of equations:,
1 = (1 − α)ηL1−η Aη knαη m(1−α)η−1n(12)

1 + µz = (1 − α)ηL1−η Aη kzαη m(1−α)η−1,z[]( α 1−α )η
Akn mnq = β αηL1−η
+q ,
kn
[
]
( α 1−α )η
Akmz z
q = β αηL1−η+ (1 + ϕµz )q ,
kz

(13)

Y =

(14)
(15)

1 η 1−ηη
A L {Z(kzα m1−α)η + (N − Z)(knα m1−αzn ) },
η(16)

C = Y − (N − Z)mn − Zmz ,(17)

K = Zkz + (N − Z)kn ,(18)

mn < ϕqkn + Vn − Vz ,(19)d = (β −1 − 1)ψqkz ,(20)

mz + d = ϕqkz ,(21)ηVn = −qkn + β{L1−α (Aknα m1−αn ) − mn + qkn + Vn },(22)

Vz = −qkz + β{L1−α (Akzα m1−α)η − mz − d + qkz + Vz },z(23)where µz is the Lagrange multiplier for (11). This system of equations is solved as follows.
First, (13), (15), (20), and (21) imply thatµz =αβ
1 − β − [ϕ − (β −1 − 1)ψ] 1−α

(24)

αβ
ϕβ + [ϕ − (β −1 − 1)ψ] 1−αGiven µz , we deﬁne Λ by
[

1−β
Λ=
1 − (1 + ϕµz )β] 1−(1−α)η
1−η

−(1 + µz )12(1−α)η
1−η

.

(25)

The macroeconomic variables are given byK,
N + (Λ − 1)Z
ΛK
kz =
,
N + (Λ − 1)Z[] 1
mn = (1 − α)ηL1−η Aη knαη 1−(1−α)η ,( αη ) 1
1−(1−α)η
Λ
mz =
mn ,
1 + µzmnαβ,q=(1 − α)(1 − β) kn(28)d = (β −1 − 1)ψqkz

(31)

kn =

β
(η −1 − 1)mn ,(1 − α)(1 − β){}β
(1 + µz )Vz =mz − mz − d − qkz .
1 − β (1 − α)η
Vn =

(26)
(27)

(29)
(30)

(32)
(33)

These variables must satisfy (8) and (19), which we check numerically in the example of
Figure 5.
Figure 5 shows the variables in the steady state equilibrium as functions of Z. The
parameter values are given in the caption of Figure 5. We set χ such that the value of
N equals unity in the stationary equilibrium where Z = 0. The value of χ is therefore
given by VSS = χ−1
SS = 3.4234, where VSS is the value of the normal ﬁrm in the stationary
equilibrium where Z = 0. The aggregate productivity of the economy is proportional toY − (N − Z)mn − Zmz,
K θ L1−θαηwhere θ = 1−(1−α)η
. As C = Y − (N − Z)mn − Zmz decreases as Z increases in thisﬁgure, the aggregate productivity is decreasing in Z. Figure 5 also justiﬁes our conjecturethat Vn < χ−1
SS (= VSS ) in the equilibrium where Z > 0. This is because Vn is decreasingin Z as Figure 5 shows. Why does Vn decrease as the debt-ridden ﬁrms increase? Since
the borrowing constraint is tight for the debt-ridden ﬁrms, the value of capital stock kt
as a collateral asset for the debt-ridden ﬁrms are higher and they purchase more capitalthan the normal ﬁrms. As a result, the price of capital increases and the increased costof capital pushes Vn down. This is how the increase of Z decreases Vn . This mechanism
is similar to the congestion eﬀect of zombie ﬁrms in Caballero, Hoshi, and Kashyap’s(2008) model. Because of this congestion eﬀect the decline of productivity is permanent.
This result forms a striking contrast to the equilibrium outcome in the case where debt
forgiveness is feasible, which is described in Appendix D. We see in Appendix D that theaggregate productivity comes back to the normal level immediately after a mass default
if the lenders and the ﬁrms can agree on debt forgiveness.

13

Next we consider the endogenous growth model, and show that the congestion eﬀect
can lowers the growth rate to zero permanently because the raised cost of capital due toemergence of debt-ridden ﬁrms discourages R&D investment.

3.4

Endogenous growth and the zero growth path

To enable endogenous growth in our model we need the following externality:
Assumption 3. There is a positive externality from the number of variety to the productivity: At = ÂNtα , where Â is an exogenous parameter. All agents take At as given,and households do not recognize the eﬀect of their choice of Nt on At .
The externality from expanding variety enables endogenous growth in this economy.
It is shown below that if Z is small the economy converges to the balanced growth path
(BGP), in which the productivity grows at a constant rate, while if Z is large the economy
falls into the zero growth path (ZGP), in which the growth rate is zero.
The BGP in this economy is an equilibrium path in which all ﬁrms are normal andthe economy grows at a constant rate, i.e., Nt+1 /Nt = Yt+1 /Yt = Ct+1 /Ct = 1 + g, whereg > 0. Since the capital K and the labor L are ﬁxed supply, the expansion of variety
Nt is the only source of economic growth in this model. We assume that the borrowing
constraint (5) does not bind in the BGP. The price of capital in the BGP is Nt q. The
variables (m, q, g) in BGP are speciﬁed by1

m = [(1 − α)ηL1−η (ÂK α )η ] 1−(1−α)η ,m
αβ,(1 − α)(1 − β) K
[
{
}
]
1
β
1
αβ
=
−1 −
m.
χ
1 + g − β (1 − α)η
(1 − α)(1 − β)q=We consider a numerical example, the parameter values of which are the same as those in
the example in the previous subsection, except for χ. We set the value of χ such that thegrowth rate of the BGP equals 1 %. The value of χ is therefore given by VBGP = χ−1
BGP =2.9307. In this example, the variables are given by m = 0.6657, q = 1.997, and g = 0.01.
We are interested in whether the emergence of the debt-ridden ﬁrms make the economy
follow the zero growth path (ZGP), in which R&D investment does not take place and theeconomy does not grow. Suppose that Nt = 1 and there emerge debt-ridden ﬁrms, whose
measure is Z, in period t. Given Z, the ZGP is speciﬁed by the system of equations in theprevious subsection (12)–(23). We consider the same numerical example as in the previous
subsection, except that we change the value of χ from χSS to χBGP . If Vn calculated by(22) satisﬁes Vn < χ−1
BGP , then the R&D does not take place and therefore Nt+j stays
at 1 for j ≥ 1. In this case the economy stays at a steady state where the productivity
stays constant, which is the ZGP. As Figure 5 shows, Vn > χ−1
BGP for 0 ≤ Z < 0.5439
14

and Vn < χ−1
BGP for 0.5439 < Z ≤ 1. The economy falls into the ZGP if Z is larger than
0.5439.
In this example, Vn calculated by (22) is larger than χ−1
BGP for Z ∈ [0, 0.4647). In
this case the equilibrium is not given by (12)–(23) since if the equilibrium were the ZGP,
the value of new entry, Vnt given by (22), would be strictly greater than the cost, χ−1
BGP ,and the R&D investment would take place, leading to Nt+1 > Nt , a contradiction. The
economy grows such that the value of new entry Vnt is equal to χ−1
BGP , and converges tothe BGP eventually. We describe the dynamics of the economy for a small Z ∈ [0, 0.5439)
in Appendix E.
Another numerical example is shown in Figure 6. In this example we set ϕ = 0.13
instead of 0.15 in the example of Figure 5. Figure 6 shows the variables in the ZGP.
With ϕ = 0.13, the value of a debt-ridden ﬁrm along with the BGP is negative: Vz,BGP =−0.5505, while it is positive along with the ZGP as the panel of Vz in Figure 6 shows. Thismeans that there exist multiple equilibria, the BGP and the ZGP, in this economy for alarge initial value of Z. First, for any value of Z the BGP can be an equilibrium. Supposethat macroeconomic expectations prevail in the initial period that the economy follows
the BGP. The all Z ﬁrms defaulted in the initial period exit immediately anticipating
their negative payoﬀ, Vz = −0.5505, and only the normal ﬁrms remain in the market.
Since the debt-ridden ﬁrms exit from the market, the capital price qt becomes suﬃciently
cheap such that the R&D investment takes place and the economy grows at the BGP rate.
Second, for Z > 0.4651 the ZGP can be an equilibrium. Suppose that macroeconomic
expectations prevail in the initial period that the economy follows the ZGP. All the Zﬁrms defaulted in the initial period stay in the market anticipating their positive payoﬀ,
Vz > 0. As Figure 6 shows that χBGP Vn,ZGP < 1 for Z > 0.4651, where Vn,ZGP is thevalue of a new ﬁrm in the ZGP, the gains from the R&D investment is strictly less thanits cost along with the ZGP when the initial value of Z is larger than 0.4651. Therefore if
Z > 0.4651 and the macroeconomic expectations are that the economy follows the ZGP,
then no households conduct the R&D investment and the economy stays at the ZGP.5
These examples in Figures 5 and 6 show that a mass default may shift the equilibrium
path from a growth path to the ZGP. This shift occurs because the debt-ridden ﬁrms
facing tighter borrowing constraints purchase collateralizable asset, i.e., physical capitalin our model, aggressively, leading to a rise of the cost of capital. The higher capital cost
reduces the expected gain of the R&D investment and discourages entry of new ﬁrms.
If there emerges a suﬃcient number of the debt-ridden ﬁrms the expected gain of the5

Note that the ZGP cannot exist in this economy if Z ≤ 0.4651. This is because χBGP Vn,ZGP ≥ 1 for

Z ≤ 0.4651, implying that the gains from the R&D investment is greater than or equal to its cost alongwith the ZGP. Since the R&D is conducted and Nt increases if the initial value of Z is no greater than
0.4651, the economy inevitably grows and the ZGP cannot be an equilibrium.

15

R&D falls below its cost and no one undertakes the R&D, leading to the zero growth of
the aggregate productivity. This negative eﬀect on the ﬁrms’ entry from higher capital
cost is similar to the negative congestion eﬀect on productivity of the zombie lending in
Caballero, Hoshi, and Kashyap (2008).

4

Conclusion

Decade-long recessions with low productivity growth are often observed after ﬁnancial
crises. We proposed a hypothesis that emergence of debt-ridden borrowers cause a persistent productivity slowdown. Economic agents become overly indebted as a result ofboom and bust of asset-price bubble. If debt reduction or debt forgiveness is not easily
implemented due to some rigidities of the market institution, the borrowers become debtridden. Analyzing the bargaining after default on the original debt, we show that thedebt-ridden borrowers are subject to tighter borrowing constraint than the normal ﬁrms
although they are under the control of the lenders. Emergence of a substantial numberof debt-ridden borrowers lowers the aggregate productivity through tightening the aggregate borrowing constraint. Emergence of debt-ridden ﬁrms also has congestion eﬀect as
in Caballero, Hoshi, and Kashyap (2008), because they purchase the collateralizable asset
aggressively and push its price up, discouraging entry of new ﬁrms. We show in a versionof our model in which the economy grows endogenously that the congestion eﬀect lowers
the growth rate of the aggregate productivity to zero if the measure of debt-ridden ﬁrms
exceeds a certain threshold level and also pessimistic expectations prevail.
Tightening of the aggregate borrowing constraints due to emergence of many debtridden borrowers may manifest itself as a “ﬁnancial shock” during or after a ﬁnancial crisis.
The mechanism of tightening of the borrowing constraint in this model is simple. We caneasily embed this model into a standard dynamic stochastic general equilibrium modeland assess qualitatively or quantitatively whether emergence of debt-ridden borrowers isa primal cause of the ﬁnancial shocks in the recent macroeconomic events. We leave this
topic for future research.

Appendix A: Derivation of borrowing constraint for normal
ﬁrms
The borrowing constraint is derived from the argument about what happens if the ﬁrm
defaults on its debt. There are the following three options for the ﬁrm and the lender todo after default.
• Liquidation: The ﬁrm manager refuses to pay and walks away. The lender conﬁscatesa part of the asset of the ﬁrm.
16

• Making the ﬁrm debt-ridden: In exchange for a continuation fee, the lender allows
the ﬁrm to continue operation, while she retains the right to liquidate it.
• Debt forgiveness: In exchange for a partial or zero repayment, the lender releases
the ﬁrm and waivers the right to liquidate it.
We need to specify the details of the institutional setting about bargaining after default.
So we make the following detailed assumption instead of Assumptions 1 and 2.
Assumption 4. In period t, ﬁrm i can default on the debt obligation of mt + bt after
the ﬁrm obtains the proceeds p(xt )xt . If ﬁrm i defaults then the ﬁrm and the lender canrenegotiate on the amount of repayment ft .
1. Once ﬁrm i defaults, the lender obtains the unilateral discretion to liquidate it. The
legal institution in this economy is such that as long as the repayment ft is strictly
less than the original debt (mt + bt ) the lender can retain the right to liquidate ﬁrm
i. And the lender can retain and exercise this right without any penalty even after
she receives ft (< mt + bt ) from ﬁrm i, implying that the lender never sell the rightto liquidate ﬁrm i at any price cheaper than the original debt.
2. If the lender liquidates the ﬁrm in the middle of the period the lender obtains a partof the physical capital (ϕ + ψ)qt kt , where 0 < ϕ + ψ < 1. When ﬁrm i is liquidated,the variety i of the intermediate goods just disappears from this economy.
3. If the lender and the ﬁrm agree on ft , which is equal to or greater than the original
debt, then the ﬁrm regains the right to choose whether or not to continue its own
business. Both the lender and the ﬁrm can verify that the lender loses the right toliquidate the ﬁrm at the moment she receives ft (≥ mt + bt ).
4. If the lender and the ﬁrm agree on ft , which is strictly less than the original debt,
then the lender retains the right to liquidate the ﬁrm. In this case at the end of
period t, the lender and the ﬁrm negotiate over the continuation fee dt+1 , which theﬁrm must pay in period t + 1 (in addition to the working capital mt+1 ).
(a) If they agree on dt+1 the lender allows the ﬁrm to continue operating in thenext period.
(b) If they do not agree on dt+1 , the lender liquidates the ﬁrm at the end of periodt. Since the bargaining over dt+1 takes place at the end of period t when apart of capital ϕqt kt has already been concealed from the lender, the lender canconﬁscate only ψqt kt by liquidating the ﬁrm at this stage.
(c) The debt-ridden ﬁrm cannot make savings inter-temporally.

17

5. Suppose that a debt-ridden ﬁrm and the lender agree on dt+1 in period t and theﬁrm continues operating in period t+1. Unpaid debt of the ﬁrm evolves to Bt+j+1 =(1 + rt+j ){mt+j + Bt+j − ft+j } in period t + j + 1 for j ≥ 0 with Bt = bt , whereft+j is realized repayment in period t + j. As long as Bt+j + mt+j > ft+j the lender
retains the right to liquidate the ﬁrm. If ft+1 ≥ mt+1 + Bt+1 the ﬁrm return to anormal ﬁrm and the lender loses the right to liquidate it.
Assumption 4 - 1 implies that debt forgiveness is infeasible because as long as repayment is strictly less than the original debt the lender chooses to retain or exercise the rightto liquidate it. Thus what can happen after default is either liquidation or continuationof the ﬁrm as a debt-ridden ﬁrm.
Suppose that ﬁrm i defaults on the debt mt + bt . After default, they negotiate over
repayment ft . If they do not reach an agreement and the lender liquidates the ﬁrm, thelender obtains (ϕ + ψ)qt kt , while the ﬁrm obtains p(xt )xt + (1 − ϕ − ψ)qt kt . If they agree
on repayment ft (≤ mt + bt ), ﬁrm i continues operation as a debt-ridden ﬁrm. If ﬁrm i
becomes a debt-ridden ﬁrm by paying ft , it obtains p(xt )xt + qt kt − ft + Vzt and the lenderobtains ft + Dt , where Dt is the present value of the expected cashﬂow which the lender
can receive from a debt-ridden ﬁrm from period t + 1 on. The bargaining over ft after
ﬁrm i defaults on mt + bt is described as a Nash bargaining:max[(ϕ + ψ)qt kt + Vzt − ft ]σ [ft + Dt − (ϕ + ψ)qt kt ]1−σ .
ft

We assume that the ﬁrm has all the bargaining power for simplicity of analysis, i.e., σ = 1.
Thus the bargaining outcome is ft = (ϕ + ψ)qt kt − Dt . Here we use Dt = ψqt kt , whichis shown in Appendix B, to have ft = ϕqt kt . Therefore, if ﬁrm i defaults on mt + bt ,the lender and the ﬁrm will agree on the repayment ft = ϕqt kt and ﬁrm i continues as adebt-ridden ﬁrm as a result of the bargaining.
Now we specify the condition for ﬁrm i not to default on mt + bt . After receiving theproceeds, p(xt )xt , if ﬁrm i does not default, it obtains p(xt )xt + qt kt − mt − bt + Vnt . On
the other hand, if it defaults, the ﬁrm and the lender bargains over repayment ft , whichleads to the agreement ft = ϕqt kt , and ﬁrm i obtains p(xt )xt + (1 − ϕ)qt kt + Vzt . The nodefault condition for ﬁrm i is p(xt )xt + qt kt − mt − bt + Vnt ≥ p(xt )xt + (1 − ϕ)qt kt + Vzt ,which can be rewritten as
mt + bt ≤ ϕqt kt + Vnt − Vzt .

Appendix B: Derivation of borrowing constraint for debtridden ﬁrms
We consider the bargaining between a debt-ridden ﬁrm and the lender over the fee dt+1
for continuation in the next period. The bargaining takes place at the end of period t.
18

If the debt-ridden ﬁrm and the lender agree on dt+1 , the ﬁrm obtains Vzt (dt+1 ), which[{}]
depends on dt+1 , and the lender obtains βEt λt+1 d˜t+1 + Dt+1 , where d˜t+1 is theλtrealized payment.6 Note that the lender and the ﬁrm take Dt+1 as given in period t
because Dt+1 is determined by the bargaining at the end of period t + 1 and they have noability to precommit to the outcome of the future bargaining. If the ﬁrm and the lender
does not agree on dt+1 , then the ﬁrm obtains (1 − ψ)qt kt and the lender obtains only
ψqt kt by liquidating the ﬁrm. This is due to Assumption 4 - 4 - (b). The Nash bargaining
between a debt-ridden ﬁrm and the lender is therefore,
[
{
}1−σ
}]
λt+1 { ˜σ.
max {Vzt (dt+1 ) − (1 − ψ)qt kt } βEt
dt+1 + Dt+1 − ψqt kt
dt+1
λt
With σ = 1, the bargaining outcome is[}]λt+1 { ˜= ψqt kt .
dt+1 + Dt+1
βEt
λt

(34)

We will describe how dt+1 and d˜t+1 are determined shortly.
After dt+1 is agreed at the end of period t, the debt-ridden ﬁrm is allowed to operatein period t + 1. Actions of the debt-ridden ﬁrm are as follows. At the end of periodt, it sells kt and buys kt+1 . At the beginning of period t + 1, it borrows working capitalmzt+1 , where mzt+1 is the material input (which is the consumer good) for the debt-ridden
α m1−α and sells it in the monopolisticallyﬁrm. It produces the intermediate good At+1 kt+1
zt+1

competitive market. After it receives the proceeds of sales, it repays the debt mzt+1 +dt+1 .7
The debt-ridden ﬁrm can default on the debt (mzt+1 + dt+1 ) after production. If theﬁrm defaults on the debt, the lender and the ﬁrm renegotiate over repayment f . The
renegotiation on f is as follows. If the debt-ridden ﬁrm and the lender reach an agreement,
the debt-ridden ﬁrm obtains p(xt+1 )xt+1 + qt+1 kt+1 + Vzt+1 − f and the lender obtains
f + Dt+1 . If there is no agreement, the lender liquidates the ﬁrm. Thus the debt-ridden
ﬁrm obtains p(xt+1 )xt+1 +(1−ϕ−ψ)qt+1 kt+1 and exits the market, and the lender obtains(ϕ+ψ)qt+1 kt+1 . As the bargaining power of the debt-ridden ﬁrm is 1 and that of the lenderis zero, we havef = (ϕ + ψ)qt+1 kt+1 − Dt+1 .
The no renegotiation condition for mzt+1 implies that mzt+1 ≤ max{0, f − dt+1 }. Therefore, the borrowing constraint that mzt+1 must satisfy is
mzt+1 ≤ max{0, (ϕ + ψ)qt+1 kt+1 − dt+1 − Dt+1 }.
6

Note that Vzt (dt+1 ) here is not Vzt in the Bellman equation (9) but the sum of Vzt and the value ofthe capital stock: Vzt (dt+1 ) = Vzt + qt kt .
7

We assume for simplicity that the redistribution shock ∆it does not hit the debt-ridden ﬁrm.

19

After the bargaining on dt+1 is over at the end of period t, the ﬁrm purchases kt+1 , knowingthat its own choice of kt+1 directly changes Dt+1 = ψqt+1 kt+1 . Therefore, the borrowing
constraint imposed on the working capital loan mzt+1 at the beginning of period t + 1 isrewritten as
mzt+1 ≤ max{ϕqt+1 kt+1 − dt+1 , 0}.

(35)

Now we specify how dt+1 and d˜t+1 are determined. The borrowing constraint (35)implies that if ϕqt+1 kt+1 < dt+1 the ﬁrm cannot borrow working capital and is forced
to set mzt+1 = 0 and d˜ = ϕqt+1 kt+1 . Although the ﬁrm defaults the lender allows theﬁrm to continue as a debt-ridden ﬁrm from the next period on. On the other hand, if
ϕqt+1 kt+1 ≥ dt+1 , then d˜t+1 = dt+1 . Thus dt+1 is the solution to[}]
λt+1 { ˜= ψqt kt ,βEt
dt+1 + ψqt+1 k t+1
λt
where d˜t+1 = qt+1 k t+1 if ϕqt+1 k t+1 < dt+1 , and d˜t+1 = dt+1 otherwise, where k t+1 is the
expected amount of capital stock that the debt-ridden ﬁrm purchases at the end of periodt after the agreement on dt+1 is reached. We denote it by k t+1 , not kt+1 , because the ﬁrm
and the lender take it as given when they bargain over dt+1 and neither the ﬁrm nor thelender can commit to the amount of capital stock (kt+1 ) which is to be purchased after
the bargaining.
In the deterministic case where d˜t+1 = dt+1 the value of dt+1 is given by the following
equation:dt+1 =

ψqt ktβ λλt+1t− ψqt+1 k t+1 .

(36)

This is because Dt+1 = ψqt+1 k t+1 .

Appendix C: Neutrality of corporate savings in the deterministic caseIn our basic model, ﬁrms cannot make savings (Assumption 2 or Assumption 4 - 4 - (c)in Appendix A). Here we consider the case where the debt-ridden ﬁrms can accumulate
ﬁnancial assets. In this appendix we focus on the deterministic case. First, we assume that
only if the lender liquidates the ﬁrm at the end of period t, she obtains ψqt kt + st , where
st is the risk-free bond, which the ﬁrm buys in period t. Suppose that the debt-ridden
ﬁrm chooses a positive amount of savings, st > 0. The bargaining over dt+1 at the end of
period t leads to Dt = β λλt+1{dt+1 + Dt+1 } = ψqt kt + st . We deﬁne the market interestt. Thus we haverate rt by (1 + rt )−1 = β λλt+1
tdt+1 = (1 + rt )(ψqt kt + st ) − Dt+1 .
20

(37)

At the beginning of period t + 1, the ﬁrm obtains (1 + rt )st from its asset and it borrows
mt+1 − (1 + rt )st as the working capital. After production, the ﬁrm has to repay mt+1 +dt+1 − (1 + rt )st . Then the ﬁrm and the lender renegotiate over the repayment f . Notethat at this stage of bargaining the ﬁrm does not own ﬁnancial asset st+1 yet. It buys
st+1 after the repayment is done. Now we consider the bargaining. If they agree on f ,the lender obtains f + D̂t+1 , where D̂t+1 = ψqt+1 kt+1 + st+1 and st+1 is the expectedvalue of st+1 at the stage of the bargaining, while she obtains (ϕ + ψ)qt+1 kt+1 if they do
not agree on f . Since the bargaining power of the ﬁrm is one, the bargaining outcome is
f = ϕqt+1 kt+1 − st+1 , which implies the borrowing constraint for the working capital:
mt+1 + dt+1 − (1 + rt )st ≤ ϕqt+1 kt+1 − st+1 ,(38)

Equations (37) and (38) imply that
mt+1 + (1 + rt )ψqt kt + st+1 − Dt+1 ≤ ϕqt+1 kt+1 .

(39)

and Dt+1 = ψqt+1 k t+1 + st+1 , the borrowing constraint (39)
Since (1 + rt )−1 = β λλt+1t
reduces to (11), because dt+1 in (11) is rewritten as (1 + rt )ψqt kt − ψqt+1 k t+1 , as (36)implies. Note that st does not appear in this constraint, implying that there is no incentive
for the debt-ridden ﬁrm to accumulate st . Thus we have shown that in the deterministic
equilibrium, the corporate savings st does not aﬀect the equilibrium prices and allocations.

Appendix D: Equilibrium with debt forgivenessIn this appendix we assume that the lender can commit to debt forgiveness and describe
the equilibrium after a mass default under the assumption that debt forgiveness is feasible.
In this Appendix we eliminate Assumption 4 - 1 and assume that the lender can forgive
the debt of defaulters. We consider two stage bargaining when the ﬁrm default on thedebt:• The lender and the ﬁrm negotiate whether or not to forgive the debt. If they agree
on repayment f , the ﬁrm continues as a normal ﬁrm after repaying f .
• If they do not agree on debt forgiveness, they bargain whether to allow the ﬁrm tocontinue as a debt-ridden ﬁrm or to liquidate it.
We show in what follows that the borrowing constraint for a normal ﬁrm is tighter whendebt forgiveness is feasible, and that if a mass default described in Section 3.3 occurs inperiod t all defaulters return to normal ﬁrms in period t + 1 by debt forgiveness.
Lemma 1. If debt forgiveness is feasible, the borrowing constraint for a normal ﬁrm is
mt ≤ max{0, (ϕ + ψ)qt kt − bt }.
21

(Proof) Suppose that the ﬁrm defaults on the debt mt + bt in period t after receiving the proceedsp(xt )xt . The outcome of the second stage bargaining whether to make the ﬁrm debt-ridden or
to liquidate it is the same as in Appendix B. The ﬁrm pays ϕqt kt to the lender and becomes a
debt-ridden ﬁrm, whereas the present value of the expected cashﬂow that the lender can receivefrom the debt-ridden ﬁrm from period t + 1 on is Dt = ψqt kt .
Now we consider the ﬁrst stage bargaining. If the ﬁrm and the lender agree on repayment f ,the ﬁrm continues as a normal ﬁrm. Thus if the agreement is reached, the ﬁrm obtains p(xt )xt −f +qt kt +Vn and the lender obtains f . If they do not agree on f , they go to the second stage bargaining,in which the ﬁrm obtains p(xt )xt +(1−ϕ)qt kt +Vzt and the lender obtains ϕqt kt +Dt = (ϕ+ψ)qt kt .
Therefore the ﬁrst stage bargaining is expressed as the following Nash bargaining:max[ϕqt kt + Vnt − Vzt − f ]σ [f − (ϕ + ψ)qt kt ]1−σ ,fwith σ = 1, which implies that f = (ϕ + ψ)qt kt . The borrowing constraint for mt is derived fromthe no default condition, mt ≤ max{0, f − bt }. (End of proof)

Note that the parameters must be chosen such that (8) holds to avoid “all default”
equilibrium. See footnote 3. Thus the borrowing constraint for the normal ﬁrms in thecase where debt forgiveness is feasible is tighter than in the case where it is not.
Next we brieﬂy describe what happens if a mass default occurs in the economy wheredebt forgiveness is feasible. Suppose that the redistribution shock ∆it is such that bit >(ϕ+ψ)qt kt for i ∈ [0, Z]. Firm i ∈ [0, Z] cannot obtain working capital and cannot produce
anything in period t. At the end of period t the ﬁrms default on bt and start bargaining
on repayment f , which is the same as above. The bargaining outcome is that the ﬁrms
pay f = (ϕ + ψ)qt kt and the lenders forgive the remaining debt bt − (ϕ + ψ)qt kt and waiver
the right to liquidate the ﬁrms. The defaulted ﬁrms return to normal in period t + 1 andthe mass default does not generate ineﬃciency from period t + 1 on.
This result presents a stark contrast to the persistent productivity decline in the casewhere debt forgiveness is infeasible (Section 3.3).

Appendix E: Dynamics when the number of debt-ridden ﬁrms
is small
We consider only the deterministic equilibrium. Suppose that N0 = 1 and there emerge
debt-ridden ﬁrms, whose measure is Z, in period 0. For small Z, the economy grows andconverges to the BGP eventually. The equilibrium path in which the R&D investment
takes place and Nt grows is described by the following system of equations. Note that22

Vnt = χ−1 in this equilibrium.
(1−α)η−1

1 = (1 − α)ηL1−η Âη (Nt knt )αη mnt,(1−α)η−1
1 + µzt = (1 − α)ηL1−η Âη (Nt kzt )αη mzt
,

]
[
η (N
αη m(1−α)η
Â
k
)
Ct
t+1
nt+1
nt+1
qt = βαηL1−η
+ qt+1 ,
Ct+1
knt+1
[
]
η
αη (1−α)η
Ct
1−η Â (Nt+1 kzt+1 ) mzt+1
qt = βαηL+ (1 + ϕµzt+1 )qt+1 ,
Ct+1
kzt+1
Yt =

}
Âη Ntαη 1−η {α 1−α ηαηLZ(kztmzt ) + (Nt − Z)(knt
m1−α
,
nt )
η

Ct + It = Yt − (Nt − Z)mnt − Zmzt ,K = Zkzt + (Nt − Z)knt ,dt =ψ Ct+1
qt kzt − ψqt+1 kzt+1 ,β Ct

mzt = ϕqt kzt − dt ,}Ct { 1−η
1−α η
α
α
−1
L (ÂNt+1 knt+1 mnt+1 ) − mnt+1 + qt+1 knt+1 + χ,
χ = −qt knt+1 + β
Ct+1}Ct { 1−ηααηVzt = −qt kzt+1 + β
L (ÂNt+1
kzt+1
m1−α
zt+1 ) − mzt+1 − dt+1 + qt+1 kzt+1 + Vzt+1 ,Ct+1−1

Nt+1 = Nt + χIt .
The following inequality must be satisﬁed:
mnt < ϕqt kt + Vnt − Vzt ,Vzt ≥ 0.
The ﬁrst inequality is necessary as we assumed that the borrowing constraint for normal
ﬁrms is not binding in the equilibrium. The second inequality is necessary since otherwise
the debt-ridden ﬁrm exits because liquidation is preferable to continuing as a debt-ridden
ﬁrm. As this equilibrium converges to the balanced growth path, it is convenient to deﬁne
and analyze the following detrended variables: ct = Ct /Nt , it = It /Nt , yt = Yt /Nt , q̃t =qt /Nt , Knt = Nt knt , Kzt = Nt kzt , zt = Z/Nt , Gt = Nt+1 /Nt . The above equations

23

are rewritten as the following system of equations of the detrended variables.
(1−α)η−1αη
1 = (1 − α)ηL1−η Âη Knt
mnt,

αη (1−α)η−1
1 + µzt = (1 − α)ηL1−η Âη Kzt
mzt,

q̃t = βq̃t = β
yt =

ct
ct+1
ct
ct+1

[
[

(1−α)ηαη
Âη Knt+1
mnt+1
αηL1−η
Knt+1(1−α)ηη αη
1−η Â Kzt+1 mzt+1

αηL

Kzt+1

]

+ q̃t+1 ,]+ (1 + ϕµzt+1 )q̃t+1 ,}Âη 1−η {α 1−α ηαηL
zt (Kztmzt ) + (1 − zt )(Knt
m1−α
,
nt )
η

ct + it = yt − (1 − zt )mnt − zt mzt ,K = zt Kzt + (1 − zt )Knt ,dt =ψ ct+1
Gt q̃t Kzt − ψ q̃t+1 Kzt+1 ,β ct

mzt = ϕq̃t Kzt − dt ,}β ct { 1−ηαη
−1
L (ÂKnt+1
m1−α,
nt+1 ) − mnt+1 + q̃t+1 Knt+1 + χGt ct+1}
β ct { 1−ηαηVzt = −q̃t Kzt+1 +
L (ÂKzt+1
m1−α
)
−
m
−
d
+
q̃
K
+
V
,
zt+1
t+1
t+1
zt+1
zt+1
zt+1
Gt ct+1χ−1 = −q̃t Knt+1 +Gt = 1 + χit ,zt+1 = zt /Gt .
The detrended variables must satisfy
mnt < ϕq̃t Knt + Vnt − Vzt ,Vzt ≥ 0.
This system of equation can be solved by the backward shooting method, in which{ct , it , yt , dt , mnt , mzt , µzt , q̃t , Knt , Kzt , zt , Gt , Vzt } are calculated, given {ct+1 , it+1 , yt+1 ,dt+1 , mnt+1 , mzt+1 , zt+1 , µzt+1 , q̃t+1 , Knt+1 , Kzt+1 , Gt , Vzt+1 }. *** The result of numerical simulation is shown in Figure ??. The economy monotonically converges to the
BGP. The theory predicts that the economy grows and converges to the BGP if and only
if Z = z0 is smaller than a threshold value, which is ∗ ∗ ∗∗ in the numerical example, andit falls into the ZGP otherwise. The numerical result conﬁrms this prediction as Figure ??
shows that there exists the path converging to the BGP only for the initial value z0 whichis smaller than ∗ ∗ ∗∗. The technical appendix that describes the solution procedures isavailable from authors upon request. ***24

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economy,” Japan and the World Economy, Vol. 18, No. 4, pp. 418–440, December.

25

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26

600

trillion yen
Real GDP

Potential GDP

550
500
450
400
350
300
80Q1
81Q3
83Q1
84Q3
86Q1
87Q3
89Q1
90Q3
92Q1
93Q3
95Q1
96Q3
98Q1
99Q3
01Q1
02Q3
04Q1
05Q3
07Q1
08Q3
10Q1

250

Source: Cabinet Oﬃce, Government of Japan, “Annual Report on National Accounts”

Figure 1: Real GDP and Potential GDP in Japan

HP

KI

JIP2011

1971–80

0.83

1981–90

1.93

2.06

1.39

1991–2000

0.36

0.35

0.04

0.48

1.13

2001–2007

1.68

Note: HP, KI, JIP2011 are from updated versions of Hayashi and Prescott(2002), Kobayashi and Inaba(2006), and Fukao and Miyagawa (2008).

Table 1: TFP growth rate in Japan

27

1980=100

600
550
500
450
400
350
300
250
200
150
100

Land value (SNA)
Nikkei 225 stock price80 82 84 86 88 90 92 94 96 98 00 02 04 06 08 10

Sources: Cabinet Oﬃce, Government of Japan; Nihon Keizai Shimbun.

Figure 2: Land value and Stock price in Japan

400

Japan

thousands
Entry

300

thousands
150
Net entry (right scale)
100

Exit

200
50

100

0

0
-100

-50

-200
-100

-300

-150
78

81

86

91

96

99

01

04

06

75

78

81

86

91

96

99

01

04

-400

～ ～ ～ ～ ～ ～ ～USthousands

thousands

200
150
100
50
0
-50
-100
-150

Entry

Exit

Net Entry (right scale)

-200

1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009

1000
800
600
400
200
0
-200
-400
-600
-800
-1000

～ ～

Note: Japan’s ﬁgures after 2001 are based on 1993-basis industry classiﬁcation.
Sources: (Japan) Ministry of Internal Aﬀairs and Communications, “Establishment and Enterprise Census”; (US) Bureau of Labor Statistics, “Business Employment Dynamics”.

Figure 3: Private sector establishment entry and exits: US and Japan28

%

Outstanding Balance of Non-perfoming Loans(Ratio to Nominal GDP)
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009

9
8
7
6
5
4
3
2
1
0

Note: The non-performing loans is the Risk Management Loans (RMLs) deﬁned in the
Banking Act in Japan, which consist of loans to bankrupt borrowers, delayed loans, 3month overdue loans, and loans with modiﬁed terms and conditions. RMLs do notinclude securitized loans.
Sources: Financial Services Agency, Status of Non-Performing Loans; Cabinet Oﬃce,
Government of Japan, Annual Report on National Accounts.
Figure 4: Development of the non-performing loans

29

k

k

n

1

z

2

1.8

0.8
1.5

1.6

0.6
0.4

1.4
0

0.5

Z

1

mn

0.7

1

0

0.5

Z

1

mz

0.5

0

0.45

1.2

0.6

0.4

1.1

0.55

0.35

1

0

0.5

Z

1

d

0.18

0

0.5

Z

1

Vn

3.5
3.42

Z

1

0.5

Z

1

Vz

0.75
3
V

0.14

0.7
0.65

0

0.5

Z

1

q

3.5

2.5

−1.2

0

0.5

Z

1

RHS−LHS (eq.7)

0
−1.4

3

−1.4

−1.6

2.5

−1.6

−1.8

2

−1.8

−2

1.5

0

0.8

0.16

0.12

0.9

0.5
C

1.3

0.65

0.5

Y

2

0

0.5

Z

1

−2

0

0.5

Z

1

−2.2

0.5

Z

1

LHS−RHS (eq.18)

0

0.5

Z

1

Parameters: α = 0.25, β = 0.9, ϕ = 0.15, η = 0.7, A = 1.9048,
K = 1, L = 1, ψ = 0.4, χ−1
SS = 3.4234,

V = χ−1
BGP = 2.9307.

Results: µz = 0.41, Λ = 1.9648, m = 0.6657, g = 0.01. If z ≥ 0.5439, V n < χ−1
BGP .
Figure 5: Equilibrium with debt-ridden ﬁrms

30

k

k

n

1

z

2.5

0.8

2

0.6

1.5

Y

2

1.5

0.4

0

0.5

Z

1

mn

0.7

1

0.4

0.6

0.35

0.55

0.3
0

0.5

Z

1

d

0.2

0.25

0.5

Z

1

mz

0.45

0.65

0.5

0

1

Z
0

0.5

1

C

1.4
1.2
1

0

0.5

Z

1

Vn

3.5
3.42

0.8

0

0.3

0.18

0.5

Z

1

Vz

0.25
3
V

0.16

0

0.5

Z

1

q

3.5

2.5

−1.5

0.2

0

0.5

Z

1

RHS−LHS (eq.7)

0
−2

0.5

Z

1

LHS−RHS (eq.18)

3
2.5

−2

−2.5

2
1.5

0

0.5

Z

1

−2.5

0

0.5

Z

1

−3

0

0.5

Z

1

Parameters: α = 0.25, β = 0.9, ϕ = 0.13, η = 0.7, A = 1.9048,
K = 1, L = 1, ψ = 0.4, χ−1
SS = 3.4234,

V = χ−1
BGP = 2.9307.

Results: µz = 0.5210, Λ = 2.1283, m = 0.6657, g = 0.01. If z ≥ 0.4651, V n < χ−1
BGP .
Figure 6: Equilibrium with debt-ridden ﬁrms: Multiple equilibria

31