Discussion Paper No.

988

CREDIT BOOMS,DEBT OVERHANG
AND
SECULAR STAGNATION

Gerhard Illing
Yoshiyasu Ono
Matthias Schlegl

December 2016
Revised November 2017

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Credit Booms, Debt Overhang and Secular Stagnation

Gerhard Illing
University of Munich

Yoshiyasu Ono
Institute of Social and Economic Research, Osaka University

Matthias Schlegl
JSPS International Research Fellow, Osaka University, University of Munich

November 2017∗

Abstract
Why do advanced economies fall into prolonged periods of economic stagnation,
particularly in the aftermath of credit booms? We present a model of persistent
aggregate demand shortage based on strong liquidity preferences of households, inwhich we incorporate financial imperfections to study the interactions between debt,
liquidity and asset prices. We show that financially more deregulated economies
are more likely to experience persistent stagnation. In the short run, credit booms
can mask this structural aggregate demand deficiency. However, the resulting debt
overhang permanently depresses spending in the long run since deleveraging becomes self-defeating because of debt deflation. These findings are in line with themacroeconomic developments in Japan during its lost decades and other advanced
economies before and during the Great Recession.

Keywords: Secular Stagnation, Aggregate Demand Deficiency, Liquidity Preferences, Financial Frictions, Leverage
JEL Classification: E13, E21, E32, E41, E51∗

We are grateful to the participants of the Annual Meeting of the “Verein für Socialpolitik” in Vienna,
the CESifo Area Conference on Macro, Money and International Finance, the Conference on “Secular
Stagnation and Wealth Preferences” at Osaka University and the Macroeconomic Research Seminar at
University of Munich for useful comments and suggestions. Financial supports by JSPS KAKENHI
Grants (JP15H05728 and JP22530178) and DFG Grants (SPP1578) are greatly acknowledged.

1

1

Introduction

Many advanced economies suffer from insufficient aggregate demand in the aftermathof the global financial crisis despite unconventional monetary policy actions of unprecedented scales. In addition, the experience of Japan shows that economic stagnation anddeflationary tendencies can prevail for decades without any natural recovery. Hence, worries that economies might permanently fail to operate at full employment are widespreadand expressed in the “secular stagnation” hypothesis.1 Proponents of this view emphasize the importance of asset prices, credit availability and private sector debt. This is forat least two reasons.
On the one hand, credit booms or asset price booms are seen as a means to temporarily
stimulate a stagnating economy. In particular, Summers (2014a,b) argues that the credit
boom in the United States in the early 2000s was masking the underlying lack of aggregate
demand by initiating unsustainable consumption spending of households. Similar effects
were at play during the stock market boom of the 1990s. Therefore, Summers (2014a)
concludes that “the difficulty that has arisen in recent years in achieving adequate growthhas been present for a long time but has been masked by unsustainable finances” and “ithas been close to 20 years since the American economy grew at a healthy pace supported
by sustainable finance”.2
On the other hand, the resulting indebtedness of the private sector is considered amajor impediment to economic recovery. For instance, Eggertsson and Krugman (2012)emphasize the reduction in private demand due to debt overhang during balance sheetrecessions.3 In addition, housing wealth and leverage are main determinants of economic
activity in the United States (cf. Mian and Sufi, 2011; Mian et al., 2013) and Japan(cf. Ogawa, 2003; Ogawa and Wan, 2007). Credit growth is also a strong indicator for
financial crises (cf. Borio and Lowe, 2002; Shin, 2013). These crises are associated with
substantially higher output losses than normal recessions despite forceful monetary policy
actions because of the decoupling of monetary aggregates and the volume of credit (cf.
Schularick and Taylor, 2012; Jordà et al., 2015).
It follows that interactions of asset prices, credit availability and private sector debt
are important factors for the emergence and the severity of economic stagnation. But
how does an economy fall into stagnation and what is the role of these factors?
1

The term “secular stagnation” itself goes back to Hansen (1938) and was taken up by Larry Summers(2013). Yet, Keynes (1936) in Chapter 17 of the General Theory already argues that permanent demand
shortage can exist as a steady state phenomenon in a monetary economy.
2
A related argument is made by Krugman (2013): “In other words, you can argue that our economy
has been trying to get into the liquidity trap for a number of years, and that it only avoided the trap fora while thanks to successive bubbles.”3
Other theoretical treatments of deleveraging shocks at the zero lower bound include Eggertsson and
Mehrotra (2015) among others.

2

In this paper, we develop a stylized dynamic macroeconomic model to analyze theinteractions between asset prices, leverage and economic stagnation. The model features
three types of assets and two types of households: Borrowers obtain funds from savers,
but their borrowing ability is limited by the value of collateral that is endogenously
determined in the housing market following Iacoviello (2005). Households gain utility
from consumption, housing and money. The last follows Sidrauski (1967) and reflects,
among other things, the demand for liquidity.
We follow the research line initiated by Ono (1994, 2001) and assume insatiable liquidity preferences: The marginal utility of money stays strictly positive even for very large
money holdings, which prevents consumption of the saver from increasing as potential
output rises. This in turn creates stagnation if consumption of the borrower is sufficiently
restricted as is the case when the economy suffers from debt overhang. Hence, economies
with a higher leverage are more prone to suffering from insufficient aggregate demand.
Our setting implies that asset price or credit booms can temporarily stimulate an economy that would otherwise suffer from demand deficiency. A credit or asset price boom,
which is triggered by financial liberalization, enables borrowers to temporarily increase
their consumption spending, stimulating aggregate demand and inflation. In addition,
housing demand is stimulated and the real house price increases, thereby reinforcing theinitial credit boom as the value of collateral increases. Yet, in the new steady state, borrowers’ consumption is depressed by interest payments to savers. Savers however do notincrease their consumption accordingly as they prefer to hoard money because of strong
liquidity preferences. As a consequence, aggregate demand falls permanently short ofpotential output and the economy experiences persistent deflation.
In this steady state, borrowers do actually delever but these efforts are self-defeating
due to deflation. Their real debt burden remains constant and permanently depresses
spending so that the economy does not recover. Monetary policy becomes ineffective
since injections of liquidity are held as cash by savers and do not stimulate spending.4
Hence, there is a temptation for policymakers to stimulate sluggish growth by initiating
lending booms that come at the cost of greater damage in the long run.
These findings are in line with the macroeconomic developments in Japan during its
lost decades and other advanced economies during the Great Recession.
We proceed as follows. In Section 2, we present some stylized facts on private sector
debt and asset prices and discuss our main assumption of insatiable liquidity preferences.
Section 3 introduces a model of economic stagnation, which is analyzed in Section 4. We
discuss the role of leverage for economic stagnation in section 5 and some extensions of
the model as well as policy recommendations in section 6. The final section concludes.
4This is consistent with the apparent ineffectiveness of the Bank of Japan’s unconventional monetary
policy actions in the late 1990s and early 2000s in stimulating inflation (see Ugai, 2007; Ueda, 2012b).

3

2

Money, Credit and Stagnation

Which empirical patterns should a model of secular stagnation be able to replicate? A
look at the transition of Japan from a high growth to a stagnating economy allows us toderive stylized facts on asset prices, credit, money and economic stagnation. We also showthat recent developments in other advanced economies are reminiscent of the Japanesesituation. In addition, we argue that our assumption of insatiable liquidity preferences isable to capture these patterns and relate our approach to the existing literature.
Stylized Facts Consider the macroeconomic development of Japan over the last three
decades as illustrated in Figure 1. The transformation from a high growth to a stagnating
economy is apparent in panels (a) and (b) which show real GDP growth and inflation:
From 1980 to 1991, the Japanese economy grew at an average rate of 4.4% in real termswith an annual inflation rate of 1.9%. In contrast, real GDP grew at only 0.9% on averagein the period since 1992 with inflation falling into negative territory.5
A distinguishing feature of the high growth and the stagnation period is the behaviorof asset prices and credit. As shown in panels (c) and (d), Japan experienced a credit
and asset price boom during its high growth period. Within ten years, the outstanding
amount of private credit to the non-financial sector as a fraction of GDP increased from
1.4 in 1980 to 2.2 in 1992 primarily driven by bank lending to small and medium-sizedcorporations and declining lending standards (see Posen, 2003). Credit to the private
sector grew on average by 7.9% in real terms during the period from 1980 to 1991 while
residential property prices (as a proxy for collateralizeable assets) increased by 5.1% inreal terms.6 Credit expansion and asset price inflation in terms of stock, land and housing
prices were at the core of Japan’s bubble economy.7
In contrast, asset prices declined and the private sector disencumbered in the stagnation period following the asset price crash of the early 1990s: Credit to the private
sector declined by 0.8% on average each year after 1991 while the real amount of credit
stagnated. Credit as a share of GDP declined by almost 20% from 2.2 in 1992 to a level of5

We measure inflation by the GDP Deflator. The patterns is the same for CPI inflation at 2.6% (19801991) and 0.25% (1992-2015). The tendency is the same when we exclude the financial crisis episodesince 2008. Then real growth is slightly higher at 1.2% (1992-2007) but still substantially below the pre1992 average. Similar developments hold for other measures of economic activity, like real consumption
expenditure growth which declines from 4.0% to 1.0%. Note that the recent increase in inflation in panel(b), as well as the spike in 1997, can be explained by an increase in the consumption tax in April 2014
(and 1997). Apart from the tax effect, there is no indication of a persistent increase in inflation.
6
Property price increases were higher for commercial property (6.0%) and in the six major cities(12.1%). Similarly, the subsequent decline was stronger for commercial property (-5.6%) and in cities(-4.8%). All housing price data comes from the Bank for International Settlement’s (BIS) “Long series
on nominal residential property prices” database, see Bank for International Settlements (2015).
7
For further discussion of the Japanese experience and the similarities with the recent developmentsin the United States, see Tsuruta (1999), Shimizu and Watanabe (2010), Ueda (2012a), among others.

4

Figure 1: Macroeconomic Development in Japan, 1980-2015(a) Real GDP Growth
9

6

6

4

3

2

0

0

−3

−2

−6

(b) Inflation

Consumption
tax increases

−4
1985

1995

2005

1985

2015

(c) Credit Growth

1995

2005

2015

(d) Asset Price Growth
16

15
10

8
5
0
0
−5

Nominal
Real
1985

1995

2005

−8

2015

Nominal
Real
1985

1995

2005

2015

Notes: Real GDP and inflation from the World Bank. Inflation is measured by the GDP
Deflator, in percent. Credit measures credit to the private non-financial sector and asset prices
refer to the residential property price index. Both series are from the Bank for International
Settlements (BIS). Real variables are deflated with the GDP Deflator.

1.6 in 2015. At the same time, nominal property prices decreased substantially by 3.1%
per year on average, while price decreases were somewhat smaller in real terms due todeflation. The decline in asset prices continued throughout the stagnation period without
any indication of a sustained recovery.
In addition, monetary policy became ineffective at stimulating output after 1991.
Panel (a) in Figure 2 illustrates the relationship between base money and nominal GDP
since 1980 for Japan. The neoclassical quantity equation seems to fit the data quite wellduring the 1980s as increases in the monetary base are associated with an increase in nominal spending. Yet, there clearly is a structural break in the early 1990s associated withthe transition to economic stagnation. Subsequent substantial increases in the money
supply - particularly during quantitative easing in the early 2000s and in the context of“Abenomics” - did not translate into higher nominal spending but simply resulted in a
decline in the circulation velocity of money.

5

Figure 2: Breakdown of the Quantity Equation(a) Japan

GDP(b) United States
GDP

400

12000
1980-1991

200

0

0
100

GDP

1980-2006

6000

200

300

M

1000

(c) United Kingdom

GDP

1400

2000

3000

M

(d) Euro Area

2000
1999-2006

700

1000

1980-2006

0

0
100

200

300

400 M

1000

2000

M

Notes: This figure shows the monetary base and nominal GDP for each country. Units aretrillion of Yen (Japan), billion of US-Dollar (United States), billion of Pound Sterling (United
Kingdom) and billion of Euro (ECB). Data from Federal Reserve Bank of St. Louis (FRED)
Database (US and UK), Bank of Japan and World Bank (Japan), Eurostat and ECB. Dataon the Euro Area includes other liquidity providing operations by national central banks.

Many advanced countries experience similar developments during the Great Recession. Panels (b) to (d) in Figure 2 show that the formerly stable relationship between
base money and nominal spending substantially changed in the United States, the United
Kingdom and the Euro Area as expansions in the money supply ceased to stimulate nominal spending. This structural change is associated with a prolonged period of depressed
spending because of persistent debt overhang in the aftermath of a credit boom. The extent of the credit expansion and the subsequent debt overhang are reminiscent of Japan’s
experience as illustrated in Figure 3.
To sum up, we observe persistent economic stagnation despite unprecedented expansions of the monetary base. The emergence of stagnation is associated with the end of
a credit boom that results in substantial debt overhang. In this paper we will present a
dynamic macroeconomic model of aggregate demand shortage that theoretically explainsthese phenomena.

6

Figure 3: Real Credit to the Private Non-Financial Sector
100

75

50

25
Japan
United States
United Kingdom
0
1985

1995

2005

2015

Notes: This figure shows real credit to the private non-financial sector. Data fromBank for International Settlements (BIS). Nominal credit deflated by GDP Deflator
and normalized to 100 in year of peak: 1996 (Japan), 2008 (US) and 2009 (UK).

Related Literature
We contribute to the literature on persistent aggregate demand
shortage based on the insatiability of liquidity or wealth preferences. This literature wasinitiated by Ono (1994, 2001) and substantially extended and microfounded by Ono and
Ishida (2014). The main modification in these models is the idea that the marginal utility
from holding real money balances has a strictly positive lower bound:
lim v 0 (m) = β > 0m→∞

As a consequence, increases in money holdings or wealth at some point cease to stimulate
consumption spending as agents prefer to hoard money or wealth instead.
The idea of a causal relationship between aggregate demand shortage and the insatiability of liquidity preferences goes back as far as Chapter 17 in Keynes (1936) as describedand analyzed in detail by Ono (2001). Moreover, Murota and Ono (2011) provide an explanation of this feature based on behavioral economics. Specifically, they show that thisproperty can be linked to relative status preferences with respect to money. From anempirical point of view, Ono et al. (2004) offer support for the insatiability of liquidity preferences based on quarterly data in Japan using parametric and non-parametricmethods.
7

Ono (1994) introduces heterogeneous households into this framework and redistributive policies are analyzed by Matsuzaki (2003) for consumption taxes and Hashimoto(2004) for intergenerational transfers.8 In these models, financial markets are assumed
to be perfect and agents are heterogeneous only with respect to their initial wealth.
We extend this framework to feature financial market imperfections and introduce
heterogeneity in time preference rates to motivate borrowing. This allows us to analyze
interactions of collateral, asset prices and aggregate demand. We implement these via
a borrowing constraint in the spirit of Kiyotaki and Moore (1997) and Iacoviello (2005)
such that the value of collateral is endogenously determined in the housing market.9
Research interest in models of secular stagnation has increased substantially in theaftermath of the financial crisis. Michaillat and Saez (2014) and Michau (2017) develop
similar models of stagnation that build upon a constant marginal utility of wealth. Eggertsson and Mehrotra (2015) analyze stagnation and the effects of deleveraging in an
OLG framework. Eggertsson et al. (2016) extend this setup to the open economy. In addition, some recent contributions analyze the effects of (the burst of) asset price bubbles
on economic growth (cf. Boullot, 2017; Hanson and Phan, 2017; Biswas et al., 2017).
Our work is linked to the idea of balance sheet recessions. In our setup, households do
continuously pay off debt. However, deflation makes these efforts self-defeating: Because
contracts are in nominal terms, deflation increases the real value of debt. In equilibrium,
the real debt burden remains unaffected and continues to impede the recovery. Balance
sheets are not restored, even in the long run. Deleveraging shocks are also discussed ina similar borrower-saver framework by Eggertsson and Krugman (2012). However, theauthors do not model persistent but only temporary stagnation.
Our approach differs from the liquidity trap literature.10 This view explains aggregate
demand shortages as the consequence of negative shocks in combination with a lower
bound on the nominal interest rate. Stagnation is a temporary phenomenon. This isin stark contrast to the experience of Japan where deflationary forces already prevail
for more than two decades. It is difficult to make the case for the prevalence of pricerigidities over such a long period. In our model, stagnation occurs in steady state despite
the possibility of continuous price adjustment.
8
Further extensions include Ono (2006, 2014) and Hashimoto and Johdo (2009) who model persistentstagnation in a two-country framework to analyze the role of FDI and international spillovers of various
policies as well as Rodrı́guez-Arana (2007) who analyzes fiscal deficits under stagnation. Moreover,
Murota and Ono (2012) explain zero nominal interest rates and excess reserve holdings by commercial
banks in a setup with preferences for deposit holdings. In addition, Ono (2015) applies this framework
to the situation of Japan to explain the transition from high growth to secular stagnation based solely
on the insatiability of wealth preferences.
9
We do not explicitly model the asymmetric information problems that give rise to the financial
friction (see Townsend, 1979; Stiglitz and Weiss, 1981, among others for a microfoundation).
10
The modern treatment of this problem started with the seminal paper of Krugman (1998), which
initiated an extensive literature, particularly since the global financial crisis.

8

3

The Model Economy

We use a continuous time model with money-in-the-utility that features competitive firms,
two types of households and a central bank but abstracts from taxation or government
expenditures. Agents have perfect foresight and there is no uncertainty in the model. We
build on Ono (1994, 2001) for the idea of permanent demand shortage based on insatiableliquidity preferences and Iacoviello (2005) for modeling endogenous borrowing constraints
with durable assets as collateral to introduce private sector debt.

3.1

Firms

The supply side is modeled as a Lucas tree. Firms are price takers and produce theamount ȳ of the consumption good without any inputs or costs. This constitutes the
economy’s production capacity or a measure of potential output. Yet, actual sales are
determined by aggregate demand Ct so that actual income yt falls short of potential
output in case of aggregate demand shortage. Firm sales are hence given by
yt = min {Ct , ȳ} .

(1)Nominal firm profits are simply given by Pt yt as production is costless. These are
distributed equally across households and show up as exogenous income in the budget
constraints. When falling short of potential output ȳ, aggregate demand determines
firm profits and household income. As a consequence, there are feedback loops between
spending and income.
In addition, we abstract from the labor market and the wage-setting process andinstead introduce a reduced-form Phillips curve for the inflation rate πt . Specifically, theprice level dynamics under full employment differ from those in the presence of aggregate
demand shortage as follows:

πt =

Ṗt
Pt




 µ if Ct = ȳ ,


=
Ct


−1
if Ct < ȳ .
 α
ȳ

(2)

Under full employment, the dynamics of the price level are similar to the standard MIU
framework. The price level adjusts to clear the money market and the inflation rate isdetermined by the growth rate of the money supply µ, such that the quantity equation
holds and money is neutral. In contrast, the output gap determines inflation in caseof aggregate demand shortage, where the parameter α > 0 governs the speed of priceadjustment. A negative output gap will result in deflation. If the output gap persists insteady state, deflation will persist.
9Similar relations are derived in standard macroeconomic models with a labor market
based on downward nominal wage rigidity.11 Specifically, Ono and Ishida (2014) and Ono(2015) provide the following microfoundation for equation (2) based on fairness concerns
in the wage setting process:12 In their model, the productivity of workers depends on theirperception of being treated in a fair way. In particular, workers withhold effort when they
are not remunerated at least with a “fair wage”. Under full employment, competition
among firms for workers determines the wage offer. Therefore, the dynamics of the pricelevel determine the wage dynamics. The former are in turn dependent on the money
supply growth. In contrast, firms have bargaining power when there is unemployment.
However, the fair wage provides a lower bound on wage offers to prevent shirking. As aconsequence, it is the dynamics of the fair wage that determine the wage and hence theprice dynamics. These are in turn related to the level of unemployment or the output
gap. Taken together, inflation is governed by an expression similar to equation (2), where
α can be interpreted as the (exogenous) job separation rate faced by workers.
In addition, the Phillips curve in equation (2) is formally equivalent to wage setting
frictions as in Eggertsson and Mehrotra (2015), Michau (2017) or Schmitt-Grohé andUribe (2016, 2017). All of these contributions introduce some form of downward nominal
wage rigidity that becomes binding in case of unemployment. Eggertsson and Mehrotra(2015) assume that wages cannot fall below a “wage norm”, which is a linear combinationof past wages and the marginal product of labor. In Michau (2017), wage demands ofworkers are guided by a reference rate of inflation, which creates an asymmetry in thewage dynamics similar to the one discussed above. Finally, Schmitt-Grohé and Uribe(2016, 2017) introduce an exogenous lower bound on the growth rate of the nominal
wage that becomes binding in case of unemployment. The same mechanism is used by
Hanson and Phan (2017) and Biswas et al. (2017).
It is worth pointing out that our conclusions on the role of asset prices and private
sector debt for economic stagnation continue to hold in the presence of a richer modeling of
the labor market. Specifically, the introduction of a production function and wage setting
frictions in the spirit of Ono and Ishida (2014) does not alter our results qualitatively but
comes at the cost of computational complexity. It is for this reason that we decided torely on the reduced-form expression for inflation introduced above.
11

As argued by Schmitt-Grohé and Uribe (2016): “There is abundant empirical evidence on downward
nominal wage rigidity stemming mostly from developed countries.” An overview of the empirical evidence
is presented in section 8 of their paper.
12
In these models, the representative household has a fixed labor endowment. In equilibrium, competitive firms make zero profits, which is why the real wage equals labor productivity, which is constant due
to the linear production function. Pt yt then is not a lump-sum transfer of profits but labor wages andthe deflation gap is related to the labor market instead of the commodity market. Yet, the implications
for the emergence of stagnation are only modestly affected.

10

3.2

Households

There is a mass one of infinitely-lived households. Each household is one of two types
based on his time preference rate ρi : A fraction n of households are savers (i = 1)
whereas the remaining fraction 1 − n are borrowers (i = 2) in the sense that ρ1 < ρ2 .13
This setting will endogenously result in differences in wealth levels and we will hencemodel an economy in which the “rich” (savers) lend to the “poor” (borrowers).14
Households have three means of savings: money Mi,t , credit contracts Bi,t and real
assets in the form of housing hi,t . Money yields an interest rate of RM = 0 whereas loans
are contracted at the non-negative nominal interest rate Rt . Let Bi,t > 0 denote savingsin the form of loans issued and Bi,t < 0 debt in the form of credit. Let Qt denote thenominal house price in period t. The return on housing depends on the resale value ofthe house in the following period.
Then total nominal wealth Ai,t is given by the sum of the household’s money holdings,
bond holdings and the value of its housing stock: Ai,t = Bi,t + Mi,t + Qt hi,t . In real terms,
total wealth is given byai,t = bi,t + mi,t + qt hi,t ,(3)where lowercase letters denote the respective variables in real terms such that qt denotes
the real house price defined as Qt = Pt qt . Households are the owners of firms and receive
firm profits Pt yt , where yt is defined in equation (1). These profits are distributed equally
across both types and considered exogenous by the households. In addition, households
receive all income from seignorage in a lump-sum transfer Zi,t . For the moment, thistransfer is not important. Later, we will assume that µ = 0 and hence Zi,t = 0. Yet, it
becomes relevant for the discussion of µ > 0 in section 6. In real terms, the flow of funds
constraint is given by15ȧi,t = rt ai,t − Rt mi,t − (rt qt − q̇t )hi,t − ci,t + yt + zi,t .
13(4)

The borrower-saver separation based on differences in time preference rates is a standard method
to introduce borrowing incentives in macroeconomic models, see Sufi (2012). Since these differences
are permanent, the roles of lenders and borrowers are static. Alternative ways of modeling include
idiosyncratic income shocks or an uneven life-cycle income distribution.
14
Alternatively, we could assume that agents differ in their initial wealth ai,0 such that a1,0 >> a2,0 .
15Equation (4) is based on the following expressions for the evolution of nominal and real wealth where
we use the composition of household assets to substitute for Bt :Ȧt = Rt Bt + Q̇t ht − Pt ct + Pt yt = Rt At − Rt Mt − Rt Qt ht + Q̇t ht − Pt ct + Pt yt + Zi,tQ̇t = Pt q̇t + qt Ṗt
 ˙ 
At
Ȧt
At Ṗtȧt ==−= (Rt − πt )at − Rt mt − (Rt qt − πt qt − q̇t )ht − ct + yt + zi,t
Pt
Pt
Pt Pt

11

The household incurs opportunity costs when holding money because of the foregone
interest income that would be associated with lending. Similar costs arise when investingin housing. Yet, housing investment involves the possibility of capital gains (or losses)
associated with changes in the real house price captured by q̇t .
Impatient households have a strong motive to borrow. However, lenders require sufficient collateral in the form of housing because of problems of asymmetric informationin the credit market. As a consequence, savers will only lend up to a fraction θ of thevalue of the borrower’s collateralizeable assets.16 In real terms, the associated borrowing
constraint takes the form:
b2,t ≥ −θqt h2,t .
(5)
In our model, housing is the only durable asset that serves as collateral. In contrast,
money is not collateralizable because it is too fungible to be effectively seized by lenders
in case of missed repayment.17
Apart from differences in time preference, households have identical preferences. They
choose consumption, real money holdings and housing to maximize their lifetime utility
function:Z ∞Ui =[u(ci,t ) + v(mi,t ) + w(hi,t )] e−ρi t dt ,(6)0where ρi denotes the subjective discount rate of the household of type i. Utility fromconsumption and housing satisfies the Inada conditions. For simplicity, we make thefollowing functional form assumptions on these instantaneous utility functions:u(ci,t ) = ln(ci,t ) ; w(hi,t ) = γln(hi,t ) ,where γ > 0 is an exogenous and positive constant. In contrast, the Inada conditions do
not hold for the utility from real money balances. As discussed in the previous section andfollowing Ono (1994, 2001), we deviate from the neoclassical assumptions and introduce
insatiable liquidity preferences. Formally, the marginal utility of real money holdings
does not converge to zero but approaches a strictly positive constant value:
lim v 0 (m) = β > 0 .

m→∞

We will explain the consequences of this assumption in the following sections.
16

We refer to the parameter θ as the loan-to-value ratio. Throughout this paper, we choose parameters
to ensure that the borrowing constraint is always binding.
17
It is easy to introduce a collateral value for money in this setting. The borrowing constraint thenbecomes b2,t ≥ −θ1 qt h2,t − θ2 m2,t , where θ2 determines the collateralizability of money. For the special
case of θ1 = θ2 = θ, this formulation, together with (3), implies that (5) becomes a pure wealth constraint
where the composition of wealth is irrelevant, i.e. b2,t ≥ −θ(1−θ)−1 a2,t . Our main results are unchanged(and even stronger) when using this formulation.

12

Rich Households (Savers): Savers maximize lifetime utility (6) subject to the wealthcomposition (3) and the flow budget constraint (4). From the Hamiltonian function:
H1 = u(c1,t ) + v(m1,t ) + w(h1,t ) + λ1,t (rt a1,t − c1,t − Rt m1,t − (rt qt − q̇t )h1,t + yt ) ,we obtain the following optimality conditions:1= λ1,t ,c1,tλ1,t Rt = v 0 (m1,t ) ,γ= λ1,t (rt qt − q̇t ) ,
h1,t

(7)
(8)
(9)

λ̇1,t = (ρ1 − rt )λ1,t ,(10)

lim λ1,t a1,t e−ρ1 t = 0 .

(11)

t→∞

Equations (7) to (11) describe the optimal consumption, money holdings, housing investment and borrowing of the rich agent as well as the transversality condition for real
wealth. For the saver, the nominal interest rate governs both the intertemporal allocationof consumption via (7) and (10) as well as the intra-temporal trade-off between money
and consumption according to (7) and (8). This yields the following expression:ċ1,t+ ρ1 + πt = Rt = v 0 (m1,t )c1,t .
c1,t

(12)

In optimum, the rich household equates the marginal rate of substitution between present
and future consumption to the marginal rate of substitution between present consumption and money holdings, i.e. the liquidity premium, which also equals the nominalinterest rate that constitutes the opportunity cost of holding money. Under neoclassicalassumptions, the liquidity premium is declining in m1,t , all else equal, thereby stimulating
consumption or decreasing the nominal interest rate.
In contrast, with insatiable liquidity preferences, the marginal utility of real money
holdings will reach the positive lower bound if the wealth of the patient households is
sufficiently high, i.e. v 0 (m1 ) = β. Then the liquidity premium no longer declines withadditional money holdings and Rt = Rt (c1,t ). As a consequence, consumption of the richhousehold is unaffected by changes in his money holdings for a given nominal interest rate.
For that reason monetary policy becomes ineffective in single agent models such as Ono(2001): Additional money is stored as cash and does no longer stimulate consumption.
The economy is trapped in a deflationary steady state despite an infinite expansion of
the real money stock.

13

Poor Households (Borrowers): Borrowers maximize lifetime utility (6) subject tothe wealth composition (3), the flow budget constraint (4) and the borrowing constraint(5). Therefore, their Hamiltonian function is given by:
H2 = u(c2,t ) + v(m2,t ) + w(h2,t ) + λ2,t (rt a2,t − c2,t − Rt m2,t − (rt qt − q̇t )h2,t + yt )+ϕt (a2,t − m2,t − (1 − θ)qt h2,t ) ,from which the following optimality conditions are obtained:1= λ2,t ,
c2,t

(13)

λ2,t Rt + ϕt = v 0 (m2,t ) ,(14)γ= λ2,t (rt qt − q̇t ) + ϕt (1 − θ)qt ,
h2,t

(15)

λ̇2,t = (ρ2 − rt )λ2,t − ϕt ,(16)

lim λ2,t a2,t e−ρ1 t = 0 .

(17)

t→∞

Equations (13) to (17) describe optimal consumption demand, money demand, housing investment and borrowing of the poor agent as well as the transversality condition.
The borrower also equates the marginal rate of substitution between present and future
consumption to the liquidity premium. This results from (13), (14) and (16) and givesċ2,t+ ρ2 + πt = Rt + ϕt c2,t = v 0 (m2,t )c2,t .
c2,t

(18)

The borrowing friction affects optimal money demand and the evolution of consumption.
Impatience creates a strong motive to borrow funds for current consumption so thatcurrent funds have a higher value to the borrowers than to the savers. When these funds
are used to increase liquidity instead of consumption, the household incurs an implicit
cost of ϕt due to the borrowing constraint facing in fact a higher implicit interest rate
than the saver. As a consequence, optimal money demand is reduced relative to the casewithout borrowing frictions.
Under neoclassical assumptions, the liquidity premium decreases with money holdings
for the borrower, i.e. v 00 (m2 ) < 0. In contrast, with insatiable liquidity preferences,v 0 (m2 ) = β > 0 if the borrower becomes sufficiently wealthy. As a consequence, ourmodel features different regions depending on the behavior of v 0 (m1 ) and v 0 (m2 ), which
we will discuss in the following sections.

14

3.3

Asset Prices, Borrowing and Leverage

What determines the dynamics of the real house price? Households incur opportunity
costs when investing in housing because of the opportunity loss of real interest incomethat is associated with the alternative of bond savings. Yet, agents gain utility fromhousing which is captured by the user cost, i.e. the marginal rate of substitution between
consumption and housing. For the saver, this follows from (7) and (9). For the borrower,
housing investment comes at a higher cost, his implicit interest rate being higher thanfor the saver. Yet, since housing serves as collateral, the associated borrowing costs arelower than those for money at (1 − θ)ϕt which can be seen in (13) and (15).
Moreover, changes in the real house price affect the costs and benefits of housing due tovaluation effects. In optimum, the real house price adjusts such that agents are indifferent
between investing in an additional unit of housing and alternative uses. Hence, the realhouse price has to appreciate if the opportunity costs from housing exceed the user costs
to compensate housing investors for the higher costs with capital gains. Similarly, thereal house price has to depreciate if the benefits of housing exceed the opportunity costs
resulting in capital losses for house owners. From equations (7), (9), (13) and (15), thedynamics of the real house price can be expressed as the difference between opportunity
costs and housing benefits for both agents asq̇t = rt qt −

γc2,t
γc1,t
= rt qt −+ ϕt (1 − θ)qt c2,t .
h1,t
h2,t

(19)

Throughout the analysis, we consider the case of a strictly binding borrowing constraint, i.e. ϕt > 0. Then, the borrower always takes on loans up to the maximum givenby (5). It follows from (3) and (5) that total real assets of the borrower consist of hismoney holdings and housing investment, a fraction θ of which serves as collateral:
a2,t = m2,t + (1 − θ)qt h2,t .

(20)

Similarly, total real assets of the saver include loans to the borrower. From (18) and(19), the consumption value of borrowing ϕt c2,t - or equivalently, the consumption costof debt-financed money holdings or housing investment - equals the difference in theliquidity premia and is proportional to the difference in the user cost of housing:1v (m2,t )c2,t − v (m1,t )c1,t = ϕt c2,t =
1−θ
0

0



γc2,t
γc1,t
−
qt h2,t qt h1,t


.

(21)

Hence, a binding borrowing constraint implies that it is more costly (in terms of consumption) for the borrower to hold money or invest in housing than for the saver.

15

3.4

Market Equilibrium Conditions

Aggregate demand Ct consists of the consumption demand of savers and borrowers:
Ct = nc1,t + (1 − n)c2,t .

(22)

Aggregate demand relative to potential output determines the output gap which in turnis related to inflation via (2). In addition, aggregate demand determines firm profits andhousehold income yt in the flow budget constraints (4).
The central bank perfectly controls the nominal money supply Mt which grows at anexogenous rate µ. Hence, the real money supply mt evolves asṁt= µ − πt .
mt

(23)

In contrast, the nominal interest rate Rt is determined endogenously in the money market.
It is related to inflation and the real interest rate via the Fisher Equation
Rt = rt + πt .

(24)

Total money demand is the weighted average of the individual money demands. Money
market clearing requires that real money demand equals the real money supply mt :mt =M
= nm1,t + (1 − n)m2,t .
Pt

(25)

Loans are financial claims among households. Hence, they are in zero net supply with
nb1,t + (1 − n)b2,t = 0 .

(26)

In contrast, housing is a real asset. Following Iacoviello (2005), we assume a fixed supplyof houses H and abstract from depreciation and construction both of which could easily
be implemented in this setting.18 Market clearing in the housing market then requires
nh1,t + (1 − n)h2,t = H .

(27)

Equations (1) to (27) fully describe the model economy. The dynamics are summarized
by a system of differential equations given by (4), (12), (18), (19) and (23) where (4)applies to both types. All other variables are derived from the solution to this system.
18
This assumption seems reasonable for an economy like Japan that is characterized by land scarcity
and a low price elasticity of the housing supply. A study by Shimizu and Watanabe (2010) concludes
that the housing supply was very price inelastic during the Japanese housing boom of the late 1980s,
partly due to the incentives given by the tax system as well as regulation on land utilization.

16

4

Analysis of the Model Economy

In the following analysis, we focus on the special case of a constant nominal money supply,µ = 0,
for simplicity. This implies a zero trend inflation rate under full employment as is evident
from (2).19 Yet, the qualitative conclusions of our analysis can be generalized and holdfor any given level of µ as we will discuss in section 6 in greater detail.
Our model framework features three regions depending on the behavior of the marginalutilities v 0 (m1 ) and v 0 (m2 ). This is in turn related to the production capacity ȳ:
1. For low levels of potential output ȳ, the economy behaves as in the standard neoclassical case. The marginal utility of money is decreasing in money holdings forboth households, i.e. v 00 (mi ) < 0, and aggregate demand equals potential output.
The price level is constant and changes proportionally with the money supply.
2. For higher levels of ȳ, there is an asymmetric steady state under stagnation. In
this region, the patient household’s marginal utility of money is constant while
the impatient household’s liquidity premium still declines with additional money
holdings, i.e. v 00 (m1 ) = 0 and v 00 (m2 ) < 0. Aggregate demand falls short of potential
output and deflation occurs.
3. For very high levels of potential output, the symmetric steady state under stagnation might occur. In this region, the marginal utility of money has reached its lower
bound for savers and borrowers, i.e. v 00 (mi ) = 0.
Among the three mentioned above, we focus on the asymmetric steady state under
stagnation for several reasons. First, this steady state features economic stagnation anddeflation unlike the neoclassical case. Secondly, indebtedness and asset prices play animportant role in affecting the severity of stagnation.20 Thirdly, it is more in conformity
with what has occurred in the Japanese economy as discussed in the introduction. The
asymmetric steady state under stagnation is defined as follows:
Asymmetric Steady State: The real and nominal interest rates are constant,the price level is declining at a constant rate, the real consumption level ofeach household is constant as is the real house price, and the borrower’s asset
level is constant while the saver’s wealth expands infinitely:ṙ = 0 , Ṙ = 0 , π < 0 , ċ1 = 0 , ċ2 = 0 , q̇ = 0 , ȧ1 > 0 , ȧ2 = 0 .
19

(28)

This parameterization also allows us to derive some expressions in closed-form that do not dependon the shape of the function v(m), which helps to provide a more intuitive understanding of our results.
20
In contrast, changes in leverage cease to affect aggregate demand in the symmetric steady state, but
simply affect asset prices and the distribution of the housing stock.

17

4.1

The Occurrence of Persistent Stagnation

Intuitively, aggregate demand shortage occurs if potential output is so high that households are no longer willing to consume the available amount of ȳ due to the insatiabilityof liquidity preferences of the saver.21 For lower levels of potential output, the economy
attains full employment at zero inflation (in the present case as µ = 0) and the pricelevel adjusts to clear the money market for a given level of the nominal money supply inequation (25). We define this full employment steady state as follows:
Neoclassical Equilibrium: The real and nominal interest rates are constant,the price level is constant, the real house price is constant and the consumption
and wealth of all households are constant:ṙ = 0 , Ṙ = 0 , π = 0 , ċ1 = 0 , ċ2 = 0 , q̇ = 0 , ȧ1 = 0 , ȧ2 = 0 .

(29)Consider first the case of homogeneous agents: Suppose there are only patient households. From (12) with ċ1 = 0, the economy attains full employment, i.e. c1 = ȳ, and zeroinflation as long as the marginal utility of money can adjust such that v 0 (m1,t )ȳ = ρ1 .
With insatiable liquidity preferences, there is a lower bound β of the marginal utility ofreal money holdings. Once the production capacity ȳ exceeds the level of ρ1 β −1 , there is
no longer a solution to (12) that is compatible with π = 0 and c1 = ȳ. This is becausehouseholds are no longer willing to consume the available output but prefer to accumulatemoney instead. As a consequence, stagnation and deflation occur in equilibrium.
Similarly, suppose there were only impatient households.22 From (18) with ċ2 = 0,
the economy attains full employment as long as higher spending can be accommodated
at zero inflation such that v 0 (m2 )ȳ = ρ2 . There is no solution to (18) consistent with full
employment once ȳ is above ρ2 β −1 . Taken together, the relevant condition for homogeneous agent models is given by ȳ > ρi β −1 where ρi refers to the representative household.
This condition is illustrated by the lower line in Figure 4.
In an economy with n savers and 1 − n borrowers, the distribution of consumption
spending under full employment determines the occurrence of stagnation. Since π = 0,
we have from (12) and (24) that R = r = ρ1 . Then, consumption levels are derived
from the flow budget constraints (4). The borrower consumes his income net of interest
payments on debt. Income in turn depends on aggregate demand which equals potential
output. His consumption in the neoclassical steady state is then given by
C
cN
=
2

ρ̄θ
ȳ , where ρ̄θ ≡ θρ1 + (1 − θ)ρ2 .
ρ̄θ + θρ1 γ

21

(30)

In addition, the liquidity premium of the borrower must still decline with additional money holdings
for asymmetric (instead of symmetric) stagnation to occur. This requirement is not important at the
moment and will be discuss in greater detail in section 6.
22
Note that in this case, the borrowing constraint would cease to be binding, i.e. ϕt = 0.

18

Figure 4: Occurrence of Persistent Stagnation

Neoclassical Case

Stagnation with Savers and Borrowers

Stagnation with Homogeneous Agents;1 n ;1 3.+7
;3- ;1 3.+n7
;3

3>0

;i
-

Potential Output y7

Notes: This figure shows the equilibrium of the model for different values of ȳ compared to homogeneous agent models. In particular, note that in the heterogeneous agent framework economic
stagnation occurs for smaller levels of potential output.

Note that ρ̄θ can be interpreted as the debt-weighted average discount rate. This follows
from (4), (20) and the requirements π = 0, q̇ = 0 and ȧ2 = 0. The rich household behaves
similarly, but receives interest income on its lending. Hence, steady state consumptionof the saver exceeds consumption of the borrower in the neoclassical steady state due tothe redistribution associated with ownership of financial assets:
C
cN
=
1

nρ̄θ + θρ1 γ N Cnρ̄θ + θρ1 γCȳ =c2 > cN.
2nρ̄θ + nθρ1 γ
nρ̄θ

(31)

It is easy to see from these expressions that aggregate demand equals potential output.
Yet, it follows from (12) and (18) with ċi = 0 that the consumption levels of both agentsin (30) and (31) are consistent with zero inflation only if the marginal utility of money
falls sufficiently. In particular, for the neoclassical case to exist it has to hold thatv 0 (m1,t ) =ρ2
ρ1
and v 0 (m2,t ) = N C .
NC
c1
c2

(32)

With insatiable liquidity preferences, there exists a lower bound on the marginal utility ofCmoney such that v 0 (mi,t ) ≥ β. Hence, the neoclassical case is not feasible once βcN> ρi .
i
So with rising levels of potential output ȳ, at some threshold the stagnation steady stateCC
will occur. Since ρ1 < ρ2 and cN> cN
1
2 , it is the saver’s marginal utility of moneythat will reach its lower bound first for rising levels of ȳ. Then, aggregate demand falls
short of the production capacity and the economy enters the stagnation steady state. We
derive the following proposition from combining (31) and (32):19

Proposition 1 The neoclassical equilibrium with full employment and zero inflation cannot be attained once potential output exceeds the following threshold:ye ≡ρ1 n θρ1 γ + ρ̄θρ1<.
β θρ1 γ + nρ̄θβ(33)

The threshold ye is affected by the model parameters as follows (see Appendix A):
∂e
y
∂e
y
∂e
y
∂e
y
∂e
y
∂e
y
<0,
>0,
>0,
<0,
>0,
<0.
∂β
∂ρ1
∂ρ2
∂θ
∂n
∂γ
Once potential output exceeds ye, the economy is in the asymmetric steady state under
stagnation defined in (28) and suffers from insufficient demand and deflation. Additional
income does no longer stimulate consumption of the saver who chooses to accumulate
wealth instead. This is represented by the upper line in Figure 4.
The lower the insatiability parameter β, the higher potential output needs to be forthe economy to enter stagnation. Similarly, increases in the time preference rate of thesaver ρ1 or in their fraction of the population n also increase the income threshold. The
same holds for a higher time preference rate ρ2 of the borrower.
What we add is the insight that financially more developed countries, i.e. countries
with higher leverage, drift into stagnation already at a lower level of potential output.
This is because the higher debt is associated with lower steady state consumption demand
from the borrower. To see this, note from (30) that if financial markets are closed and noborrowing is possible, i.e. if θ = 0 or γ = 0, the consumption levels of both households areequal and given by ci = ȳ under full employment. Once we allow for borrowing, housing
investment is associated with an increase in indebtedness of the borrower. This in turnresults in a higher real interest burden on poor households and reduces their affordable
consumption. This gives rise to a more unequal income distribution but does not affectaggregate demand as long as the rich households expand their consumption accordingly.
If they invest in liquidity holdings instead, aggregate demand falls short of the economy’s
production capacity and stagnation occurs.23
Let us contrast this condition with the existence condition in single-agent models
without lending and borrowing as in Ono (2001), which was discussed above. Condition(33) is reduced to this expression if we abstract from housing (γ = 0), if we do not allow
for borrowing (θ = 0) or if there are only rich households (n = 1). In all other cases, ye isbelow the threshold of the single-agent model. Hence, the economy enters stagnation inan earlier stage, which is illustrated in Figure 4. The reason is that consumption of thesaver is higher due to additional income associated with interest payments on loans.
23

The same effect arises for a higher value of γ. Economies that invest more heavily in assets in fixed
supply are hence more prone to stagnation. Also note that the effects of θ and γ are mutually reinforcing.

20

4.2

The Asymmetric Steady State under Persistent Stagnation

Under asymmetric stagnation, higher consumption spending of the borrower stimulates
consumption spending of the saver. This follows from (12) with ċ1 = 0 and (2) tosubstitute for π. The reasoning is as follows: An increase in consumption of the borrower
expands aggregate demand and mitigates deflation. Less deflation in turn increases thenominal interest rate via (24) since the real rate equals the saver’s time preference rate
ρ1 as obtained from (10) with λ˙1 = 0. The nominal rate has to equal the marginal rateof substitution between consumption and money holdings for the saver and since themarginal utility of money is constant due to insatiability, consumption c∗1 has to increase.
This is the same relation as in Ono (1994) and Matsuzaki (2003) and results from theinsatiability of liquidity preferences in combination with sluggish price adjustment as
manifested in the Phillips curve relation in (2):24c∗1 =(ρ1 − α)ȳ α(1 − n) ∗+c .
β ȳ − αnβ ȳ − αn 2(34)Spillovers from aggregate demand are stronger the higher the share of spending constrained households (1 − n) and the higher the speed of price adjustment α. In particular,
steady state consumption c∗1 is not directly affected by the borrowing decision or asset
composition of the impatient household. Yet, there are indirect effects via c∗2 .
In contrast, consumption spending of the borrower is affected by his money holdingsin equilibrium. This follows from optimality condition (18) with ċ2 = 0 and (2) and (34)
to substitute for π ∗ and c∗1 and impliesc∗2 =χ
v 0 (m∗2 )(β ȳ − αn) − βα(1 − n)

,

(35)

where χ ≡ ρ2 (β ȳ − αn) − α(β ȳ − ρ1 n) .
More money induces more consumption of the borrower. This is necessary to equalize
the liquidity premium to the nominal interest rate which is itself a function of c∗2 via (12)and (34). Note that this channel does not exist for the saver who accumulates money
without expanding consumption.
Importantly, we require parameter restrictions to guarantee positive consumption levels c∗1 and c∗2 in steady state. From (34) and the denominator of (35), since v 0 (m∗2 ) > β,we make the following assumptions throughout this paper:(i) ρ1 > α and (ii) β ȳ > α .
24

(36)

Steady state values of the respective variables will be characterized by “*” for notational convenience.

21

In addition, the shadow value of borrowing is determined by the difference in discountrates. From (12) and (18) with ċ1,t = ċ2,t = 0, the Lagrange parameter on the borrowing
constraint is given byρ2 − ρ1
ϕ∗ =
>0.
(37)
c∗2
Hence, the borrowing constraint is binding in equilibrium. Higher money holdings of theborrower reduce the value of additional funds since c∗2 increases in m∗2 as discussed above.
Total wealth of the borrower consists of money holdings and housing investment netof loans via (20). The steady state value of housing investment of each agent in turn isa constant fraction of its consumption level given byq ∗ h∗1 =γ ∗γ
c1 and q ∗ h∗2 = c∗2 , where ρ̄θ ≡ θρ1 + (1 − θ)ρ2 .
ρ1
ρ̄θ

(38)

This follows from (16) with λ̇2 = 0 and from (19). Increases in θ provide higher incentives
for borrowers to invest in housing because of its role as collateral. Note that (5) and (38)
imply that the real level of debt is constant. Hence, borrowers do delever in nominal
terms, which however is self-defeating due to deflation. Substituting (35) solved for m∗2
and (38) into (20) implies the following expression for the real wealth of the borrower insteady state:(1 − θ)γ ∗a∗2 =c2 + m∗2 ,(39)ρ̄θ
where m∗2 is a function of c∗2 given by (35). It follows that real wealth and consumption
demand are positively related for the borrower. An increase in consumption induces both
higher money holdings and higher housing investment, and hence higher real wealth.
Higher consumption demand implies higher housing demand, as is clear from (38),because they are substitutes. This is true for both types of households. Market clearing
in the housing market then requires a higher equilibrium house price in response to anincrease in aggregate demand. This follows from the market clearing condition (27) incombination with steady state housing demands. The real house price obtained from (27)and (38) is given by
γ n ∗ 1−n ∗
∗
q =
c +
c .
(40)
H ρ1 1
ρ̄θ 2
From (34), c∗1 = c∗1 (c∗2 ) and hence q ∗ = q ∗ (c∗2 ). An increase in consumption of the borrower
increases the real house price in steady state. Also note that the house price increases
with the housing preference γ and decreases with a higher supply H, all else equal.
The borrower’s real assets are constant in the asymmetric steady state. From thebudget constraint (4) with ȧ2 = 0 and (20), (22) and (38), we getnc∗1 + (1 − n)c∗2 + (ρ1 − βc∗1 )m∗2 =
22




θρ1 γ
+ 1 c∗2 ,
ρ̄θ

(41)

where c∗1 = c∗1 (c∗2 ) from (34) and m∗2 = m∗2 (c∗2 ) from (35). The borrower obtains real
income from two sources: First, the household receives firm profits which are determinedby aggregate demand. This is reflected in the term nc∗1 + (1 − n)c∗2 of (41). Secondly,
inflation affects the real return on money. Since money does not pay interest, the real
return is given by the rate of deflation, i.e. −π ∗ = ρ1 −βc∗1 . This income is used to financeconsumption expenditures c∗2 and to pay interest on debt. These interest payments dependon the household’s borrowing capacity which is determined by the value of its housing
collateral via (5). The collateral value in turn is related to consumption as is clear from(38). In equilibrium, real interest payments are a fraction θρ1 γ ρ̄−1θ of consumption andincrease with θ, γ and ρ1 but decrease with ρ2 , because ρ̄θ = θρ1 + (1 − θ)ρ2 .
Finally, it follows from (25) that the real money stock becomes infinitely high becauseof the effects of deflation. The increase in the real money supply exclusively benefits
the saver in the asymmetric steady state. Hence, his real wealth expands at the rate ofdeflation. However, this does not violate the transversality condition (13) since it holds
that ρ1 > α > −π ∗ . In contrast, increases in the real money stock accrue to both agentsin the symmetric steady state, again without violating the transversality conditions.
Equations (34) to (41) define the asymmetric steady state under persistent stagnation.
Once m∗2 , c∗1 and c∗2 are jointly determined from the combination of (34), (35) and (41),all other variables are derived from these values.
Model Dynamics under Asymmetric Stagnation The model dynamics are represented by a system of six differential equations for consumption and real assets of savers
and borrowers, the real house price and the real money supply. All other variables canbe derived from this system: From (2) and (22), it follows that πt = π(c1,t , c2,t ). From(12) with v 0 (m1,t ) = β, we have Rt = R(c1,t ) and hence rt = r(c1,t , c2,t ) from (24). Given
c1,t , c2,t , qt and a2,t , equations (20), (21) with v 0 (m1,t ) = β and (27) determine money holdings m2,t and housing investments h1,t and h2,t . From (21), it follows that the implicit
cost due to the borrowing constraint, ϕt , is a function of the same four variables.
The evolution of the saver’s consumption is determined by (12) with v 0 (m1,t ) = β:ċ1,t
= βc1,t − πt − ρ1 .
c1,t

(42)

Since πt = π(c1,t , c2,t ), it follows that ċ1,t = ċ1 (c1,t , c2,t ). Similarly, the evolution of theborrower’s consumption is determined by (18) with v 0 (m2,t ) > β:ċ2,t
= v 0 (m2,t )c2,t − πt − ρ2 .
c2,t

(43)

Since πt = π(c1,t , c2,t ) and m2,t = m2 (c1,t , c2,t , qt , a2,t ), we have ċ2,t = ċ2 (c1,t , c2,t , qt , a2,t ).
23

The evolution of the real house price is determined by (19):
q̇t
γc1,t
= rt −.
qt
qt h1,t(44)With rt = r(c1,t , c2,t ) and h1,t = h1 (c1,t , c2,t , qt , a2,t ), we have q̇t = q̇(c1,t , c2,t , qt , a2,t ). The
evolution of the real wealth of both agents is determined by (4) where we use (19) tosubstitute for q̇t − rt qt , (18) for ϕt in (19) and (22) for yt :ȧ1,t = −πt a1,t + βc1,t θn−1 qt H + (1 − θ)qt h1,t − (1 − n + γ)c1,t + (1 − n)c2,t ,ȧ2,t = −πt a2,t + v 0 (m2,t )c2,t (1 − θ)qt h2,t − (n + γ)c2,t + nc1,t .

(45)
(46)

It is easy to see that ȧ1,t = ȧ1 (c1,t , c2,t , qt , a1,t , a2,t ) and ȧ2,t = ȧ2 (c1,t , c2,t , qt , a2,t ). Finally,
the real money supply decreases with the inflation rate as is clear from (25):ṁt= −πt .
m1

(47)

Therefore, it holds that ṁt = ṁ(c1,t , c2,t , mt ). Equations (42) to (47) fully describe
the economy together with the initial asset levels a1,0 and a2,0 . Given paths for thesevariables, we can derive the associated paths of all other variables. This system satisfies
saddle-point stability around the asymmetric steady state as shown in Appendix B. In
the following section, we analyze the dynamic and static properties of this steady state.

5

Asset Prices and Leverage Under Stagnation

As shown above, private sector debt affects the occurrence of persistent stagnation. In
addition, leverage and asset prices also affect aggregate demand under stagnation. Specifically, we show that credit booms can temporarily mask aggregate demand insufficiency.
However, this comes at the cost of more severe stagnation in the new steady state.

5.1

Credit and Asset Price Booms under Stagnation

We have argued that an economy can enter an equilibrium of persistent stagnation asa consequence of the debt burden of some households. However, an expansion of debt
via financial liberalization can in the short run mask aggregate demand shortage by
creating a temporary credit and asset price boom. Specifically, consider an economy thatis suffering from insufficient aggregate demand. Suppose that lending standards loosensuch that borrowers can take on more loans per unit of housing net worth. This setup
is in line with the claims of Larry Summers about the U.S. economy during the early24

2000s and also mirrors several features of the situation of Japan in the late 1980s that
we described in the introduction.25
Figure 5 shows the associated model dynamics as deviations from the initial steadystate for two values of the housing preference parameter. The increase in the loan-to-value
ratio triggers a substantial credit boom. Borrowers can acquire new funds for a given
collateral value some of which they consume and some of which they hold as money orinvest in new housing. These funds are provided by savers and financed by their money
holdings and the sale of houses. What follows is a temporary boom in both the real
economy and the housing market.
The credit boom stimulates aggregate demand as borrowers increase their consumption. This creates inflation which lowers the real interest rate and stimulates consumptionof the savers as well. As a consequence, the nominal interest rate increases. If the credit
boom is sufficiently strong, the economy can temporarily return to full employment withaggregate spending being constrained by potential output.
In addition, an asset price boom ensues since the real house price surges as housing
demand of impatient households increases. The initial jump in the house price has apositive valuation effect on the housing holdings of both agents, which increases thereal value of their assets. A feedback loop sets in with higher house prices increasing
the collateral value of borrowers which in turn enhances their borrowing ability thereby
reinforcing the initial credit boom. The housing allocation shifts in favor of the impatient
households, which further strengthens the value of their collateral.26 .
The allocation of new funds among consumption, money and housing investment isguided by the parameters in the utility function of the borrower. Higher impatience
implies a stronger increase in consumption and hence aggregate demand and inflation.
In contrast, higher preferences for housing imply that more of the newly available funds
are spent on purchasing fixed supply assets. As a consequence, higher preferences forhousing imply a stronger amplification of the dynamics because of more pronounced
collateral effects. In fact, aggregate consumption might actually fall during the credit
boom for very high values of γ. Figure 5 illustrates the dependency of the dynamic
responses on γ.
25
Note that we proxy the credit boom by variations in θ but do not make explicit claims about theorigin of this variation. The sources of the Japanese credit boom are still up to debate. Yet, Posen(2003) argues that both partial deregulation in corporate finance and a relaxation of lending standards
in the mortgage market with mortgage limits rising from 65% of the home value on average to 100%
played a major role for the Japanese credit boom. According to Posen (2003), “there is a consensus view
among economists on how partial financial deregulation in Japan in the 1980s led to a lending boom”.
The effects of deregulation and financial liberalization are also well-documented in Tsuruta (1999).
26
This is the same propagation mechanism as described in Kiyotaki and Moore (1997) and Iacoviello(2005) among others, which creates amplification and persistence of shocks. Note that we only consider
the case of a binding borrowing constraint. For a treatment of occasionally binding constraints, see
Guerrieri and Iacoviello (2015).

25

Figure 5: Dynamic Effects of Financial Liberalization
Output Gap (in Percent)
0

75

−6
−12
−18

t0

time

0

25

−10

0

−20

−25

−30

−50

Housing Wealth (Rich HH)
0

30

−5

15

−10

0
time

t0

−40

time

Housing Wealth (Poor HH)
45

t0

10

50

5

−15

Money Holdings m∗2−15

Total Wealth a∗2

t0

time

Total Housing Wealth
10
5
0

t0

−5

time
γ = 0.1

t0

time

γ = 0.5

Notes: This figure shows the dynamics associated with a permanent increase in the loan-to-value
ratio from θ = 0.15 to θ = 0.5. The output gap is given in percentage points. All other variables
are depicted as deviations from the initial steady state in percent. We assume the following utility
from money for the borrower: v(m2,t ) = βm2,t + δln(m2,t ). The figure is based on the following
calibration: β = 0.0005, y = 100, ρ1 = 0.05, ρ2 = 0.1, α = 0.01, n = 0.5, H = 1 and δ = 0.1.
Simulations are based on a modification of the relaxation algorithm of Trimborn et al. (2008).

Over time, some of the newly acquired assets are sold by the borrower to smooth its
consumption and for interest payments. Therefore, the allocation of the housing stock
reverts in favor of the saver and aggregate demand remains above its new equilibrium fora prolonged period, thereby masking the underlying demand deficiency. Yet, eventually
the resulting debt overhang pushes the economy into persistent stagnation, which is worse
than before the credit boom, as we will show in the next subsection.
Also note that a house price boom, which is typically modeled by an increase in γ (cf.
Iacoviello, 2005), can temporarily stimulate the stagnating economy though at the cost of
more severe stagnation in the long run. The argument is similar: Higher housing demand
creates an immediate increase in the real house price resulting in valuation gains forboth households. In addition, the value of collateral that borrowers can pledge for funds
increases, which initiates a credit boom. These funds are used to increase consumption,
money holdings and housing investment, the last of which feeds back into the value ofborrowers’ collateral. The dynamics are similar to those in Figure 5.

26

5.2

Debt Overhang and Stagnation

While increases in θ and γ can temporarily stimulate aggregate demand by initiating acredit boom, they also affect the properties of the stagnation steady state. The former
represents financial liberalization - or the degree of sustainable finance - whereas the latter
is a proxy for the level of asset prices. Higher leverage θ and a higher house price reduce
aggregate demand in the asymmetric steady state and hence worsen economic stagnation.
This is summarized in the following proposition (see Appendix C for the proof):
Proposition 2 In the asymmetric steady state under stagnation, an increase in the loanto-value ratio reduces aggregate demand and worsens deflation. It holds that
dc∗1
dc∗2
dm∗2
da∗2
dπ ∗
dC ∗
<0,
<0,
<0,
<0,
<0,
<0.
dθ
dθ
dθ
dθ
dθ
dθ
The same effects arise from an increase in the housing preference γ. It holds that
dC ∗
dc∗1
dc∗2
dm∗2
da∗2
dπ ∗
<0,
<0,
<0,
<0,
<0,
<0.
dγ
dγ
dγ
dγ
dγ
dγ
Consider intuitively the effects of an increase in the loan-to-value ratio θ. Initially,
the borrowing constraint (5) is relaxed allowing the borrower to acquire new funds, as
described above. However, the new steady state is associated with higher debt and hence
higher real interest payments as the steady state real interest rate is not affected. These
payments are a fraction θρ1 γ ρ̄−1of the borrower’s consumption spending where ρ̄θ isθ
decreasing in θ. This can be seen from (41). Therefore, higher leverage is associated withhigher interest costs per unit of consumption which implies that the borrower’s income is
not sufficient to cover expenditures for a given c∗2 once θ increases. As this would violate
the lifetime budget constraint of the borrower, his consumption spending has to decline.
This implies that the expenditures of the borrower are reduced (“spending effect”),
raising disposable income. However, the lower spending negatively affects the borrower’s
income since aggregate demand declines (“demand effect”). This partially offsets the first
effect. In addition, the real return on money holdings is affected (“capital gains effect”):
Higher deflation increases the return on money. Yet, lower consumption discourages
money holdings. The first effect is stronger, the higher money holdings, but the net effectis always negative in the asymmetric stagnation case (see Appendix B). This implies thata decrease in consumption reduces expenditures. Hence, consumption must decline in
response to an increase in θ. These effects can be seen from the total differential of (41):dc∗n ∗1 + 1 − n
dc2
|
{z




Demand Effect



 ∗ 
∗
∗
dc∗2
θρ1 γ dc∗2
ρ1 ρ2 γ ∗
dc2
∗ dm2
∗ dπ
+ −π
− m2 ∗− 1+=
c . (48)∗dθ
dc2
dc2 dθρ̄θdθρ̄2θ 2
} |
{z
} |
{z
}
| {z }
Capital Gains Effect27

Spending EffectInterest Cost

The decrease in consumption of the borrower feeds back into the other variables ofthe model. Aggregate demand decreases, which aggravates deflation via (2). Deflationin turn reduces the nominal interest rate via (24) since the real rate is determined by ρ1 .
This reduces consumption of the saver, which can be seen in (34). In addition, money
demand of the borrower declines, as is clear from (35), as does the borrower’s real wealth,which can be seen in (39).
The effects of a higher θ on the real house price and the distribution of the housing
stock are ambiguous because of two opposing effects on q ∗ : On the one hand, investment
in housing becomes more attractive for a given level of consumption c∗2 since housing
becomes more collateralizeable. Higher housing demand bids up the house price. On
the other hand, there is an indirect effect on the house price because lower consumption
spending decreases housing demand of both agents which in turn lowers the real house
price. This can be seen from (40).27
The preference for housing γ determines the relative strength of these effects. The
higher γ, the weaker the effects associated with the higher collateral value relative to thenegative effect on consumption. If γ is sufficiently high, the indebtedness of the borrower
might actually decline in response to financial liberalization since housing is reallocatedto the saver. The reason is that higher levels of γ are associated with a higher collateralvalue and hence higher household debt. A given change in θ needs to be balanced by astronger decline in consumption and hence a stronger reduction in housing demand.
Figure 6 illustrates the effects of a rise in the loan-to-value ratio θ on the steady state.
Each subplot shows the elasticity of the respective variable to a rise in θ as a functionof the housing preference parameter γ for three different values of n. We set parameterssuch that the economy is at full employment for θ = 0. In particular, note the negative
effect on the borrower’s housing investment and the real house price for large values of γ.
This in turn implies that financial liberalization is associated with a substantial decreasein the real wealth of the poor household. Also, the responses of consumption and asset
prices are stronger the higher the share of poor households.
In the literature, γ is typically calibrated to match empirical observations on the
housing market. In a similar framework, Iacoviello (2005) chooses a value of γ = 0.1
to match the value of residential housing to output in the United States. Guerrieri and
Iacoviello (2015) follow a similar approach in a model with endogenous housing supply
and select a value of γ = 0.04. When we apply the same criterion, the implied valueof γ ranges between 0.08 and 0.1 and is substantially below unity. This implies thedominance of the collateral channel and hence financial liberalization raises asset prices
and credit-financed housing investment.
27

What is clear from (38), is that the equilibrium value of the saver’s housing investment declines withθ while the effect on the value of the borrower’s housing investment is unclear.

28

Figure 6: Elasticities with Respect to θConsumption c∗20

Money Holdings m∗2

0

−0.4

−0.4

−0.6

−0.8

−0.8

−1.2

−1.2

0

0.5

1

−1.2
γ

Borrowing b∗2

1.5

0

0.5

1

−1.8
γ

Housing h∗2

0.4

Total Wealth a∗2

0

0

0.5

1

γ

1

γ

House Price q∗

0.4

1
0

0

0.5
0

0

0.5

1

−0.4
γ

0

0.5

n = 0.2

1
n = 0.5

−0.4
γ

0

0.5

n = 0.8

Notes: This figure shows the elasticities of the model variables with respect to the loan-to-value
ratio θ as a function of the housing preference γ and for three different values of n. The y-axis
shows the %−change in each variable in response to a 1% increase in θ in steady state. We assume
the following utility from money for the borrower: v(m2,t ) = βm2,t + δln(m2,t ). The calibration is
as follows: β = 0.0005, ȳ = 100, ρ1 = 0.05, ρ2 = 0.1, α = 0.01, δ = 0.1, θ = 0.5 and H = 1.

These effects are in stark contrast to the standard neoclassical case with v 00 (mi ) < 0
for both types of households. From (30), it is clear that an increase in indebtedness
reduces the consumption demand of the borrower in the neoclassical steady state since
this agent faces higher real interest payments. Yet, aggregate demand is unaffected byvariations in θ or γ because the saver increases his consumption level accordingly as longas his liquidity premium is decreasing in real money holdings, which can be seen from(31). As a consequence, changes in these parameters do result in a redistribution ofavailable income and hence of consumption spending and housing investment. However,they do not trigger deviations from full employment because aggregate demand is notaffected by these changes. In addition, the price level will adjust to clear the money
market, which can be inferred from (25).
Similarly, aggregate demand is no longer affected by variations in these parameters
once the model economy is in the symmetric steady state under stagnation. Then, variations in θ or γ cease to affect the consumption spending of both agents and simply leadto a redistribution of the housing stock and changes in the real house price. This case as
well as other extensions of the model will be discussed in the next section.

29

6Model Extensions and Discussion

In this section, we analyze two extensions of the model that have been turned off so far inorder to focus on the core mechanism. In addition, we discuss policy recommendations.

6.1

Asymmetric and Symmetric Stagnation

From Proposition 1, we know that stagnation does not occur for sufficiently low levelsof potential output. In addition, it is clear from (32) and the discussion in the previous
section that the borrower will also eventually choose to accumulate money holdings if
his consumption level is sufficiently high. More specifically, it follows from (35) thatsymmetric stagnation will occur once the borrower’s consumption level has reached thecritical threshold of χ/β(β ȳ − α). We first derive a sufficient condition for asymmetricstagnation to prevail and then give an intuition for the occurrence of the symmetric case.
Under symmetric stagnation, v 0 (m1,t ) = v 0 (m2,t ) = β and both households accumulate wealth infinitely. Consumption of neither type is stimulated by additional money.
Formally, the symmetric steady state is defined as follows:
Symmetric Steady State: The real and nominal interest rates are constant,the price level is declining at a constant rate, the real house price is constant,the real consumption levels are constant but the wealth of each household
expands infinitely:ṙ = 0 , Ṙ = 0 , π < 0 , ċ1 = 0 , ċ2 = 0 , q̇ = 0 , ȧ1 > 0 , ȧ2 > 0 .

(49)

The economy enters stagnation once potential output exceeds ye defined in (33). Then,v (m1,t ) = β and there is deflation and demand shortage, i.e. π < 0 and C < ȳ from (2)and (22). Consider the population-weighed average of (12) and (18) with ċ1 = ċ2 = 0:0nρ1 + (1 − n)ρ2 = βnc1 + v 0 (m2 )(1 − n)c2 − π .

(50)Symmetric stagnation cannot occur if β ȳ < nρ1 + (1 − n)ρ2 . To see this, suppose we havev 0 (m2,t ) = β and β ȳ < nρ1 + (1 − n)ρ2 . Then from (2), (22) and (50), we get
β ȳ − α < nρ1 + (1 − n)ρ2 − α = β[nc1 + (1 − n)c2 ] − π − α = (β ȳ − α)C.
ȳThis only holds for C > ȳ which is not the case. Hence, we always have v 0 (m2,t ) > β for
β ȳ < nρ1 + (1 − n)ρ2 . Together with Proposition 1 and Condition (36), this yields thefollowing proposition:

30

Proposition 3 Given the parameter restrictions ρ1 > α and β ȳ > α, the following
condition is sufficient for the asymmetric steady state under stagnation to occur:nρ1θρ1 γ + ρ̄θθρ1 γ + nρ̄θ< β ȳ < nρ1 + (1 − n)ρ2 .

(51)

The first inequality in (51) follows from (33) and ensures that aggregate demand
falls short of potential output and the second inequality ensures asymmetry. Intuitively,
the second condition requires that the time preference rate ρ2 is sufficiently high so thatborrowers still strive for higher consumption. Yet, note that an increase in ρ2 also tightens
the first inequality, which is clear from Proposition 1.
Importantly, (51) is a sufficient condition for the existence of the asymmetric steadystate but not a necessary condition. Under certain conditions, the asymmetric stagnationcase will prevail for higher values of potential output. This is the case when further
increases in potential output do not stimulate the borrower’s consumption to exceed
the threshold discussed above. We discuss the necessary existence condition in detail in
Appendix B and only provide some intuition here.
Intuitively, the borrower’s consumption depends on two factors as can be seen from(41): Income from firm profits which are determined by aggregate demand and capitalgains on money holdings which depend on the rate of deflation. Under stagnation, an
increase in the economy’s production capacity worsens deflation which has two effects on
the borrower’s income. On the one hand, deflation reduces the consumption incentivesof the saver. This reduces the income of the borrower since aggregate demand declines(“aggregate demand effect”). On the other hand, the purchasing power of money holdings
rises which stimulates the borrower’s consumption (“capital gains effect”). The second
effect is stronger the higher his money holdings. If the capital gains effect dominates,
the borrower’s consumption increases with a higher production capacity as do his money
holdings. Then, the marginal utility of money eventually reaches the lower bound andsymmetric stagnation occurs.28 But the asymmetric case may persist even for high levelsof potential output ȳ as long as the capital gains effect is weak or negative.
To summarize, our model features three regions depending on ȳ: If potential output
is below the threshold ye given by (33), the neoclassical case applies and there is no
demand shortage. In contrast, stagnation occurs for ȳ > ye because of the insatiabilityof liquidity preferences. The asymmetric case always occurs if condition (51) holds andmight prevail for even higher values of potential output. Finally, the symmetric caseoccurs if consumption of the borrower under stagnation becomes sufficiently high.
28

Thus, there will be an implicit threshold ŷ such that there is symmetric stagnation for ȳ > ŷ. This
threshold depends on the model parameters, particularly on those affecting equilibrium money holdingsof the borrower. These in turn depend on the shape of the utility function v(m). Therefore, we cannot
give a closed-form expression for this threshold.

31

6.2

Stagnation with Positive Money Growth

So far, we have focused on the case of zero trend inflation as a result of a constant money
supply, i.e. µ = 0, under full employment. Two considerations need to be taken into
account when considering the case of µ > 0 that affect the occurrence of stagnation as
well as the existence of the stagnation steady state. For the general conditions and proofs,
we refer to Ono and Ishida (2014) for the case of homogeneous households. Here, we will
provide an intuitive discussion of the effects of µ > 0 for the case of heterogeneous agentsin the borrower-saver framework.
First, as argued above, stagnation occurs once one of the households is no longer
willing to consume the amount consistent with full employment because of his insatiable
desire for holding liquidity, i.e. once the following threshold is reached for any household:
C
c̃N
>
i

ρi + µ.
β(52)This is a generalization of condition (32) for the case of positive money growth. Twoeffects emerge relative to the case of µ = 0 that has been discussed so far.
Positive nominal money growth raises the nominal interest rate under full employment, due to the Fisher equation (24). This increases the opportunity cost of holding
money for both agents, which stimulates their consumption, thereby increasing the liquidity premium. As a consequence, full employment can be sustained for higher levelsof potential output and stagnation occurs at a later stage. In fact, for every level of potential output ȳ there exists a nominal money growth rate µ such that full employment
prevails. However, this comes at the cost of higher inflation.
In addition, there is a more subtle effect as positive money growth might affect bothChouseholds’ consumption levels c̃Nunder full employment. This crucially depends on theiassumption about the distribution of seignorage profits zt = µmt . If these are distributed
in proportion to each agents money holdings, there is no effect on the full employment
levels of consumption, given by (30) and (31).29 However, if seignorage income is distributed equally across households, the household with lower money holdings benefits
at the expense of the household with higher money holdings. For reasonable parameter
specifications, it will be the saver whose consumption will be lowered by this effect, while
the borrower benefits. This further increases the income threshold for stagnation.30
29
The intuition is simple: Each household incurs implicit costs of money holdings due to inflation.
In turn, the household benefits from inflation via the seignorage profits. If profits are distributed inproportion to money holdings, these effects exactly offset each other.
30
It could even be the case that the saver’s consumption is actually lower under full employment thanthe borrower’s consumption level because of the redistributive effect of inflation. Yet, note that thiseffect only occurs for a very restrictive parameterization. Specifically, both the difference in discountrates and the money growth rate need to be sufficiently high. In addition, the loan-to-value ratio or the
housing preference parameter need to be sufficiently low.

32

Secondly, the existence condition of the asymmetric steady state is affected. Becauseof persistent deflation, the money supply expands indefinitely and so does the wealthlevel of the saver. With µ = 0, the rate of expansion is given by the rate of deflation as
is clear from (23). Since the deflation rate is below the real interest rate, as we assume
ρ1 > α, the transversality condition (11) holds despite this expansion. With positive
nominal money growth, however, the expansion of the real money supply increases toµ − π as does the growth rate of household wealth. For the transversality condition tohold, we need to require that this rate of expansion is below the time preference rate of
the saver that determines the real interest rate. Specifically, for a steady state to exist,
it has to hold thatṁt0>
− ρ1 = µ − βc̃∗1 ,(53)
mt
where c̃∗1 denotes the saver’s consumption in the asymmetric steady state with µ > 0.
On top of that, the occurrence condition of the symmetric stagnation steady state isaffected by introducing positive money growth. The effects depend again on the assumption on the distribution of seignorage income. If this income is distributed in proportion
to each household’s money holdings, then there are no effects as the borrower’s consumption under asymmetric stagnation is not affected. In contrast, if this income is distributed
equally across households, the borrower’s consumption will be stimulated under asymmetric stagnation. As the money supply expands, so does his exogenous income, whichallows for higher consumption. Then, the symmetric stagnation case will eventually occur
if condition (53) holds.
In conclusion, the equilibrium of the economy is conditional on the money growthrate. A sufficiently high rate of money growth may help to restore full employment.
Since this comes at the cost of high inflation, policymakers are likely to be inclined toprefer a scenario of persistent stagnation and take measures to improve aggregate demand
within that equilibrium. However, for any rate of money growth µ, there exists a level ofpotential output above which stagnation occurs. Sustainable full employment will hence
require an ever-increasing expansion of the money growth rate. Even worse, the interplayof conditions (52) and (53) also implies that multiple equilibria can emerge with bothstagnation and full employment as steady state equilibria for the same parameterization.
It might also be the case that no equilibrium exists at all. So once the economy hasreached stagnation, it will be very hard and costly in terms of high inflation to movetowards the full employment steady state.
For that reason, our analysis has focused primarily on the stagnation case with µ = 0.
Note, however, that the conclusions also hold for a low inflation scenario which requires
sufficiently low levels of monetary growth.31
31

This is similar to the assumptions of Michaillat and Saez (2014) and Michau (2017) that the central
bank follows a sufficiently low inflation target.

33

6.3

Policy Recommendations and the Nature of the Friction

Two features in our model prevent the economy from reaching full employment - insatiable liquidity preferences and debt overhang. Insatiable liquidity preferences imply
that stagnation always occurs for sufficiently high levels of potential output, even in theabsence of financial frictions. The reason is that agents prefer to hold excessive money
instead of consumption. This implies that expansionary monetary policy is ineffectivein the stagnation steady state of our model.32 In fact, the deflationary steady state ischaracterized by an infinite expansion of the real money stock.
In contrast, the case for fiscal policy as a potential cure to stagnation is straightforward. The government is not constrained by the same liquidity motives as the private sector and can expand its spending.33 Redistributive policies work by transferring resourcesfrom rich agents to poor ones. The latter expand their consumption while spending of
the former is not directly affected (unless at the margin). Therefore, targeted redistributive interventions can help to stimulate the economy. In reality, targeted transfers might
not be feasible though. Yet, Matsuzaki (2003) shows in a similar setting that lump-sumtransfers financed by a consumption tax can increase aggregate demand if the fraction ofpoor households is sufficiently small.34
Private debt overhang is another factor that depresses aggregate demand since indebted households reduce their consumption spending. In fact, borrowers do deleverin nominal terms in the deflationary steady state. But this is self-defeating because of
the effects of debt deflation on the real value of their outstanding obligations. Hence,
policies that limit household indebtedness and help to repair balance sheets of spendingconstrained households are another option to expand aggregate demand. Yet, they include a potentially costly adjustment process in the short run.
Similar conclusions hold when we impose the borrowing constraint on the supply side.
Although this analysis is beyond the scope of this paper, the following thought experiment clarifies this point: Suppose the collateralizeable asset is a factor of production andproducers are constrained in their borrowing ability. As above, financial liberalization
is associated with higher equilibrium collateral holdings by the borrower under certain
parameter constellations. These in turn imply a higher equilibrium production capacity.
Therefore, financial liberalization may improve equilibrium output under neoclassical as32

Yet, a sufficiently large expansion of the money supply might restore the full employment case though
at the cost of inflation as discussed before.
33
Note that the expansionary effect of government spending has nothing to do with deficit-budget
financing or balanced-budget financing. It works through a direct creation of demand. We refer to Ono(1994, 2001) for an explicit modeling of government spending. We abstract from public debt in ourframework since we focus primarily on private debt.
34
Note that this discussion uses aggregate demand as the relevant policy criterion rather than a welfare
function based on individual utilities. The latter is complicated by household heterogeneity and theinfinite expansion of the saver’s money holdings in steady state.

34

sumptions. However, the economy is demand-constrained in our model because of theinsatiability of liquidity preferences so that the implied improvements in the supply side
actually worsen the output gap and deflation. An increase in indebtedness hence deteriorates equilibrium income for reasonable parameter ranges, irrespective of the modelingof the borrowing friction on the demand side or supply side.
Finally, our results continue to hold with insatiable wealth preferences instead ofliquidity preferences. Unlike the latter, wealth preferences affect the equilibrium realinterest rate by encouraging household savings (cf. Ono, 2015; Kumhof et al., 2015). As
a consequence, the natural real rate of interest can turn negative in steady state (cf.
Michau, 2017). In our setting, this would imply a redistribution from savers to borrowers
as the real cost of debt becomes negative. However, the very existence of housing as adurable asset without depreciation prevents the real rate from turning negative in our
setup. This can be easily seen from (19), which is unaffected by the introduction of wealthpreferences. Housing yields a positive “dividend” stream in the form of the user cost of
housing while the cost of housing investment are given by the real opportunity cost, since
there is no depreciation. The real house price adjusts to make agents indifferent between
housing investment and other uses of funds. Hence, from (19) a negative real rate ofinterest would require a decline of the real house price in steady state:r∗ < 0 ⇔γc∗q̇< − ∗1 < 0 .
qh1This is not consistent with a stationary steady state. Moreover, it would imply that thereal house price eventually converges to zero and hence that the current asset price itself
is not well-defined. We can therefore exclude the possibility of a negative real rate in ourmodel under wealth preferences. Hence, there cannot be a redistribution from savers toborrowers via negative interest cost of debt in steady state.

7

Conclusion

Many developed countries, e.g. Japan, EU and the USA, have been suffering from persistent stagnation of aggregate demand under which some households do not increase
consumption and keep wealth while others do not increase consumption because they areseverely indebted. It typically occurred after a credit and stock price boom. To analyze
this phenomenon, we have introduced private indebtedness into a model with two typesof agents that have different time patience and insatiable preferences for money holding.
The less patient households borrow funds from the more patient ones but face aborrowing constraint that depends on the value of their housing. Therefore their consumption is restricted by this constraint. The more patient households earn interests35

from the lending and hence can expand consumption, but in fact do not because of highpreference for money holding. Thus, aggregate demand shortages arise and deflation occurs. The deflation makes it more advantageous for the lenders to reduce consumption
and hold money. It in turn expands the real value of debt of the borrowers and decreases
their consumption because they have to pay high interests to the lenders.
If the borrowers could consume more, deflation would mitigate and stimulate thelenders consumption as well, leading to an expansion of total income. Thus, a governmentthat faces this situation may be tempted to ease the borrowing constraint. It will indeed
enable the borrowers to consume more and mitigate deflation, which also stimulates thelenders consumption by lowering the advantage of holding money. Moreover, easing theborrowing constraint makes the borrowers think housing investment to be more valuable
because an increase in the value of housing enables them to borrow more for consumption.
Thus, it triggers a housing price boom.
However, those positive effects occur only in the short run. In the long run the
borrowers are more indebted so that they have to reduce consumption, which worsens
deflation and makes the lenders to decrease consumption and save more because money
holding is more profitable. The decrease in total consumption stops the housing priceboom. The economy eventually falls into secular stagnation of aggregate demand. Thus,
direct transfers from the richer to the poorer, which does not create debt overhang, will
be more promising.

Appendices
Appendix A: Proof of Proposition 1
The model parameters affect the stagnation threshold ye in (33) as follows:nρ1 γθρ1 + ρ̄θ
∂e
y
=− 2
<0,
∂β
β γθρ1 + nρ̄θ∂eyn[(1 + γ)(γ + n)θ2 ρ21 + 2(1 + γ)nθ(1 − θ)ρ1 ρ2 + (1 − θ)2 nρ22 ]
=
>0,
∂ρ1
β[γθρ1 + nρ̄θ ]2∂eynρ1 (1 − n)γθ(1 − θ)ρ1
>0,
=
∂ρ2
β
[γθρ1 + nρ̄θ ]2∂eynρ1 ρ1 ρ2 γ(1 − n)
=−
<0,
∂θ
β [γθρ1 + nρ̄θ ]2∂eyγθρ21 [γθρ1 + ρ̄θ ]=>0,∂nβ[γθρ1 + nρ̄θ ]2∂eynρ1 (1 − n)ρ1 θρ̄θ
=−
<0.
∂γ
β [γθρ1 + nρ̄θ ]236Appendix B: Existence and Stability of the Asymmetric Steady StateIn this appendix, we derive the necessary existence condition for the asymmetricsteady state under stagnation and show that it satisfies saddle-point stability.
(i) Existence: Using (34) to substitute for c∗1 and (35) for c∗2 , we rewrite (41) asF (m2 ) ≡ α[(β ȳ − ρ1 n)v 0 (m2 ) − (1 − n)βρ2 ]m2 + n(ρ1 − α)ȳ[v 0 (m2 ) − β] = A ,where A ≡

(B.1)

θρ1 γ
χ + n(β ȳ − α)(ρ2 − ρ1 ) > 0 ,ρ̄θand χ ≡ ρ2 (β ȳ − αn) − α(β ȳ − ρ1 n) > 0 .
The asymmetric steady state under stagnation exists for ȳ > ye if there exists a finite andstrictly positive value of m2 as a solution to this equation. Note that the RHS of thisexpression is a positive constant that is independent of m2 . In contrast, the LHS of thisexpression is a function of m2 . It holds that:limm2 →0 F (m2 ) = ∞; −∞ if β ȳ < nρ1 + (1 − n)ρ2 ,lim α(β ȳ − ρ1 n)(v 0 (m2 ) − β)m2 if β ȳ = nρ1 + (1 − n)ρ2 ,
limm2 →∞ F (m2 ) =
m
2 →∞


 +∞ if β ȳ > nρ + (1 − n)ρ .
1

2

Under otherwise standard assumptions on v(m), it follows that there is a unique solutionof (B.1) if β ȳ < nρ1 + (1 − n)ρ2 . This is the sufficient condition in Proposition 3. For
higher values of ȳ, there may be two solutions, exactly one solution or no solution to(B.1). Existence of the asymmetric steady state then requires that the minimum (orlimit if β ȳ = nρ1 + (1 − n)ρ2 ) of F (m2 ) is smaller than or equal to the RHS:
min F (m2 ) <
m2

θρ1 γ
χ + n(β ȳ − α)(ρ2 − ρ1 ) .
ρ̄θ

(B.2)

This condition guarantees the existence of at least one solution to (B.1). In case ofmultiple solutions, we choose the solution that satisfies F 0 (m∗2 ) < 0. This is for two
reasons: First, it is consistent with continuous variations in ȳ. Second, this solution satisfies saddle-point stability, whereas the other solution is unstable. Therefore, a necessary
condition for the asymmetric steady state under stagnation to occur is given by∂F (m2 )
∂m2

<0.

(B.3)

m∗2

Finally, if there is no finite value of m2 that solves (B.1), we must have ȧ2,t > 0 which
implies that the economy is in the symmetric stagnation steady state.
37

Figure 7: Existence of the Asymmetric Steady State under Stagnation

β ȳ > nρ1 + (1 − n)ρ2

β ȳ > nρ1 + (1 − n)ρ2

β ȳ < nρ1 + (1 − n)ρ2m2

Note: This figure illustrates the LHS (solid line) and RHS (dotted line) of (B.1) for different
values of potential output ȳ and for the specific utility function v(m2 ) = βm2 + δln(m2 ).

To summarize: The asymmetric steady state under stagnation exists for β ȳ > nρ1 +(1 − n)ρ2 if there exists a finite, positive value of m2 that solves (B.1). Moreover, (B.2)
is a sufficient condition for the existence of the asymmetric steady state under stagnation
given β ȳ − nρ1 − (1 − n)ρ2 ≥ 0. In addition, (B.3) is a necessary condition for theasymmetric steady state under stagnation to occur.
For illustration, consider the specific utility function v(m2 ) = βm2 + δln(m2 ). Figure
7 shows the behavior of the two sides of (B.1) as a function of m2 , which is given byαβ(β ȳ − ρ1 n − (1 − n)ρ2 )m2 + δα(β ȳ − ρ1 n) +δn(ρ1 − α)ȳ
θρ1 γ=
χ + n(β ȳ − α)(ρ2 − ρ1 ) .
m2ρ̄θFor β ȳ − nρ1 − (1 − n)ρ2 = 0, the existence of the asymmetric steady state requires asufficiently low value of δ:


1
θρ1 γ
χ + n(β ȳ − α)(ρ2 − ρ1 ) .
δ < δ̄ ≡α(β ȳ − ρ1 n) ρ̄θ

(B.4)

For β ȳ − nρ1 − (1 − n)ρ2 > 0, we require in addition that (B.2) holds which implies
θρ1 γ
χ + n(β ȳ − α)(ρ2 − ρ1 ) − αδ(β ȳ − ρ1 n)ρ̄θ

2> 4αβ(β ȳ − nρ1 − (1 − n)ρ2 )n(ρ1 − α)δ ȳ .

38

(B.5)

(ii) Stability: The dynamic system is characterized by six differential equations for
c1 , c2 , q, a1 , a2 and m given by (42), (43), (44), (46), (45) and (47) and by the static
equations (20), (21) and (27) for m2 , h1 and h2 . The asymmetric steady state under
stagnation is characterized by a diverging real money supply and real assets of the saver.
−1
Define z1,t ≡ a−1
1,t and z2,t ≡ mt . Then the steady state of {c1,t , c2,t , qt , a2,t , z1,t , z2,t } isgiven by {c∗1 , c∗2 , q ∗ , a∗2 , 0, 0}. We linearize the system around this steady state using afirst-order Taylor approximation:





v11
  
 ċ2,t  v21   q̇  v
 t   31
 =ȧ2,t  v41
  
  
 ż1,t   0
ż2,t
0
ċ1,t

v12

0



c1,t − c∗1



0

0

0

v22 v23 v240v32 v33 v340v42 v43 v44

0

0

0

0

v55

0

0

0

0



0   c2,t − c∗2 



∗ 
0 
  qt − q 

 ,
∗

0 
 a2,t − a2 
0   z1,t − z1∗ 
v66
z2,t − z2∗

where the entries vij in the transition matrix V refer to the respective terms in thelinearized system. The eigenvalues ξi of V determine the stability of this system and are
given by the solution to(v11 − ξ)v22 − ξ
v32
v42

v23

v24

v21

v33 − ξv34

− v12 v31 v33 − ξv43v44 − ξ

v41

v23
v43

v24



v34


 (v55 − ξ)(v66 − ξ) = 0 ,v44 − ξwhere Q is the determinant of Q. Since only c1,t , c2,t and qt are jumpable, there must
be three positive and three negative eigenvalues for the system to exhibit saddlepoint
stability. ξi = βc∗1 − ρ1 = π ∗ is a solution and under stagnation π ∗ < 0. Thus, these two
eigenvalues are negative. We use a numerical analysis for the other solutions.
Based on the functional form v(m2 ) = βm2 + δln(m2 ), we simulate V for three cases
determined by β ȳ − nρ1 − (1 − n)ρ2 . For each case, we vary δ (and implicitly m∗2 ),which determines the strength of the capital gains channel in (41). We then determine
the number of negative eigenvalues. The results are summarized in Figure 8 which also
highlights the threshold parameter δ̄ in (B.4) or (B.5).
For β ȳ − nρ1 − (1 − n)ρ2 < 0 (case 1), the system is saddlepoint-stable for all δ > 0.
This corresponds to condition (51). For β ȳ − nρ1 − (1 − n)ρ2 = 0 (case 2), the systemis saddlepoint-stable for 0 < δ < δ̄. Hence, under existence condition (B.4) the steadystate exhibits saddle-point stability. For β ȳ − nρ1 − (1 − n)ρ2 > 0, there are two solutions
to (B.1) shown in cases 3 and 4. Both solutions require condition (B.5) to hold. Yet,
only one of these solutions shows saddle-point stability. This is the solution that fulfills
condition (B.3). We therefore conclude that the model is saddlepoint-stable around theasymmetric stagnation steady state under conditions (B.1), (B.2) and (B.3).

39

Figure 8: Stability of the Saddle Path

5

Case 1: βȳ − nρ1 − (1 − n)ρ2 < 0

5

4

4

3

3

2

2

1

1

0

Case 2: βȳ − nρ1 − (1 − n)ρ2 = 0

0

δ
5

δ

Case 3: βȳ − nρ1 − (1 − n)ρ2 > 0

5

4

4

3

3

2

2

1

1

0

Case 4: βȳ − nρ1 − (1 − n)ρ2 > 0

0

δ

δ

Notes: This figure shows the number of negative eigenvalues in V for the function v(m2 ) =βm2 + δln(m2 ). Case 1 refers to existence condition (51) and cases 2 to 4 refer to conditions(B.4) and (B.5) which are represented by vertical lines. Variations in δ are shown on thex-axis. The calibration is as follows: β = 0.0005; ρ1 = 0.05; ρ2 = 0.1; α = 0.01; n = 0.5; H = 1,
θ = 0.5 and ȳ = 120 (case 1), ȳ = 150 (case 2) and ȳ = 200 (cases 3 and 4). In the dashedareas, the existence conditions for the asymmetric steady state are fulfilled.

Appendix C: Proof of Proposition 2
The effects of variations in the model parameters on the asymmetric steady state arederived from the total differential of (B.1). Define Ω(m2 , x) ≡ 0 where x is any parameterin the model asΩ(m2 , x) = α[(β ȳ − ρ1 n)v 0 (m∗2 ) − (1 − n)βρ2 ]m∗2 + n(ρ1 − α)ȳ[v 0 (m∗2 ) − β] − A ,where A ≡

θρ1 γ
χ + n(β ȳ − α)(ρ2 − ρ1 ) > 0 ,ρ̄θand χ ≡ ρ2 (β ȳ − αn) − α(β ȳ − ρ1 n) > 0 .
From this expression, we can recover the effect on money demand of the borrower asfollows:∂Ω(m2 ,x)∂Ω(m2 , x)∂Ω(m2 , x)
dm2
∂x
dx +
dm2 = 0 ⇔= − ∂Ω(m
.
(C.1)
2 ,x)
∂θ
∂m2
dx
∂m2

From (B.3) and the discussion in Appendix B, it follows that∂Ω(m2 , x)
= F 0 (m2 ) < 0 .
∂m2
40

(C.2)

Consider the effects of variations in the loan-to-value ratio θ and the housing preference
parameter γ on the asymmetric steady state under stagnation:
ρ1 ρ2 γ∂Ω(m2 , θ)=− 2 χ<0,
∂θ
ρ̄θ

(C.3)

θρ1
∂Ω(m2 , γ)
=−
χ<0.
∂γ
ρ̄θ

(C.4)

It hence follows from (C.1), (C.2), (C.3) and (C.4) that
dm∗2
dm∗2
<0,
<0.
dθ
dγ
These results imply together with (35) that
dc∗2
dc∗2
<0,
<0.
dθ
dγ
The effects on the steady state value of the other variables can be derived from theirrelation with c∗2 and m∗2 . We get from (34) that
dc∗1
dc∗1
<0,
<0,
dθ
dγ
and from (39) that

da∗2
da∗2
<0,
<0.
dθ
dγ

Finally, these results imply in combination with (2) and (22) that increases in θ or γ
reduce aggregate demand and worsen deflation in the asymmetric steady state:
dC ∗
dC ∗
<0,
<0,
dθ
dγ
dπ ∗
dπ ∗
<0,
<0.
dθ
dγ
Also note that the cross-derivative is strictly negative which implies mutually reinforcing effects of γ and θ as illustrated in Figure 6:∂ 2 Ω(m2 , θ)ρ1 ρ2=− 2 χ<0.
∂θ ∂γ
ρ̄θ

41

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