CIGS Working Paper Series No. 22-008E

Debt-Ridden Borrowers and Economic Slowdown

Keiichiro Kobayashi (Keio University / The Canon Institute for Global Studies / RIETI)Daichi Shirai (Tohoku-gakuin University / The Canon Institute for Global Studies)2022.12※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.
※CIGS Working Paper Series is circulated in order to stimulate lively discussion and comments.
※Copyright belongs to the author(s) of each paper unless stated otherwise.

General Incorporated Foundation

The Canon Institute for Global Studies
一般財団法人 キヤノングローバル戦略研究所

Phone: +81-3-6213-0550

https://cigs.canon/Debt-Ridden Borrowers and Economic Slowdown
Keiichiro Kobayashi and Daichi Shirai ∗June 2021 (Substantial revision. First version: March 2012)

Abstract
Economic growth slows for an extended period after a ﬁnancial crisis. We constructa model in which a one-time buildup of debt can depress the economy persistently, evenwhen there is no ﬁnancial technology shock. We consider the debt dynamics of ﬁrms
under endogenous borrowing constraints, with lenders having an option to forgive
defaulting borrowers. A ﬁrm is referred to as debt-ridden when it owes maximum
debt and pays all income in each period as an interest payment. In the deterministic
case, a debt-ridden ﬁrm continues ineﬃcient production permanently. Further, if theinitial debt exceeds a certain threshold, the ﬁrm intentionally chooses to increase
borrowing and, thus, to become debt-ridden. The emergence of a substantial numberof debt-ridden ﬁrms lowers economic growth persistently. A debt restructuring policyor the relief of debt-ridden borrowers from excessive debt may be able to restore theireﬃciency and economic growth.

JEL Classiﬁcation Numbers: E30, G01, O40
Keywords: borrowing constraint, debt overhang, debt supercycle, labor wedge, secularstagnation∗

Kobayashi: Corresponding author. Faculty of Economics, Keio University, CIGS, and RIETI; Address:

2-15-45 Mita, Minato-ku, Tokyo, 108-8345 Japan; E-mail: kobayasi@econ.keio.ac.jp; Phone: +81-3-54186703; Fax: +81-3-3217-1251.

Shirai: Tohoku-gakuin University and The Canon Institute for Global

Studies (CIGS); E-mail: shirai.daichi@gmail.com. Previous versions of this paper were circulated under
the title “Debt-Ridden Borrowers and Productivity Slowdown.” We thank Tomoyuki Nakajima for very
helpful comments and fruitful discussions on this project and other related topics. We also thank Kosuke
Aoki, Hiroki Arato, R. Anton Braun, Yasuo Hirose, Selahattin Imrohoroglu, Masaru Inaba, Nobuhiro
Kiyotaki, Huiyu Li, Daisuke Miyakawa, Tsutomu Miyagawa, Kengo Nutahara, Ryoji Ohdoi, Vincenzo
Quadrini, Yosuke Takeda, Hajime Tomura, Ichihiro Uchida, Yuichiro Waki, and the seminar participantsat the CIGS Conference on Macroeconomic Theory and Policy 2012, the 2012 JEA autumn meeting, Aichi
University, the 2012 GRIPS macro conference, the Summer Workshop on Economic Theory (SWET 2015)
at Otaru University of Commerce, the 11th Dynare conference, the 2015 JEA autumn meeting, 2016 AMES
(Kyoto), 2016 ESEM (Geneva), the 18th Macro Conference (Osaka), 2017 CEF (New York), and 2017 EEA(Lisbon) for their insightful comments and valuable discussions. Kobayashi gratefully acknowledges theﬁnancial support from the JSPS Grant-in-Aid for Scientiﬁc Research (C) (No. 23330061) for the project“Analysis of dynamic models with systemic ﬁnancial crises.” All remaining errors are ours.

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). A widely held concern, which has
been growing in the aftermath of the Great Recession of 2007–2012, is the hypothesis ofsecular stagnation (Summers, 2013), namely that the US and European economies could
stagnate persistently in the coming decades. Further, ﬁnancial constraints were tightened
both during and after the Great Recession (e.g., Altavilla, Darracq Paries and Nicoletti, 2015). However, which factors caused the tightening of these ﬁnancial constraints
and whether this tightening can cause a persistent slowdown in economic growth remain
unclear.
In this study, we propose a theoretical model in which the buildup of debt tightens
borrowing constraints and lowers growth persistently, even if there is no technology shock.
Our theory demonstrates that ineﬃciency due to the buildup of debt can continue persistently, which is consistent with the debt supercycle hypothesis (Rogoﬀ, 2016; Lo andRogoﬀ, 2015), whereas existing theories of secular stagnation posit that a permanent or
persistent technological shock causes persistent stagnation (see Eggertsson and Mehrotra,
2014; Gordon, 2012). Thus, our theory provides a rationale for heterodox policy recommendations to restore economic growth, that is, government interventions that facilitate
partial debt forgiveness in the private sector (see Geanakoplos, 2014).
Our theoretical contribution to the literature is to show that a one-time buildup of debt
can persistently tighten borrowing constraints and cause the aggregate ineﬃciency thatcan continue indeﬁnitely. In standard models of ﬁnancial frictions such as Carlstrom andFuerst (1997) and Bernanke, Gertler and Gilchrist (1999), the buildup of debt generates
ineﬃciency only for a few periods. Jermann and Quadrini (2012) and Albuquerque andHopenhayn (2004) show in their models of long-term debt that ineﬃciency due to thebuildup of debt can continue, but still for ﬁnite periods. Our result that ineﬃciency cancontinue indeﬁnitely thus contrasts sharply with the ﬁndings in the literature and suggests
new causality from a ﬁnancial crisis to persistent stagnation.
Recent empirical studies show that large corporate debt has a negative eﬀect on GDP
growth (e.g., Cecchetti, Mohanty and Zampolli, 2011; Mian, Suﬁ and Verner, 2017).
Giroud and Mueller (2017) ﬁnd that the establishments of more highly leveraged ﬁrms
experienced larger employment losses during the Great Recession in the United States.
Duval, Hong and Timmer (2017) also show that highly leveraged ﬁrms experienced large
and persistent drops in total factor productivity (TFP) growth in the aftermath of theGreat Recession. These ﬁndings seem to be consistent with our hypothesis.
Our model of ﬁnancial contracts has endogenous borrowing constraints that arise because of borrowers’ lack of commitment and lenders can choose whether to liquidate de-2faulting ﬁrms or forgive them. The market is incomplete, and debt and equity are the only
available ﬁnancial instruments. Firms cannot relax the borrowing constraints by raisingfunds from outside investors because of market frictions that prevent them from issuing
new equity quickly. There is a distinction between inter-period and intra-period loans
in this economy. The borrowing constraint binds more tightly as the initial amount ofinter-period debt increases. As the borrowing constraint tightens, ﬁrms cannot raise sufﬁcient intra-period debt for working capital, which leads to ineﬃcient production. Whenthe initial amount of inter-period debt reaches the maximum amount, ﬁrms fall into a
debt-ridden state in which they can repay only the interest even though they pay all of
their income in each period. As a result, the amount of debt does not decrease. Therefore,
debt-ridden ﬁrms continue ineﬃcient production permanently (in the deterministic case).
Moreover, when the initial debt exceeds a certain threshold, a ﬁrm may choose to increase
borrowing and intentionally become debt-ridden because the gain from additional borrowing can exceed the ineﬃciency of the additional tightening of the borrowing constraint.
This result implies that an overly indebted ﬁrm may rationally choose to become and,then, stay debt-ridden. Although our model is a simple modiﬁcation of that of Jermann
and Quadrini (2012), there is a signiﬁcant diﬀerence in that the debt-ridden state arises
naturally in our model, whereas it does not exist in Jermann and Quadrini (2012). Thisdistinction is due to a diﬀerence in settings: a portion of output can serve as the collateralfor borrowing in our model, whereas it cannot in Jermann and Quadrini’s model.
We embed the model of ﬁrms’ debt into a general equilibrium model of endogenous
growth in which productivity grows as a result of ﬁrms’ R&D activities. If a substantial number of ﬁrms become debt-ridden, their borrowing capacities decline persistently.
Tighter borrowing constraints discourage ﬁrms’ R&D activities and, thus, the entries of
new ﬁrms. Then, productivity stagnates persistently, as does the labor wedge.1 Thesefeatures of our model seem to be consistent with the facts observed in persistent recessions after ﬁnancial crises. See, among many others, Chari, Kehoe and McGrattan (2007)
for the Great Depression and Brinca, Chari, Kehoe and McGrattan (2016) for the GreatRecession.
In our model, persistent ineﬃciency is not caused by technological shocks, whereas
in existing models, persistent recessions are usually caused by persistent technologicalshocks. See, for example, Christiano, Eichenbaum and Trabandt (2015) and Bianchi,
Kung and Morales (2019) for the Great Recession, Cole and Ohanian (2004) for the Great
Depression, and Kaihatsu and Kurozumi (2014) for the lost decade of Japan. Many authors
have argued that persistent shocks that cause persistent recessions are exogenous changes
in the ﬁnancial technology parameters, such as the risk shock in Christiano, Motto and1

The labor wedge, 1 − τL , represents market frictions that manifest as an imaginary labor income tax

with tax rate τL in the literature on business cycle accounting (Chari, Kehoe and McGrattan, 2007).

3

Rostagno (2014) and the ﬁnancial shock in Jermann and Quadrini (2012). In this study,
we consider an exogenous one-time redistribution of wealth from borrowers to lenders that
leads to the buildup of debt, whereas there is no change in the technological parameters.2
One example of such a redistribution shock is the boom and bust of the asset-price bubble,
which changes the value of collateral for debt.
Thus, our model implies a policy recommendation distinct from those of most existingmodels in which exogenous shocks on technological parameters cause persistent recessions
and the policymaker can only mitigate these shocks by setting accommodative monetaryand ﬁscal policies or by designing ex-ante ﬁnancial regulations. In our model, debt restructuring or debt forgiveness for overly indebted borrowers restores aggregate eﬃciency
and enhances economic growth. Note that restoring economic growth does not necessitate
the physical liquidation of debt-ridden ﬁrms but rather their relief from excessive debt.
This argument is in line with the policy recommendations of partial debt forgiveness byGeanakoplos (2014).
Intuitive example:

To illustrate how persistent ineﬃciency can arise in our model,

we consider a simple model of a ﬁrm that produces output f (σ) from input σ, wheref ′ (σ) > 0 and f ′′ (σ) < 0. The ﬁrst-best solution that maximizes the social surplus,
f (σ) − σ, is attained by σ ∗ , where σ ∗ solves f ′ (σ) = 1. This ﬁrm initially holds debt b−1
and then borrows new debt Rb , where R is the loan rate. Thus, the ﬁrm’s cash ﬂow is
π = f (σ) − σ − b−1 + Rb . Here, we assume that b−1 is given exogenously, and the ﬁrm
chooses b such that π ≥ 0. The dividend π cannot be negative because current equityholders are protected by limited liability. The ﬁrm chooses σ and b to maximize f (σ) − σ,subject to the non-negativity constraint, π ≥ 0, and the borrowing constraintbσ ≤ ϕf (σ) + max ξS − , 0 ,Rwhere ϕ and ξS are constants that satisfy 0 ≤ ϕ < 1 and ξS > 0. This borrowing
constraint is derived in Section 2. Considering the Lagrangean L(σ, µ) = f (σ) − σ +µ ϕf (σ) − σ + max{ξS − Rb , 0} , the ﬁrst-order condition (FOC) is equal tof ′ (σ) =1+µ.
1 + ϕµWhen the Lagrange multiplier with respect to the borrowing constraint is positive (µ > 0),
the above FOC implies that f ′ (σ) > 1, which means that the input is smaller than σ ∗and production becomes ineﬃcient. To make this example interesting, we assume thatthe constants ϕ and ξS satisfy σ ∗ < ϕf (σ ∗ ) + ξS. If Rb is smaller than ξS, the borrowing
2

Kobayashi and Shirai (2016) analyze the eﬀects of wealth redistribution in a model of a borrowing-

constrained economy.

4constraint becomesσ ≤ ϕf (σ) + ξS −

b
.
R

(1)

This borrowing constraint is qualitatively similar to those in Jermann and Quadrini (2012)
and Kiyotaki and Moore (1997). If b is small, such that 0 ≤ b ≤ b∗ , where b∗ = [ϕf (σ ∗ ) −σ ∗ +ξS]R, then the ﬁrst-best outcome is achieved: σ = σ ∗ and µ = 0. If b∗ < b < RξS, theborrowing constraint binds tightly (µ > 0) and production becomes ineﬃcient. However,in the dynamic model, the ﬁrm repays as much of its debt as possible to reduce b andrelax the borrowing constraint. Thus, when the borrowing constraint is (1), ineﬃciency
is temporary. This result is similar to those of existing models such as Bernanke et al.
(1999). When debt Rb is larger than ξS, the borrowing constraint becomesσ ≤ ϕf (σ).

(2)

We deﬁne σz as the solution to σ = ϕf (σ). Assuming that ϕ is suﬃciently small, we positthat σz < σ ∗ and production is ineﬃcient. The Lagrange multiplier µz is calculated from
1+µz(> 1). We can show that ineﬃciency can continue permanently if b−1 isf ′ (σz ) = 1+ϕµzz )−σzsuﬃciently large, as follows. We deﬁne bz = f (σ1−and assume that bRz > ξS. Suppose
1
R

that b−1 = bz . Then, we can show as follows that b = bz and σ = σz . On the premise thatσ = σz , cash ﬂow is π = f (σz ) − σz − b−1 + Rb . Then, b must be chosen such that b ≥ bz tosatisfy π ≥ 0. Given b ≥ bz , it holds in turn that σ = σz . Therefore, in the dynamic model,
once the debt to be repaid in the current period (b−1 ) is equal to bz , the debt to be repaid
in the next period (b) must also be equal to bz , and this chain continues forever. Then,the borrowing constraint continues to be (2), and production is permanently ineﬃcient(σ = σz ). In summary, when b−1 < bz , ineﬃciency continues temporarily, but it continues
permanently if b−1 = bz . Permanent ineﬃciency arises from the accumulation of debt, notfrom changes in ﬁnancial or production technology. A caveat is that the constraint π ≥ 0
is necessary for permanent ineﬃciency. This constraint means that equity ﬁnance fromthe incumbent ﬁrm owner or outside investors is infeasible because of unspeciﬁed market
frictions.
Related literature:

Our theory is related to the literature on debt overhang, such as

Myers (1977), Krugman (1988), and Lamont (1995). Debt overhang is an ineﬃciency thatis typically due to the coordination failure between incumbent lenders and new lenders,
whereas ineﬃciency is generated in our model even though incumbent lenders also provide
new money. Debt overhang typically causes ineﬃciency in the short run. However, in
our study, ineﬃciency can continue permanently. Our study is also closely related to thework of Caballero, Hoshi and Kashyap (2008). They analyze “zombie lending,” deﬁned
as the provision of a de facto subsidy from banks to unproductive ﬁrms, and argue that5

congestion by zombie ﬁrms hinders the entry of more productive ﬁrms and lowers aggregate
productivity. In this study, we make the complementary point to their argument thateven an intrinsically productive ﬁrm can become unproductive when it is debt-ridden.
This point results in a notably diﬀerent policy implication. Caballero et al. (2008) imply
that the physical liquidation of zombie ﬁrms is desirable, whereas our theory implies thatzombie ﬁrms can restore high productivity if they are relieved of their excessive debt.
Fukuda and Nakamura (2011) report that the majority of ﬁrms identiﬁed as zombies
by Caballero et al. (2008) recovered substantially in the ﬁrst half of the 2000s. Thisobservation seems consistent with our model. In the macroeconomic literature, endogenous
borrowing constraints are introduced by the seminal works of Kiyotaki and Moore (1997),
Carlstrom and Fuerst (1997), and Bernanke et al. (1999), which spawned the large bodyof the literature on dynamic stochastic general equilibrium (DSGE) models with ﬁnancial
frictions. The borrowing constraints in an economy in which intra-period and inter-periodloans exist are analyzed by Albuquerque and Hopenhayn (2004), Cooley, Marimon andQuadrini (2004), and Jermann and Quadrini (2006, 2007, 2012). The modeling methodin this study is closest to that of Jermann and Quadrini (2012). Our model is also closely
related to that of Kobayashi and Nakajima (2017), who analyze endogenous borrowing
constraints and non-performing loans (NPLs). Furthermore, our model is similar to thatof Guerrón-Quintana and Jinnai (2014) in that a temporary shock persistently aﬀects
economic growth, although there is a signiﬁcant diﬀerence in the policy implications. In
our model, the emergence of debt-ridden borrowers due to a redistribution shock causes
a persistent recession. Thus, debt restructuring (i.e., wealth redistribution from lenders
to borrowers) restores aggregate eﬃciency. By contrast, debt restructuring has no eﬀect
in the model of Guerron-Quintana and Jinnai because, in their model, the ﬁnancial crisis
is caused by a shock to the parameters of ﬁnancial technology. Another study closely
related to ours is Ikeda and Kurozumi (2014). They build a medium-scale DSGE modelwith ﬁnancial friction à la Jermann and Quadrini (2012) and endogenous productivity
growth à la Comin and Gertler (2006). Their study is diﬀerent from ours in that Ikeda
and Kurozumi (2014) also posit that a ﬁnancial crisis is an exogenous technological shock.
The remainder of this paper is organized as follows. In the next section, we present
the partial equilibrium model of the lender–borrower relationship and analyze the debt
dynamics. In Section 3, we construct the full model by embedding the ﬁnancial frictionsof the previous section into an endogenous growth model, showing that stagnation cancontinue persistently when a substantial number of debt-ridden borrowers emerge. Section
4 presents our concluding remarks.

6

2

A model of corporate debt with borrowing constraints

In this section, we consider the partial equilibrium model of debt contracts. We derive theborrowing constraint and analyze the debt dynamics under exogenously given prices. We
then embed this model into the endogenous growth model in Section 3.

2.1

Setup

Time is discrete and continues from zero to inﬁnity: t = 0, 1, 2, · · · , ∞. There are three
agents in this model: a bank (lender), a ﬁrm (borrower), and a household (worker). The
main players are the bank and the ﬁrm; the household only supplies labor and capital atmarket prices and buys consumer goods from the ﬁrm. Real prices {wt , rtK , rt , mt } aretaken as given, where wt is the wage rate, rtK is the rental rate of capital, rt is the intertemporal rate of interest for safe assets, and mt is the stochastic discount factor. These
prices are later determined in the general equilibrium model. The stochastic discountfactor satisﬁes
1= Et mt+1
1 + rtwhere Et is the expectation operator at t.
Consumer goods are produced by the ﬁrm from the labor and capital inputs. The
ﬁrm’s gross revenue in period t is given by(1−α)η

F (At , kt , lt ) = At ktαη lt,

where kt and lt are the capital and labor inputs, respectively, chosen in period t, At is atime-variant revenue parameter, and 0 < η < 1. Firms use equity and debt, where debt
is not state-contingent. We focus on the case with initial debt stock b−1 at t = 0, whereb−1
R−1

is the amount of inter-period debt at the end of the previous period and Rt is thegross rate of corporate loans. In this study, following Jermann and Quadrini (2012), we
assume that ﬁrms hold inter-period debt because it oﬀers tax advantages.3 Thus, Rt isdetermined by
Rt = 1 + (1 − τ )rt ,where τ represents the tax beneﬁt. The debt Rbt−1
at the end of period t − 1 grows at the
t−1gross rate Rt−1 to become bt−1 at the beginning of period t. The amount of initial debt
b−1 is given as an exogenous shock in this model. This amount may be ineﬃciently large,
as we see in the following sections. One explanation, which falls outside our model, for3This assumption is a shortcut to formulate the motivation for holding debt. As is well known, withasymmetric information and costly state veriﬁcation, the optimal contract takes the form of debt (e.g.,
Townsend, 1979; Gale and Hellwig, 1985).

7

why b−1 can be ineﬃciently large is as follows. The debt in this model can be interpreted
as unsecured debt. Suppose that a ﬁrm puts up a valuable asset (e.g., real estate or
corporate stocks) as collateral and the corporate debt b−1 is initially completely securedby this collateral. Then, the asset price collapses in period 0 when the ﬁnancial crisis
breaks out, and the value of the collateral asset decreases to, say, zero, meaning thatb−1 becomes unsecured debt. In this way, the asset-price collapse can make the initial(unsecured) debt b−1 ineﬃciently large.
In period t, the ﬁrm employs labor lt and capital kt from the household to produce(1−α)η

and sell consumer goods, and it earns revenue F (At , kt , lt ) = At ktαη lt. The cost of

the capital and labor inputs for the ﬁrm is
σt ≥ rtK kt + wt lt .
The ﬁrm needs to borrow working capital, σt , from the bank as an intra-period loan andpay the household in advance of production, as in Albuquerque and Hopenhayn (2004),
Cooley, Marimon and Quadrini (2004), and Jermann and Quadrini (2006, 2007, 2012).
When production is completed, the ﬁrm receives revenue F (At , kt , lt ). Let us denote theexogenous state variables at t by xt , where xt = {At , rtK , wt , rt , mt }. We denote the spaceof xt by Λ (i.e., xt ∈ Λ). State xt follows a Markov process, which we do not specify
further as the exact process is irrelevant for now. We deﬁne f (σt , xt ) byf (σt , xt ) = max F (At , k, l),k,lsubject to rtK k + wt l ≤ σt .
Thus, the solution impliesf (σt , xt ) = At

α
rtK

αη 

1−α
wt

(1−α)η
σtη .

The budget constraint for the ﬁrm is given byπt ≤ f (σt , xt ) − σt − bt−1 +

bt
,
Rt

(3)

where πt is the payment to the ﬁrm’s owner as a dividend. The payment of the intraperiod loan σt = wt lt + rtK kt is subject to the following borrowing constraint (derived in
the next subsection):bt
σt ≤ ϕf (σt , xt ) + max ξS(xt ) −
, 0 ,
Rt

(4)

where 0 ≤ ϕ < 1, 0 ≤ ξ ≤ 1, and S(xt ) is deﬁned by (7), which is the value that thelender can obtain by taking control of the ﬁrm. Throughout this analysis, we assume thatϕ < η,
8

which means that production becomes ineﬃcient when the borrowing constraint is σt ≤ϕf (σt ). The ﬁrm’s owner has no liquid assets in hand and is protected by limited liability,
as in Albuquerque and Hopenhayn (2004). Therefore, the dividend must be non-negative:4

πt ≥ 0.

(5)

The ﬁrm cannot avoid the limited liability constraint (5) by soliciting equity investment
from outside investors because of market frictions. The details of the market frictions are
not speciﬁed in this analysis, and we simply assume that the ﬁrm cannot issue new equities
in a timely manner, even if the new money can generate a positive surplus by relaxing
the borrowing constraint and even if outside investors are willing to buy new equities.
This assumption can be justiﬁed by market frictions such as a lack of commitment andasymmetric information.5
Now, we can describe the optimization problem for the ﬁrm. Denoting the value of the
ﬁrm with debt stock bt−1 at the beginning of period t as V (bt−1 , xt ), the ﬁrm’s problem iswritten as the following Bellman equation:
V (bt−1 , xt ) = max πt + Et [mt+1 V (bt , xt+1 )] ,bt ,σt ,πt(6)subject to the budget constraint (3), borrowing constraint (4), limited liability constraint(5), and participation constraint of the bank (i.e., the no-Ponzi condition):bt ≤ bz (xt ),where bzR(xtt ) is the upper limit of the amount that the bank agrees to lend in period t,
given by (8) below.
In the above problem (6), the ﬁrm takes {bz (xt ), S(xt )}∞t=1 as given. In equilibrium,
S(x) is determined endogenously from the solution to the ﬁrm’s problem. S(xt ) is theliquidation value of the ﬁrm. Thus, in equilibrium, the following must be satisﬁed:
S(xt ) = max Et [mt+1 V (b, xt+1 )] +
b

4

b
.
Rt

(7)

Even if we allow π < 0, the ineﬃciency continues permanently, when b−1 is suﬃciently large. Thus,

relaxing this assumption does not change the qualitative results presented in the paper.
5
The following are two examples of market frictions. The ﬁrst example is a lack of commitment. Supposethat the human capital of the ﬁrm’s owner is indispensable to the ﬁrm’s operations and that she cannot
commit to pay the promised amount to the outside investors. In this case, the ﬁrm’s owner renegotiates
the payment down to zero after the outside investors provide additional money. Anticipating this outcome,the outside investors do not invest in the ﬁrm. The second example is informational frictions. Supposethat the bargaining over the surplus between the ﬁrm and outside investors needs to be settled before theﬁrm can issue new equities and assume that this bargaining is associated with informational frictions à
la Abreu and Gul (2000) that result in a prohibitively delayed settlement. Abreu and Gul (2000) showthat if the two players in a Rubinstein-type bargaining game each suspect that their counterparty may beirrational, the bargaining experiences substantial delay.

9The upper limit of debt bz,t = bz (xt ) is the upper bound of repayable debt under6
any future realizations of the states {xt+j }∞j=1 given that xt is the current state. We

derive bz (xt ) on the premise that bzR(xtt ) > ξS(xt ), which is later justiﬁed by the parameter
restriction (9). Under this assumption, the borrowing constraint becomes σt ≤ f (σt , xt )when the outstanding debt is bz (xt ). We deﬁne σz,t = σz (xt ) as the solution to σ =ϕf (σ, xt ). Deﬁne, recursively, that for a given value of xt−1 ,bz (xt−1 ) =(1 − ϕ)f (σz (xt ), xt ) +inf

xt ∈Λ(xt−1 )bz (xt ),Rt(8)

where Λ(xt−1 ) is the domain for xt given that the state of time t − 1 is xt−1 . If Λ(xt−1 ) =Λ, then the upper limit is a constant (i.e., bz (xt−1 ) = bz ). If the condition, bzR(xtt ) >ξS(xt ) for all xt , is satisﬁed, then it is justiﬁed that bz (xt ) is truly the maximum repayabledebt. To ensure the above condition, we assume the following restriction (9) on theparameters.
∂
The ﬁrst-best amount of input, σ ∗ (x), is deﬁned as the solution to ∂σf (σ, x) = 1. The

upper limit of the total surplus of the match between the bank and ﬁrm, given that thecurrent state is x, is denoted by ω(x), for x ∈ Λ, and the unconditional upper limit isdenoted by ω̄. These limits are deﬁned as follows:ω(xt ) = [1 + τ r(xt )] {f (σ ∗ (xt ), xt ) − σ ∗ (xt )} + Et [mt+1 ω(xt+1 )] ,ω̄ ≡ sup ω(x).
x∈Λ

Since the maximum amount of the tax advantage is τ r(x) {f (σ ∗ (x), x) − σ ∗ (x)}, it isstraightforward that S(x) ≤ ω(x) because S(x) is the value to the bank of operating the
seized ﬁrm. Thus, we assume the following restriction on the parameter values:ξω(xt ) <bz (xt ),Rt

for all xt .

(9)This inequality ensures that the maximum repayable debt bzR(xtt ) is strictly larger thanξS(xt ).
Diﬀerence between inter- and intra-period debt:

We have the following diﬀerence

between inter-period debt bt−1 and intra-period debt σt . The ﬁrm has the chance to default
on its inter-period debt bt−1 at the beginning of period t, and it will do so if and only if thecontinuation value of the ﬁrm is negative, V (bt−1 , xt ) < 0. However, this outcome never6bWhen the amount of debt is larger than Rz,t
, there is a positive probability of default. Thus, the bankt

agrees to lend only if the amount of debt is no greater than

bz,t
Rt

given that the loan rate is equal to themarket rate for safe assets, 1 + rt . In the general case in which the loan rate can be set larger than 1 + rt ,b

the bank may agree to lend a larger amount than Rz,tas long as the expected rate of return is no smallertthan 1 + rt . In this case, the upper limit of debt can be larger than

bz,t
.
Rt

However, even with a larger

upper limit, the analyses and results in this study would not change qualitatively.

10

occurs because when bt−1 ≤ bz (xt−1 ), the ﬁrm’s dividend is non-negative (π(xt ) ≥ 0), as is
the continuation value (V (bt−1 , xt ) ≥ 0). Thus, as long as bt−1 ≤ bz (xt−1 ), the ﬁrm neverdefaults on its inter-period debt bt−1 . The ﬁrm has the chance to default on intra-period
debt σt at the end of period t, which we analyze in the next subsection, in which theborrowing constraint (4) for σt is given as the no-default condition. Thus, the ﬁrm does
not default on σt in equilibrium.
Timing of events:

The events in a given period t occur in the following way. The ﬁrm

and bank enter period t with outstanding debt of bt−1 .7 At the beginning of the period, theﬁrm has the chance to default on bt−1 , and it will do so if the continuation value is negative(which never happens because bt−1 ≤ bz (xt−1 )). Then, the ﬁrm borrows intra-period debt
σt , employs labor and capital by paying σt , and produces output f (σt , xt ). The ﬁrm
repays bt−1 and borrows new inter-period debt Rbtt by paying bt−1 − Rbtt . Finally, it repays
intra-period debt σt to the bank. At this point, the ﬁrm has the chance to default on σt .
After repaying σt , the ﬁrm pays out the remaining amount, πt = f (σt , xt ) − σt − bt−1 + Rbtt ,to the ﬁrm owner as a dividend.

2.2

Derivation of the borrowing constraint

In this subsection, we describe the events that follow a counterfactual default on σt andderive the borrowing constraint (4) as the no-default condition. Our argument is similar
to that of Jermann and Quadrini (2012).
As described in the previous subsection, the ﬁrm owes inter-period debt Rbtt and intraperiod debt σt at the end of period t, where bt is to be repaid in period t + 1 and σt is tobe repaid in period t. At the end of period t, the ﬁrm has the chance to default on σt .
Now, we consider what would happen if the ﬁrm defaults on σt . Once the ﬁrm defaults,
the bank unilaterally seizes a part of the ﬁrm’s revenue, ϕf (σt , xt ), where 0 ≤ ϕ < 1.8 The
seized amount, ϕf (σt , xt ), may be interpreted as a collateral that the bank can legitimately
7bMore accurately, the ﬁrm owes (1 + rt−1 ) Rt−1
to the bank. Hence, the ﬁrm has to pay this amount tot−1b

the bank, whereas it obtains a transfer from the government as a tax advantage, amounting to τ rt−1 Rt−1
.
t−1
b

b

Thus, the net payment by the ﬁrm is (1 + rt−1 ) Rt−1
− τ rt−1 Rt−1
= bt−1 .
t−1
t−1
bt
8
Because the ﬁrm has paid bt−1 − Rt , the remaining value of the resources it possesses is f (σt , xt ) −bt−1 + Rbtt after defaulting on σt . Thus, if the bank were to seize ϕf (σ, x) from the remaining output only,
then the seizure should have been feasible only if
ϕf (σt , xt ) ≤ f (σt , xt ) − bt−1 +

bt
.
Rt

(10)

However, we assume that the bank can take ϕf (σ, x) from the ﬁrm owner’s pocket and not just from theremaining output of the ﬁrm. Because the ﬁrm owner does not have liquid assets in hand, the bank may
be unable to collect ϕf (σ, x) immediately, whereas it can recover this present value from the ﬁrm owner’s
illiquid assets within some ﬁnite periods. Thus, in this study, we assume that the bank seizure is notconstrained by (10).

11seize when the ﬁrm defaults. Then, the ﬁrm and bank renegotiate over the conditions forthe ﬁrm to continue to operate. Following Jermann and Quadrini (2012), we assume thatthe ﬁrm has all of the bargaining power in the renegotiation. At this stage, the bank has
acquired the right to liquidate the ﬁrm. Here, liquidation is the bank’s taking control of
the ﬁrm. When the bank chooses liquidation, it successfully operates the ﬁrm by itself andrecovers value St with probability ξ, whereas the ﬁrm is destroyed with probability 1 − ξ.
Thus, the expected value that the bank can obtain by liquidation is ξSt . By contrast, if
the bank decides to allow the ﬁrm to continue to operate, it can recover its inter-period
debt in the next period, the present value of which is Rbtt . The renegotiation agreement
depends on whether ξSt is larger or smaller than Rbtt .
• Case where ξSt > Rbtt :The ﬁrm has to make a payment that leaves the bankindiﬀerent between liquidation and allowing the ﬁrm to continue to operate. Thus,
the ﬁrm has to make payment ξSt − Rbtt and promise to pay (1+rt ) Rbtt at the beginningof the next period. Therefore, the ex-post default value for the ﬁrm is


bt
bt
(1 − ϕ)f (σt , xt ) − bt−1 +− ξS(xt ) −+ Et [mt+1 V (bt , xt+1 )] .
Rt
Rt
• Case where ξSt ≤ Rbtt : In this case, the optimal choice for the bank is to wait until
the next period, when (1 + rt ) Rbtt is due. In period t, the bank receives no further
payments. Thus, the ex-post default value is(1 − ϕ)f (σt , xt ) − bt−1 +bt+ Et [mt+1 V (bt , xt+1 )] .
Rt

Therefore, the default value is expressed as


bt
bt
− max ξS(xt ) −, 0 + Et [mt+1 V (bt , xt+1 )] .
(1 − ϕ)f (σt , xt ) − bt−1 +Rt
Rt
Enforcement requires that the value of not defaulting is no smaller than the value of
defaulting, that is,



bt
bt
bt
− σt ≥ (1 − ϕ)f (σt , xt ) − bt−1 +− max ξS(xt ) −, 0 ,f (σt , xt ) − bt−1 +
Rt
Rt
Rt

which can be rearranged as (4).

2.3

Equilibrium debt dynamics

In this subsection, for a given variable qt , we denote the variables in the previous period,in the current period, and in the next period by q−1 , q, and q+1 , respectively. Here,we characterize the equilibrium path that solves (6), taking S(x) as given exogenously.
We prove the existence of the equilibrium that satisﬁes (7) in Section 2.4. Let G be thefollowing closed, bounded, and convex set of non-negative continuous functions, S(xt ):G = {S(x)|0 ≤ S(x) ≤ ω̄, x ∈ Λ, S(x) ∈ C(Λ)}.
12

Proposition 1. Let S(x) (∈ G) be given exogenously. There exists a solution V (b, x; S)
to the Bellman equation (6), and V (b, x; S) is continuous in (b, x).
The proof is omitted because this proposition follows directly from Theorem 9.6 in
Stokey and Lucas with Prescott (1989).9
Here, we brieﬂy describe the outline of the equilibrium dynamics. The detailed description is presented in Appendix A.
Given S(xt ), there exists thresholds B ce (x; S), Bz (x; S), B c (x; S), B̄c (x; S), and B̄z (x)such that
0 < B ce (x; S) < Bz (x; S) < B c (x; S) ≤ B̄c (x; S) ≤ B̄z (x).
• If the initial debt b−1 satisﬁes b−1 ≤ B ce (x; S), then the economy stays in theconstrained-eﬃcient equilibrium, where σt = σ ce (xt ), bt = bce (xt ), and πt ≥ 0 for
all t ≥ 0. σ ce (x) and bce (x) are deﬁned in Appendix A. In the constrained-eﬃcientequilibrium, bce (x) is determined such that the marginal gain from the tax advantageof additional borrowing equals the marginal cost from tightening the borrowingµconstraint, i.e., E mm+1 rτ
R = R , where µ is the Lagrange multiplier associated withthe borrowing constraint.
• If B ce (x; S) < b−1 ≤ Bz (x; S), the ﬁrm repays as much debt as possible by setting
the dividend to zero, i.e., πt = 0, to move into the constrained-eﬃcient equilibrium
as quickly as possible. The borrowing constraint is σ ≤ ϕf (σ, x) + ξS(x) − Rb andthis is tighter than in the constrained-eﬃcient equilibrium. Thus, σt < σ ce (xt ).
• If Bz (x; S) < b−1 , the borrowing constraint becomes σ ≤ ϕf (σ, x), implying thatσt = σz (xt ).
– If Bz (x; S) < b−1 ≤ B c (x; S), the ﬁrm sets πt = 0 to reduce the remaining debt
as quickly as possible.
– If b−1 ∈ (B̄c (x; S), B̄z (x)), then the ﬁrm intentionally borrows to increase thedebt to bzR(x) . This is because the marginal gain of additional borrowing fromthe tax advantage is strictly larger than the marginal cost from tightening theborrowing constraint. Note that b−1 ∈ (B̄c (x; S), B̄z (x)) may not exist, as
B̄c (x; S) may be equal to B̄z (x) for a certain x in this stochastic case.
9

To see that Theorem 9.6 in Stokey and Lucas with Prescott (1989) is applicable to the problem in(6), it is useful to change the variables by m̃t ≡ β −t mt and Ṽ (bt−1 , xt ) ≡ m̃t V (bt−1 , xt ). The Bellman
equation (6) can be rewritten as[]Ṽ (bt−1 , xt ) = max m̃t πt + βEt Ṽ (bt , xt+1 ) ,bt ,σt ,πt

subject to the same constraints. Assumptions 9.4–9.7 in Stokey and Lucas with Prescott (1989) are clearlysatisﬁed in this problem, which allows their Theorem 9.6 to apply.

13

Debt-ridden ﬁrms:

A debt-ridden ﬁrm is deﬁned as a ﬁrm with outstanding debt

bt−1 = bz (xt−1 ). From the deﬁnition of bz (xt ), the borrowing constraint is σt ≤ ϕf (σt , xt )for all t in the case where xt evolves deterministically. In the case where xt evolves
stochastically, there may be a non-zero probability with which the ﬁrm repays its debt
and eventually returns to the constrained-eﬃcient equilibrium.

2.4

Existence of an equilibrium

The Schauder ﬁxed point theorem implies the existence of an equilibrium in which theﬁrm solves (6) and the equilibrium condition (7) is satisﬁed.
ηProposition 2. Let the parameters satisfy 2−η≤ ϕ. Let Assumptions 1 and 2 in Appendix

A and Assumption 3 in Appendix B be satisﬁed. There exists an equilibrium in which theﬁrm solves the optimization problem (6) taking S(x) as given and (7) is satisﬁed.
The proof is given in Appendix B. We have established the existence of an equilibrium
but not the uniqueness. It may not be possible to show such uniqueness analytically.
However, in the deterministic case, where state xt evolves deterministically, we can use
a numerical simulation to show that the equilibrium is unique for a range of relevant
parameter values.

2.5

A deterministic caseIn this subsection, we focus on the deterministic equilibrium, where state xt evolves deterministically. As shown in Proposition 2, S(x) is given endogenously by (7). In thedeterministic case, the state of nature can be indicated by time. Thus, in this subsection,
we use the time subscript t to represent state xt . The detailed description is presented inAppendix C.
Although the equilibrium dynamics in the deterministic case are similar to those in
the stochastic case, two features are unique to the deterministic case.
• First, once a ﬁrm falls into the debt-ridden state, it stays there forever. That is,once bt = bz,t for a certain t, then bt+j = bz,t+j and σt+j = σz,t+j for all j ≥ 0.
• Second, we can show B̄c,t is strictly smaller than bz,t , where bz,t = B̄z,t in the deterministic case (see Appendices A and C). Thus, there always exists bt−1 ∈ (B̄c,t , bz,t−1 )such that if the initial debt is this bt−1 , the ﬁrm intentionally borrows to increaseb

the debt to Rz,tt and stays debt-ridden permanently.
It is likely that
Bc,t = B̄c,t ≡ Bc,tfor a wide range of parameter values. Thus, we assume this in what follows.
14

Why does a heavily indebted ﬁrm choose to become debt-ridden? Intuitively, suppose,for simplicity, that all prices are constant over time. Then, suppose the initial debt b islarge, meaning that RξS < b < bz . Then, it takes n periods to reduce the debt to RξS if
the ﬁrm continues repaying as much as possible in every period. The value of n is uniquelydetermined as a function of b (i.e., n = n(b)), n is increasing in b, and limb↑bz n(b) = ∞.
Now, we can calculate the gain and loss of choosing to become debt-ridden at t = 0 incomparison with choosing to repay as much debt as possible. It is shown that the gain is
1
proportional to R1n , whereas the loss is proportional to (1+r)
n , where n = n(b). We know

1 + r > R because τ > 0. Therefore, there exists nc such that if n > nc , then the gainof choosing to be debt-ridden exceeds the loss. We deﬁne Bc by nc = n(Bc ). Then, theoptimal choice for a ﬁrm with b (> Bc ) is to become debt-ridden.
Policy functions and value function:

Figure 1 shows the policy functions bt =b(bt−1 ) and σt = σ(bt−1 ) and the value function Vt = V (bt−1 ) in the deterministic case inwhich the prices are invariant over time. The policy functions and value function have akink at bt−1 = Bz = (1 − ϕ)f (σz ) + ξS, which is the boundary of the borrowing constraint
between σt ≤ ϕf (σt ) and σt ≤ ϕf (σt ) + ξS − bRt . The policy function b(bt−1 ) showsthat debt decreases rapidly in the region where bt−1 ≤ Bz , whereas in the region whereBz < bt−1 ≤ Bc , the slope of the graph bt = b(bt−1 ) is R, which is close to the slope of
the 45-degree line in the standard parameter setting. Thus, the speed of the decrease in
debt is extremely slow in the region where Bz < bt−1 ≤ Bc . This ﬁgure indicates that theeconomy can suﬀer from extremely persistent ineﬃciency if bt−1 falls into the region whereBz < bt−1 ≤ Bc . Debt jumps to bz and stays there permanently for bt−1 > Bc . The valuefunction has a kink at bt−1 = Bc . V (bt−1 ) > 0 for Bc < bt−1 < bz and V (bz ) = 0 because
V (bt−1 ) = bz − bt−1 in this case (see Appendix C). The policy function σ(bt−1 ) showsthat production becomes ineﬃcient when the dividend is 0. Production becomes the mostineﬃcient (i.e., σt = σz ) when bt−1 ≥ Bz . σt stays at σz permanently for bt−1 > Bc .
A note on the kinks and jumps in the policy function:

As Figure 1 shows, thepolicy function σt = σ(bt−1 ) in this example is continuous at bt−1 = Bz even though itηη
has a kink. However, it is continuous only because we assume 2−η≤ ϕ. If ϕ < 2−η, thepolicy function is no longer continuous, and it has a jump at bt−1 = Bz . The existenceof the jump is explained as follows. The binding non-negativity condition πt = 0 and theborrowing constraint (σt = ϕf (σt ) + ξS − bRt ) imply that σt solves the following equation:bt−1 − ξS = (1 + ϕ)f (σt ) − 2σt ,which may have two solutions. For bt−1 = Bz = (1−ϕ)f (σz )+ξS, there exist two solutions,ηη
σ1 and σ2 , such that σ1 = σz and σ2 > σ1 if ϕ < 2−η. If ϕ > 2−η, then σ2 < σ1 = σz , and15

Figure 1: Policy functions and value function

16

η
η
if ϕ = 2−η
, then σ2 = σ1 = σz . Therefore, if ϕ < 2−η, the policy function σ(bt−1 ) jumps

from σz to σ2 (> σz ) as bt−1 decreases slightly from Bz . Because the continuity of σ(bt−1 )ηat bt−1 = Bz is necessary to prove Proposition 2, we assume 2−η≤ ϕ.

3Full model

In this section, we embed the partial equilibrium model of borrowing constraints into a
general equilibrium model of endogenous growth. We consider a closed economy in which
the ﬁnal good is produced competitively from varieties of intermediate goods. The ﬁrms
are monopolistic competitors, and they produce their respective varieties of intermediate
goods from the capital and labor inputs. This is a version of the expanding variety model,
in which the new entry of ﬁrms increases aggregate productivity (Rivera-Batiz and Romer,
1991; Acemoglu, 2009). We follow Benassy (1998) in that only labor is used to conduct
R&D activities that expand the variety of goods. We assume that the monopolisticallycompetitive ﬁrms, which are subject to borrowing constraints, produce intermediate goods
and conduct R&D activities.

3.1

Basic setup

A representative household owns a mass of ﬁrms, indexed by i ∈ [0, Nt−1 ], that produce
intermediate goods, where Nt−1 measures the varieties of intermediate goods in period t.
Firm i produces variety i monopolistically and can borrow funds from the household. In
what follows, we omit the bank for simplicity. The ﬁnal good is produced competitively
from intermediate goods yi,t by the following production function:
Z Nt−1
Yt =
0

 η1

η
yi,t
di

,

where 0 < η < 1. Because the ﬁnal good producer maximizes Yt −R Nt−10pi,t yi,t di, where

pi,t is the real price of intermediate good i, perfect competition in the ﬁnal goods market
implies thatη−1pi,t = p(yi,t ) = At yi,t,where At ≡ Yt1−η . Firm i produces intermediate good i from capital ki,t and labor li,p,t by
the following production function:
α 1−αyi,t = ki,tli,p,t .

Each ﬁrm i employs labor li,t and capital ki,t , produces intermediate goods yi,t from
li,p,t (≤ li,t ) and ki,t , and conducts R&D with labor input li,t − li,p,t . The R&D activity
creates κN̄t−1 (li,t − li,p,t ) units of new varieties of intermediate goods, where κ is the17parameter that represents the eﬃciency of R&D activity and N̄t−1 is the social level of thevariety of the good, which represents the externality from the stock of knowledge on the
R&D activity. This externality ensures the existence of the balanced growth path (BGP).
The law of motion for the measure of varieties is written as follows:
Nt = Nt−1 + κN̄t−1 (Lt − Lp,t ),RN
RN
where Lt = 0 t−1 li,t di and Lp,t = 0 t−1 li,p,t di. When a new variety is created, a newmonopolistic ﬁrm that produces the variety is also born. Each new variety is produced by a
newborn ﬁrm. The parent ﬁrm creates newborn ﬁrms and then treats them as members of
its own dynasty. As the parent and newborn ﬁrms are technologically identical, the burdenof inter-period debt for the parent ﬁrm is shared equally by all ﬁrms in the dynasty. InNt−1
this general equilibrium model, the state of nature xt is given by xt = (Nt−1 , {bi,t−1 }i=0)because prices are the equilibrium outcomes. Exogenous redistribution shocks may hit
Nt−1
the distribution of debt, {bi,t−1 }i=0
, and make xt evolve stochastically. The transition ofstates is a Markov process, which is determined endogenously in equilibrium, and is taken
as given by the individual households and ﬁrms. To simplify the notation, we use the time
subscript t instead of the state xt and simply omit xt in what follows on the understandingthat all variables and functions with subscript t depend on state xt . Thus, the value ofthe ﬁrm is determined by the following dynamic programming equation in which we omit
the subscript i for simplicity:
Vt (bt−1 ) = max πt + Et mt+1 1 + κN̄t−1 (lt − lp,t ) Vt+1 (bt ) ,subject to(1−α)η

πt = At ktαη lp,t


 bt
,
− wt lt − rtK kt − bt−1 + 1 + κN̄t−1 (lt − lp,t )Rt(1−α)η

wt lt + rtK kt ≤ ϕAt ktαη lp,t






bt
+ max ξSt −, 0 1 + κN̄t−1 (lt − lp,t ) ,
Rt

(11)
(12)

πt ≥ 0,(13)

lt ≥ lp,t ,(14)

bt ≤ bz,t ,(15)

where St and bz,t are taken as given and are the equilibrium outcomes, as we see shortly.
The Lagrange multipliers, λt , µt , λπ,t , and λl,t , are associated with the budget constraint(11), the borrowing constraint (12), the non-negativity of dividends (13), and the nonnegativity of the labor input for R&D (14), respectively. In equilibrium, the liquidation
value St and the upper limit of borrowing bz,t are given in the same manner as in theprevious section. Thus, the equilibrium condition that determines St is
St = max Et [mt+1 Vt+1 (b)] +
b

18

b
.
Rt

The value of bz,t = bz (xt ) is given bybz (xt−1 ) =inf(1 − ϕ)ft (σz (xt ), xt ) +xt ∈Λ(xt−1 )bz (xt )
,
Rt

(16)

where f (σ, x) and σz (x) are the same as in Section 2, and we assume the parameter
restriction (9).
A representative household solves the following problem:"∞#X
max E0β t U (Ct , Lt ) ,Ct ,Lt ,Bt ,Kt

t=0

subject to the budget constraint
Ct + Kt +Bt≤ wt Lt + (rtK + 1 − ρ)Kt−1 + Bt−1 + Tt ,1 + rtwhere β is the subjective discount factor, Ct is consumption, Lt is total labor supply, Kt iscapital stock, ρ is the depreciation rate of capital, Bt is inter-period lending to the ﬁrms,
and Tt is a lump-sum transfer that consists of the tax and dividends. The period utility is

U (C, L) =h 1
γ i1−θ
C 1+γ (1 − L) 1+γ
1−θ.

Let mt be the Lagrange multiplier associated with the budget constraint for the representative household, which is given by the FOC with respect to Ct :mt+1 =

β t+1 ∂U (Ct+1 , Lt+1 )/∂Ct+1
β t ∂U (Ct , Lt )/∂Ct

The FOC with respect to Kt and Bt implies"#
11= Et K= Et [mt+1 ] .
1 + rtrt+1 + 1 − ρThus, mt is the stochastic discount factor in the partial equilibrium model of Section
RN
2. The market-clearing conditions are Ct + Kt − (1 − ρ)Kt−1 = Yt , 0 t−1 li,t di = Lt ,R Nt−1 bi,t
R Nt−1
R Nt−1
Bt
k
di
=
K
,
l
di
=
L
,
i,t
t−1
i,p,t
p,t
Rt di = 1+rt , Nt = Nt−1 +κN̄t−1 (Lt −Lp,t ).
000The equilibrium condition is N̄t = Nt .
Competitive equilibrium.— A competitive equilibrium consists of sequences of prices{rt , rtK , wt , mt }, a household’s decisions {Ct , Lt , Kt , Bt }, ﬁrms’ decisions {πt , lt , lp,t , kt , bt },and a measure of varieties Nt , such that (i) the representative household and ﬁrms solve
their respective optimization problems, taking prices and Nt−1 as given, and (ii) themarket-clearing conditions and equilibrium condition are all satisﬁed.
A ﬁrm that owes the maximum amount bz,t is called a debt-ridden ﬁrm in a similarmanner to the previous section.

19

3.2

BGP without debt-ridden ﬁrms

In what follows, we assume for simplicity that θ → 1 andU (C, L) = ln C + γ ln(1 − L).
In this subsection, we focus on the deterministic case where state xt evolves deterministically and then characterize the BGP on which all ﬁrms are normal. The ﬁrms are normalwhen the dividend is positive, that is, πt > 0 for all t. On the BGP, labor and the growthrate are constant, that is, Lt = L and Nt /Nt−1 = g. We deﬁne
E≡

1−η
.
(1 − α)η
(1−η)E

E , A = A × N
We guess that Yt = Y × Nt−1
tt−1

E , K = K × NE,, Ct = C × Nt−1
t
t

E−1
E−1
E−1
E , l = L/N
wt = w × Nt−1tt−1 , kt = k × Nt−1 , Vt = V × Nt−1 , and πt = π × Nt−1 . The

FOCs and constraints imply that there exists a unique BGP, which is given in Appendix
D. For the numerical simulation, we set the parameters to the values for Japan, theUnited States, and the European Union (EU),10 shown in Table 1. The parameters of theborrowing constraints, ϕ and ξ, and the size of shock z are calibrated to ﬁt the data ofJapan, the United States, and the EU according to the data in Appendix E and the method
described in Appendix F, where we identify and assess the parameter region used in our
numerical experiments. It is an annual model. We adopt a simulation strategy underwhich the parameters γ and κ are calculated endogenously such that the two variables L
and gT F P attain the target values calculated from the data. The economic growth rate on
the BGP is 2.51% per year for the Japanese economy.11 It is easily conﬁrmed numericallythat there exists a BGP for a wide range of parameter settings.
A debt-ridden ﬁrm does not conduct R&D activities on the BGP under a wide rangeof parameter settings. See Claim 3 in Appendix G for details.

3.3

Low growth equilibrium with debt-ridden ﬁrms

Now, we consider the equilibrium in which some ﬁrms are debt-ridden and others arenormal. In this subsection, we again focus on the deterministic case in which state xt
evolves deterministically, and bz,t = bz (x) = bz is given and determined by the steadystate.12 We assume that ﬁrms i ∈ [0, Zt ] are debt-ridden and ﬁrms i ∈ (Zt , Nt ] are normal.
10

The EU comprises the following 28 countries: Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech

Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Ireland, Italy, Latvia, Lithuania,
Luxembourg, Malta, the Netherlands, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, andthe United Kingdom.
1−η

11

η

1

1−η
The economic growth rate on the BGP is given by g E = gT(1−α)η
= gT1−αF P = 1.0251, where g is theFP1−ηgrowth rate of Nt and E = (1−α)η
. See Appendix H.
12
The upper bound of debt constraint (15) is always binding for debt-ridden ﬁrms. Their debt bz isconstant over time.

20Common parameters
Parameter

Economic interpretation

value

β

the subjective discount factor

0.98

ρ

the depreciation rate of capital

0.1

ηthe parameter for the aggregation function

0.7Country-speciﬁc parameters
Parameter

Economic interpretation

Japan

US

EU

α

the share of labor in production

0.31

0.34

0.37

γ

the inverse of the elasticity of labor supply

2.49

3.25

3.60

κ

the eﬃciency of R&D

0.55

0.42

0.39

ϕ

the collateral ratio of revenue

0.51

0.64

0.63

τ

the tax advantage for debt

0.30

0.35

0.35

ξ

the collateral ratio of the foreclosure value

0.08

0.94

0.92

L

the total labor supply in the steady state

0.25

0.21

0.19

gT F P

the growth rate of TFP in the steady state

1.017

1.013

1.011

z

the ratio of debt-ridden ﬁrms at the shock period

0.880

0.649

0.701

Table 1: Parameter settings
We assume and justify numerically in the following simulation that only normal ﬁrms
conduct conduct the R&D activity and debt-ridden ﬁrms do not on the entire equilibrium
path.
Zt. The initial value of zt , denoted by z, is given exogenously. Although zt
Deﬁne zt ≡ N
teventually converges to zero at BGP, the fraction of ﬁrms that can conduct R&D decreases
to (1−zt ), so the new entry ﬁrms decrease, and zt decreases only slowly. In our simulation,b

we assume z < 1 and Rz,tt > ξSt .13
Numerical experiment: We can calculate the equilibrium dynamics numerically using
a full non-linear method. Linearization is not necessary for the deterministic simulation(see Appendix H for detrending and Appendix I for the details of the calculation of the
dynamics).
Figure 2 shows the results of the numerical simulation in which the economy is initiallyon the BGP, where Zt = 0, and an unexpected redistribution shock hits the economyb

13

We cannot ﬁnd the parameter set that makes z = 1 and Rz,t> ξSt simultaneously. If both z = 1 and
t

bz,t
Rt

> ξSt hold simultaneously, there would exist a zero growth path on which all ﬁrms are debt-ridden

and the economic growth rate is zero, because no ﬁrms conduct R&D activities. In a simpliﬁed modelin which capital k does not exist and labor is the only input, it is proven analytically that z = 1 andbz,tRt> ξSt cannot hold simultaneously for any parameter set.

21

in period 10, making z10 = z = 0.880. In other words, the sudden redistribution of
wealth from ﬁrms to households makes 88.0% of all ﬁrms debt-ridden in period 10. The
parameter values for Figure 2 are given as those for Japan in Table 1. In this counterfactual
experiment, we assume that all stochastic shocks except redistributive shocks are zero.
Counterfactually, Japan’s long-term stagnation after the collapse of the bubble economy
was caused solely by the redistribution shock. It should be noted that the size of theredistribution shock is large because other shocks are assumed to be zero. However, while
we argue that the mechanism due to the redistribution shock is important, we do notbelieve this shock to be the single cause. The numerical experiment is intended to showthat excessive debt can cause long-term stagnation. A richer model, i.e., a medium-scale
DSGE model, would allow for experiments that take into account other shocks, but themechanism of the redistribution shock would be too complex, so we use the simple model
to consider the results of our counterfactual experiments. The features of the equilibrium
path shown in Figure 2 are as follows:• Slowdown of economic growth: Borrowing constraints are tighter after the buildup
of debt. Thus, aggregate inputs decrease and economic growth slows for an extended
period.
• Persistently lower rates of interest and wages: These features are observed in theaftermath of the Great Recession and are the focus of the recent literature on secularstagnation and unconventional monetary policy.
• Decrease in TFP and net entry: The growth rate of the number of ﬁrms, gt =Nt /Nt−1 , decreases. TFP also slows, where TFP in the model is deﬁned by
T F Pt =Yt.
αKt−1 Lt1−αThis feature is consistent with the observation that TFP and the net entry of ﬁrms
decreased in Japan in the 1990s.
• Buildup of NPLs: In this example, there are Zt debt-ridden ﬁrms, and their debt
stays at an ineﬃciently high level. This feature is consistent with the historicalepisodes of persistent stagnation with overly indebted ﬁrms and/or households, such
as Japan in the 1990s.
• Labor wedge reduction: In this example, the labor wedge, 1 − τL , diminishes persistently as a direct consequence of the tightening of the aggregate borrowing constraint
on working capital loans for wage payments. This tighter borrowing constraint creates a larger gap between the wage rate and marginal product of labor. The gap

22

1
0.8

0.4

0.8

0.35

0.75

0.6

0.7

0.3
0.4

0.65
0.25

0.2
0

0.6
0

10

20

30

0.2

0.26

0.18

0.24

0.17

0.22

0.16

0.2

0.15

0.18

0.14

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

1
0.9

0.16

0

10

20

30

0.13

0.8
0.7
0

10

20

30

0.05

0.4

0.24
0.38

0.04
0.22

0.36
0.03
0.34

0.2
0.02

0.32

0.18
0

10

20

30

0.01

0

10

20

30

1.4

0.45

1.04

Simulation
Trend

0.4

Simulation
Trend

1.035

1.2

0.35

0.3

1.03

0.3

1

1.025

0.25
0.2

1.02

0.8
0

10

20

30

0

10

20

30

Figure 2: Responses to a buildup of debt (Japan)23

is measured by τL . In this way, the persistent reduction in the labor wedge observed in the aftermath of a ﬁnancial crisis can be accounted for by the emergenceof debt-ridden ﬁrms.14
The redistribution shock causes a fraction of zt ﬁrms to become debt-ridden ﬁrms. Theseﬁrms are permanently ineﬃcient in their production and do not conduct R&D. Normal
ﬁrms, which are not directly aﬀected by the redistribution shock, are also negatively affected by the production ineﬃciency of debt-ridden ﬁrms. Production ineﬃciency reduces
aggregate demand and lowers prices. As a result, the sales of normal ﬁrms decrease, andthe borrowing constraint becomes tighter than before the shock. As a result, R&D by
normal ﬁrms also are sluggish, and the new entry of ﬁrms decreases, resulting in a decline
in the growth rate of TFP.
Next, we calibrate and conduct numerical simulations for the United States and the
EU. Table 1 provides the parameter values. Figure 3 compares TFP in the model withthe actual TFP in Japan, the United States, and the EU. Similarly, Figure 4 compares the
GDP of the numerical experiment with the actual real GDP per capita of the working age
population (15–64 years old). We assume that the unexpected shock hits the economy inperiod 10 of the simulation, which corresponds to the asset-bubble collapse in 1991 in thecase of Japan and the ﬁnancial crisis in 2009 in the cases of the United States and the EU.
For the United States and the EU, we extend the observed variables to 2025. We positthat the variables grow in future periods by constant growth rates, which are equal to theaverage growth rates in 2011–2016 for the United States and the EU. These extensions
are based on the implicit assumption that the US and EU economies have fallen into
decade-long stagnation. We compare our simulation results with the extended data on the
United States and the EU because a goal of our numerical experiment is to examine thecapability of our model to account for decade-long recessions in the aftermath of ﬁnancial
crises. The ﬁgures show that the model ﬁts the growth rate data fairly well. Figure 5
compares the labor wedge of the numerical experiment with the actual labor wedge. The
simulated deterioration of the labor wedge is fairly consistent with the data from Japan,
the United States, and the EU.
In any case, the numerical simulation of our model shows the overall slowdown of
14

M RSt
, where M RSt =As Chari et al. (2007) posit, the labor wedge, 1 − τL,t , is deﬁned by 1 − τL,t = M
P Lt

γCt
1−Lt

t
t Lt
= wt and M P Lt = αYin our model. Thus, the labor wedge can be calculated by 1 − τL,t = wαY
. In
Lt
t

our model, the labor wedge 1−τL is proportional to the labor share. Thus, both an economic slowdown anda shrinkage of the labor share (from the buildup of debt) are observed simultaneously in our model. Thisfeature of our model contrasts with the countercyclicality of the labor share in business cycle frequencies(Schneider, 2011). However, our model seems compatible with countercyclicality in the short run. In
our model, the buildup of debt causes the long-term variations in the labor wedge, whereas short-run
countercyclicality can be caused by factors such as productivity shocks in business cycle frequencies (Rı́osRull and Santaeulàlia-Llopis, 2010).

24

economic growth resulting from the debt shock, indicating the usefulness of our model inaccounting for persistent recessions in the aftermath of ﬁnancial crises.

3.4

Policy implications

The policy implications of the results presented in this paper seem noteworthy from apractical point of view. The shocks that cause persistent stagnation are exogenous technological changes in the existing literature. In our model, the one-time buildup of debt
tightens the borrowing constraint and causes a persistent slowdown in economic growtheven though there is no technological change. Thus, our model implies that reducing
overly accumulated debt can restore economic growth. Note that the physical liquidationof debt-ridden borrowers is not necessary, but relieving them from excessive debt restores
their eﬃciency and high economic growth at the aggregate level. This policy implication
contrasts sharply with those of prior studies, in which debt reduction, per se, is not on
the table, and policymakers can only mitigate recessions by implementing accommodative
monetary and ﬁscal policies or designing ex-ante ﬁnancial regulations.
Government intervention in debt reduction is justiﬁed as follows. In our model, theineﬃciency of persistent stagnation cannot be resolved by the market mechanism for thefollowing three reasons. First, constraint (5) implies that outside investors cannot relax theborrowing constraint of debt-ridden ﬁrms by purchasing new equity. Constraint (5) is an
exogenous assumption in this analysis, but it may be justiﬁed by plausible market frictions
such as a lack of commitment and coordination failures (see footnote 5). If constraint (5)did not exist, outside investors would purchase new equity of debt-ridden ﬁrms and makethem constrained-eﬃcient because they can earn strictly positive proﬁts by investing newmoney in debt-ridden ﬁrms. Second, there is no free entry of new ﬁrms in our model. The
entry of a new ﬁrm is equal to that of a new variety, which occurs as a result of R&D
activities by incumbent ﬁrms. Thus, new entries of new varieties decrease as debt-ridden
ﬁrms increase, and the growth in productivity slows. The productivity slowdown continues
persistently because there is no entry by outside ﬁrms.15 Third, it is optimal for lenders tokeep borrowers debt-ridden if the outstanding debt is large. Because lenders can be repaid
in full even when borrowers are debt-ridden, they have no incentive to reduce their loans,
whereas their inaction protracts aggregate ineﬃciency. Thus, policy interventions by thegovernment can be eﬀective in restoring economic growth by promoting debt restructuring
or wealth redistribution from lenders to borrowers. Policy measures may include regulatory
reforms to make bankruptcy procedures less costly and debtor friendly and to promote
15

The persistence of the productivity slowdown can be preserved even if we relax our assumption of noentry from outsiders as long as the cost of R&D is suﬃciently higher for outsiders than it is for incumbent
ﬁrms. It seems plausible to posit that the cost of R&D is substantially higher for outsiders than forincumbents in any industry.

25

Japan6%, TFP growth rate

TFP (1992 = 1)

1.3
1.2
1.1
1
0.9
0.8
0.7
1980 1985 1990 1995 2000 2005

4
2
0
-2
-4
1980 1985 1990 1995 2000 2005

Year

Year

United States
1.4%, TFP growth rate

TFP (2009 = 1)

1.3

3
Data
Prediction
Simulation

1.2
1.1
1
0.9

2
1
0
-1

0.8
2000 2005 2010 2015 2020 2025

-22000 2005 2010 2015 2020 2025

Year

Year

1.4

2

1.3

1

%, TFP growth rate

TFP (2009 = 1)

EU

1.2
1.1
1
0.9

0
-1
-2
-3

0.8
2000 2005 2010 2015 2020 2025

-42000 2005 2010 2015 2020 2025

Year

Year

Figure 3: TFP for Japan, the United States, and the EU: Comparison between the data
and the simulation
Note: In Japan, TFP is classiﬁed as the “market economy” sectors, which excludes education, medical
services, government activities, and imputed housing rent.
Sources: Our calculation; The Research Institute of Economy, Trade and Industry, JIP 2014 database;
Fernald (2012); European Commission, AMECO

26

Japan%, GDP per capita growth rate

GDP per capita (1992 = 1)

1.6
1.4
1.2
1
0.8
0.6
1980 1985 1990 1995 2000 2005

10
5
0
-5
-10
-15
-20
1980 1985 1990 1995 2000 2005

Year

Year

GDP per capita (2009 = 1)

1.6
1.4

Data
Prediction
Simulation

1.2
1
0.8
0.6
2000 2005 2010 2015 2020 2025%, GDP per capita growth rate

United States
50-52000 2005 2010 2015 2020 2025

Year

Year

GDP per capita (2009 = 1)

1.6
1.4
1.2
1
0.8
0.6
2000 2005 2010 2015 2020 2025

Year%, GDP per capita growth rate

EU
4
2
0
-2
-4
-6
2000 2005 2010 2015 2020 2025

Year

Figure 4: GDP for Japan, the United States, and the EU: Comparison between the data
and the simulation
Sources:

Our calculation; Cabinet Oﬃce, Government of Japan, Annual Report on National Accounts;

Statistics Bureau of Japan, Labour Force Survey; U.S. Bureau of Economic Analysis, National Income andProduct Accounts; U.S. Bureau of Labor Statistics, “Current Employment Status”; European Commission,
AMECO

27

1
0.9
0.8
0.7
0.6
1980

1990

2000

Year

1.2
1.15

United States
Data
Prediction
Simulation

1.1
1.05
1
0.95

Labor wedge (2000:2008 = 1)

Labor wedge (2000:2008 = 1)

Labor wedge (1980:1989 = 1)

Japan

1.1

1.03

EU

1.02
1.01
1
0.99
0.98
0.97

0.9
2000 2005 2010 2015 2020 2025

0.962000 2005 2010 2015 2020 2025

Year

Year

Figure 5: Labor wedge for Japan, the United States, and the EU: Comparison between
the data and the simulation
Sources:

Our calculation; Cabinet Oﬃce, Government of Japan, Annual Report on National Accounts;

Statistics Bureau of Japan, Labour Force Survey; The Research Institute of Economy, Trade, and Industry,JIP 2014 database; U.S. Bureau of Economic Analysis, National Income and Product Accounts; U.S. Bureauof Labor Statistics, Current Employment Status; European Commission, AMECO

debt-for-equity swaps to reduce outstanding debt as well as the injection of funds as a
subsidy or equity to banks that forgive debt and write oﬀ NPLs. The injection of a banksubsidy or equity is usually interpreted as bank recapitalization because the banks become
insolvent in most cases when a substantial number of their borrowers are in distress. This
policy implication is straightforward and robust in our model and seems reasonable fromour experience of Japan’s lost decade of the 1990s, the Great Recession in the UnitedStates, and the subsequent debt crises in Europe, whereas existing models may not clarify
whether borrowers’ relief from excessively accumulated debt is good for an economy hit
by a crisis.

4

Conclusion

Decade-long recessions are often observed after ﬁnancial crises. In particular, the “secularstagnation” hypothesis has drawn much attention recently. In this study, we hypothesizedthat the buildup of large debt in the private sector causes a persistent economic slowdown even without technology shocks. This model may be considered to reﬂect a “debtsupercycle” rather than secular stagnation, as ineﬃciency can continue persistently but isremoved if debt is reduced. Economic agents can become overly indebted because of, forexample, the boom and bust of asset-price bubbles. We showed that borrowers who owe
the maximum repayable debt fall into a debt-ridden state in which they can repay only
the interest and cannot reduce the principal of the debt, which means that they continue
ineﬃcient production forever in the deterministic case.
The emergence of a substantial number of debt-ridden borrowers lowers economic
growth by tightening the aggregate borrowing constraint. This tightening of aggregate
28

borrowing constraints owing to the mass emergence of debt-ridden borrowers may manifest as a “ﬁnancial shock” during or after a ﬁnancial crisis. Because lenders have no
incentive to reduce their loans to debt-ridden borrowers, government intervention to facilitate debt restructuring (i.e., relief for debt-ridden borrowers from their excessive debt)
may be necessary to enhance economic growth in the aftermath of a ﬁnancial crisis. Ourpolicy recommendations are in line with those of partial debt forgiveness by Geanakoplos(2014).
The endogenous borrowing constraint of this study has a unique feature in that debt
that exceeds a threshold generates persistent ineﬃciency. Therefore, it may serve as a
useful building block for business cycle models, thereby enriching aggregate dynamics.
Broader applications are left for future research.

References
Abreu, Dilip and Faruk Gul (2000) “Bargaining and Reputation,” Econometrica, Vol. 68,
No. 1, pp. 85-118, January.
Acemoglu, Daron (2009) Introduction To Modern Economic Growth, Princeton: Princeton
University Press.
Albuquerque, Rui and Hugo A. Hopenhayn (2004) “Optimal Lending Contracts and Firm
Dynamics,” Review of Economic Studies, Vol. 71, No. 2, pp. 285-315, April.
Altavilla, Carlo, Matthieu Darracq Paries, and Giulio Nicoletti (2015) “Loan supply, credit
markets and the euro area ﬁnancial crisis,” Working Paper Series 1861, European Central Bank.
Benassy, Jean-Pascal (1998) “Is there always too little research in endogenous growth withexpanding product variety?” European Economic Review, Vol. 42, No. 1, pp. 61 - 69,
URL: http://www.sciencedirect.com/science/article/pii/S0014292197000615.
Bernanke, Ben S., Mark Gertler, and Simon Gilchrist (1999) “The Financial Accelerator ina Quantitative Business Cycle Framework,” in Taylor, John B. and Michael Woodford
eds. Handbook of Macroeconomics, Vol. 1: Elsevier, Chap. 21, pp. 1341-1393.
Bianchi, Francesco, Howard Kung, and Gonzalo Morales (2019) “Growth, slowdowns,
and recoveries,”

Journal of Monetary Economics, Vol. 101, pp. 47-63, DOI:

https://doi.org/10.1016/j.jmoneco.2018.07.001.
Brinca, Pedro, V. V. Chari, Patrick J. Kehoe, and Ellen McGrattan (2016) “Accounting for Business Cycles,” in Taylor, John B. and Harald Uhlig eds.

29

Handbook ofMacroeconomics, Vol. 2A, Amsterdam:

Elsevier, Chap. 13, pp. 1013-1063, DOI:

10.1016/bs.hesmac.2016.05.
Caballero, Ricardo J., Takeo Hoshi, and Anil K. Kashyap (2008) “Zombie Lending andDepressed Restructuring in Japan,” American Economic Review, Vol. 98, No. 5, pp.
1943-1977, December.
Carlstrom, Charles T. and Timothy S. Fuerst (1997) “Agency Costs, Net Worth, and Business Fluctuations: A Computable General Equilibrium Analysis,” American Economic
Review, Vol. 87, No. 5, pp. 893-910, December.
Cecchetti, Stephen G., Sunil Mohanty, and Fabrizio Zampolli (2011) “Achieving growthamid ﬁscal imbalances: the real eﬀects of debt,” Proceedings - Economic Policy Symposium — Jackson Hole, pp. 145-196.
Chari, Varadarajan V., Patrick J. Kehoe, and Ellen R. McGrattan (2007) “Business Cycle
Accounting,” Econometrica, Vol. 75, No. 3, pp. 781-836, May.
Christiano, Lawrence J., Roberto Motto, and Massimo Rostagno (2014) “Risk Shocks,”
American Economic Review, Vol. 104, No. 1, pp. 27-65, January.
Christiano, Lawrence J., Martin S. Eichenbaum, and Mathias Trabandt (2015) “Understanding the Great Recession,” American Economic Journal: Macroeconomics, Vol. 7,
No. 1, pp. 110-167, January.
Cole, Harold L. and Lee E. Ohanian (2004) “New Deal Policies and the Persistence of theGreat Depression: A General Equilibrium Analysis,” Journal of Political Economy,
Vol. 112, No. 4, pp. 779-816, August, URL: http://ideas.repec.org/a/ucp/jpolec/
v112y2004i4p779-816.html.
Comin, Diego and Mark Gertler (2006) “Medium-Term Business Cycles,” American Economic Review, Vol. 96, No. 3, pp. 523-551, June.
Cooley, Thomas, Ramon Marimon, and Vincenzo Quadrini (2004) “Aggregate Consequences of Limited Contract Enforceability,” Journal of Political Economy, Vol. 112,
No. 4, pp. 817-847, August.
Duval, Romain A, Gee Hee Hong, and Yannick Timmer (2017) “Financial Frictions and theGreat Productivity Slowdown,” IMF Working Papers 17/129, International Monetary
Fund.
Eggertsson, Gauti B. and Neil R. Mehrotra (2014) “A Model of Secular Stagnation,”
NBER Working Papers 20574, National Bureau of Economic Research, Inc.

30

Fernald, John (2012) “A quarterly, utilization-adjusted series on total factor productivity,”
Working Paper Series 2012-19, Federal Reserve Bank of San Francisco, (updated March2014).
Fukuda, Shin-ichi and Jun-ichi Nakamura (2011) “Why Did ‘Zombie’ Firms Recover inJapan?” The World Economy, Vol. 34, No. 7, pp. 1124-1137, July.
Gale, Douglas and Martin Hellwig (1985) “Incentive-Compatible Debt Contracts: The
One-Period Problem,” Review of Economic Studies, Vol. 52, No. 4, pp. 647-663, October.
Geanakoplos, John (2014) “Leverage, Default, and Forgiveness: Lessons from the American and European Crises,” Journal of Macroeconomics, Vol. 39, No. PB, pp. 313-333.
Giroud, Xavier and Holger M. Mueller (2017) “Firm Leverage, Consumer Demand, andEmployment Losses during the Great Recession,” Quarterly Journal of Economics, Vol.
132, No. 1, pp. 271-316, February.
Gordon, Robert J. (2012) “Is U.S. Economic Growth Over? Faltering Innovation Confronts the Six Headwinds,” NBER Working Papers 18315, National Bureau of Economic
Research, Inc.
Guerrón-Quintana, Pablo and Ryo Jinnai (2014) “Liquidity, trends, and the great recession,” Working Papers 14-24, Federal Reserve Bank of Philadelphia.
Havik, Karel, Kieran Mc Morrow, Fabrice Orlandi, Christophe Planas, Rafal Raciborski,
Werner Roeger, Alessandro Rossi, Anna Thum-Thysen, and Valerie Vandermeulen(2014) “The Production Function Methodology for Calculating Potential Growth Rates& Output Gaps,” European Economy - Economic Papers 535, Directorate General Economic and Financial Aﬀairs (DG ECFIN), European Commission.
Ikeda, Daisuke and Takushi Kurozumi (2014) “Post-Crisis Slow Recovery and Monetary
Policy,” IMES Discussion Paper Series 14-E-16, Institute for Monetary and Economic
Studies, Bank of Japan.
Jermann, Urban J. and Vincenzo Quadrini (2006) “Financial Innovations and Macroeconomic Volatility,” NBER Working Papers, No. 12308, June.
(2007) “Stock market boom and the productivity gains of the 1990s,” Journal of
Monetary Economics, Vol. 54, No. 2, pp. 413-432, March.
(2012) “Macroeconomic Eﬀects of Financial Shocks,” American Economic Review, Vol. 102, No. 1, pp. 238-271, February.

31

Juillard, Michel (1996) “Dynare : a program for the resolution and simulation of dynamic
models with forward variables through the use of a relaxation algorithm,” CEPREMAP
Working Papers (Couverture Orange) 9602, CEPREMAP.
Kaihatsu, Sohei and Takushi Kurozumi (2014) “What caused Japan’s Great Stagnationin the 1990s? Evidence from an estimated DSGE model,” Journal of the Japanese andInternational Economies, Vol. 34, No. C, pp. 217-235.
Kiyotaki, Nobuhiro and John Moore (1997) “Credit Cycles,” Journal of Political Economy, Vol. 105, No. 2, pp. 211-248.
Kobayashi, Keiichiro and Tomoyuki Nakajima (2017) “A Theory of Non-performing Loans
and Debt Restructuring,” mimeo.
Kobayashi, Keiichiro and Daichi Shirai (2016) “Heterogeneity and redistribution in ﬁnancial crises,” Macroeconomic Dynamics, Vol. 20, No. 6, pp. 1527-1549, September.
Krugman, Paul R. (1988) “Financing vs. forgiving a debt overhang,” Journal of Development Economics, Vol. 29, No. 3, pp. 253-268, November.
Lamont, Owen (1995) “Corporate-Debt Overhang and Macroeconomic Expectations,”
American Economic Review, Vol. 85, No. 5, pp. 1106-1117, December.
Lo, Stephanie and Kenneth Rogoﬀ (2015) “Secular stagnation, debt overhang and otherrationales for sluggish growth, six years on,” BIS Working Papers 482, Bank for International Settlements.
Mian, Atif, Amir Suﬁ, and Emil Verner (2017) “Household Debt and Business Cycles
Worldwide,” The Quarterly Journal of Economics, Vol. 132, No. 4, pp. 1755-1817.
Myers, Stewart C. (1977) “Determinants of corporate borrowing,” Journal of Financial
Economics, Vol. 5, No. 2, pp. 147-175, November.
Reinhart, Carmen M. and Vincent R. Reinhart (2010) “After the Fall,” NBER Working
Papers, No. 16334, September.
Reinhart, Carmen M. and Kenneth S. Rogoﬀ (2009) This Time Is Diﬀerent: Eight Centuries of Financial Folly, Princeton: Princeton University Press.
Rı́os-Rull, José-Vı́ctor and Raül Santaeulàlia-Llopis (2010) “Redistributive shocks andproductivity shocks,” Journal of Monetary Economics, Vol. 57, No. 8, pp. 931-948,
November.
Rivera-Batiz, Luis A. and Paul M. Romer (1991) “Economic Integration and Endogenous
Growth,” The Quarterly Journal of Economics, Vol. 106, No. 2, pp. 531-555, May.
32

Rogoﬀ, Kenneth (2016) “Debt Supercycle, Not Secular Stagnation,” in Blanchard, Olivier,Raghuram Rajan, Kenneth Rogoﬀ, and Lawrence H. Summers eds. Progress and Confusion: The State of Macroeconomic Policy: MIT Press, Chap. 2, pp. 19-28.
Rotemberg, Julio J. and Michael Woodford (1997) “An Optimization-based Econometric
Framework for the Evaluation of Monetary Policy,” in Bernanke, Ben S. and Julio J.
Rotemberg eds. NBER Macroeconomics Annual 1997, Cambridge, MA: MIT Press, pp.
297-346.
Schneider, Dorothee (2011) “The Labor Share: A Review of Theory and Evidence,” SFB
649 Discussion Papers SFB649DP2011-069, Humboldt University, Collaborative Research Center 649.
Stokey, Nancy L. and Robert E. Lucas, Jr., with Edward C. Prescott (1989) Recursive
Methods in Economic Dynamics, Cambridge, MA: Harvard University Press.
Summers, Lawrence H. (2013) “IMF Economic Forum: Policy Responses to Crises,” speechat the IMF Fourteenth Annual Research Conference, Washington, DC, 9 November.
Townsend, Robert M. (1979) “Optimal contracts and competitive markets with costly
state veriﬁcation,” Journal of Economic Theory, Vol. 21, No. 2, pp. 265-293, October.

For Online Publication: AppendicesA

Details of equilibrium dynamics in Section 2.3

We make the following two assumptions.
Assumption 1. The parameters satisfy the following condition:ξ ω̄ <(1 − ϕ)fz,
rmax

where fz ≡ inf x∈Λ f (σz (x), x) and rmax ≡ supx∈Λ r(x).
This assumption implies that once the amount of debt Rbtt becomes smaller than ξ ω̄,
it can reach the constrained-eﬃcient level bce (x) (deﬁned below) within a ﬁnite number(1−ϕ)fof periods. This implication is because the ﬁrm can repay at least rmax z by paying all
income to the bank in all future periods. Deﬁneµce (x) ≡τ r(x).
1 + r(x)Then, σ ce (x) is deﬁned as the solution to∂1 + µce (x)f (σ, x) =.
∂σ
1 + ϕµce (x)33Deﬁne bce (x; S), where S = S(x), bybce (x; S)≡ ϕf (σ ce (x), x) − σ ce (x) + ξS(x).
R(x)
Assumption 2. The parameters satisfy the following conditions:
sup ϕf (σ ce (x), x) − σ ce (x) < 0,
x∈Λ
ce

(x;S)
such that b R(x)< ξS(x) for all x, andbce (x; S)> 0.
inff (σ ce (x), x) − σ ce (x) − bce (x−1 ; S) +x,x−1 ∈Λ,S∈G
R(x)The ﬁrst condition of this assumption is satisﬁed for small τ .16 The second condition ofthis assumption implies that once the economy enters the constrained-eﬃcient equilibrium,
it stays there forever. Constrained eﬃciency is deﬁned and explained in the following
paragraph “Constrained-eﬃcient equilibrium with small debt.” We deﬁne the total surplusof the match of the ﬁrm and bank, W (b−1 , x), as
W (b−1 , x) =1 + r−1b−1 + V (b−1 , x).
R−1−1
Note that the bank receives 1+r
R−1 b−1 from the ﬁrm, whereas the net payment for the ﬁrm−1
is b−1 because the government provides it with a tax advantage τRr−1
b−1 . Thus, the ﬁrm’s

problem is equivalent to maximizing W (b−1 , x) given b−1 . Function W (b−1 , x) satisﬁes
the following dynamic programming equation:
i
hm
r−1 τ
+1
W (b, x+1 ) ,b−1 + f (σ, x) − σ + E
σ,b R−1mbsubject to f (σ, x) − σ − b−1 + ≥ 0,Rbσ ≤ ϕf (σ, x) + max ξS(x) − , 0 ,RW (b−1 , x) = maxb ≤ bz (x).

(17)
(18)
(19)
(20)

By denoting the Lagrange multipliers for (18), (19), and (20) as λπ , µ, and ν, respectively,
the FOCs and envelope condition areµ
1 + 1+λ∂π
f (σ, x) =µ ,∂σ
1 + ϕ 1+λπ(
m+1 ∂RE[ m ∂b W (b, x+1 )] + λπ − ν − µ = 0,

if ξS(x) ≥ Rb ,∂RE[ mm+1 ∂b
W (b, x+1 )] + λπ − ν = 0,

if ξS(x) < Rb ,

(21)

∂
r−1 τ
− λπ .
W (b−1 , x) =
∂b
R−1

(22)
(23)

∂
The equilibrium values of the ﬁve endogenous variables {σ, b, λπ , µ, ν}, and ∂b
W (b−1 , x)

are determined by the constraints (18)–(20) and the conditions (21)–(23).
16

η−ϕ
τr
In the steady state, where the state is time-invariant, this condition is equivalent to 1+r< (1−η)ϕ
.

34

A.1

Special case with τ = 0

In the special case in which τ = 0, the ﬁrm is indiﬀerent between debt and equity as a tool
for funding as long as the amount of debt is suﬃciently small that the borrowing constraint
is non-binding. When τ = 0, the variables satisfy σ ce (x) = σ ∗ (x) and bce (x) = b∗ (x) in
Assumption 2, which implies that once the economy enters the ﬁrst-best equilibrium, where{σt , bt } = {σ ∗ (xt ), b∗ (xt )}, it stays there forever: {σt+j , bt+j } = {σ ∗ (xt+j ), b∗ (xt+j )}, forall j ≥ 1. This result is because for any b∗ (x−1 ), the ﬁrm can choose σ = σ ∗ (x) because
π can be non-negative when it chooses σ = σ ∗ (x) for all x ∈ Λ(x−1 ).
We deﬁne W ∗ (x), b∗ (x; S), B ∗ (x; S), Bz (x; S), and B̄z (x) byW ∗ (x) ≡ f (σ ∗ (x), x) − σ ∗ (x) + E

hm

+1

i
W ∗ (x+1 ) ,mb∗ (x; S)∗∗≡ ϕf (σ (x), x) − σ (x) + ξS(x),R(x)
b∗ (x; S)B ∗ (x; S) ≡ f (σ ∗ (x), x) − σ ∗ (x) +,R(x)Bz (x; S) ≡ f (σz (x), x) − σz (x) + ξS(x),B̄z (x) ≡ f (σz (x), x) − σz (x) +bz (x).
R(x)The meanings of the above variables are as follows: W ∗ (x) is the ﬁrst-best value of thematch between the bank and ﬁrm in the case where τ = 0; b∗ (xt ) is the maximum amountof debt bt feasible in the ﬁrst-best equilibrium, where σt = σ ∗ (xt );B ∗ (xt ; S) is the

maximum amount of debt bt−1 repayable in the ﬁrst-best equilibrium where σt = σ ∗ (xt );Bz (xt ; S) is the minimum amount of debt bt−1 that makes Rbtt ≥ ξS(xt ); and B̄z (xt ) is
the maximum amount of debt bt−1 repayable with certainty given that the current statein period t is xt . Capital B is used to represent the debt threshold.
In the case where τ = 0, the equilibrium dynamics are qualitatively the same as thosein Albuquerque and Hopenhayn (2004). We can show the following lemma, which isequivalent to Lemma 3 in Albuquerque and Hopenhayn (2004). The diﬀerence is that the
debt in our model is not state-contingent, whereas it is state-contingent in their model.
Lemma 3. For a given state x,(i) W (b−1 , x) is weakly decreasing in b−1 .
(ii) For all b−1 ≤ B ∗ (x; S), W (b−1 , x) = W ∗ (x).
(iii) For all b−1 > B ∗ (x; S), W (b−1 , x) < W ∗ (x) and π = 0.
Proof. The proof of (i) is as follows. As τ = 0, the envelope condition (23) implies∂
that ∂b
W (b, x) = −λπ ≤ 0, that is, W (b−1 , x) is weakly decreasing in b−1 . (ii) follows

immediately from the facts that B ∗ (x; S) − b−1 is the dividend when b−1 < B ∗ (x; S) and35

the ﬁrm chooses σ = σ ∗ (x) and that Assumption 2 implies that if σt = σ ∗ (xt ), then
σt+j = σ ∗ (xt+j ) for all j. (iii) is shown as follows. When B ∗ (x; S) < b−1 ≤ Bz (x; S), we
∗

(x;S)
∗
have b R(x)< b(x;S)R(x) ≤ ξS(x). This inequality and the deﬁnition of b (x; S) imply that theborrowing constraint is tight and σ(x) < σ ∗ (x). Thus, W (b−1 , x) < W ∗ (x). Now, supposethat π > 0 in this case. Then, the ﬁrm can relax the borrowing constraint by reducing π
and b and can increase W (b−1 , x). This is a contradiction. Therefore, π should be zero.
When b−1 > Bz (x; S), it is clear that W (b−1 , x) < W ∗ (x), and there exists T such thatthe borrowing constraint is σt ≤ ϕf (σt , xt ) for 0 ≤ t < T , and σt ≤ ϕf (σt , xt ) + ξSt − Rbtt
for t ≥ T in the sequential problem corresponding to the Bellman equation (17). Suppose
π0 > 0. Then, the ﬁrm can relax the borrowing constraint at T by reducing π0 and b0 and
can increase W (b−1 , x). This is a contradiction. Thus, π0 should be zero.
This lemma implies that if b−1 > B ∗ (x; S), then the ﬁrm repays as much debt as
possible by setting the dividend to zero: π = 0. This ﬁrm’s behavior is the same as thatin Albuquerque and Hopenhayn (2004). If b−1 > Bz (x; S), then σ = σz (x), where σz (x)is the solution to σ = ϕf (σ, x). Now, we turn to the general case where τ > 0.

A.2

Case with τ > 0

The features of the equilibrium path in the case where τ > 0 are characterized by the
initial amount of inter-period debt b−1 . We deﬁne
B ce (x; S) ≡ f (σ ce (x), x) − σ ce (x) +bce (x; S).
R(x)Note that the deﬁnition of bce (x; S) implies that B ce (x; S) = (1 − ϕ)f (σ ce (x), x) + ξS(x).
Thus, the deﬁnition of σz (x) and (9) imply that
B ce (x; S) < Bz (x; S) < bz (x).

(24)

Constrained-eﬃcient equilibrium with small debt:

We focus on the case where

initial debt b−1 is suﬃciently small (i.e., b−1< B ce (x; S)).

In this case, the borrowing

constraint is written asσ ≤ ϕf (σ, x) + ξS(x) −

b
.
R

(25)

∂
∂
When µ > 0, the FOCs imply that ∂k
F (A, k, l) > rK and ∂l
F (A, k, l) > w. It immediately

follows that inputs k and l are ineﬃciently smaller than their eﬃcient levels, k ∗ and l∗ ,∂∂F (A, k ∗ , l∗ ) = rK and ∂l
F (A, k ∗ , l∗ ) = w, respectively. Note thatwhich are given by ∂k

the borrowing constraint is always binding (i.e., µ > 0) because the ﬁrm borrows intertemporal debt to exploit the tax advantage. We call the equilibrium constrained-eﬃcientif λπ,t = νt = 0 for all t. We assume and justify later that λπ = λπ,+1 = ν = 0 whenµb−1 < B ce (x; S). In this case, conditions (21)–(23) imply E[ mm+1 ] rτ
R = R , which can

36be rewritten as µ = µce (x). Therefore, the tightness of the borrowing constraint (19)(or (25)) in the constrained-eﬃcient equilibrium, µce (x), is not dependent on b−1 andis decided solely by the tax advantage. Therefore, the value of σ in the constrainedeﬃcient equilibrium, σ ce (x), is also independent of b−1 . The value of b should be bce (x; S).
Note that Assumption 2 guarantees that once the economy enters the constrained-eﬃcientequilibrium, it stays there forever. This result is because for any bce (x−1 ; S), the ﬁrm
can choose σ = σ ce (x) because π can be non-negative when it chooses σ = σ ce (x) for allx ∈ Λ(x−1 ). Thus, our assumption that λπ = λπ,+1 = ν = 0 is justiﬁed.
Medium-sized debt:

Consider the case where initial debt b−1 is medium-sized (i.e.,

B ce (x; S) ≤ b−1 < Bz (x; S)). In this case, the borrowing constraint is still (25). We focus
on the cases in which the parameter values are chosen such that ν = 0 in equilibrium whenb−1 is medium-sized.17 Then, we have the following lemma.
Lemma 4. Consider the case where b−1 is medium-sized and σ < σ ce (x) in equilibrium.
Then, λπ > 0, and the dividend is equal to zero, π = 0, in equilibrium.
Proof. The proof is by contradiction. Suppose that λπ = 0. Then, conditions (22) andrτce(23) imply that µ = RE[ mm+1 ( rτR − λπ+1 )] ≤ 1+r = µ , which, together with (21), implies

that σ ≥ σ ce (x). This is a contradiction. Thus, λπ = 0 cannot hold in equilibrium. Hence,
λπ > 0 and π = 0 in equilibrium.
The values of {k, l, b, µ} are determined by the budget constraint (3) with π = 0, theborrowing constraint (25), the FOC (21), and


k
α
w
=
,
l
1 − α rK
where the last equation is derived from the conditions for k and l to maximize F (A, k, l)
subject to rK k+wl ≤ σ. Given that outstanding debt b−1 is medium-sized, the ﬁrm repays
as much debt as possible by setting π = 0 and eventually returns to the constrainedeﬃcient equilibrium. Assumption 1 implies that it takes a ﬁnite number of periods toreturn to the constrained-eﬃcient equilibrium. These features are qualitatively the sameas those of debt repayment in Albuquerque and Hopenhayn (2004).
Large debt:

If b−1 ≥ Bz (x), the borrowing constraint becomesσ ≤ ϕf (σ, x).

Then, the FOC (22) implies




m+1 ∂
ν − λπE
W (b, x+1 ) =
.
m ∂b
R

It immediately follows that17

ν = 0 holds if the parameters satisfy (27).

37

(26)

• If λπ > 0, then π = 0, b = R{b−1 − f (σz (x), x) + σz (x)}, and ν = 0.
• If ν > 0, debt jumps to the upper limit (i.e., b = bz (x)), λπ = 0, and π > 0.
Note that although the FOCs should be satisﬁed at the solution, the choice between
b = R{b−1 − f (σz (x), x) + σz (x)} and b = bz (x) cannot be determined by the FOCs
alone because they are the local maxima. The ﬁrm compares the total surplus whenb = R{b−1 − f (σz (x), x) + σz (x)} with that when b = bz (x) and then chooses the higher
surplus. Deﬁne W (1) (b−1 , x) and W (2) (b−1 , x) by
hm
i
τ r−1
+1
b−1 + f (σz (x), x) − σz (x) + E
W (bz (x), x+1 ) ,
R−1
m
hm
i
τ
r
−1
+1
W (2) (b−1 , x) =b−1 + f (σz (x), x) − σz (x) + E
W (R{b−1 − f (σz , x) + σz }, x+1 ) .
R−1m
W (1) (b−1 , x) =Here, W (1) (b−1 , x) is the total surplus when the ﬁrm chooses b = bz (x) and W (2) (b−1 , x)is that when the ﬁrm chooses b = R{b−1 − f (σz (x), x) + σz (x)}. Then, W (b−1 , x) is givenbyW (b−1 , x) = max {W (1) (b−1 , x), W (2) (b−1 , x)}.
We assume that the values of the parameters are chosen such that
W (1) (Bz (x; S), x) < W (2) (Bz (x; S), x),

for all x.

(27)

From the deﬁnition of B̄z (x), the following equality holds:
W (1) (B̄z (x), x) = W (2) (B̄z (x), x),

for all x.

(28)Then, the following lemma holds.
Lemma 5. There exist Bc (x; S) and B̄c (x; S) such that Bz (x; S) < Bc (x; S) ≤ B̄c (x; S) ≤B̄z (x) and• If b−1 ∈ [Bz (x; S), Bc (x; S)), then W (1) (b−1 , x) < W (2) (b−1 , x), π = 0, and b =R{b−1 − f (σz (x), x) + σz (x)}.
• If b−1 ∈ [B̄c (x; S), B̄z (x)], then W (1) (b−1 , x) ≥ W (2) (b−1 , x) and b = bz (x).
Proof. The lemma follows immediately from the continuity of W (1) (b−1 , x) and W (2) (b−1 , x)together with (27) and (28).
In the general case with stochastic shocks, we can only say that B̄c (x; S) ≤ B̄z (x),whereas, in the deterministic case, where xt evolves deterministically, it is shown thatB̄c,t < B̄z,t . Thus, the ﬁrm intentionally increases borrowing to bz,t when outstanding
debt bt−1 is suﬃciently large, such that B̄c,t ≤ bt−1 < B̄z,t .

38

B

Proof of Proposition 2

We prove Proposition 2 using the Schauder ﬁxed point theorem (see, for example, Theorem
17.4 in Stokey and Lucas with Prescott, 1989). First, we posit the following assumption.
Assumption 3. The values of the parameters satisfy inf x∈Λ (1−ϕ)f (σz (x), x) > ∇, where∇ is deﬁned by∇=supx∈Λ,x+1 ∈Λ(x),S∈G

B ce (x; S) −B ce (x+1 ; S).
R(x)Note that this assumption is always satisﬁed in the case where the state and prices
do not change over time. In this case, inf x∈Λ (1 − ϕ)f (σz (x), x) > ∇ can be rewrittence
as (1 − ϕ)f (σz ) = rbz > R−1
R B , which is clearly satisﬁed because 1 < R < 1 + r andbz > B ce , as (24) in Appendix A implies.
Now, we deﬁne several parameters.
Deﬁnition 1. Deﬁne rmin , Rmax , and fmax as follows:rmin = inf r(x),x∈ΛRmax = sup R(x),x∈Λ∂f (σz (x), x).
∂σx∈Λfmax = sup∂f (σ, x) because σ ≥ σz (x) for all x, and f (σ, x)Note that fmax is the upper bound for ∂σ

is increasing and concave in σ. It is easily shown that
fmax =

η.
ϕ

Proposition 1 shows that, given S(xt ), there exists a solution to (6) in which the valuefunction can be denoted as V (b, x; S). Deﬁne the operator T that maps S(xt ) to T S(xt ),where
T S(xt ) ≡ max Et
b


b
mt+1
V (b, xt+1 ; S) +.
mt
Rt

If the mapping T : G → G is continuous and the family T (G) is equicontinuous, then the
Schauder ﬁxed point theorem applies, and it is shown that T has a ﬁxed point in G. Then,
Proposition 2 is proven.
T is clearly a mapping from G to G, because T (G) ⊆ G. The continuity of T follows
directly from the continuity of V (b, x; S) with respect to S, which is shown in the following
lemma.
Lemma 6. V (b, x; S) is continuous with respect to S, that is, for all ε > 0, there exists
δ > 0 such that if |S − S ′ | < δ for any S, S ′ ∈ G, then |V (b, x; S) − V (b, x; S ′ )| < ε, where
the norms are the sup norm.
39

Proof. The equivalence between the functional equation (6) and the corresponding inﬁnite
horizon problem shown in Theorem 9.2 of Stokey and Lucas with Prescott (1989) impliesthat
V (b, x; S) =subject tomax{σt ,bt }∞
t=0

E0

"∞
X mt
t=0

m0

#
πt ,

bt
πt = f (σt , xt ) − σt − bt−1 +
,
Rt


bt
σt ≤ ϕf (σt , xt ) + max ξS(xt ) −, 0 ,Rtbt ≤ bz (xt ),πt ≥ 0,b−1 = b,x0 = x.

The solution to the above problem is a path: {σt , bt }∞t=0 . Assumptions 1 and 2 directly
imply that there are three stages in this path, that is, given b−1 , there exist two integers,
t1 and t2 , where 0 ≤ t1 ≤ t2 ≤ ∞, such that
• Stage 1 (Bz,t < bt−1 ≤ B̄z,t ): ξS(xt ) < Rbtt for 0 ≤ t < t1 ;• Stage 2 (Btce < bt−1 ≤ Bz,t ): ξS(xt ) ≥ Rbtt and σt < σ ce (xt ) for t1 ≤ t < t2 ;• Stage 3 (bt−1 ≤ Btce ): ξS(xt ) ≥ Rbtt and σt = σ ce (xt ) for t ≥ t2 .
Note that the values of t1 and t2 are history-dependent,18 but history dependence does
not aﬀect the following proof. Assumption 1 implies that once the economy enters Stage2, it never returns to Stage 1. Assumption 2 implies that once the economy enters Stage
3, it never returns to Stage 2. As we saw in Section 2.3, πt = 0 in Stages 1 and 2, andπt > 0 in Stage 3.
Now, we evaluate |V (b, x; S) − V (b, x; S + δ/ξ)|, where S + δ/ξ = S(x) + δ/ξ for a small∞
δ > 0. We use {σt (δ), bt (δ)}∞
t=0 = {σt + σ̂t (δ), bt − b̂t (δ)}t=0 to denote the solution to theabove inﬁnite horizon problem, with S replaced by S + δ/ξ. The corresponding dividend∞
stream is written as {πt (δ)}∞
t=0 = {πt + π̂t (δ)}t=0 . The boundary of stages, {t1 , t2 }, changes

to {t1 (δ), t2 (δ)}. We assume that δ is suﬃciently small, such that(Θ + 1)δ < inf (1 − ϕ)f (σz (x), x) − ∇,
x∈Λ

(29)

where Θ is deﬁned in (36) below. Note that Assumption 3 warrants the existence ofδ (> 0).
18′′ ∞History dependence is described as follows. Deﬁne x̃ = {xt }∞t=0 and x̃ = {xt }t=0 . The history-

dependent ti is a function of x̃, which is written as ti (x̃), for i = 1, 2. It satisﬁes that ti (x̃) = ti (x̃′ ) if
xt = x′t for all t ∈ {0, 1, 2, · · · , ti }, for i = 1, 2.

40

Stage 1:

For t ≤ t1 (δ), Rbtt > ξSt + δ.

Thus, for t ≤ t1 (δ), the change fromξS to ξS + δ does not change the variables; (σt (δ), bt (δ)) = (σt , bt ) or (σ̂t (δ), b̂t (δ)) =(0, 0). The dividend is also zero: πt (δ) = πt = 0. Therefore, as (29) implies that
δ < inf x∈Λ f (σz (x), x) − σz (x), it is clear that t1 − 1 ≤ t1 (δ) ≤ t1 . We evaluate thevalue of bt1 −1 (δ), which is the initial debt at the beginning of t1 , in the case where ξStchanges to ξSt + δ in the above borrowing constraint.
• Case where t1 (δ) = t1 :In this case, it is obvious that bt (δ) = bt for all t ≤ t1 − 1. Thus, bt1 −1 (δ) = bt1 −1 .
• Case where t1 (δ) = t1 − 1 :In this case, bt1 −2 (δ) = bt1 −2 and ξSt1 −1 ≥constraints in period t1 − 1 imply

bt1 −1 (δ)
Rt1 −1 .

The budget and borrowing

bt1 −2 − ξSt1 −1 − δ = (1 + ϕ)f (σt1 −1 (δ), xt1 −1 ) − 2σt1 −1 (δ).

(30)

In the original problem with ξSt , the budget and borrowing constraints imply
bt1 −2 − ξSt1 −1 > (1 + ϕ)f (σz,t1 −1 , xt1 −1 ) − 2σz,t1 −1 .

(31)

η
∂
The parameter restriction that 2−η< ϕ in Assumption 3 implies that (1+ϕ) ∂σ
f (σ, x)−2 ≤ (1 + ϕ)fmax − 2 < 0, which implies that function (1 + ϕ)f (σ, x) − 2σ is decreasingin σ. Thus, σ̂t1 −1 (δ) > 0, and (30) and (31) imply
0 ≤ (1 + ϕ)f (σz,t1 −1 , xt1 −1 ) − 2σz,t1 −1 − [(1 + ϕ)f (σt1 −1 (δ), xt1 −1 ) − 2σt1 −1 (δ)] ≤ δ.
(32)∂
The concavity of f (σ, x) and (1 + ϕ) ∂σ
f (σ, x) − 2 < 0 imply that

(1 + ϕ)f (σt1 −1 (δ), xt1 −1 ) − 2σt1 −1 (δ) <(33)(1 + ϕ)f (σz,t1 −1 , xt1 −1 ) − 2σz,t1 −1 − [2 − (1 + ϕ)fmax ]σ̂t1 −1 (δ).
Inequalities (32) and (33) imply that0 ≤ σ̂t1 −1 (δ) ≤
Then,

δ
.
2 − (1 + ϕ)fmax

bt1 −1 (δ)
Rt1 −1 = ϕf (σt1 −1 (δ), xt1 −1 ) − σt1 −1 (δ) + ξSt1 −1 + δ must satisfy






1 − (1 + ϕ)fmax
bt1 −1 (δ)
bt1 −1
2 − fmaxδ≤−≤δ.
2 − (1 + ϕ)fmax
Rt1 −1Rt1 −12 − (1 + ϕ)fmax

Therefore, there exists a positive number C−1 such that bt1 −1 − C−1 δ < bt1 −1 (δ) <bt1 −1 + C−1 δ.

41

Stage 2:

Note that πt (δ) = 0 for t ≤ t2 (δ) − 1. We denote t = t1 + n and counttime by n. The above analysis of Stage 1 implies that bt1 −1 (δ) satisﬁes bt1 −1 − C−1 δ <bt1 −1 (δ) < bt1 −1 + C−1 δ. We change the subscript of the variables from t to n. Deﬁne n̄
as the solution ton̄X(1 − ϕ)fzξ ω̄ =.
(1 + rmax )ii=1

Note that Assumption 1 warrants that this equation has a ﬁnite solution n̄ and that n̄
does not depend on S(x). Then,t2 (δ) − t1 (δ) ≤ n̄.
In period n = 0, which corresponds to t = t1 , we have f (σ0 (δ)) − σ0 (δ) ≤ f (σ0 ) − σ0 +(fmax − 1)σ̂0 (δ). The budget constraint implies that b̂0R(δ)≤ (fmax − 1)σ̂0 (δ) + C−1 δ. Then,0
the borrowing constraint implies that σ̂0 (δ) ≤ ϕfmax σ̂0 (δ) + δ + (fmax − 1)σ̂0 (δ) + C−1 δ,(1+C−1 )δ
which can be rewritten as σ̂0 (δ) = 2−(1+ϕ)f. We deﬁne
max

A0 ≡

fmax − 1(> 0).
2 − (1 + ϕ)fmax

Then, (fmax − 1)σ̂0 (δ) = (1 + C−1 )A0 δ and b0R(δ)≥ Rb00 − (1 + C−1 )A0 δ. In period n = 1,0f (σ1 (δ)) − σ1 (δ) ≤ f (σ1 ) − σ1 + (fmax − 1)σ̂1 (δ). The budget constraint implies thatb̂1 (δ)R1 ≤ (fmax − 1)σ̂1 (δ) + (1 + C−1 )Rmax A0 δ. Then, the borrowing constraint implies thatσ̂1 (δ) ≤ ϕfmax σ̂1 (δ) + δ + (fmax − 1)σ̂1 (δ) + (1 + C−1 )Rmax A0 δ, which can be rewritten as−1 )Rmax A0 ]
δ. We deﬁneσ̂1 (δ) = [1+(1+C
2−(1+ϕ)fmax

A1 ≡ A0 [1 + (1 + C−1 )Rmax A0 ].
Then, (fmax − 1)σ̂1 (δ) = A1 δ and b1R(δ)≥ Rb11 − [(1 + C−1 )Rmax A0 + A1 ]δ. In period n = 2,
1
f (σ2 (δ)) − σ2 (δ) ≤ f (σ2 ) − σ2 + (fmax − 1)σ̂2 (δ). The budget constraint implies thatb̂2 (δ)2
R2 ≤ (fmax − 1)σ̂2 (δ) + [(1 + C−1 )Rmax A0 + Rmax A1 ]δ. Then, the borrowing constraint
2 A +R
implies that σ̂2 (δ) ≤ ϕfmax σ̂2 (δ) + δ + (fmax − 1)σ̂2 (δ) + [(1 + C−1 )Rmax
0
max A1 ]δ,2[1+(1+C−1 )Rmax A0 +Rmax A1 ]which can be rewritten as σ̂2 (δ) =δ. We deﬁne2−(1+ϕ)fmax
2
A2 ≡ A0 [1 + (1 + C−1 )Rmax
A0 + Rmax A1 ].
2 A + R
Then, (fmax − 1)σ̂2 (δ) = A2 δ and b2R(δ)≥ Rb22 − [(1 + C−1 )Rmax
0
max A1 + A2 ]δ.
2Similarly, the following claim is proven by induction. To simplify the notation, we deﬁne
A−1 ≡C−1 A0.
Rmax

3
2 A ) and A = A (1 + R
2Then, A1 = A0 (1 + Rmax A0 + Rmax
−1
20
max A1 + Rmax A0 + Rmax A−1 ).

Claim 1. Suppose the economy is in Stage 2 in period n (for 0 ≤ n ≤ n̄). Then,!
n+1
Xb̂n (δ)i≤
Rmax An−i δ,
Rn
i=0

42

(34)

where An is deﬁned by

"

n+1
X

An ≡ A0 1 + Rmax

!#
i
Rmax
An−i

.

i=0

Proof. This claim holds true for n = 0, 1, 2,, as we see above. Suppose (34) holds true forn = j. Then, for n = j + 1, the budget constraint and borrowing constraint imply that!
j+1
Xb̂j+1 (δ)
i
0 ≤ (fmax − 1)σ̂j+1 (δ) + Rmax
Rmax
Aj−i δ −
,
(35)
Rj+1
i=0

σ̂j+1 (δ) ≤ ϕfmax σ̂j+1 (δ) + δ +b̂j+1 (δ).
Rj+1

These equations imply that"
(fmax − 1)σ̂j+1 (δ) ≤ A0 1 + Rmax

j
X

!#
i
Rmax
Aj−i

δ.

i=0

h
P
i
j
i
Deﬁning Aj+1 ≡ A0 1 + Rmax
RA
, this equation is written as (fmax −j−i
i=0 max
P

b̂
(δ)
j
i
≤
A
δ
+
R
1)σ̂j+1 (δ) = Aj+1 δ, and (35) implies that j+1
R
A
j+1
max
i=0 max j−i δ =
Rj+1
P

j+1 i
i=0 Rmax Aj+1−i δ. Therefore, it has been shown that (34) holds true for 0 ≤ n ≤ n̄
as long as period n is in Stage 2. This proves Claim 1.
Now, as t2 (δ) − t1 (δ) ≤ n̄, it holds true that for n ≤ t2 (δ) − t1 (δ),
b̂n (δ)≤ Θδ,Rnwhere the constant, Θ, is deﬁned by
Θ≡

n̄+1
X

!
i
Rmax
An̄−i

.

(36)

i=0

Stage 3:

Now, we use subscript t instead of n. We can show that t2 − 1 ≤ t2 (δ) ≤ t2 as
bce (δ)

bce
ce
ce
t (δ)
Rt ≡ ϕf (σt , xt ) − σt + ξSt + δ.
bce (δ)bce
Thus, Btce (δ) = Btce + δ and tRt = Rt t + δ. We know that bt2 −1 < Btce2 andfollows. Deﬁne Btce (δ) ≡ f (σtce , xt ) − σtce + tRt

cebt > Bt+1,

and

for t ≤ t2 − 2.

(37)

ce (δ) for t (δ) ≤ t ≤ t − 3 satisfy the following inequality:
The variables bt (δ) and Bt+1
1
2
ce
ce
bt (δ) − Bt+1(δ) ≥ bt − Bt+1
− (Θ + 1)δ

bt+1
ce− Bt+1− (Θ + 1)δ
Rt+1
B ce
ce
> inf (1 − ϕ)f (σz (x), x) + t+2 − Bt+1
− (Θ + 1)δ
x∈Λ
Rt+1> inf (1 − ϕ)f (σz (x), x) +x∈Λ> inf (1 − ϕ)f (σz (x), x) − ∇ − (Θ + 1)δ > 0.
x∈Λ43The ﬁrst inequality is due to bt (δ) ≥ bt −Θδ and Btce (δ) = Btce +δ. The second inequality is
due to bt = Rbt+1+ f (σt+1 , xt+1 ) − σt+1 ≥ Rbt+1+ inf(1 − ϕ)f (σz,t , xt ). The third inequality
t+1
t+1
is due to (37). The fourth inequality is due to the deﬁnition of ∇ in Assumption 3. The
ﬁnal inequality is due to (29). Thus, we have shown that the economy is in Stage 2
for t ≤ t2 − 2. Therefore, it must be the case that t2 (δ) = t2 − 1 or t2 . We evaluate|V (b, x; S) − V (b, x; S + δ/ξ)| for each case.
• Case where t2 (δ) = t2 :In this case,bcet2 (δ)− bt2 −1 (δ)
Rt2bce≤ f (σtce2 , xt2 ) − σtce2 + t2 + δ − bt2 −1 + ΘδRt2

πt2 (δ) = f (σtce2 , xt2 ) − σtce2 += πt2 + (Θ + 1)δ.
The inequality is due tobcebcet2 (δ)t2=
Rt2
Rt2 + δ and bt2 −1 (δ) ≥ bt2 −1 − Θδ. For t ≥ t2 + 1,

πt (δ) = f (σtce , xt ) − σtce +

bce
t (δ)
− bce
t−1 (δ)Rt= πt + (1 − Rt−1 )δ.
In this case, using the fact that t2 (δ) ≥ 0,
|V (b, x; S) − V (b, x; S + δ/ξ)| ≤ E0

"∞
X mt
t=0

m0

≤ (Θ + 1)δ +#|πt − πt (δ)|∞
X
Rmax − 1(1 + rmin )t
Rmax − 1= Θ+1+
δ.
rmin

δ

t=1

• Case where t2 (δ) = t2 − 1:
In the original ﬁrm’s problem with S, it holds true that bt2 −2 > Btce2 −1 and bt2 −1 ≤Btce2 , whereas in the modiﬁed ﬁrm’s problem with S+δ/ξ, it holds true that bt2 −2 (δ) ≤Btce2 −1 (δ), where bt2 −2 (δ) ≥ bt2 −2 −Θδ. In the original problem, the budget constraint
with πt2 −1 = 0 and the borrowing constraint imply(1 + ϕ)f (σt2 −1 , xt2 −1 ) − σt2 −1 = bt2 −2 − ξSt2 −1 .

(38)

In the modiﬁed problem, there exists Ĉ(bt2 −2 ) such that 0 < Ĉ(bt2 −2 ) ≤ Θ andbt2 −2 − Ĉ(bt2 −2 )δ = Btce2 −1 (δ).
This equation and the borrowing constraint σtce2 −1 = ϕf (σtce2 −1 , xt2 −1 ) + ξSt2 −1 + δ −bcet2 −1 (δ)/Rt2 −1 together imply that
(1 + ϕ)f (σtce2 −1 , xt2 −1 ) − σtce2 −1 = bt2 −2 − ξSt2 −1 − (Ĉ(bt2 −2 ) + 1)δ.
44

(39)

The derivative of function (1 + ϕ)f (σ, x) − 2σ satisﬁes−2 < (1 + ϕ)∂f (σ, x) − 2 < (1 + ϕ)fmax − 2 < 0.
∂σ

(40)

η
The rightmost inequality is due to fmax = η/ϕ and 2−η< ϕ. Then, because (1 +ϕ)f (σ, x) − 2σ is concave and continuously diﬀerentiable with respect to σ, theequations (38) and (39), together with inequality (40), imply that
σtce2 −1 −Ĉ(bt2 −2 ) + 1
Ĉ(bt2 −2 ) + 1
δ ≤ σt2 −1 ≤ σtce2 −1 −δ.
2 − (1 + ϕ)fmax
2

The fact that 0 < Ĉ(bt2 −2 ) ≤ Θ implies that1
σtce2 −1 − Φδ ≤ σt2 −1 ≤ σtce2 −1 − δ,
2

(41)

Θ+1
where Φ ≡ 2−(1+ϕ)f. This inequality and the concavity and diﬀerentiability ofmax

f (σ, x) imply that
bce
bce
(δ)
bt2 −1
t2 −1 (δ)+ (ηΦ + 1)δ << t2 −1+ (Φ − 1)δ
Rt2 −1Rt2 −1Rt2 −1(42)

because bt2 −1 /Rt2 −1 = ϕf (σt2 −1 , xt2 −1 ) − σt2 −1 + ξSt2 −1 . Now, we evaluate πt (δ) fort = t2 −1, t2 , t2 +1, · · · . We know πt2 −1 = 0 in the original problem. The inequalities(41) and (42), together with bt2 −2 (δ) ≥ bt2 −2 − Θδ, imply thatbcet2 −1 (δ)− bt2 −2 + ΘδRt2 −11
bt −1≤ f (σt2 −1 + Φδ, xt2 −1 ) − σt2 −1 − δ + 2− (ηΦ + 1)δ − bt2 −2 + Θδ2
Rt2 −1
3
δ.
< (fmax − η)Φ + Θ −2

πt2 −1 (δ) ≤ f (σtce2 −1 , xt2 −1 ) − σtce2 −1 +Similarly,bcet2 (δ)− bce
t2 −1 (δ)
Rt2bce
≤ f (σtce2 , xt2 ) − σtce2 + t2 + δ − bt2 −1 + Rt2 −1 (Φ − 1)δ
Rt2

πt2 (δ) = f (σtce2 , xt2 ) − σtce2 += πt2 + [1 + Rt2 −1 (Φ − 1)]δ.
For t ≥ t2 + 1, the same result holds as in the case where t2 (δ) = t2 :
πt (δ) = πt + (1 − Rt−1 )δ.

45

Deﬁne Ψ ≡ supx∈Λ |1 + R(x)(Φ − 1)|. Then, using t2 − 1 ≥ 0,
|V (b, x; S) − V (b, x; S + δ/ξ)|
"∞
#
X mt
≤ E0
|πt − πt (δ)|
m0
t=0


∞
X
3
Ψ
Rmax − 1≤ (fmax − η)Φ + Θ −
δ+
δ+
δ
2
1 + rmin(1 + rmin )t
t=2


3
Ψ
Rmax − 1= (fmax − η)Φ + Θ − ++δ.
2 1 + rmin (1 + rmin )rmin
The above analysis of Stages 1–3 implies that there exists a positive number C that
does not depend on (ε, δ, S) such that for a suﬃciently small and positive number δ thatsatisﬁes (29),|V (b, x; S + δ/ξ) − V (b, x; S)| < Cδ.
This inequality implies that, for any small ε > 0, there exists δ ≤ Cε , such that |V (b, x; S ′ )−V (b, x; S)| < ε if |S ′ −S| < δ. Thus, V (b, x; S) is continuous with respect to S. This proves
Lemma 6.
Given the continuity of V , it is clear that T is continuous with respect to S; that is, forany ε > 0, there exists δ > 0 such that |T S ′ − T S| < ε if |S ′ − S| < δ. The equicontinuityof T (G) is also shown as follows.
Lemma 7. T (G) is an equicontinuous family; that is, for all ε > 0, there exists δ > 0such that for all T S(x) ∈ T (G), |T S(x′ ) − T S(x)| < ε if |x′ − x| < δ.
Proof. Pick a small ε (> 0) arbitrarily. We ﬁx ε throughout this proof. As T S is deﬁned
h ′
i
m
b
as T S(x) = maxb E m(x)V (b, x′ , S) + R(x), the continuity of V with respect to x (seeProposition 1) implies that there exists δ > 0 such that |T S(x′ )−T S(x)| < 2ε , if |x′ −x| < δ.
Note that the maximum value of δ that satisﬁes the above condition must depend on S ∈ G.
Deﬁne δ(S) byδ(S) ≡ sup{δ| |T S(x′ ) − T S(x)| < 2ε , if |x′ − x| < δ}.
Pick an arbitrary sequence of positive real numbers {δn }∞n=0 that satisﬁes 0 < δn+1 < δnand limn→∞ δn = 0. Deﬁne
Gn ≡ {S|δ(S) ≥ δn , S ∈ G}.
Clearly,Gn ⊆ Gn+1 ⊆ G,

andNow, the following claim holds.
46

lim Gn = G.

n→∞Claim 2. For all δ > 0, there exists n such that for all S ∈ G, there exists Sn ∈ Gn suchthat |S − Sn | ≤ δ.
Proof. Suppose that this claim is false. Then, there exists δ > 0 such that for all n > 0,
there exists Snc ∈ G such that |Snc − Sn | > δ for all Sn ∈ Gn . We denote the set of Snc
by Gcn (δ). The assumption implies that Gcn (δ) ̸= ∅, for all n. Clearly, Gn ∩ Gcn (δ) = ∅,Gn ∪ Gcn (δ) ⊆ G, and Gcn+1 (δ) ⊆ Gcn (δ) ⊆ G. Taking the limit of n → ∞ implies thatlimn→∞ Gcn (δ) ̸= ∅, which contradicts limn→∞ Gn = G. This proves Claim 2.
The continuity of T implies that there exists δ1 such that |T S ′ −T S| < 4ε if |S ′ −S| ≤ δ1 .
The above claim implies that there exists n such that, for all S ∈ G, there exists Sn ∈ Gn
that satisfy |S − Sn | ≤ δ1 . By deﬁnition, |T Sn (x′ ) − T Sn (x)| ≤ 2ε if |x′ − x| < δn . Now,
we show the equicontinuity of T (G). For a given ε(> 0), we choose δ1 and Gn as above.
Given (ε, δ1 , δn ), it holds that for all S ∈ G, there exists Sn ∈ Gn such that the following
inequality holds for any x and x′ (∈ Λ) that satisfy |x − x′ | ≤ δn :|T S(x′ ) − T S(x)| = |T S(x′ ) − T Sn (x′ ) − T S(x) + T Sn (x) + T Sn (x′ ) − T Sn (x)|≤ |T S(x′ ) − T Sn (x′ )| + |T S(x) − T Sn (x)| + |T Sn (x′ ) − T Sn (x)|ε ε ε≤ + + = ε.
4 4 2Thus, it has been shown that T (G) is equicontinuous. This proves Lemma 7.
As the mapping T : G → G is continuous and the family T (G) is equicontinuous, then,given that G is non-empty, closed, bounded, and convex, the Schauder ﬁxed point theorem
implies that T has a ﬁxed point in G.

C

Details of the deterministic case in Section 2.5

In this Appendix, we use prime (′ ) to indicate diﬀerentiation.
The ﬁrm’s problem isrt−1 τ
mt+1
bt−1 + ft (σt ) − σt +Wt+1 (bt ),
Rt−1
mt
bt
ft (σt ) − σt − bt−1 +
≥ 0,
R
t

bt
σt ≤ ϕft (σt ) + max ξSt −
, 0 ,
Rt

Wt (bt−1 ) = max
σt ,btsubject to

bt ≤ bz,t .

47

The FOCs and envelope condition areft′ (σt ) =( mµt
1 + 1+λπ,tµt
1 + ϕ 1+λπ,t,λπ,t −νt −µt
t+1
′
= 0,
mt Wt+1 (bt ) +Rtλπ,t −νt
mt+1
′
= 0,
mt Wt+1 (bt ) +Rt

Wt′ (bt−1 ) =if ξSt ≥ Rbtt ,if ξSt < Rbtt ,τ rt−1− λπ,t .
Rt−1

Permanent ineﬃciency of a debt-ridden ﬁrm:

If a ﬁrm is debt-ridden (i.e., bt−1 =bz,t−1 ), ineﬃciency continues permanently because the ﬁrm’s debt does not decrease even
if it sets πt = 0 and repays as much debt as possible for all t. The following lemma isestablished immediately from the deﬁnition of bz,t in the deterministic economy (see (8)).
Lemma 8. Once bt = bz,t , then bt+j = bz,t+j and σt+j = σz,t+j for all j ≥ 0.
This lemma means that once a ﬁrm becomes debt-ridden, it continues to be debt-ridden
and production is ineﬃcient forever. Note that the value of a debt-ridden ﬁrm is zero
because πt = 0 for all t, that is,
Vt (bz,t−1 ) = 0

and

Wt (bz,t−1 ) =1 + rt−1bz,t−1 .
Rt−1

Note that the deterministic version of our model has two steady states, the constrainedeﬃcient and debt-ridden steady states, whereas Jermann and Quadrini’s (2012) model has
only the constrained-eﬃcient steady state. The debt-ridden steady state does not existb∗

in Jermann and Quadrini’s economy because ϕ = 0 in their model. We deﬁne Wt∗ , Rtt ,
Bt∗ , Bz,t , B̄z,t , and Btce using the same deﬁnitions as those of their counterparts in thestochastic case. Note that B̄z,t = bz,t .

C.1

Special case where τ = 0

In this case, the optimal equilibrium is characterized by λπ,t = νt = µt = 0 and ft′ (σt ) = 1.
The FOCs and envelope condition imply the following lemma.
Lemma 9. If λπ,t = 0, the economy stays in the optimal equilibrium from t onward, that
′ (σis, λπ,t+j = νt+j = µt+j = 0 and ft+jt+j ) = 1 for all j ≥ 0.

Lemma 10.

(i) Wt (b) is weakly decreasing in b.

(ii) For all bt−1 ≤ Bt∗ , the economy is in the optimal equilibrium (i.e., Wt (bt−1 ) = Wt∗ ).
(iii) For all bt−1 > Bt∗ , Wt (bt−1 ) < Wt∗ and πt = 0.
The above item (i) follows from Wt′ (bt−1 ) = −λπ,t and λπ,t ≥ 0. This lemma implies
that if bt−1 ∈ (Bt∗ , B̄z,t ), where B̄z,t = bz,t−1 , production is ineﬃcient (ft′ (σt ) > 1), and the48

ﬁrm sets πt = 0 and repays as much debt as possible to reduce the amount of remaining
debt bt . When bt−1 becomes small and satisﬁes bt−1 ≤ Bt∗ , production becomes eﬃcient(ft′ (σt ) = 1) and the ﬁrm is indiﬀerent about the amount of debt because its value nolonger changes by repaying debt. If a ﬁrm is debt-ridden (i.e., bt−1 = bz,t−1 ), ineﬃciency
continues permanently (see Lemma 8). Thus, in the case where τ = 0, the ﬁrm repays
debt and returns to normal eventually as long as the initial debt, b−1 , is strictly smaller
than the upper limit, bz,−1 . However, it remains debt-ridden and continues ineﬃcient
production permanently if the initial debt is bz,−1 .

C.2

Case with τ > 0

In the case where τ > 0, the constrained-eﬃcient equilibrium, characterized by λπ,t = νt =
1+µce

τ rt
′ ce
ce
t
0, πt > 0, µce
t = 1+rt , f (σt ) = 1+ϕµce , and bt = bt for all t, is attained when the debt ist

suﬃciently small (i.e., bt−1 < Btce ). For medium-sized debt (i.e., bt−1 ∈ [Btce , Bz,t )), theﬁrm sets the dividend to zero (πt = 0) and repays as much debt as possible. Assumption
1 guarantees that it goes to the constrained-eﬃcient equilibrium within ﬁnite periods.
This result is the same as in Albuquerque and Hopenhayn (2004). For large debt (i.e.,bt−1 ∈ [Bz,t , B̄z,t ]), the borrowing constraint becomes (26). Note that B̄z,t = bz,t−1 in thedeterministic economy. When debt is large, the following two cases exist in equilibrium:(i) νt = 0: Either λπ,t > 0 or λπ,t = 0. If λπ,t > 0, then bt+1 < bt , and the FOC andenvelope condition imply that λπ,t+j > 0 for j = 1, 2, · · · as long as the borrowing
constraint at t + j is (26). If λπ,t = 0, then the FOC and envelope condition imply
that λπ,t+1 > 0 and, thus, that λπ,t+j > 0 for j = 2, 3, · · · , as long as the borrowing
constraint at t+j is (26). In any case, the ﬁrm eventually returns to the constrainedeﬃcient equilibrium by setting πt = 0 to repay as much debt as possible.
(ii) νt > 0: In this case, bt = bz,t , that is, the ﬁrm intentionally chooses to increase
borrowing and become debt-ridden. Once the ﬁrm becomes debt-ridden, it stays
there forever (Lemma 8).
For bt−1 ∈ (Bz,t , B̄z,t ), where B̄z,t = bz,t−1 , the equilibrium path is determined as follows.
The ﬁrm compares the total surplus when bt = Rt {bt−1 − ft (σz,t ) + σz,t } with that whenbt = bz,t and then chooses the higher surplus. For Bz,t ≤ bt−1 < bz,t−1 , the Bellman
equation reduces to
(1)

(2)

Wt (bt−1 ) = max{Wt (bt−1 ), Wt (bt−1 )},1τ rt−1(1)
bt−1 + ft (σz,t ) − σz,t +Wt+1 (bz,t ),where Wt (bt−1 ) =Rt−1
1 + rtτ rt−1
1(2)
Wt (bt−1 ) =bt−1 + ft (σz,t ) − σz,t +Wt+1 (Rt {bt−1 − ft (σz,t ) + σz,t }).
Rt−1
1 + rt

49

(1)

(2)

We show that there exists B̄c,t such that B̄c,t < bz,t−1 and Wt (bt−1 ) > Wt (bt−1 ) for
bt−1 ∈ (B̄c,t , bz,t−1 ). To prove this result, we need the following restriction on the parametervalues.
inft

1 + rt
> 1.
Rt

(43)

Proposition 11. In the deterministic case, there exist Bc,t and B̄c,t , where Bz,t ≤ Bc,t ≤B̄c,t < bz,t−1 , such that
(1)

(2)

• If Bz,t ≤ bt−1 < Bc,t , then Wt (bt−1 ) < Wt (bt−1 ). Here, the ﬁrm sets the dividend
to zero (i.e., πt = 0) to repay as much debt as possible and goes to the constrainedeﬃcient equilibrium in ﬁnite periods.
(1)

(2)

• If bt−1 > B̄c,t , then Wt (bt−1 ) > Wt (bt−1 ). In this case, the ﬁrm intentionally
increases debt to the upper limit (i.e., bt = bz,t ) and remains debt-ridden forever.
Proof. As Lemma 5 holds in the deterministic case, it is suﬃcient to show that B̄c,t isstrictly smaller than B̄z,t = bz,t−1 to prove Proposition 11. Deﬁne ∆t ≡ ft (σz,t ) − σz,t . We
(i)

(i)

(i)

t−1
also deﬁne Vt (bt−1 ) for i = 1, 2 by Vt (bt−1 ) = Wt (bt−1 ) − 1+r
Rt−1 bt−1 . These functions

are written as follows when the borrowing constraint is (26) for periods t and t + 1:
(1)

Vt (bt−1 ) = ∆zt − bt−1 +(2)

Vt (bt−1 ) =

bz,t
bz,t
1
+
Vt+1 (bz,t ) = ∆zt − bt−1 +,Rt
1 + rtRt1
Vt+1 (Rt {bt−1 − ∆zt }),1 + rt(1)(2)

Vt (bt−1 ) = max {Vt (bt−1 ), Vt (bt−1 )}.
We can show the following lemma.
(1)

(2)

(2)

Lemma 12. If Vt (bt−1 ) < Vt (bt−1 ), then Vt+j (bt+j−1 ) = Vt+j (bt+j−1 ) for all j ∈{1, 2, · · · , J}, where bt+j−1 = Rt+j−1 {bt+j−2 −∆zt+j−1 } and J is deﬁned by J = min{j|bt+j <Bz,t+j+1 }.
(1)

(2)

Proof. The proof is by contradiction. Suppose, on the contrary, that Vt (bt−1 ) < Vt (bt−1 )(1)and Vt+1 (bt ) = Vt+1 (bt ), where bt = Rt {bt−1 − ∆zt }. It is calculated that
(2)

1
1
(1)
Vt+1 (Rt {bt−1 − ∆zt }) =V (Rt {bt−1 − ∆zt })1 + rt1 + rt t+1

bz,t+1
bz,t
1
Rt
z
z
z
=
∆t+1 − Rt {bt−1 − ∆t } +=∆t − bt−1 +1 + rtRt+1
1 + rtRt
Rt(1)
V (1) (bt−1 ) < Vt (bt−1 ).
=1 + rtVt (bt−1 ) =Rt
The last inequality is due to 1+r< 1. This inequality contradicts the assumption. Thus,
t
(1)

(2)

(2)

(1)

it has been shown that if Vt (bt−1 ) < Vt (bt−1 ), then Vt+1 (bt ) ≥ Vt+1 (bt ) and Vt+1 (bt ) =(2)

Vt+1 (bt ). The above argument continues to hold in period t + j as long as the borrowing
constraint is (26), which means that j = 1, 2, · · · , J. This proves Lemma 12.
50

(1)

(2)

(2)

(2)

This lemma implies that if Vt (bt−1 ) < Vt (bt−1 ), then Vt+j (bt+j−1 ) = 1+r1t+j Vt+j+1 (bt+j )
(2)

(2)

for j = 1, 2, · · · , J, and, therefore, Vt (bt−1 ) = Ṽt (bt−1 ), where1(2)Ṽt (bt−1 ) ≡ QJ

j=0 (1 + rt+j )Vt+J+1 (bt+J ),where J and bt+J are deﬁned in the above lemma. First, we restrict our attention to thecase where debt bt−1 takes discrete values. We assume that bt−1 ∈ Γt−1 , where Γt−1 =
[n]

[n]

{bt−1 |n = 1, 2, · · · , ∞}, where bt−1 is speciﬁed below shortly. Under this assumption, we[n∗ ]show that there exists n∗ such that bt−1 is the closest to B̄c,t among all of the elements ofΓt−1 . We then proceed to the case where bt−1 takes a continuous value. We deﬁne
BtS = Rt ξSt .
BS

[n]

t+n−1
in period t + n − 1 if the ﬁrm repays
The deﬁnition of bt−1 is the debt that becomes Rt+n−1∆zt+j in period t + j for j = 0, 1, · · · , n − 1. Thus, it is written as follows:[n]bt−1 =
where we deﬁne

n−1
X

∆zt+j

S
Bt+n−1
,
+
Qn−1
Qj−1
R
R
t+j
t+i
j=0
i=0
j=0

Q−1

j=0 Rt+j = 1. The upper limit of the debt is also written as follows:bz,t−1 =

∞
X

∆zt+j

Qj−1

j=0

i=0 Rt+i

.
(1)

[n]

[n]

Now, let us consider the discrete case where bt−1 ∈ Γt−1 . Then, Vt (bt−1 ) = ∆zt − bt−1 +[n]bz,tRt = bz,t−1 − bt−1 is written as follows:1(1) [n]Vt (bt−1 ) = Qn−1

j=0 Rt+j(2)

S[bz,t+n−1 − Bt+n−1].

[n]Ṽt (bt−1 ) is written as follows:(2)

1
S
Vt+n (Bt+n−1
).
(1
+
r
)
t+j
j=0

[n]

Ṽt (bt−1 ) = Qn−1
Therefore,"# n−1 S
Y 1 + rt+j 
bz,t+n−1 − Bt+n−1=.
S(2) [n]
Rt+j)
Vt+n (Bt+n−1
Ṽt (bt−1 )
j=0
(1)

[n]

Vt (bt−1 )As it is always the case that BtS = ξRt St ≤ ξRt ωt and Vt (b) ≤ ωt , the above equation
implies that
 n−1 
bz,t+n−1 − ξRt+n−1 ωt+n−1 Y 1 + rt+j≥.
(2) [n]ωt+nRt+jṼt (bt−1 )
j=0
(1)

[n]

Vt (bt−1 )



51

(44)

Then, it is immediate from (9), (43), and (44) that there exists a ﬁnite integer n∗ (> 0)such that Vt (bt−1 ) > Ṽt (bt−1 ) for all n ≥ n∗ . Because the parameters are chosen such
(1)

[n]

(2)

[n]

[n]

[n∗ ]

that BtS < bz,t (see (9)), it is immediate from the deﬁnition of bt−1 that bt−1 < bz,t−1 .
[n∗ ]Thus, we have established that the threshold value bt−1 exists and is strictly smaller thanbz,t−1 in the discrete case where bt−1 ∈ Γt−1 . Let us proceed to the continuous case wherebt−1 is not restricted to an element of Γt−1 but is a positive real number. We know from
Lemma 5 that B̄c,t ≤ bz,t−1 . We prove B̄c,t < bz,t−1 by contradiction. Suppose that(2)B̄c,t = bz,t−1 . We know that Vt (bt−1 ) is continuous and decreasing in bt−1 and that(1)

Vt (bt−1 ) is a linear function of bt−1 . These facts imply that there must exist a small
(1)

(2)

positive number ε (> 0) such that for all bt−1 ∈ [bz,t−1 − ε, bz,t−1 ), Vt (bt−1 ) ≤ Ṽt (bt−1 ).
However, this result obviously contradicts Vt (bt−1 ) > Ṽt (bt−1 ) for all n ≥ n∗ , because
(1)

[n]

(2)

[n]

there exists n(> n∗ ) such that b[n] ∈ [bz,t−1 − ε, bz,t−1 ), as limn→∞ bt−1 = bz,t−1 . Thus, it[n]cannot hold that B̄c,t = bz,t−1 . Therefore, B̄c,t < bz,t−1 . This proves Proposition 11.
Note that the threshold B̄c,t is strictly smaller than bz,t−1 .
Numerical example of the deterministic case:

Figure 6 shows the numericalexample of the deterministic case. We assume that prices are invariant over time. The
values of the parameters are chosen such that α = 0.31, τ = 0.3, A = 0.73, w = 0.65, r =0.02, rK = 0.12, R = 1.014, η = 0.72, ϕ = 0.5, and ξ = 0.08. Most of these parametervalues are taken from Table 1 (see Section 3.3 for details). Figure 6 shows the responseof the economy to a buildup of debt. Initially, the economy is in the steady state, where
the level of debt is 0.30. At t = 0, debt b−1 suddenly increases to 0.53 because of an
exogenous shock (e.g., the breakout of a ﬁnancial crisis). The value of b−1 is chosen suchthat Bz < b−1 < bz , where Bz and bz are 0.44 and 4.63, respectively. The ﬁrm repays
as much debt as possible, meaning that the dividend is zero for periods 0–2. Production
is ineﬃcient for periods 0–2 and returns to the constrained-eﬃcient level in period 3.
In this example, the borrowing constraint is σt ≤ ϕf (σt ) for periods 0 and 1, and it isσt ≤ ϕf (σt ) + ξS − bRt from period 2 onward. Note that σ0 = σ1 = 0.065 (= σz ). In thecase where the initial increase in debt b−1 is larger, the borrowing constraint continues tostay at σt ≤ ϕf (σt ) for arbitrarily longer periods, thereby allowing ineﬃcient production,σt = σz , to continue persistently. This example shows that bRz = 4.56 and ξS = 0.37,
meaning that ξS < bRz .

52

4.5

0.25

4.4

0.2

4.3

0.15

4.2

0.1

4.1
-2

0

2

0.1

0.5
0.05
0.4

0.05
-2

4

0.6

0

2

0
-2

4

0

2

4

0.3
-2

0

2

4

Figure 6: Responses to a buildup of debt

D

The BGPThe detrended variables of the BGP are determined by the following system of equations:
A = Y 1−η ,Y = K α L1−α
= AK αη L(1−α)η
,pp
C + (g E − 1 + ρ)K = Y,
γC,
1−L
(1 − α)η(1 + ϕµ) Y
w=
,
1+µ
Lp
αη(1 + ϕµ) Y
,
rK =
1+µ K
gE
,
1+r =
β
w=

R = 1 + (1 − τ )r,
µ=1−
1=

β
R,
gE


β
rK + 1 − ρ ,Egg = [1 + κ(L − Lp )] ,gE
π = Y − wL − r K − B 1 −
,
R
π
Vn =
,
1−β
β
g E−1
S = Vn +
B,
g
R


gE B
K
,
wL + r K = ϕY + gξS −Rβ
g E−1 B
E−1 B.
κVn = (1 + µ)w − κg− µκ ξS −g
R
RK

where µ is the Lagrange multiplier associated with the borrowing constraint and B is thetotal amount of corporate lending, which satisﬁes B = b, where b is the outstanding debt
53

for a ﬁrm. There are 16 equations for 16 unknowns: {g, A, Y, K, L, Lp , C, w, µ, r, rK , R, π, B, Vn , S}.
The following condition must be satisﬁed on the BGP:ξS > g E−1

B
.
R
λ

Then, the variables for debt-ridden ﬁrms on the BGP, {kz , lz , bz , µλzz , λl,z}, are calculatedz
as follows:w=(1 − α)η 1 + ϕ µλzz

rK =1 + µλzzαη 1 + ϕ µλzz
1 + µλzz

Akzαη lz(1−α)η−1 ,Akzαη−1 lz(1−α)η ,wlz + rK kz = ϕAkzαη lz(1−α)η ,bz
0 = Akzαη lz(1−α)η − wlz − rK kz + g E−1 − bz ,
R


λ
µz
b
l,z
z
0= 1+
w − κg E−1 −
.
λz
R
λz
The following conditions must be satisﬁed on the BGP:
λl,z
> 0,
λz
ξS < g E−1

E

bz
< S.
R

Data

Table 2 provides data sources that are used for calibration and ﬁgures. Note that the European Commission’s Annual macro-economic (AMECO) database is constructed by Havik,
Morrow, Orlandi, Planas, Raciborski, Roeger, Rossi, Thum-Thysen and Vandermeulen(2014).
The actual data of Lt is used for the labor wedge in the ﬁgures and is constructed by
Lt =Average annual hours worked per employee
Employees
×,
Total hours
15-64 aged population

where “total hours” is deﬁned as 24 hours × 12 months × 20 days = 5760 hours.
The average annual hours worked per employee for the EU is only available startingfrom 2008 because Estonia and Malta are only available starting from 2000 and Croatia isonly available starting from 2008. We calculate the average annual hours as the weighted
average excluding these countries in the years with deﬁcient data, where the weight is
proportional to the number of employees.
Similarly, Eurostat provides the population aged 15–64 for the EU only from 2001
because of missing data in Croatia before 2001. However, the World Bank World Development Indicators provide “Population ages 15–64 (% of total)”. We complement theEurostat data with World Bank data.
54

Variable(s)Japan

United States

EU

Y, C

SNA

NIPA

AMECO

TFP

JIP

Fernald (2012)

AMECO

α

JIP

Fernald (2012)

Havik et al. (2014)

Average hours worked

JIP

NIPA

AMECO

Employees

LFS

BLS

AMECO

15–64 aged population

OECD

OECD

Eurostat; WDI

Table 2: Data sources
Note: SNA: Cabinet Oﬃce, Government of Japan, Annual Report on National Accounts
NIPA: U.S. Bureau of Economic Analysis, National Income and Product Accounts
AMECO: European Commission, Macro-economic database AMECO
JIP: The Research Institute of Economy, Trade, and Industry, JIP 2014 database
LFS: Statistics Bureau of Japan, Labor Force Survey
BLS: U.S. Bureau of Labor Statistics, Current Employment Status
OECD: OECD, Working age population (indicator). doi: 10.1787/d339918b-en
WDI: World Bank, World Development Indicators

F

Calibration and parameter region

Table 1 reports the values of the calibrated parameters. First, the parameters β, ρ, andη are common values for all countries. We set the discount factor β equal to 0.98, thedepreciation rate ρ equal to 0.1, and the parameter for the aggregation function η equal
to 0.7. These are the standard settings for an annual model in the literature.
Second, we calibrate the country-speciﬁc parameters and some BGP values. The tax
advantage τ is set equal to 0.3 for Japan and 0.35 for the United States and the EU. These
numbers are the corporate income tax rates in Japan and the United States, respectively.
The share of labor in production (α), the total labor supply on the BGP (L), and thegrowth rate of TFP on the BGP (gT F P ) are set from the data. Then, gT F P is deﬁned by1−η

gT F P = g η , as we see in Appendix E, and L is set to the ratio of total labor supply.
The sample period is from 1981 to 2008 for Japan and from 1997 to 2016 for the UnitedStates and the EU. We assume that the economy is on the BGP before a ﬁnancial crisis.
In the case of Japan, the ﬁnancial crisis starts in 1991, whereas, in the case of the UnitedStates and EU, it starts in 2009. Then, α, L, and gT F P in Japan are taken as the averagefor 1981–1989, and for the United States and the EU they are taken as the average for
1997–2007. When we calculate the variables on the BGP, L and gT F P are given by thedata. Hence, the inverse of the elasticity of labor supply γ and the eﬃciency of R&D κ
are determined endogenously in the system of the BGP, which is given in Appendix D.
Lastly, the parameters (ϕ, ξ, z) are identiﬁed by the simulated least squares criterion;that is, they are determined as the solution to the following minimization of the residual

55

sum of squares (RSS) using a grid search method:min (Xt − X̂t )′ (Xt − X̂t )ϕ, ξ, zsubject to
λlz
bz> 0, ξS < g E−1 < S, bn > 0,
λz
Rt = 1990-2008 in Japan; t = 2009-2025 in EU and the United States

0 < ϕ < η, 0 < ξ < 1,
g E > mean(gtE ),where Xt is the observed variables vector and X̂t is the simulation-generated variables
vector. Xt = [TFP growth rate; Real GDP per caita growth rate]. We extend the observed variables to 2025 by extrapolating the average growth rates in 2011–2016 for theUnited States and the EU. The simulation-generated variables, X̂t = [T F Pt /T F Pt−1 −1; Yt /Yt−1 −1], are calculated by the method presented in Appendix I, taking (γ, κ, ϕ, ξ, χ, z)
as the given exogenous parameters, where the debt shock hits the economy in period 10(i.e., zt = 0 for t ≤ 9 and z10 = z). Period 10 corresponds to 1991 for Japan and 2009 for
the United States and the EU. Minimization is done for the period 1990–2008 in the caseof Japan and for 2009–2025 in the case of the United States and the EU.
We restrict the domain of the parameters (ϕ, ξ) for the above minimization to the region
that enables all constraints to be satisﬁed on the BGP. Figure 7 denotes the domain of
the parameters (ϕ, ξ) as the white region, whereas the values of the other parameters are
given as those for Japan in Table 1.
To calibrate the parameters, we choose the optimum parameters to minimize the distance between the simulation implied by our model and the actual data. This procedure
is similar to impulse response matching, as described in, for example, Rotemberg andWoodford (1997), who choose the parameters to minimize the distance between the impulse responses implied by a reduced-form VAR and those implied by a DSGE model.19
As we saw in Figures 2–5, our model with the parameter values chosen by the above
minimization exhibits an economic slowdown. However, economic growth may not slow
down if the values of the parameters are diﬀerent. The white region in Figure 8 shows
the values of (ϕ, ξ), for which economic growth is accelerated when the economy is hit bya debt shock with maximum z. An economic slowdown occurs only in the red region of
Figure 8. In the white region, economic growth is accelerated because a larger amount oflabor is used in the R&D sector than before. More labor is used for R&D activities because
the marginal product of labor becomes lower in the production sector. When a debt shockhits the economy, the marginal product of labor in the R&D sector can become higher orlower than that in the production sector, depending on the values of the parameters (ϕ,ξ).
19

Our method can be regarded as a variant of impulse response matching if the actual data are interpreted

as a response to a one-time shock of debt buildup.

56

1

0.8

0.6

0.4

BGP exists
0.2

0
0

0.2

0.4

0.6

Figure 7: Domain of (ϕ, ξ) for the BGP

G

Condition of no R&D

The following claim provides the condition for a debt-ridden ﬁrm on the BGP to refrainfrom conducting the R&D activity.
Claim 3. There exists a threshold ϕ̄ ∈ (0, 1] such that for ϕ < ϕ̄, a debt-ridden ﬁrm does
not conduct R&D on the BGP (i.e., lz = lz,p ) given that the BGP exists.
Proof. As we saw in Appendix D, the FOC with respect to lz for a debt-ridden ﬁrm is


λl,z
µz
bz
= 1+
w − κg E−1 .
(45)
λz
λz
R
The ﬁrst term on the right-hand side of (45) is the marginal cost of labor input. The
second term is the marginal gain from R&D activity. If λl,z /λz > 0, then the constraint
lz,p ≤ lz is binding, where lz,p is the labor input for production, and the debt-ridden ﬁrm

57

Japanthe United States

EU

0.9

0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0

0

0.2

0.4

0

0.6

0

0.2

0.4

0.6

0

0

0.2

0.4

0.6

Economic slowdown
Accelelated growth
No BGP

Figure 8: Domain of (ϕ, ξ) for an economic slowdown
does not conduct R&D. With some tedious calculation, we can rewrite (45) to


 

 αη 1+ϕµ1+µEK
ϕ − 1 + 1 + γ(1 − [g − 1 + ρ]) − r
W
rK
λl,z
η(1 − ϕ)=Yλz
ϕ(1 − η)W − (1+µ)Z
|
{z U
}
"
κg E−1 (1 − ϕ)
−

≡Ωw

# η

1−η

( ϕ
η

1+ϕµ
1+µ

)

Y,
(46)
R − g E−1βκ(1+µξ)
gE
g E−1
ξβ
gE
where X ≡ g(1−β) , U ≡ X 1 − R − κ R [1 − µ(1 − ξ)], Z ≡ 1−β 1 − R + (1 −EEξβξ) gR , W = 1−β+ 1 − XUZ , R = 1 + (1 − τ )r, µ = 1 − gβE R, and rK = gβ − 1 + ρ. R − g E−1

is always positive under a standard parameter setting.20 The deﬁnition of Ωw ≡ Yw implies
that there exist Ω and Ω̄ such that 0 < Ω ≤ Ωw ≤ Ω̄ < ∞ for any ϕ ∈ [0, 1]. We divide

20

E

E−1

R − g E−1 > 0 can be written as τ < g g−βg
E −β

1−η
≡ f (g), where E = (1−α)η
> 0. It holds that E < 1,

under a standard parameter setting. Thus,
f (g) = 1 + β

1
1 − g1−E

gE − β> 1,

for all g > 1. As τ < 1, the inequality τ < f (g) is always satisﬁed, meaning that R − g E−1 > 0.

58

(46) by Y and take the limit of ϕ → 0 using the fact that Ω ≤ Ωw ≤ Ω̄:η(1 − ϕ)Ωw = ∞,
ϕ→0 ϕ(1 − η)"limκg E−1 (1 − ϕ)
lim

( ϕ
η

# η

1−η

1+ϕµ
1+µ

)

= 0.

R − g E−1

ϕ→0

These results imply that there exists ϕ̄ ∈ (0, 1] such that if ϕ < ϕ̄, then λl,z /λz > 0.
The intuitive explanation is as follows. Debt-ridden ﬁrms’ borrowing constraints aretight because of the pressure from the huge amount of inter-temporal debt, and thus theireﬀective costs of labor are high. With our parameter values, the marginal product of laborin production for debt-ridden ﬁrms can be equalized to the marginal cost of labor, whichis strictly larger than the marginal product of labor in R&D. For this reason, these ﬁrms
do not conduct R&D. Figure 7 in Appendix F shows the parameter region in the (ϕ, ξ)
space in which the BGP exists given that the other parameters are ﬁxed. This ﬁgurenumerically conﬁrms that ϕ < ϕ̄ is satisﬁed in the region in which the BGP exists. Thus,
the debt-ridden ﬁrms in our simulation do not conduct the R&D on the BGP.

H

Detrending for a ﬁrm’s problems(1−η)E

E , C = C̃ N E , K =We detrend the variables as follows: At = Ãt Nt−1
, Yt = Ỹt Nt−1
t
t t−1
t
E−1
E−1
E
E
˜
˜
, lz,t = lz,t /Nt−1 , ln,t = ln,t /Nt−1 , wt = w̃t N ,, kn,t = k̃n,t N
K̃t N , kz,t = k̃z,t N
t

t−1

t−1

t−1

E−1
, bz,t= b̃n,t Nt−1

E−1
, Vn,t= b̃z,t Nt−1

E−1
π̃t , bn,tπt = Nt−1

E−1zt= Ṽn,t Nt−1

= Zt /Nt−1 , andgt = Nt /Nt−1 .
Detrending TFP:Suppose that T F P = T F˜ P · N a . The equation Y = T F P · K α L1−α

implies that(1−η)α1−ηỸ N (1−α)η = T F˜ P · N a K̃ α N (1−α)η L1−α ,which implies that a = 1−ηη and, thus,1−η

T F P = T F˜ P · N η .

I

Transition dynamics with z > 0

In this appendix, we describe the transition dynamics in the case where the economy isinitially on the BGP and a proportion z of ﬁrms are suddenly imposed with the maximum
debt bz at time 10, where 0 < z < 1. The economy eventually converges to the BGP.
All agents have perfect foresight on the path after the one-time buildup of debt, whichis the only unexpected event. In our model, the borrowing constraint is always binding59

both in transition and on the BGP. The equilibrium is calculated by solving a full nonlinear system of simultaneous equations using a modiﬁed Newton–Raphson algorithm. The
details of the algorithm can be found in Juillard (1996). This algorithm solves 31 × 300
simultaneous equations, where 31 is the number of endogenous variables and 300 is thenumber of simulation periods.
Altogether, 31 variables—{Ãt , Ỹt , K̃t , C̃t , Lt , Rt , ˜ln,p,t , ˜lz,p,t , ˜lz,t , ˜ln,t , k̃z,t , k̃n,t , b̃z,t , b̃n,t , ỹz,t ,
µ

λ

µ

λ

n,t
z,t
, λn,t, λl,n,t, λl,z,t, λπ,n,t , λn,t }—are calculated fromỹn,t , gt , rt , rtK , w̃t , zt , Ṽn,t , S̃t , π̃n,t , π̃z,t , λz,tn,tz,tthe following 31 equations:Ãt = Ỹt1−η ,1ηηỸt = {zt ỹz,t+ (1 − zt )ỹn,t}η ,Lt = zt ˜lz,t + (1 − zt )˜ln,t ,K̃t−1 = zt k̃z,t + (1 − zt )k̃n,t ,C̃t + gtE K̃t − (1 − ρ)K̃t−1 = Ỹt ,γ C̃t,
1 − Lt

β C̃t
K
1= E
rt+1
+1−ρ ,
gt C̃t+1w̃t =1 + rt =gtE C̃t+1,
β C̃tRt = 1 + (1 − τ )rt ,higt = 1 + κ (1 − zt ) ˜ln,t − ˜ln,p,t ,zt = gt zt+1 ,"S̃t = gtE−1#
Ṽn,t+1 b̃n,t+,
1 + rtRtequations for both normal ﬁrms and debt-ridden ﬁrms, i = n, z:µi,t(1 − α)η 1 + ϕ λi,tαη ˜(1−α)η−1
Ãt k̃i,tli,p,t,w̃t =µi,t
1 + λi,tµi,tαη 1 + ϕ λi,tαη−1 ˜(1−α)η
rtK =Ãt k̃i,tli,p,t ,µi,t1 + λi,tα ˜1−αỹi,t = k̃i,tli,p,t ,60

equations for normal ﬁrms only:ib̃n,t hαη ˜(1−α)η
π̃n,t = Ãt k̃n,tln,p,t − w̃t ˜ln,t − rtK k̃n,t +1 + κ ˜ln,t − ˜ln,p,t gtE−1 − b̃n,t−1 ,Rt˜ln,p,t ≤ ˜ln,t , λl,n,t ˜ln,t − ˜ln,p,t = 0,
λn,t
(
)


E−1
λ
µn,t
b̃
µ
g
b̃
β C̃t κṼn,t+1
n,t
l,n,t
n,t
n,t
= 1+
w̃t − κgtE−1−−κmax ξ S̃t − t
, 0 ,
λn,t
Rt
λn,t
λn,t
Rt
gt C̃t+1 λn,t if ξ S̃ − gtE−1 b̃n,t ≤ 0, 1 = β C̃t λn,t+1 R ,

t
t

Rt
gtE C̃t+1 λn,t
µn,t
β C̃t λn,t+1
gtE−1 b̃n,t


= E
Rt ,
> 0, 1 − if ξ S̃t − Rtλn,tgt C̃t+1 λn,tihαη (1−α)η
K k̃

˜
˜
˜

=
Ã
1
+
−
if
µ
>
0,
w̃
l
+
r
ϕ
k̃
l
+
l
l
κ
n,t
t n,t
t n,t n,p,t
n,t
n,p,t
t n,t


n
o



 if µ· max ξ S̃t −gtE−1 b̃n,t
, 0
Rt

,

n,t ≤ 0,µn,t = 0,i
β C̃t hṼn,t = π̃n,t +1 + κ ˜ln,t − ˜ln,p,t Ṽn,t+1 ,gt C̃t+1(if λπ,n,t ≤ 0, λn,t = 1,if λπ,n,t > 0, π̃n,t = 0,
λn,t = 1 + λπ,n,t ,equations for debt-ridden ﬁrms only:b̃z,t = bz ,˜lz,t = ˜lz,p,t ,
π̃z,t = 0,(1−α)ηαηw̃t ˜lz,t + rtK k̃z,t = ϕÃt k̃z,tlz,p,t

,

µz,t
η−ϕ
=
,
λz,t
ϕ(1 − η)
λl,z,t
µz,t
b̃z,t
= 1+
w̃t − κgtE−1
> 0.
λz,t
λz,t
Rt
The equilibrium variables must satisfy the following conditions:gtE−1 b̃n,t< ξ S̃t ,Rt
g E−1 b̃z,tξ S̃t < t
< S̃t .
Rt

61