CIGS Working Paper Series No. 14-005E

Debt-Ridden Borrowers and Productivity
Slowdown

Keiichiro Kobayashi
Keio University/The Canon Institute for Global Studies

Daichi Shirai
The Canon Institute for Global Studies

August 2012 (First version: March 2012)※Opinions expressed or implied in the CIGS Working Paper Series are solely those of theauthor, and do not necessarily represent the views of the CIGS or its sponsor.
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Debt-Ridden Borrowers and Productivity Slowdown ∗Keiichiro Kobayashi† and Daichi Shirai‡
August 2012 (First version: March 2012)

Abstract
Many authors argue that ﬁnancial constraints have been tightened in several countries since the Great Recession in 2007–2009. To explain this, we construct a model inwhich borrowing constraints for ﬁrms are tightened as a result of mass default due toa bubble collapse. In Jermann and Quadrini’s (2012) model, a defaulting ﬁrm either
goes back to being a normal ﬁrm by (partially) repaying its debt or is liquidated. We
assume that there is an intermediate status: a “debt-ridden” ﬁrm, deﬁned as a ﬁrm
whose lender retains the right to liquidate it. 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 cannot go back to being a normal ﬁrm, unless it repays all its debt.
Prohibition of debt forgiveness can be justiﬁed as a collective choice of the society, inorder to expand the borrowing limit for normal ﬁrms.
It is shown that borrowing constraints are tighter for debt-ridden than for normal ﬁrms. This implies that the emergence of a large mass of debt-ridden borrowers
may be a cause of the “ﬁnancial shocks” discussed in recent macroeconomic literature.
Tightened borrowing constraints due to the emergence of debt-ridden ﬁrms lower theaggregate productivity. This 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.

∗

We thank Hiroki Arato, Masaru Inaba, Daisuke Miyakawa, Tsutomu Miyagawa, Kengo Nutahara,
Yosuke Takeda, Ichihiro Uchida, and seminar participants at CIGS Conference on Macroeconomic Theory
and Policy 2012, the 2012 JEA autumn meeting, Aichi University, and the 2012 GRIPS macro conference
for helpful comments and discussions. Kobayashi gratefully acknowledges ﬁnancial support from the JSPS
Grant-in-Aid for Scientiﬁc Research (C) (No. 23330061) for the project, “Analysis of dynamic models withsystemic ﬁnancial crises.” All the remaining errors are ours.
†

Faculty of Economics, Keio University; CIGS; and RIETI. Email: keiichirokbys@gmail.com‡

Corresponding author. The Canon Institute for Global Studies. 11th Floor, ShinMarunouchi Building,
5-1 Marunouchi 1-chome, Chiyoda-ku, Tokyo, 100-6511 Japan. TEL: +81-3-6213-0550, FAX: +81-3-32171251, Email: shirai.daichi@canon-igs.org

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 growth of
total factor productivity slows down or even becomes negative for a decade (Kehoe andPrescott 2007). Why does productivity growth slow down persistently after a ﬁnancial
crisis? Many authors argue that ﬁnancial constraints were tightened during and after theGreat Recession in 2007–2009. Why are ﬁnancial constraints tightened after a ﬁnancial
crisis or the collapse of an asset-price bubble? Can tightening of ﬁnancial constraints cause
a persistent slowdown in economic growth? We consider these questions in this paper andpropose a theoretical model, in which the emergence of debt-ridden borrowers lowers theaggregate productivity persistently through tightening ﬁnancial constraints.
We construct a general equilibrium model, in which an exogenous shock makes asubstantial number of ﬁrms default on their debt. The model is based on Jermann andQuadrini’s (2006, 2012) models, which explicitly consider the bargaining process after adefault to derive a borrowing constraint, à la Kiyotaki and Moore (1997). In Jermann
and Quadrini, ﬁrms’ ability to borrow is bounded by the limited enforceability of thedebt contract. The borrowing ﬁrm can default on the debt obligation and renegotiate
repayment. Borrowing is limited such that the amount borrowed is renegotiation-proof.
In the hypothetical renegotiation in Jermann and Quadrini, the defaulting ﬁrm can goback to being a normal ﬁrm if the lender and the ﬁrm agree on repayment; otherwise,
the ﬁrm is liquidated if the renegotiation breaks down. Thus, there are two statuses for aﬁrm: being a normal ﬁrm and being liquidated.
A novel feature of our model is that we assume an intermediate status: being a “debtridden” ﬁrm. We deﬁne a debt-ridden ﬁrm as one whose lender retains the right toliquidate it. The lender allows the debt-ridden ﬁrm to continue operations, if she agrees tothe 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 being normal, unless it repays all of the original debt.
The defaulting ﬁrm is either liquidated or kept as a debt-ridden ﬁrm.
We analyze the borrowing constraint for debt-ridden ﬁrms and show that it is tighter
than for normal ﬁrms in the case of working capital loans. This result seems counterintuitive, because debt-ridden ﬁrms are under the control of their lenders, whereas normal
ﬁrms are not. The reason for this result is that in our model, if a normal ﬁrm defaults2

on its debt, it inevitably becomes a debt-ridden ﬁrm unless it repays all of the originaldebt. Thus, if a normal ﬁrm defaults, it loses its status of a normal ﬁrm, whereas if a
debt-ridden ﬁrm defaults, it continues as a debt-ridden ﬁrm after renegotiation. Therefore,
a normal ﬁrm loses more by defaulting than a debt-ridden ﬁrm does. Because of this, theincentive-compatibility condition (or no-default condition) implies that a normal ﬁrm canborrow more than a debt-ridden ﬁrm can.
Tighter borrowing constraints for working capital loans of debt-ridden ﬁrms makes
their production activity ineﬃcient. If a substantial number of ﬁrms become debt-ridden,
both the aggregate borrowing capacity and productivity decline. This implies that the
emergence of debt-ridden borrowers may be a cause of the “ﬁnancial shocks” discussed
in the recent macroeconomic literature. After the Great Recession in 2007–2009, many
authors argue that a shock in the ﬁnancial sector can cause a severe recession (e.g., arisk shock in Christiano, Motto, and Rostagno 2011, and a ﬁnancial shock in Jermann
and Quadrini 2012). In our model, the emergence of a substantial number of debt-ridden
ﬁrms manifests itself as a tightening of the aggregate borrowing constraint, which can beinterpreted as a ﬁnancial shock.
We also show that the emergence of debt-ridden ﬁrms has a persistent negative eﬀect
on productivity. Higher ineﬃciency due to tighter borrowing constraints lowers the valueof new entry to the market for a potential entrant, and discourages R&D activity. The
decrease in R&D activity leads the economy into a steady state with low productivity. We
then consider a modiﬁed version of the model where the economy grows endogenously, andshow 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 observed 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 The debt overhang
problem typically occurs when a ﬁrm cannot borrow new money for productive projects
because its existing debt is too large. Debt overhang arises if the existing debt holder is1

See also Krugman (1988) on debt overhang in international ﬁnance. See Moyen (2007) and Chen andManso (2010), for example.

3

diﬀerent from the potential lender who would lend new money. In this paper, we take asmall step forward by proposing a new theory that ineﬃciency can arise even if the lenderof new money is the existing debt holder. This paper is also close to Caballero, Hoshi,
and Kashyap (2008). They analyze “zombie lending,” deﬁned as a de facto subsidy tounproductive ﬁrms from banks. They argue that congesting zombie ﬁrms hinder the entryof highly productive ﬁrms and lower aggregate productivity. In this paper, we make acomplementary 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 constraints for working capital loans.
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 themodel to an endogenous growth model in Section 3.4. Section 4 presents our concluding
remarks.

2

Facts on persistent recessions after ﬁnancial crises

Numerous examples of productivity slowdown after a ﬁnancial crisis have been observed.
The most notable is the Great Depression in the 1930s in the US and similar depressions inthat period in the major nations. Ohanian (2001) shows that there was a large productivity
decline during the US Great Depression that is unexplained by capital utilization or laborhoarding. Kehoe and Prescott (2007) drew our attention to the fact that many countries
experienced decade-long severe recessions, which they call the “great depressions” of thetwentieth century. Papers in their book unanimously emphasize that declines in the growthrate of total factor productivity were the primary cause of these great depressions.
Another example of a decade-long recession after a ﬁnancial crisis is the 1990s in Japan.
The growth rates of gross domestic product (GDP) and total factor productivity (TFP) in
the 1990s were both lower than in the 1980s. Figure 1 shows the GDP and the potential
capacity in Japan. The kink in the beginning of the 1990s is apparent, when huge assetprice bubbles burst in the stock and real estate markets. See Figure 2 for asset prices inJapan in the 1990s. Table 1 shows various estimates of the TFP growth rate in Japan.
Hayashi and Prescott (2002) emphasize that the growth of TFP slowed down in the 1990s
in Japan.2 One notable feature in the 1990s is the signiﬁcant decrease in entries and
2

There is substantial debate on whether the TFP slowdown in Japan is truly a slowdown of technical4

increase in exits of ﬁrms. See Figure 3 for a 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 growth for US
manufacturing establishments (Bartelsman and Doms 2000). Nishimura, Nakajima, andKiyota (2005) argue that the malfunctioning of entry and exit contributes substantially toa fall in Japan’s TFP in the late 1990s. Another characteristic of Japan in the 1990s was
the persistently lingering nonperforming loans (NPLs) in the banking sector. The NPLs
were the excess debt of nonﬁnancial ﬁrms, mainly in real estate, wholesale, retail, andconstruction. Figure 4 shows the amount of NPLs in Japan from 1992 through 2009. The
delayed disposal of huge NPLs was seen as a de facto subsidy to nonviable ﬁrms (“zombie
lending”). This zombie lending has also been named as the cause of Japan’s persistentrecession (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 from labor
input and varieties of intermediate goods. The ﬁrms are monopolistic competitors andproduce respective varieties of intermediate goods from material input, which is the ﬁnal
good, and capital input. In our model, when a ﬁrm defaults on its debt, the lender cannot
forgive the debt. Instead, the lender can choose whether to liquidate the ﬁrm or allow it tocontinue operations as a “debt-ridden ﬁrm.” In this paper, a debt-ridden ﬁrm is one whose
lender retains the unilateral discretion to liquidate the ﬁrm. We later clarify the diﬀerence
between normal and debt-ridden ﬁrms by formally deﬁning their respective optimization
problems. The model is a version of the expanding variety model, in which new entryof ﬁrms increases aggregate productivity. The expanding variety model was proposed byRivera-Batiz and Romer (1991) and simpliﬁed by Acemoglu (2009). We follow the latter’s
exposition. In Sections 3.1–3.3, we analyze the model in which the economy converges
to a steady state. In Section 3.4, we introduce a positive externality from the variety ofgoods to aggregate productivity that enables endogenous growth. We show in Section 3.4
that the growth rate falls to zero if there is a suﬃcient number of debt-ridden ﬁrms.

progress or simply a measurement error (see Kawamoto 2005, Fukao and Miyagawa 2008). The tentative
conclusion on this issue in the literature is that there was a slowdown in technical progress in Japan,
though it may not be as severe as Hayashi and Prescott originally measured.

5

3.1

Basic setup

Supplies of capital K and labor L are ﬁxed. Labor is used for producing the ﬁnal good,whereas capital is used for the production of intermediate goods. A mass of ﬁrms, indexedby i ∈ [0, Nt ], produces intermediate goods, where Nt is the measure of the varieties ofintermediate goods in period t. The ﬁrms are either normal or debt-ridden ﬁrms; thestatus of being “normal” or “debt-ridden” is clariﬁed later. A representative householdowns these ﬁrms and solves the following problem:
[∞
]
∑
max
E0
β t ln Ct ,Ct ,It ,bt+1

subject to

t=0

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

Nt+1 = (1 − δ)Nt + χIt ,It ≥ 0.
Here, β is the subjective discount factor, Ct is consumption, L is the ﬁxed amount of labor
supply, It is the R&D investment, wt is the wage rate, rt is the market interest rate, bt
is a 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 its recovery
rate, where ξt = 1 if no ﬁrm defaults and ξt < 1 if some ﬁrms default. The cost of R&D
investment is measured in units of the ﬁnal good; one unit of R&D investment creates χunits of a new variety of intermediate goods. The δ fraction of the varieties of intermediate
goods depreciates every period. The ﬁrst-order conditions (FOCs) with respect to It andNt+1 imply that R&D investment takes place if χVnt = 1; it does not take place if χVnt < 1,
where Vnt is the value of a normal ﬁrm. The cost of R&D investment is unity and the
expected gain from it is χVnt . The amount of R&D investment is determined such thatthe cost and gain are equalized: χVnt = 1.
The ﬁnal good is produced competitively from the intermediate goods xit , i ∈ [0, Nt ]and labor by the following production function:
1
Yt =
η

(∫ N t0)xηit diL1−η ,

where 0 < η < 1. Because the ﬁnal good producer maximizes Yt −∫ Nt0

pit xit di − wt Lt ,where pit is the real price of the intermediate good i, perfect competition in the ﬁnal good6

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 analysis, we assume that productivity is identical across ﬁrms and constant over time, that is,
Ait = Afor all i and t.

Firm i needs to buy kit at the price of qt−1 in period t − 1 and mit at the price ofunity in period t. We assume for simplicity that on the 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 from the representative household to buy the material input; this debt is subject tothe ﬁnancial constraint that we specify below. Firm i incurs both intertemporal debt b′it
and intra-temporal debt (working capital) mit . The intertemporal 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
sole shock in this economy is the idiosyncratic redistribution shock ∆it that changes theamount of the intertemporal debt to be repaid in period t. We deﬁne bit bybit ≡ b′it + ∆it .
This is the amount that ﬁrm i must repay in period t to the lender of the intertemporal
debt. All agents know the probability distribution function of ∆it and take the expectation
over the distribution of ∆it when they decide their actions.
The 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 intertemporal debt becomes bit = b′it + ∆it .
The ﬁrm then 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 there7

is no stochastic shock during the short span when the ﬁrm borrows the working capital.
Therefore, the gross interest rate for the intra-period working capital is 1.
We now 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 following 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; we will
later 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, because ﬁrms
can default on their debt obligations. The basic logic is the same as Jermann and Quadrini(2012). The decision to 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 Jermann–
Quadrini model, the lender and the ﬁrm renegotiate on repayment if the ﬁrm defaults. If
the renegotiation breaks down and the ﬁrm is liquidated at this point, the lender obtains(ϕ + ψ)qt kt by conﬁscating a part of capital stock (kt ), where 0 < ϕ + ψ < 1. A departurefrom the Jermann–Quadrini model is the following assumption:
Assumption 1. The institutional environment of this economy is such that a lendercannot forgive the debt of a borrowing ﬁrm that defaults. Once the ﬁrm defaults, thelender 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 the ﬁrm. Furthermore, the lender can retainand exercise this right without any penalty even after she receives ft (< mt + bt ) from theﬁrm. This implies that the lender never waivers the right to liquidate the ﬁrm in exchange
for any repayment cheaper than the original debt. Instead, the lender either liquidates
the defaulting ﬁrm immediately or allows it to continue operations as a debt-ridden ﬁrm.
The debt-ridden ﬁrm is one whose lender retains the right to liquidate it and allows it tocontinue operations if she agrees on the continuation fee that the ﬁrm pays to the lender.
The lender and the debt-ridden ﬁrm negotiate at most twice in each period. The ﬁrst
negotiation arises if the ﬁrm defaults on the debt obligation in the middle of the period.
In this negotiation, the lender and ﬁrm negotiate on the repayment f , whereas the lender8

obtains (ϕ + ψ)qt kt if she liquidates the ﬁrm at this stage. The second negotiation, whicharises at the end of the period, is over the continuation fee dt+1 that the ﬁrm should pay
in the next period. If the lender liquidates the ﬁrm in the second negotiation, she obtains
only ψqt kt .
The diﬀerence in the liquidation value of the ﬁrm in the ﬁrst and second negotiations
can be justiﬁed as follows. The ﬁrst negotiation takes place in the middle of the period
and the second at the end of the period; the ﬁrm can conceal or sell oﬀ a part of its
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
its debt. The defaulting ﬁrm loses Vnt − Vzt inevitably. Appendix A shows that thedefaulting ﬁrm and the lender renegotiate on repayment and agree that the 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 by defaulting,(ϕqt kt + Vnt − Vzt ). Thus, a normal ﬁrm solves the following Bellman equation:[}]b′
λt+1 {
Vnt = max
− qt k + Et max β
p(x)x − m − b̃ + qt+1 k + Ṽt+1 ,mk,b′ 1 + rt
λt
subject to(2)

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 intertemporal debt b. Constraint (5) says that if
the realization of ∆ is such that ϕqt+1 k + Vnt+1 − Vzt+1 − b < 0, the ﬁrm sets m = 0.
Because 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 default after borrowing too
much bt , we assume that the values of the parameters are chosen such that∀t,

Vnt − Vzt ≥ ψqt kt

9(8)

in the equilibrium path.3
Why is debt forgiveness prohibited?

Our institutional setting of no debt forgivenessis justiﬁed as a collective choice of the society to expand the borrowing limit. Given thatthe parameter values satisfy (8), the borrowing limit for a normal ﬁrm in an economy
where debt forgiveness is feasible is (ϕ + ψ)qt kt , as we show in Appendix D. However, thelimit is ϕqt kt + Vnt − Vzt in an economy where debt forgiveness is prohibited. (8) implies
that the borrowing limit is higher in the economy where debt forgiveness is prohibitedthan in the economy where it is allowed. Thus, the prohibition of debt forgiveness can bejustiﬁed as a social choice to expand the borrowing limit.
We now 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 latter should pay the former in period t + 1. If theyagree on dt+1 , the lender allows the ﬁrm to continue operations in period t + 1 and theﬁrm maximizes its net proﬁt after paying dt+1 . The debt-ridden ﬁrm cannot accumulate
ﬁnancial assets intertemporally.
The last sentence of this assumption implies that the debt-ridden ﬁrm cannot makesavings. As we show in Appendix C, this assumption is not necessary for our results inthe case of the deterministic equilibrium.
The bargaining over dt+1 takes place at the end of period t. 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 the capital stock. We assume that the lender cannot
conﬁscate (ϕ+ψ)qt kt , because ϕqt kt has been concealed from her at the end of the period.4
Therefore, if the debt-ridden ﬁrm continues operations, the lender can obtain Dt ≡[{}]˜t+1 + Dt+1 , where d˜t ≤ dt is the realized payment, whereas the promised
βEt λλt+1dtNote 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. Thus, 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 periodt onward. This equilibrium path is self-consistent but does not seem to be relevant to reality. Condition(8) excludes the possibility of the emergence of this “all default” equilibrium.
3

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. However, because they have a second chance to renegotiate at the end of period t, thevalue of dt+1 must be identical to the value that is determined in the second renegotiation. Otherwise,
dt+1 is revised in the second renegotiation at the end of period t.

10amount is dt ; and Dt is the present value of the ﬂow of payments from the ﬁrm. On theother hand, the lender can obtain ψqt kt if she liquidates the ﬁrm immediately. As we seein 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 a
detailed description of the values of dt+1 and d˜t+1 . After dt+1 is agreed upon at the end of
period t, the debt-ridden ﬁrm is allowed to operate in period t + 1. The debt-ridden ﬁrm’s
actions are as follows. At the end of period t, after dt+1 is agreed upon, it sells kt and buys
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 , whereas 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 }) imply that
mzt+1 ≤ max{0, (ϕ + ψ)qt+1 kt+1 − dt+1 − Dt+1 }.
Because 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 , 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 , because the lender and theﬁrm agree on dt+1 at the end of t, taking the expected value of kt+1 as given.
11

One may suspect that prohibiting corporate savings by 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 may be possible to relax the borrowing
constraint eventually. However, we show in Appendix C that at least in the deterministic
equilibrium, the borrowing constraint is identical to (11), even if the debt-ridden ﬁrm cansave, under the assumption that the lender can conﬁscate the savings when she liquidates
the ﬁrm. Thus, there is no incentive for the debt-ridden ﬁrm to save 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.
We consider the economy in which all ﬁrms are normal and there is no debt-ridden
ﬁrm. The resource constraints of the economy are∫ Nt
mit = Yt ,Ct + It +0∫ NtK=
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 capitaland material inputs, respectively, for one ﬁrm. We focus on the parameter values thatmake the borrowing constraint (5) non-binding in the stationary equilibrium. In this case,
α m1−α )Aα1−α − m. That the borrowingm = m̂, where m̂ = arg maxm p(Ait+1 kt+1
it+1 kt+1 m

constraint is non-binding implies thatm̂ ≤ ϕqt+1 kt+1 + Vnt+1 − Vzt+1 .
In this case, bt+1 is indeterminate. However, we assume an inﬁnitesimally small tax beneﬁt
for issuing intertemporal debt, such that ﬁrms are willing to borrow intertemporal debt
12

up to the borrowing limit. In the case of a deterministic equilibrium, the amount of bt+1
is determined by bt+1 = ϕqt+1 kt+1 − m̂ + Vnt+1 − Vzt+1 . Note that Vnt does not dependon bt+1 , because the tax beneﬁt is inﬁnitesimal, and that the loan rate and market rateare 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
all ﬁrms 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] that makebiτ > ϕqτ kτ + Vnτ − Vzτ for i ∈ [0, Z] and biτ ≤ ϕqτ kτ + Vnτ − Vzτ for i ∈ (Z, N ]. Thus,ﬁrm i ∈ [0, Z] chooses to default on biτ and pays ϕqt kt , whereas ﬁrm i ∈ (Z, N ] does not
default on its debt. At the end of period τ , the ﬁrms in [0, Z] and their lenders negotiate
on the amount of the continuation fee dτ +1 . We conjecture and verify later that Vnt < χ1along the equilibrium path. Since Vnt < χ1 , no R&D investment takes place and Nt does
not change from N . Since N , K, and L are all constant over time, the equilibrium is thesteady state. We focus on the equilibrium where the borrowing constraint (5) for normal
ﬁrms is not binding and (11) for debt-ridden ﬁrms is binding. Denoting the variables fordebt-ridden and normal ﬁrms with the subscripts z and n, respectively, the equilibrium is13

described by the following system of equations:
1 = (1 − α)ηL1−η Aη knαη m(1−α)η−1
,
n

(12)

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

Given µz , we deﬁne Λ by
[

1−β
Λ=
1 − (1 + ϕµz )β] 1−(1−α)η
1−η

−(1 + µz )14(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−θαη. C = Y − (N − Z)mn − Zmz decreases as Z increases in thiswhere θ = 1−(1−α)η

ﬁgure. Therefore, the aggregate productivity is decreasing in Z. Figure 5 also justiﬁes ourconjecture that Vn < χ−1
SS (= VSS ) in the equilibrium where Z > 0. This is because Vn isdecreasing in Z, as Figure 5 shows. Why does Vn decrease as the number of debt-ridden
ﬁrms increases? Because the borrowing constraint is tight for debt-ridden ﬁrms, the valueof capital stock kt as a collateral asset for these ﬁrms is higher and they purchase more
capital than the normal ﬁrms. As a result, the price of capital increases and the increased
cost of capital pushes Vn down. This mechanism is similar to the congestion eﬀect ofzombie ﬁrms in Caballero, Hoshi, and Kashyap’s (2008) model. Because of this congestioneﬀect, the decline in productivity is permanent. This result forms a striking contrast to
the equilibrium outcome in the case where debt forgiveness is feasible, which is described15

in Appendix D. We see in Appendix D that aggregate productivity comes back to thenormal level immediately after a mass default, if the lenders and ﬁrms can agree on debt
forgiveness.
Next, we consider the endogenous growth model and show that the congestion eﬀect
can lower the growth rate to zero permanently, because the raised cost of capital due tothe emergence 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 growth in varieties of intermediate
goods to productivity: At = ÂNtα , where Â is an exogenous parameter. All agents takeAt 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 productivity grows at a constant rate. However, 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 and theeconomy grows at a constant rate, that is, Nt+1 /Nt = Yt+1 /Yt = Ct+1 /Ct = 1 + g, whereg > 0. Because the capital K and labor L are ﬁxed in 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 the BGP are speciﬁed by1

m = [(1 − α)ηL1−η (ÂK α )η ] 1−(1−α)η ,αβm,(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 debt-ridden ﬁrms makes the economy
follow the ZGP, in which R&D investment does not take place and the economy does16

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 R&D investment does not take place and 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
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 thiscase, the equilibrium is not given by (12)–(23); for if the equilibrium were the ZGP, thevalue of new entry Vnt , given by (22), would be strictly greater than the cost, χ−1
BGP , andR&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 to the BGP
eventually. We describe the dynamics of the economy for a small Z ∈ [0, 0.5439) inAppendix E.
Another numerical example is shown in Figure 6. In this example, we set ϕ = 0.13,
instead of 0.15 as in the example of Figure 5. Figure 6 shows the variables in the ZGP. Withϕ = 0.13, the value of a debt-ridden ﬁrm along the BGP is negative: Vz,BGP = −0.5505,
whereas it is positive along the ZGP, as the panel of Vz in Figure 6 shows. This meansthat there exist multiple equilibria, the BGP and the ZGP, in this economy for a large
initial value of Z. First, for any value of Z, the BGP can be an equilibrium. Suppose thatmacroeconomic expectations prevail in the initial period that the economy would follow the
BGP. Then, all Z ﬁrms that defaulted in the initial period exit immediately, anticipating
their negative payoﬀ, Vz = −0.5505, and only the normal ﬁrms remain in the market.
Because of the exit of debt-ridden ﬁrms, the capital price qt becomes suﬃciently cheap
such that 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 would follow the ZGP. Then, all the Z ﬁrms
that defaulted in the initial period stay in the market, anticipating their positive payoﬀ,
Vz > 0. Figure 6 shows that χBGP Vn,ZGP < 1 for Z > 0.4651, where Vn,ZGP is the value ofa new ﬁrm in the ZGP. In this case, the gains from R&D investment are strictly less than17

its cost along the ZGP. Therefore, if Z > 0.4651 and the macroeconomic expectations arethat the economy would follow the ZGP, then no households conduct 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 equilibriumfrom a growth path to the ZGP. This shift occurs because debt-ridden ﬁrms facing tighterborrowing constraints purchase collateral assets, that is, physical capital in our model,
aggressively, leading to a rise in the cost of capital. The higher capital cost reduces the
expected gains from R&D investment and discourages entry of new ﬁrms. If there emerge
a suﬃcient number of debt-ridden ﬁrms, the expected gains from R&D fall below its costand no one undertakes R&D, leading to zero growth in aggregate productivity. This
negative eﬀect of higher capital cost on ﬁrms’ entry is similar to the negative congestioneﬀect of zombie lending on productivity 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 the emergence of debt-ridden borrowers causes apersistent productivity slowdown. Economic agents become overly indebted as a resultof the boom and bust of asset-price bubbles. If debt reduction or debt forgiveness is noteasily implemented owing to rigidities in the market institution, the borrowers becomedebt-ridden. Analyzing the bargaining after default, we show that debt-ridden borrowers
are subject to tighter borrowing constraints than normal ﬁrms, although they are under
their lenders’ control. The emergence of a substantial number of debt-ridden borrowers
lowers the aggregate productivity through tightening the aggregate borrowing constraint.
The emergence of debt-ridden ﬁrms also has a congestion eﬀect, as discussed in Caballero,
Hoshi, and Kashyap (2008), because these ﬁrms purchase the collateral asset aggressively
and push its price up, discouraging the entry of new ﬁrms. In a version of our model inwhich the economy grows endogenously, we also show that the congestion eﬀect lowers thegrowth rate of aggregate productivity to zero, if the measure of debt-ridden ﬁrms exceeds
a certain threshold level and pessimistic expectations prevail.

5
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 R&D investment are greater than or equal to its cost along the
ZGP. Because R&D is conducted and Nt increases if the initial value of Z is no greater than 0.4651, theeconomy inevitably grows and the ZGP cannot be an equilibrium.

18Tightening of aggregate borrowing constraints owing to the 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 can easily embed this model into a standard dynamic stochastic general equilibrium
model and assess, qualitatively or quantitatively, whether the emergence of debt-ridden
borrowers is a primary cause of the ﬁnancial shocks in 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 of what happens if a ﬁrm defaults
on its debt. The ﬁrm and the lender have the following three options after default.
• Liquidation: The ﬁrm manager refuses to pay and walks away. The lender conﬁscatesa part of the ﬁrm’s assets.
• Making the ﬁrm debt-ridden: In exchange for a continuation fee, the lender allows
the ﬁrm to continue operations but 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 for bargaining after default. For
this, 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. Furthermore, 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 neversells the right to liquidate ﬁrm i at any price cheaper than the original debt.

19

2. If the lender liquidates the ﬁrm in the middle of the period, she obtains a part ofthe physical capital (ϕ + ψ)qt kt , where 0 < ϕ + ψ < 1. When ﬁrm i is liquidated,the variety i of the intermediate goods disappears from this economy.
3. If the lender and the ﬁrm agree on ft that is equal to or greater than the original
debt, 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 to liquidate
the ﬁrm the moment she receives ft (≥ mt + bt ).
4. If the lender and the ﬁrm agree on ft that is strictly less than the original debt, thelender retains the right to liquidate the ﬁrm. In this case, at the end of period t, thelender and the ﬁrm negotiate over the continuation fee dt+1 that the ﬁrm 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 period
t. Because 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 intertemporally.
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. The ﬁrm’s unpaid debt 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 , where ft+j
is the 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 returns 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, after default, the two possibilities for the ﬁrm are either liquidationor continuation as a debt-ridden ﬁrm.
Suppose that ﬁrm i defaults on the debt mt + bt . After default, the ﬁrm and the lendernegotiate over repayment ft . If they do not reach an agreement and the lender liquidates
20

the ﬁrm, the lender obtains (ϕ + ψ)qt kt , while the ﬁrm obtains p(xt )xt + (1 − ϕ − ψ)qt kt . Ifthey agree on repayment ft (≤ mt + bt ), ﬁrm i continues operations 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 andthe lender obtains ft + Dt , where Dt is the present value of the expected cash ﬂow thatthe lender can receive from a debt-ridden ﬁrm from period t + 1 onward. The bargaining
over ft after ﬁrm i defaults on mt + bt is described as a Nash bargaining as follows:max[(ϕ + ψ)qt kt + Vzt − ft ]σ [ft + Dt − (ϕ + ψ)qt kt ]1−σ .
ft

For simplicity of analysis, we assume that the ﬁrm has all the bargaining power, that is,σ = 1. Thus, the bargaining outcome is ft = (ϕ + ψ)qt kt − Dt . Here, we use Dt = ψqt kt ,shown in Appendix B, to have ft = ϕqt kt . Therefore, if ﬁrm i defaults on mt + bt , thelender and the ﬁrm will agree on the repayment ft = ϕqt kt . Thus, ﬁrm i continues as adebt-ridden ﬁrm as a result of the bargaining.
We now 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 theother hand, if it defaults, the ﬁrm and the lender bargain over repayment ft . This leads
to the agreement ft = ϕqt kt and ﬁrm i obtains p(xt )xt + (1 − ϕ)qt kt + Vzt . The no-default
condition for ﬁrm i is p(xt )xt + qt kt − mt − bt + Vnt ≥ p(xt )xt + (1 − ϕ)qt kt + Vzt . Thiscan 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.
If the debt-ridden ﬁrm and the lender agree on dt+1 , the ﬁrm obtains Vzt (dt+1 ), which[{}]
˜t+1 + Dt+1 , where d˜t+1 is thedepends on dt+1 , and the lender obtains βEt λλt+1
dt
realized 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 haveno ability to pre-commit to the outcome of future bargaining. If the ﬁrm and the lender
6

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

21

do 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 because of Assumption 4–4–(b). The Nash bargaining
between a debt-ridden ﬁrm and the lender is, therefore,
{
[
}1−σ
}]
λt+1 { ˜
max {Vzt (dt+1 ) − (1 − ψ)qt kt } βEt
dt+1 + Dt+1 − ψqt kt
.
dt+1
λt
σ

With σ = 1, the bargaining outcome is
[
βEt

}]
λt+1 { ˜= ψqt kt .
dt+1 + Dt+1
λt

(34)

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

monopolistically competitive market. After it receives the proceeds of sales, it repays thedebt mzt+1 + dt+1 .7 The debt-ridden ﬁrm can default on the debt (mzt+1 + dt+1 ) after
production. If the ﬁrm 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 andthe 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, andthe lender obtains (ϕ + ψ)qt+1 kt+1 . Because the bargaining power of the debt-ridden ﬁrm
is 1 and that of the lender is 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 }.
After the bargaining on dt+1 ends 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
7

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

22

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 the ﬁrm
to continue as a debt-ridden ﬁrm from the next period onward. On the other hand, if
ϕqt+1 kt+1 ≥ dt+1 , then d˜t+1 = dt+1 . Thus dt+1 is the solution to
[
βEt

}]
λt+1 { ˜
dt+1 + ψqt+1 k t+1= ψqt kt ,λtwhere d˜t+1 = qt+1 k t+1 if ϕqt+1 k t+1 < dt+1 and d˜t+1 = dt+1 otherwise; k t+1 is the expectedamount of capital stock that the debt-ridden ﬁrm purchases at the end of period t after
the agreement on dt+1 is reached. We denote it by k t+1 , not kt+1 , because the ﬁrm andthe lender take it as given when they bargain over dt+1 . Furthermore, neither the ﬁrm
nor the lender can commit to the amount of capital stock (kt+1 ) that 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 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 therisk-free bond that 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 interest rate rt byt23

(1 + rt )−1 = β λλt+1. Thus, we havet
dt+1 = (1 + rt )(ψqt kt + st ) − Dt+1 .

(37)

At the beginning of period t + 1, the ﬁrm obtains (1 + rt )st from its asset and 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 the ﬁnancial asset st+1 yet. It buys
st+1 after the repayment is done. The bargaining then proceeds as follows. 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 this stage of bargaining, whereas she obtains (ϕ + ψ)qt+1 kt+1 if they do
not agree on f . Because the bargaining power of the ﬁrm is 1, the bargaining outcome is
f = ϕqt+1 kt+1 − st+1 . This implies the following borrowing constraint for 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)

Because (1 + rt )−1 = β λλt+1
and Dt+1 = ψqt+1 k t+1 + st+1 , the borrowing constraint (39)treduces to (11). That is, 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, corporate savings st do not aﬀect equilibrium prices and allocations.

Appendix D: Equilibrium with debt forgivenessIn this appendix, we assume that the lender can commit to debt forgiveness. We thendescribe the equilibrium after a mass default.
In this appendix, we eliminate Assumption 4–1 and assume that the lender can forgive
the debt of defaulters. We consider the following two-stage bargaining when the ﬁrm
defaults on its debt:• 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 .

24• 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. Furthermore, if the mass default described in Section 3.3
occurs in period t, all defaulters return to normal ﬁrms in period t + 1 through 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 }.
(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, that is, 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 cash ﬂow that the lender can receivefrom the debt-ridden ﬁrm from period t + 1 onward is Dt = ψqt kt .
We now 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 move to thesecond stage of bargaining, in which the ﬁrm obtains p(xt )xt + (1 − ϕ)qt kt + Vzt and the lenderobtains 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, in order to avoid the “alldefault” equilibrium (see footnote 3). Thus, the borrowing constraint for normal ﬁrms in
the case where debt forgiveness is feasible is tighter than in the case where it is not.
Next, we brieﬂy describe what happens in the case of a mass default in an economy
where debt forgiveness is feasible. Suppose that the redistribution shock ∆it is such thatbit > (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 startbargaining on repayment f ; this process is the same as above. The bargaining outcome isthat the ﬁrms pay f = (ϕ + ψ)qt kt ; the lenders forgive the remaining debt bt − (ϕ + ψ)qt kt
and waiver the right to liquidate the ﬁrms. The defaulting ﬁrms return to normal in periodt + 1 and the mass default does not generate ineﬃciency from period t + 1 onward.
This result presents a stark contrast to the persistent productivity decline in the casewhere debt forgiveness is infeasible (Section 3.3).
25

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 a small Z, the economy grows andeventually converges to the BGP. The equilibrium path in which R&D investment takes
place and Nt grows is described by the following system of equations. Note that 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,[]ηαη (1−α)η
Ct
1−η Â (Nt+1 knt+1 ) mnt+1
qt = βαηL+ qt+1 ,
Ct+1
knt+1
]
[
η (N
αη m(1−α)η
Â
k
)
Ct
t+1
zt+1
zt+1
qt = β+ (1 + ϕµzt+1 )qt+1 ,αηL1−η
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−ηααη−1L (ÂNt+1
knt+1
m1−α
,
nt+1 ) − mnt+1 + qt+1 knt+1 + χ
Ct+1}Ct { 1−ηααηVzt = −qt kzt+1 + β
L (ÂNt+1
kzt+1
m1−α
)
−
m
−
d
+
q
k
+
V
,
zt+1
t+1
t+1
zt+1
zt+1
zt+1
Ct+1

χ−1 = −qt knt+1 + β

Nt+1 = Nt + χIt .
The following inequality must be satisﬁed:
mnt < ϕqt kt + Vnt − Vzt ,Vzt ≥ 0.
The ﬁrst inequality is necessary because we assumed that the borrowing constraint fornormal ﬁrms is not binding in equilibrium. The second inequality is necessary, because
otherwise, the debt-ridden ﬁrm exits because liquidation is preferable to continuing as adebt-ridden ﬁrm. As this equilibrium converges to the BGP, it is convenient to deﬁne andanalyze the following detrended variables: ct = Ct /Nt , it = It /Nt , yt = Yt /Nt , q̃t =26

qt /Nt , Knt = Nt knt , Kzt = Nt kzt , zt = Z/Nt , Gt = Nt+1 /Nt . The above equations
are rewritten as the following system of equations of the detrended variables.
(1−α)η−1αη
1 = (1 − α)ηL1−η Âη Knt
mnt,(1−α)η−1αη
1 + µzt = (1 − α)ηL1−η Âη Kzt
mzt
,
[
]
(1−α)η
αη
ηK
m
Â
ct
nt+1
nt+1
q̃t = βαηL1−η
+ q̃t+1 ,
ct+1
Knt+1
[
]
η K αη m(1−α)η
Â
ct
zt+1 zt+1αηL1−η
q̃t = β+ (1 + ϕµzt+1 )q̃t+1 ,
ct+1
Kzt+1

yt =

}
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
)
−
m
+
q̃
K
+
χ
,
nt+1
t+1
nt+1
nt+1
Gt ct+1}
β ct { 1−ηαη
L (ÂKzt+1
m1−α
Vzt = −q̃t Kzt+1 +
)
−
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.
After linearizing around the BGP (steady state), we can solve for the variables{ct , it , yt , dt , mnt , mzt , µzt , q̃t , Knt , Kzt , zt , Gt , Vzt } as linear functions of thestates, zt and dt . The parameters are set at the same values as in the text. The impulse
response to the productivity shock is shown in Figures 7 and 8. Figure 7 shows the IRF
for 1,000 periods, while Figure 8 shows the same for 40 periods. According to both ﬁgures,
most of the macroeconomic variables converge to the BGP monotonically, except for thecontinuation fee dt and the present value of its ﬂow, Dt . These variables become higher
and then lower than their BGP values after the shock. This non-monotonicity can be
explained as follows: β λtλ+1
declines largely on impact and then monotonically convergest27

to the BGP value. Because ψqt kzt = Dt =∑∞ii λt +ii=0 β (λt+i /λt ) dt+i , we can interpret β λtas the shadow price of dt+i for the debt-ridden ﬁrm. Because the total value of Dt is ﬁxed
is lower for smaller i than their BGP values, theat ψqt kzt and the shadow price β i λλt +it
debt-ridden ﬁrm increases dt+i for smaller i and decreases it for larger i.

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30

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 and potential GDP in Japan

1971–80
1981–90
1991–2000
2001–2007

HP
0.83
1.93
0.36

KI
2.06
0.35
0.48

JIP2011
1.68
1.39
0.04
1.13

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

Table 1: TFP growth rate in Japan

31

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 prices 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: Entry and exit of private sector establishments: US and Japan32

%

Outstanding Balance of Non-performing 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 are the Risk Management Loans (RMLs) deﬁned in the
Banking Act in Japan. These consist of loans to bankrupt borrowers, delayed loans,
three-month 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 non-performing loans

33

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

3

−1.4

2.5

−1.6

2

−1.8

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

RHS−LHS (eq.7)

0
−1.5

0.5

Z

1

LHS−RHS (eq.19)

−2

0

0.5

Z

1

−2.5

0

0.5

Z

1

Parameters: α = 0.25, β = 0.9, ϕ = 0.15, η = 0.7, A = 1.9048,
V = χ−1
K = 1, L = 1, ψ = 0.4, χ−1
BGP = 2.9307.
SS = 3.4234,
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

34

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

0

0.5

Z

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.19)

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,
V = χ−1
K = 1, L = 1, ψ = 0.4, χ−1
BGP = 2.9307.
SS = 3.4234,
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

35

c

i

t

0

−0.1

−0.2

0

500

t

dt

1000

0

0

0.2

−0.2

0
mnt

500

t

1000

0
µzt

500

t

1000

−0.2
0

500

t

qt

1

1000

−1

0
mzt

500

0
Kzt

500

0

500

t

1000

−0.05

0
Knt

500

t

1000

−0.1
0

−0.5

0

−0.4
0

−0.1

0.1

0

−0.04

t

0

−0.02

0.2

−0.2

y

t

0

t

1000

−0.5

0

500

t

1000

−1

z

t

0.4

t

1000

G

t

0
−0.005

0.5

0.2
−0.01

0

1

0

500

t

1000

Vzt

0

1

0

500

t

1000

Dt

−0.015

0

0

500

−3
x 10 , β λt+1/λt=β ct/ct+1

t

1000

−0.5
0.5

0
−1

0

0

500

t

1000

−1

0

500

t

1000

−1.5

0

500

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
V = χ−1
SS = 3.4234,
BGP = 2.9307.
Figure 7: Impulse responses to the productivity shock

36

t

1000

c

i

t

−0.08
−0.1

−0.025

−0.12

−0.03

y

t

−0.02

t

−0.2

−0.3

−0.14

0

10

20

30

dt

0.2

t

40

−0.035
−0.06

0
mnt

10

20

30

t

40

−0.4
−0.05

−0.08

−0.06

−0.1

−0.07

0
mzt

10

20

30

0
Kzt

10

20

30

10

20

30

20

30

t

40

0

−0.2
0.2

0
µzt

10

20

30

t

40

−0.12
0

0
Knt

10

20

30

t

40

−0.08
0

t

40

−0.2
0.1

−0.5
−0.4

0

0

10

20

30

qt

0.5

t

40

−0.6

0

10

20

30

z

t

40

0
−3

t

0.4

−1

−6

t

40

x 10 , Gt

0.45
−8

0.4

0.3
−10

0.35
0
1

10

20

30

Vzt

t

40

0.2

1

0

10

20

30

t

40

Dt

−12

−6

0

10

−4
x 10 , β λt+1/λt=β ct/ct+1

t

40

−8
0.5

0
−10

0

0

10

20

30

t

40

−1

0

10

20

30

t

40

−12

0

10

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
V = χ−1
SS = 3.4234,
BGP = 2.9307.
Figure 8: Impulse responses to the productivity shock

37

20

30

t

40