ISSN (Print) 0473-453X

Discussion Paper No. 1174

ISSN (Online) 2435-0982

ACCOUNTING FOR THE SLOWDOWNIN OUTPUT GROWTH AFTERTHE GREAT RECESSION:A WEALTH PREFERENCE APPROACH
Kazuma Inagaki
Yoshiyasu Ono
Takayuki Tsuruga

Revised March 2023
May 2022

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

Accounting for the slowdown in output growth after the GreatRecession: A wealth preference approach∗
Kazuma Inagaki†, Yoshiyasu Ono‡, and Takayuki Tsuruga§
March 16, 2023

Abstract
Previous studies have argued that US output growth declined persistently after the GreatRecession. To explain the persistent slowdown in output growth, we develop a simple model thatincorporates wealth preferences and downward nominal wage rigidity into a standard monetary
growth model. Our model predicts that output initially grows at a constant steady rate andslows endogenously afterward. In the model, persistent stagnation occurs together with the
declining real interest rate. Applying our model to the US data, we show that it successfully
explains the slowdown in output growth along with the declines in the real interest rate. We
also examine the model with the Japanese data. The model replicates the persistent stagnationthat has been observed since the 1990s.
JEL Classification: E41, E47, O10
Keywords: Secular stagnation, Wealth preferences, Liquidity preferences, the Great Recession, Downward nominal wage rigidity∗

The authors would like to thank Ryo Kato, Matthias Schlegl, and Etsuro Shioji for their helpful comments andShuhei Kishimoto for providing excellent research assistance. The authors also gratefully acknowledge the financial
support of the Joint Usage/Research Centers at the Institute of Social and Economic Research and Kyoto Instituteof Economic Research, and Grants-in-Aid for Scientific Research (20H05631, 20H05633, 21H00700, and 21H04397).
This paper is a substantially extended version of Ono’s 2015 working paper entitled “Growth, secular stagnation, andwealth preference.” (ISER Discussion Paper No. 946)†
University of Rochester‡
Osaka University§
Osaka University; Correspondence to: Institute of Social and Economic Research, Osaka University 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. E-mail address: tsuruga@iser.osaka-u.ac.jp1

1

Introduction

The persistent decline in output growth in the United States (US) after the Great Recession has
formed one of the most well-known debates in macroeconomics. In his “secular stagnation hypothesis,” Summers (2014) argues that while long-run US output was expected to grow steadily before
the Great Recession, the actual output after the Great Recession failed to catch up with the expected output trend. Figure 1 plots two estimated trends for log real gross domestic product (GDP)per capita along with the actual data. The upper panel shows the US output trends together withactual output. The linear trend (the dot-dashed line) is estimated using log real GDP data from1990:Q1 to 2007:Q1 and extrapolated after 2007:Q1. While output was expected to grow based on
the data up to 2007, actual real GDP after 2007:Q1 did not grow as fast as the expected output
trend. Using the data that includes the period after 2007:Q1, we estimate the cubic trend of output(the dashed line). The cubic trend falls below the linear trend, exhibiting a slowdown in output
growth.1 A similar observation can be made for Japan in the 1990s, as shown in the lower panel ofthe figure. We estimate the linear trend of log real GDP from 1980:Q1 to 1991:Q1 and the cubic
trend from 1980:Q1 to 2019:Q4.2 Comparing the linear and cubic trends, we observe that Japan
experienced a significant decline in output growth.
[Figure 1 about here.]In this paper, we develop a model with wealth preferences and assess the model’s ability to
account for the slowdown in output growth. The literature on wealth preferences has found thatstrong wealth preferences could lead to an inherently stagnant economy in the steady state.3 In thestandard model without wealth preferences, households receive market interest from savings. In themodel with wealth preferences, however, households receive additional benefits of savings, namely
holding wealth. Under our preference assumptions, the additional benefits incentivize households
to give up more consumption to enjoy holding more wealth. In turn, the permanent shortage inaggregate demand or the strong desire for savings leads to the permanently low real interest rateand low inflation, which well characterize the secular stagnation observed in advanced economies.
1

We confirmed a similar pattern in the potential GDP series by the Congressional Budget Office.
We use 1991:Q1 as the end of the sample period in calculating the linear trend of log GDP because a persistentdecline in output growth started after this year.
3
The earliest examples of these studies are Ono (1994) and Ono (2001), among others. In a recent paper, Michau(2018) also proposes a model with wealth preferences that leads to secular stagnation in the steady state.
22

We incorporate two components into the standard monetary growth model. As discussed, thefirst component is wealth preferences. Following the literature (e.g., Michau (2018), Michaillat
and Saez (2021), Hashimoto, Ono, and Schlegl (2023)), we introduce wealth preferences with astrictly positive marginal utility in equilibrium. These preferences lead to a strong desire forsavings compared with the case without wealth preferences. The second component is downward
nominal wage rigidity (DNWR), which is widely discussed in recent studies.4 Together with wealthpreferences, DNWR plays an important role in generating secular stagnation in the monetary
growth model.
We demonstrate that our model endogenously generates a slowdown in output growth in thetransition path to the steady state. In particular, we theoretically show that output initially equals
the first-best allocation, but later falls below the first-best allocation. In our model, strong wealthpreferences lead to disinflation, while DNWR is not binding. However, once DNWR binds, inflation
no longer decreases and the household with wealth preferences accumulates wealth rather thanconsuming enough to reach the first-best allocation in output. Then, aggregate demand determines
output, an aggregate demand shortage occurs, and the growth rate of output is lower than that
under the first-best allocation.
In our numerical simulations, we focus on the slowdown in output growth in the US after theGreat Recession. For the US data, our model explains 95.6 percent of the slowdown in output
growth in 2017. Our simple growth model explains surprisingly well the deviations between therealized output trend and the expected output trend before the Great Recession. We also conduct
numerical simulations for Japan. The simulation results also perform well for the Japanese datain the 1990s. In particular, the model explains 77.5 percent of the slowdown in output growth in2001.
Our model also predicts the permanent decline in the real interest rate, which is linked to thesecular stagnation hypothesis.5 In the standard consumption Euler equation, the real interest rateis determined by the household’s subjective discount rate and the growth rate of consumption.
However, as pointed out by Michaillat and Saez (2021), wealth preferences in the consumption
4

See Barattieri, Basu, and Gottschalk (2014), Sigurdsson and Sigurdardottir (2016), Schmitt-Grohé and Uribe(2016), Fallick, Lettau, and Wascher (2016), Hazell and Taska (2020), and Grigsby, Hurst, and Yildirmaz (2021),
among others.
5
Characterizations of secular stagnation appear in Baldwin and Teulings (2014) and Krugman (2014).

3

Euler equation create discounting in the real interest rate. This discounting leads to a persistently
low real interest rate, consistent with the data under secular stagnation.
In our model, inflation also declines persistently until the output trend deviates from the productivity trend. This observation is consistent with inflation in the US before and after the GreatRecession. Hall (2011) points out that US inflation declined during the 1990s but became stableeven in the presence of long-lasting slack in the economy from the Great Recession. Our modelinterprets the missing deflation as being a consequence of the binding DNWR, where inflation stops
declining even if aggregate demand falls short of aggregate supply.
Previous studies have fallen into one of four groups in explaining secular stagnation. The
first focuses on the productivity slowdown (e.g., Fernald (2015), Gordon (2015), Takahashi and
Takayama (2022)). This group emphasizes the decline in productivity growth as a source of thesecular stagnation. In our model, (labor) productivity growth also declines and the decline isdriven by the decline in aggregate demand growth. Moreover, we intentionally remove exogenous
productivity in the production function from the model because we highlight the degree to which
the model with wealth preferences alone explains the observed slowdown in output growth. The
second group focuses on the impact of demographic changes on savings in explaining the declining
real interest rate. (e.g., Carvalho, Ferrero, and Nechio (2016), Gagnon, Johannsen, and LopezSalido (2021), Jones (2022)). We exclude this potentially important factor from our model because
we focus on the mechanism behind the impact of wealth preferences on savings. The third grouprelies on debt deleveraging (e.g., Hall (2011), Eggertsson and Krugman (2012), Mian and Sufi
(2014), Guerrieri and Lorenzoni (2017), Eggertsson, Mehrotra, and Robbins (2019)). Among thesestudies, Eggertsson, Mehrotra, and Robbins (2019) introduce debt deleveraging into a model thatincorporates declines in productivity and changes in demography. They numerically evaluate theirmodel of the secular stagnation and discuss its policy implications.6 f Our paper is categorized into
the fourth group, which introduces wealth preferences into a standard macroeconomic model. This
group assumes a strong desire for liquidity or wealth (e.g., Michau (2018), Illing, Ono, and Schlegl(2018)).7 The study closest to ours is Michau (2018), who incorporates wealth preferences and
6

Ikeda and Kurozumi (2019) discuss monetary policy rules to prevent secular stagnation in a model with financial
frictions and endogenous total factor productivity growth. Kobayashi and Ueda (2022) argue that a fear of large-scale
taxation and capital misallocation arising from a debt crisis may be a driving force of the output slowdown in Japan.
7
In recent studies, models with wealth preferences are analyzed using the New Keynesian framework (Michaillat
and Saez (2021)) and search models (Michaillat and Saez (2022) and Hashimoto, Ono, and Schlegl (2023)).

4

DNWR into the standard neoclassical growth model and shows the existence of both the neoclassicaland stagnation steady states in his model. By contrast, our model has a unique steady stateand endogenously generates a regime change from efficient allocation in the neoclassical economy
to inefficient allocation in the stagnant economy.8 Furthermore, we implement a quantitative
assessment of whether the model with wealth preferences explains the slowdown in output growth
together with permanent trend declines in the real interest rate and inflation.
The paper is organized as follows. Section 2 presents our simple growth model. Section 3 studies
the model dynamics and presents the main analytical results. In Section 4, we simulate the modeland show that its predictions are consistent with the data. Section 5 concludes.

2

The modelThis section first presents the setup of the monetary growth model along with two important model
assumptions: wealth preferences and DNWR. We then discuss the implications of DNWR for the
goods market. Finally, we define the competitive equilibrium in this model.

2.1

Setup

2.1.1

The household

The representative household solves the following maximization problem:
Z ∞exp(−ρt) [u(ct ) + v(mt ) − φ(et ) + β(at )] dt,

max

ct ,mt ,et ,at ,ht

s.t.

(1)

0

ȧt = rt (at − mt ) − πt mt + wt et ht − ct + τt ,(2)

ht ≤ 1,(3)where a0 is given and the subjective discount rate ρ is strictly positive. Here ct is consumption,
mt is real money balances (mt = Mt /Pt , where Mt is nominal money balances and Pt is the pricelevel), et is the labor effort (or the quality of labor), and at is total real asset holdings. In the
8

In this sense, our analysis also differs from Benigno and Fornaro (2018), who develop an endogenous growth modelwith DNWR. They show that weak growth depresses aggregate demand and that the resulting aggregate demand
shortage may lead to the stagnation steady state. In contrast to our study, stagnation arises as a self-fulfilling
equilibrium.

5budget constraint, rt is the real interest rate (at − mt = bt represents the illiquid asset holdingsof the household), πt is inflation or the opportunity cost of holding money, wt is real wages, ht ishours worked and τt is lump-sum transfers from government. Here, wage income is proportional
to effective labor nt (= et ht ). The budget constraint (2) indicates that the sources of consumption
and saving (ȧt ) equal income from asset holdings (rt (at − mt ) − πt mt ), labor income (wt nt ), andthe lump-sum transfers from government (τt ). In this maximization problem, the effective labor
supply is decomposed into ht ≤ 1 and 0 ≤ et < ∞. We also assume no capital in the economy;
therefore, at = mt holds for all t (i.e., bt = 0 in equilibrium).
The utility functions u(ct ), v(mt ), and φ(et ) take a constant relative risk aversion form. These
functions are expressed as follows:(4)u(ct ) = ln ct ,v(mt ) = v
φ(et ) =

m1−η
t

,
1−η
(φt et )2,2

v > 0,

η > 0,

φt > 0,(5)(6)

where, with a slight abuse of notations, v > 0 represents a parameter for the function v(mt ). The
time-varying parameter φt follows a deterministic process given by

φt = φ0 exp(−gt),

g > 0,(7)

where φ0 > 0 denotes the initial value of φt . Here, φt decreases at the rate g. As we show below, φt
enables us to consider endogenous variations in labor productivity through the household’s effort.
Our preference assumptions on the utility from wealth are critical. We assume that the household has an insatiable desire for wealth. The utility from wealth satisfies β 0 (at ) > 0, β 00 (at ) ≤ 0,
and β 0 (at ) is strictly positive and constant in equilibrium. The simplest specification that satisfies
these conditions for β(at ) is a linear function:β(at ) = β × at ,β > 0,(8)

where β is a strictly positive parameter, again with a slight abuse of notations, for the function6

β(at ). This specification follows Michau (2018) and extends Ono (1994) and Ono (2001), in which
the household has an insatiable desire for liquidity.
We assume (8) for simplicity, not for the necessity of our main results. A necessary condition forour main results is that marginal utility from wealth is strictly positive and constant in equilibrium.
As argued by Michau (2018) and Michaillat and Saez (2021), there are a variety of alternative
specifications for the utility from wealth that generate positive constant marginal utility. In thesestudies, while the concavity of the utility function is ensured, marginal utility from wealth isconstant in equilibrium.9 We employ linear utility (8) because these specifications lead to the same
results in this paper.10
The first-order conditions are:v 0 (mt )u0 (ct )ċtct
0φ (et )/ht
u0 (ct )(9)= rt + πt ,= rt − ρ +β 0 (at ),u0 (ct )= wt , if the DNWR is not binding,

(10)
(11)
(12)

ht = 1,and the transversality condition is limt→∞ exp(−ρt)u0 (ct )at = 0. In (9), the household pays theopportunity cost of holding money, rt + πt , to receive the marginal benefits v 0 (mt ) (or v 0 (mt )/u0 (ct )when measured in units of consumption goods). In (10), the household pays the marginal costof savings, ρ + ċt /ct . This is the household’s consumption discount rate, which allows for thehousehold’s risk aversion.11 Regarding the marginal benefits of savings, the household receives
market returns on illiquid assets rt and the marginal utility from wealth β 0 (at ) (or β 0 (at )/u0 (ct ) when9

Michau (2018) and Hashimoto, Ono, and Schlegl (2023) consider the preferences for wealth excluding money,β(at −mst ), where mst is the real money supply and the household takes it as given. They assume that β 0 (at −mst ) > 0,
β 00 (at − mst ) < 0, and limat →∞ β 0 (at − mst ) = 0, but β 0 (0) > 0. Thus, marginal utility from wealth is constant inequilibrium, where bt = at − mst = 0. Ono and Yamada (2018) and Michaillat and Saez (2021) allow for utility from
relative wealth at (i) − ãt . Here, at (i) denotes wealth at the individual household level, and ãt is average wealthin the economy, which the household takes as given. They assume that β 0 (at (i) − ãt ) > 0, β 00 (at (i) − ãt ) < 0 andlimat (i)→∞ β 0 (at (i) − ãt ) = 0, but again β 0 (0) > 0 where at (i) = ãt .
10Constant marginal utility results in equilibrium money holdings beyond the amount the consumers use for theirtransactions. Nevertheless, constant marginal utility is not necessarily inconsistent with the neuroscientific evidence.
Based on lab experiments, Camerer, Loewenstein, and Prelec (2005) argue that “people value money without carefully
computing what they plan to buy with it.” (p. 35)
11
The consumption discount rate is generally represented by the sum of the steady-state discount rate and thegrowth rate of marginal utility. In the equation, it is given by ρ − [du0 (ct )/dt][1/u0 (ct )] = ρ + ċt /ct .

7

measured in units of consumption goods). When the wealth preferences are absent, (10) reduces
to the standard Euler equation ċt /ct = rt − ρ and savings only yield market returns rt . When
the wealth preferences are present, however, savings generate additional benefits of β 0 (at )/u0 (ct ).
Thus, the household would give up more consumption and accept a lower interest to enjoy holding
more wealth. Equation (11) has the standard interpretation that the marginal rate of substitution
between effort and consumption (per unit of hours worked) equals the real wage. If we incorporate(12) into (11), the equation reduces to the standard first-order condition for labor supply. As wewill discuss in Section 2.2, however, this equation only holds with equality when DNWR is notbinding.
Eliminating rt from (9) and (10) and allowing for bt = 0 in equilibrium yields the condition for
aggregate demand:Ω(mt , ct ) = ρ +
where

ċt
+ πt ,ctΩ(mt , ct ) ≡(13)v 0 (mt ) + β 0 (mt ).
u0 (ct )Here Ω(mt , ct ) denotes the marginal benefits of holding wealth (measured in units of consumption
goods). In Ω(mt , ct ), v 0 (mt ) is benefits of increasing money, and β 0 (mt ) = β 0 (at ) is the additional
benefit of increasing wealth. The right-hand side of (13) represents the opportunity cost of holding
wealth. Thus, (13) determines the rate at which the household substitutes wealth for consumption.
To hold an additional unit of assets, the household must give up consumption goods by an amountequal to the household’s consumption discount rate (ρ + ċt /ct ) and the inflation rate (πt ).
2.1.2

The firm

There is a representative firm in a competitive market in the economy. In our model, the firm’s
technology is linear in effective labor nt :yt = nt ,(14)

where yt is output. With this production function, the firm’s effective labor demand condition is:

wt = 1.

8(15)

2.1.3The downward nominal wage rigidity

One of the most important assumptions in our model is the DNWR in the labor market. Following
the literature (e.g., Schmitt-Grohé and Uribe (2016)), we assume that nominal wage inflationẆt /Wt cannot be lower than the lower bound γ:
Ẇt
≥ γ.
Wt
2.1.4

(16)

The government

The government has a budget constraint τt = µmst , where mst = Mts /Pt and Mts is the nominal
money supply. Throughout this paper, we assume that the money growth rate is strictly positive(µ > 0) and sufficiently high:µ > γ,(17)

which means that the money growth rate always exceeds the lowest level of inflation.

2.2

Implications of DNWR for the goods market

To explore the implications of DNWR for the goods market, we first rewrite the equation of DNWR
as a complementary slackness condition in the labor market:
Ẇt
−γ
Wt

!
(nft − nt ) = 0,

where nft is the first-best allocation of effective labor when DNWR is not binding. If the nominal
wage inflation exceeds γ, the labor market achieves the first-best allocation of effective labor. If
Ẇt /Wt is equal to γ, DNWR is binding. As a result, the demand for effective labor determines the
allocation of nt in the labor market.
We translate the complementary slackness condition in the labor market into that in the goods
market. Note that DNWR equates to downward nominal price rigidity πt ≥ γ because real wages
wt = Wt /Pt are constant from (15). Furthermore, the production function is yt = nt . Combining

9these relations with the above condition yields:(πt − γ)(ytf − yt ) = 0,(18)where ytf is the first-best allocation of output.
The economy has two regimes. If inflation is high (πt > γ), the goods market achieves thefirst-best allocation yt = ytf . We refer to this regime as the high-inflation regime. Alternatively, if
the goods market fails to achieve the first-best allocation, namely yt < ytf , inflation hits the lower
bound (πt = γ).12 We refer to this regime as the low-inflation regime.
When πt > γ, output grows at an exogenous rate of g > 0. As φt decreases at the rateg > 0, marginal disutility of effort also decreases. This decline in φt raises the household’s effort.
Furthermore, (11) holds with equality. The assumptions in the model allow us to derive:−1
ytf = φ−1t = φ0 exp(gt),(19)

which means that output grows at the exogenous rate g > 0.13
When πt = γ, (11) no longer holds with equality. In particular, we obtain:wt >

φ0 (et )/ht
, if the DNWR is binding.
u0 (ct )(20)

Here, while ht = 1 continues to hold, DNWR introduces a wedge into the supply of effort. We provethat (20) is equivalent to yt < ytf .14 Thus, (11) and (20) are consistent with the complementaryslackness condition (18).
Labor productivity varies endogenously depending on whether DNWR is binding. In this model,labor productivity is simply yt /ht = et because of the production function yt = nt = et ht . Increased
effort improves the quality of labor, and the improved quality of labor enhances labor productivity.
12

We exclude the case of yt > ytf because it does not achieve feasibility of production.
To derive this equation, note that (11) and (15) imply that φ0 (yt ) = u0 (yt ) because ct = yt = et ht and ht = 1
for all t. The utility functions (4) and (6) lead to φ2t yt = 1/yt . Given that the goods market achieves the first-best
allocation (i.e., yt = ytf ), this condition leads to ytf = 1/φt = φ−10 exp(gt).
14Note that wt = 1 and ht = 1. Together with the goods market-clearing condition ct = yt = et , (20) becomes(1 − yt2 φ2t ) > 0. This inequality is further rewritten as (1 − yt φt )(1 + yt φt ) > 0. As yt and φt are both strictly positive,
this inequality is simplified to 1/φt − yt > 0. Noting the definition of ytf given by (19), ytf = 1/φt . Therefore, wehave yt < ytf .
13

10

When DNWR is not binding, labor productivity et = ytf /ht grows at the rate g as seen from (19).
When DNWR binds, labor productivity is no longer determined by ytf .

2.3

The competitive equilibrium

We are ready to discuss the competitive equilibrium. The market-clearing conditions are:
1. Goods market ct = yt ,2. Labor market (πt − γ)(ytf − yt ) = 0,
3. Money market mt = mst ,4. Bond market at − mt = 0.
A competitive equilibrium of the model is the set of allocations {ct , yt , mt , at } and prices {wt , rt , πt }that satisfy the following: (i) the representative household maximizes (1) subject to (2) and (3);(ii) the representative firm maximizes profits; (iii) the government’s transfers and money supply are
specified as above; and (iv) all markets clear except for the labor market. The labor market-clearing
condition depends on the complementary slackness condition (18).

3

The model dynamics

This section investigates the model dynamics from t = 0 to t = ∞, given the initial state at t = 0.
We first show the system of equilibrium conditions under the two inflation regimes. Then, we
demonstrate that a unique transition path to the “stagnation” steady state exists under wealthpreferences. Along the transition path to the stagnation steady state, slowdown in output growthoccurs endogenously. Finally, we show that the model without wealth preferences fails to generatea slowdown in output growth.
Throughout this section, we rely on the equation for aggregate demand, which is the key equation for understanding the model:Ω(mt , yt ) = ρ +ẏt+ πt ,
yt

where we integrate the goods market-clearing condition with (13).
11

(21)

It is convenient to define the threshold value of Ω(mt , yt ) for which output grows at the rate g(i.e., the first-best allocation) but inflation is as low as γ (i.e., πt = γ). Substituting ẏt /yt = g and
πt = γ into (13) gives the threshold value of Ω(mt , yt ):Ω∗ = ρ + g + γ.

(22)

We use the threshold Ω∗ to evaluate the allocation and prices in the two inflation regimes.

3.1

High-inflation regime

The high-inflation regime is characterized by the first-best allocation yt = ytf and high inflation
πt > γ, where DNWR is not binding in (18). Using the goods and bond market-clearing conditions
ct = yt and bt = 0 (or at = mt ), we summarize the system of equilibrium conditions as follows:
ẏt
yt
ṁt
mt

= g,

(23)

= µ − γ − [Ω(mt , yt ) − Ω∗ ],(24)

πt = γ + [Ω(mt , yt ) − Ω∗ ],rt = ρ + g −β 0 (mt ).
u0 (yt )

(25)
(26)

Equation (23) immediately follows from (19) because yt = ytf holds under the first-best allocation.
Next, we derive (26) from (10) using ẏt /yt = g in the high-inflation regime. Equation (25) is derived
from (21) and (22) to obtain (25). Finally, (24) is derived from the definition of mt = Mt /Pt ,ṁt /mt = µ − πt . In solving the model, we compute output from yt = ytf = φ−10 exp(gt), given φ0 .
Given m0 = a0 , mt can be solved numerically for mt from (24).
3.1.1

Low-inflation regime

The low-inflation regime is characterized by inefficient allocation yt < ytf and the lower bound of
inflation πt = γ, where DNWR is binding in (18). Using the threshold value Ω∗ given by (22), we

12

summarize the equilibrium conditions as follows:
ẏt
yt
ṁt
mt

= g − [Ω∗ − Ω(mt , yt )] ,(27)

= µ − γ,

(28)
(29)

πt = γ,rt =β 0 (mt )ρ+g− 0− [Ω∗ − Ω(mt , yt )] .
u (yt )

(30)

Equation (29) shows that DNWR is binding. Equation (28) immediately follows from (29) becauseṁt /mt = µ − πt . To obtain (27), we use (21) and (22).15 We derive (30) from (10) using ẏt /yt =g − [Ω∗ − Ω(mt , yt )] in the low-inflation regime.
Comparisons between the two regimes reveal that the difference between Ω(mt , yt ) and Ω∗
matters for the model dynamics. For example, (27) indicates that output growth is lower than gunder the low-inflation regime if and only if Ω(mt , yt ) < Ω∗ . Furthermore, (25) suggests that by
how much inflation under the high-inflation regime exceeds γ depends on the size of Ω(mt , yt ) − Ω∗ .
The real interest rate under the low-inflation regime depends on the difference between Ω(mt , yt )and Ω∗ . If Ω(mt , yt ) < Ω∗ , the difference generates downward pressure on the real interest rate as
is seen from (30).

3.2

The stagnation steady state

We characterize the steady state before analyzing the model dynamics. The steady state in ourmodel is the “stagnation” steady state where (i) DNWR binds and (ii) output converges to a
constant value. We show that the stagnation steady state arises under our preference assumptions.
Let t∗ be the period in which Ω(mt , yt ) becomes equal to Ω∗ . When Ω(mt , yt ) = Ω∗ , output
growth is g and inflation is as low as γ. Equation (28) implies that mt goes to ∞ as t → ∞ becauseof (17). In particular, mt can be solved as:
mt = mt∗ exp[(µ − γ)(t − t∗ )] ⇒ lim mt = ∞,
t→∞

(31)

15
In particular, we use (21) to obtain ẏt /yt = Ω(mt , yt ) − ρ − πt . Because πt = γ, ẏt /yt = Ω(mt , yt ) − ρ − γ. By
adding and subtracting g in this equation, we obtain ẏt /yt = g − Ω∗ + Ω(mt , yt ) from (22).

13where mt∗ is the real money balances evaluated at t = t∗ .
To prove the existence of the stagnation steady state, we use the transversality condition:
limt→∞ u0 (yt )at exp(−ρt) = limt→∞ mt /yt exp(−ρt) = 0. Taking the time derivative of this condition translates the condition into ṁt /mt − ẏt /yt − ρ < 0.

In the stagnation steady state,ṁt /mt = µ − πt = µ − γ (because (i) DNWR binds) and ẏt /yt = 0 (because (ii) output converges to a constant value). Thus, µ − γ < ρ ensures the transversality condition. Note that if theassumption µ > 0 holds, the transversality condition implies that ρ + γ > 0.
Now, the allocations and prices in the steady state are characterized as follows. Because (i)DNWR binds, inflation equals its lower bound: π ss = γ. Output growth is zero in the stagnation
steady state because (ii) output converges to a constant value. These results imply that the righthand side of (21) is ρ + γ in the stagnation steady state. Next, the left-hand side of (21) in thestagnation steady state is:Ωss ≡ lim Ω(mt , yt )t→∞

= lim [v 0 (mt ) + β 0 (mt )]yt
t→∞

(32)

= βy ss .
Combining these results, we have the steady-state level of output y ss :y ss =

ρ+γ
,
β

(33)

which is strictly positive because ρ + γ > 0. For the remaining variables, the steady-state growthrate of real money balances is µ − γ because (28) holds under binding DNWR. The nominal interest
rate in the stagnation steady state is zero because of (9): limt→∞ (rt + πt ) = limt→∞ [v 0 (mt )yt ] = 0.
The real interest rate in the stagnation steady state is rss = −π ss = −γ.

3.3Transition dynamics to the stagnation steady state

We are now ready to discuss the transition path to the stagnation steady state given the initialstate at t = 0. We first present two lemmas. The first lemma describes the uniqueness of thetransition path:14

Lemma 1. Suppose that g > 0, µ > 0, µ > γ, and µ − γ < ρ. Under the preference assumptions
specified by (4)–(8) with a strictly positive β, we have a unique dynamic path of output toward thestagnation steady state. Furthermore, assume that π0 > γ at the initial state. Then, the economy
experiences a regime change from a high-inflation regime to a low-inflation regime at t = t∗ .
Proof. See Appendix A.1.
The second lemma discusses the properties of Ω(mt , yt ):Lemma 2. Under the assumptions in Lemma 1, the marginal benefits of holding wealth Ω(mt , yt )satisfy the following:Ω(mt , yt ) > Ω∗ ,for 0 ≤ t < t∗ ,Ω(mt , yt ) < Ω∗ ,for t > t∗ ,Proof. See Appendix A.2.
Using Lemmas 1 and 2, we demonstrate that the model exhibits an endogenous slowdown in
output growth.
Proposition 1. Under the assumptions in Lemma 1, the slowdown in output growth occurs at
t = t∗ . For 0 ≤ t ≤ t∗ , output growth is equal to g. For t∗ < t < ∞, output growth is strictly
positive but lower than g.
Proof. It is obvious from (23) and (27). Equation (23) shows that output growth is g in the highinflation regime. Equation (27) and Lemma 2 suggest that output growth is lower than g in the
low-inflation regime.
The slowdown in output growth results from a regime change from a high-inflation to lowinflation regime. In our model, steady growth of output persists as long as DNWR is not binding(i.e., 0 ≤ t < t∗ ). However, after DNWR binds at t = t∗ , the growth rate becomes lower thang for t > t∗ . As we suggested earlier, output growth in the low-inflation regime is lower than gif Ω(mt , yt ) < Ω∗ . Lemma 2 proves this inequality. Furthermore, output continues to slow until

15

output growth converges to zero in the stagnation steady state at t = ∞:

lim

ẏt

t→∞ yt= lim [g + Ω(mt , yt ) − Ω∗ ] = βy ss − ρ − γ = 0
t→∞

(34)

where the first equality is from (27) and the second equality is from (33).
The prediction in Proposition 1 is consistent with the fact emphasized in the literature on
secular stagnation. That is, output growth in the US was high prior to the Great Recession but
was low after the Great Recession. The slowdown in output growth is not transitory in the US
data. Similar but more notable observations are confirmed for Japan after the early 1990s.
To better understand the model dynamics, it is useful to investigate the marginal benefitsof holding wealth Ω(mt , yt ) shown in (21). Lemma 2 means that Ω(mt , yt ) declines between thehigh- and low-inflation regimes. While Ω(mt , yt ) is always larger than Ω∗ under the high-inflation
regime, it is always smaller than Ω∗ under the low-inflation regime. Thus, Ω(mt , yt ) after t∗
must be lower than Ω(mt , yt ) before t∗ . In simulations implemented in the subsequent section,
we confirm that Ω(mt , yt ) decreases smoothly over time. Under our specification of preferences,Ω(mt , yt ) = (β + vm−ηt )yt is decreasing in mt and increasing in yt so that there are two offsetting
effects on Ω(mt , yt ). Numerically, Ω(mt , yt ) tends to decrease over time because the effect of mt onΩ(mt , yt ) overwhelms the effect of yt on Ω(mt , yt ). Intuitively, this is because strong preferences
for wealth make real money balances grow faster than aggregate demand.
The declining marginal benefits of holding wealth and DNWR give rise to the slowdown in
output growth. Equation (21) shows that the marginal cost represented by the right-hand sideof the equation must decline in equilibrium. Namely, either output growth (ẏt /yt ) or inflation(πt ) must decline, given a constant ρ. If the economy is initially in the high-inflation regime (i.e.,
π0 > γ), output growth can be kept at ẏt /yt = g. As long as DNWR is not binding, decreases in
inflation can lower the marginal cost of holding wealth. By contrast, if the economy turns to be ina low-inflation regime (i.e., πt = γ), inflation no longer decreases. Only through the slowdown in
output growth can the marginal cost of holding wealth decrease.
We emphasize that the aggregate demand shortage drives this slowdown in output growth. In
our model, the household’s wealth preferences weaken aggregate demand growth by substituting
wealth for consumption goods. Initially, because DNWR is not binding, the weakened aggregate
16demand leads to disinflation and the economy achieves the first-best allocation in output under
flexible prices. However, when DNWR makes price adjustment rigid, the weakened aggregate
demand determines equilibrium output. As a result, the growth rate of output is g for t ≤ t∗ andbecomes lower than g for t > t∗ .
There are some remarks on the transition path. First, the initial state of the economy mattersfor the slowdown in output growth. In Lemma 1, we assume that inflation exceeds γ at t = 0 (i.e.,
π0 > γ). This inequality is satisfied when wealth accumulation at t = 0 is low enough to ensurethat the marginal benefits of holding wealth is high. When the marginal benefits of holding wealthis high, inflation is also high (see (21)). As wealth deepening proceeds, the marginal benefits ofholding wealth decrease and thus disinflation occurs. When inflation hits the lower bound, output
growth starts declining. The timing of the slowdown in output growth depends on preference
parameters such as β and the degree of DNWR γ.
Second, the declining marginal benefits of holding wealth also account for the declining realinterest rate over time. While Section 4 will demonstrate that the simulated real interest rate
declines over time, the following proposition provides the analytical results of the declining realinterest rate between the high- and the low-inflation regimes.
Proposition 2. Let rt∗ be the real interest rate when the slowdown in output growth starts, wherert∗ = ρ + g − βyt∗ . Under the assumptions in Lemma 1, the real interest rate is strictly higher thanrt∗ for 0 < t < t∗ , and the real interest rate is strictly lower than rt∗ for t > t∗ . In the stagnation
steady state, rt converges to −π ss .
Proof. It immediately follows from (26) that rt = ρ + g − βyt > rt∗ = ρ + g − βyt∗ for t < t∗ . Thisis because Proposition 1 implies that output growth is positive; therefore, yt < yt∗ for t < t∗ . It is
straightforward to obtain rt = ρ + g − βyt − [Ω∗ − Ω(mt , yt )] < rt∗ = ρ + g − βyt∗ for t > t∗ from(30). This is because Ω∗ > Ω(mt , yt ) from Lemma 2 and yt > yt∗ from Proposition 1. For proof ofthe steady-state real interest rate, note that (9) with the goods market-clearing condition implies
that m−ηt yt = rt + πt . While mt goes to infinity (see (31)), output converges to the constant givenby (33). As t → ∞, the above equation converges to 0 = rt + πss , resulting in rt = −π ss .
Finally, our model also predicts the slowdown in labor productivity growth as discussed by
Fernald (2015). He argues that labor productivity slowed in the middle of the 2000s. Recall that17

labor productivity in our model is et = yt /ht where ht = 1. Thus, labor productivity initially grows
at g and later slows at a rate lower than g.

3.4

The role of wealth preferences

As we emphasized in the previous subsections, strong wealth preferences of β > 0 are a key
assumption for generating the results of Proposition 1. To crystallize the role of wealth preferences,
this subsection discusses the model’s prediction under no wealth preferences (i.e., β = 0). The
following proposition shows that, if the wealth preferences are absent, Ω(mt , yt ) is constant inequilibrium.
Proposition 3. Suppose that the assumptions in Lemma 1 hold except for β > 0. If β = 0,Ω(mt , yt ) is constant in equilibrium. The regime is predetermined and regime change never occurs.
There is no endogenous slowdown in output growth.
Proof. See the Appendix A.3
The model without wealth preferences (β = 0) fails to generate an endogenous slowdown in output growth because Ω(mt , yt ) is constant in equilibrium. The parameters in the model predetermine
the regime in equilibrium. For example, if the economy is initially in a high-inflation regime, theeconomy is on the balanced growth path where the goods market achieves the first-best allocation
with output growth of g. If the economy instead starts from a low-inflation regime, the economy
experiences low economic growth given the rigid wage adjustment arising from DNWR. In this case,the allocation is inefficient because the aggregate demand shortage makes economic growth slower
than g. In either case, however, we do not observe a slowdown in output growth, in contrast to thecase of β > 0. Moreover, the growth rate of output with the binding DNWR does not converge tozero. Even if DNWR is binding, there is no stagnation steady state under β = 0. Therefore, theessential ingredient for generating a slowdown in output growth and stagnation steady state is astrictly positive β in the wealth preferences.

3.5

The role of DNWR

The other key assumption in the model is DNWR. To understand the role of DNWR, we next
consider what happens if DNWR is absent, but β is strictly positive. Using (21), Section 3.3
18

explained that, after πt hits the lower bound, output growth must decline in response to decreasesin Ω(mt , yt ). However, a slowdown in output growth can be explained when both ẏt /yt and πtdecrease in response to decreases in Ω(mt , yt ). In this case, DNWR may not be necessary.
If DNWR is removed from the model, however, there is no monetary equilibrium with a strictly
positive β. To see this, assume that γ = −∞. This assumption ensures that DNWR never binds. In
this case, the transversality condition µ − γ < ρ is not satisfied and steady-state output is negativefrom (33). As we discussed, when µ > 0, the transversality condition becomes ρ + γ > 0. Thus, γ
is at least strictly larger than −ρ to guarantee the equilibrium in this model.

4

Simulating the model

While our model is qualitatively consistent with the observed slowdown in output growth, it is
not necessarily clear whether the model numerically explains the data. It is also worth assessing
macroeconomic variables other than output growth. In this section, we simulate the model for the
US and Japan. We will explore whether our model quantitatively explains the observed slowdown
in output growth as well as the declining real interest rate and inflation.

4.1

Calibration

Before simulating the model, we calibrate the parameters for the simulations. We first discuss thetiming of the regime change from the high-inflation to the low-inflation regime. Comparing thelinear trend and the cubic trend shown in Figure 1, we interpret that t∗ is the period in which the
cubic trend (dashed line in the figure) starts falling below the linear trend (dot-dashed line). Aftera closer look at Figure 1, we set t∗ at 2001:Q3 for the US and at 1989:Q1 for Japan.
Some deep parameters are assumed to be common between the US and Japan. The subjective
discount factor ρ is 0.04. The degree of relative risk aversion for ct is one (σ = 1) and that for mt
is four (η = 4).
Other deep parameters differ between the US and Japan, and we use the data to parameterize
them. We first calibrate the parameters for the US economy. Recall that we estimated the linear
trend (dot-dashed line in the upper panel of Figure 1) from log real GDP over 1990:Q1 and 2007:Q1.
The mean growth rate over the sample period (or the slope of the linear trend) is 0.022 at the19annual rate. We take this value as the growth rate of output under the first-best allocation,g = 0.022. The money growth rate µ is 0.043, using the mean growth rate of M2 stock (per capita)
over 1990:Q1–2019:Q4. For calibrating γ, we use πt = γ after t∗ (see (29)). We extract trend
inflation using a cubic trend over 1990:Q1–2019:Q4 and calculate the mean of the cubic trend after
2001:Q3, where the period of 2001:Q3 corresponds to t∗ in our model. The resulting mean trend
inflation is 0.017. Here, the actual inflation is the year-on-year inflation calculated from the Personal
Consumption Expenditure Price Index Excluding Food and Energy. To parameterize v and β inthe utility functions v(mt ) and β(at ), we target the 2001:Q3 values of the cubic trend for the realinterest rate and the velocity of money. For the real interest rate, we use 1-year real Treasury yields
from the database of the Cleveland Federal Reserve Bank and estimated by Haubrich, Pennacchi,
and Ritchken (2012).16 The real interest rate in 2001:Q3 is 1.56 percent in the cubic trend. For thevelocity of money, we employ the velocity of M2. The velocity of money in 2001:Q3 is 2.10 in the
cubic trend. The values of v and β that achieve these target values are v = 0.054 and β = 0.014,
respectively.
We next calibrate the parameters for the Japanese economy. We set g = 0.039 from the growthrate of real GDP per capita between 1980:Q1 and 1991:Q1 (corresponding to the linear trend in
Figure 1). We parameterize µ = 0.041 from per capita M2 growth averaged over 1980:Q2 –2019:Q4.
As before, we calibrate γ using πt = γ after t∗ . Given t∗ corresponds to 1989:Q1 for Japan, we
calibrate γ at 0.003 based on trend inflation after 1989:Q1. Here, we calculate the trend inflation
after 1989:Q1 from the Consumer Price Index for All items less fresh food.17 To obtain v andβ in v(mt ) and β(at ) for Japan, we target the real interest rate and the velocity in 1989:Q1. We
construct the real interest rate using the nominal interest rate and the actual inflation from 1986:Q3
to 2019:Q4.18 We use velocity of M2 as in the calibration for the US. We calibrate v and β to hit
the target of the 1989:Q1 values of the real interest rate (3.78 percent in cubic trend) and velocity
of money (1.07 in cubic trend). We obtained v = 0.063 and β = 0.027, respectively.
16

The most recent data are available at https://www.clevelandfed.org/our-research/indicators-and-data/
inflation-expectations/background-and-resources.aspx#research.
17
The price index is adjusted for consumption tax hikes in 1997 and 2014.
18
To compute the 1-year real interest rate, we use the 12-month Japanese yen London Inter-Bank Offered Rate (JPYLIBOR) as an interest rate benchmark over 1986:Q3–2019:Q4. We combine core inflation from 1986:Q3 to 1989:Q4
and tax-adjusted core inflation from 1990:Q1 to 2019:Q4 because the consumption tax-adjusted core inflation is only
available from 1990. Note that inflation is not adjusted for the introduction of the consumption tax in 1989:Q2 andthe increase in 2019:Q4.

20

We solve the model using a method similar to the shooting algorithm. We calculate the transition path forward from t∗ to t̄, where t̄ is a sufficiently large number to approximate t = ∞. In thistransition path, the economy moves toward the stagnation steady state. We guess output in the
period of a regime change (denoted by yt∗ ) and compute real money balances in the same period(denoted by mt∗ ).19 Together with yt∗ and mt∗ , we use (27) and (28) to obtain future variables
yt+∆ and mt+∆ , where ∆ is a small increment of time. We iterate this calculation forward until wehave yt̄ ' yt̄+∆ and define yt̄ as a candidate of the steady-state output y ss . If yt̄ ' y ss , we conclude
that the transition path to the stagnation steady state is obtained. If not, we update the guess ofthe output yt∗ and iterate computations until we have yt̄ ' y ss . Regarding the transition path fort < t∗ , output growth always equals g. We compute the transition path backward from t∗ to t = 0,
where t = 0 refers to the first period of the sample.20

4.2

Simulation results

Figures 2 and 3 report our simulation results for the US and Japan, respectively. The upper panelpresents log real GDP per capita, the middle panel shows the real interest rate, and the lower panel
is inflation. Each panel contains the simulated data (solid line), estimated cubic trend (dashedline), and actual data (dotted line). Our model has no stochastic shocks. Therefore, our model
aims to explain the estimated cubic trend rather than the actual data.
4.2.1

The USThe upper panel of Figure 2 presents US output. In this panel, the simulated output at t∗ isequalized to the linear trend in the same period. We investigate how closely the simulated output
after t∗ tracks the cubic trend. Note that, because the simulated output grows at g when t ≤ t∗ ,the simulated output is designed to match perfectly with the linear trend, which grows at the samerate of g.
[Figure 2 about here.]
19

Here we use Ω(mt∗ , yt∗ ) = Ω∗ to obtain mt∗ . The threshold value Ω∗ is given by (22).
All variables including ẏt /yt and ṁt /mt are smoothly connected between the high-inflation and the low-inflation
regimes. We can confirm this smooth transition by evaluating Ω(mt , yt ) in both systems of equations (23)–(26) and(27)–(30) at Ω∗ . At t = t∗ , the marginal benefits of holding wealth are equal to the threshold level, and (27)–(30)are equivalent to (23)–(26), respectively.
20

21

Comparing the solid and dashed lines in the upper panel of Figure 2 reveals that the modelaccounts for a substantial fraction of the slowdown in output growth relative to the linear trend(dotted line). For example, the model explains 75.8 percent of the slowdown in output growthin 2012:Q1 (5 years after the start of the Great Recession) and 95.6 percent of the slowdown in
output growth in 2017:Q1 (10 years after the start of the Great Recession).21 Thus, the model
almost entirely explains the slowdown in output growth in 2017.
The middle panel shows the simulated real interest rate. Overall, the model successfully replicates the real interest rate that has decreased since the 1990s. The real interest rate in the cubic
trend (dashed line) peaks at 2.43 percent in 1994:Q3 and reaches a minimum at -0.70 percent in2014:Q1. The simulated real interest rate exhibits a similar pattern to the data. The simulated
real interest rate was 2.34 percent in 1994:Q3 and -0.60 percent in 2014:Q1.
As discussed in the previous section, the decline in trend results from decreases in Ω(mt , yt )together with wealth preferences. Under the high-inflation regime, the additional benefit of holding
wealth β 0 (mt )/u0 (yt ) = βyt in (30) lowers the real interest rate. Once DNWR binds and regimechange occurs, the real interest rate further declines because of additional downward pressure on
the real interest rate represented by Ω∗ − Ω(mt , yt ) (see (30)).
The lower panel of Figure 2 plots inflation.22 The simulated inflation replicates the estimatedcubic trend quite well, especially in replicating the decline in the 1990s. The figure indicates thattrend inflation (dashed line) decreased steadily until 2003:Q2 and became stable at around 1.5–2
percent. In our simulation, inflation also decreases steadily until 2001:Q3 and becomes constant at
γ (see (29)).
4.2.2

Japan

Figure 3 conducts the same exercise with the Japanese data. As in the US case, the model performs
well. Recall that we set the timing of the regime change to 1989:Q1. Thus, changes in model
21

In the case of 2012:Q1, the linear trend of output is 43.6 percent higher in 2012:Q1 than the actual output in
1990:Q1. The cubic trend is 33.1 percent higher in 2012:Q1 than the actual output in 1990:Q1. Thus, the differencein the percentage, namely the decrease in output growth in the data, is 9.9(= 43.6 − 33.4) percentage points between
the linear and cubic trends. Next, the simulated output trend is 36.1 percent higher in 2012:Q1 than the actual
output in 1990:Q1, so the decrease in output growth explained by the model is 7.5(= 43.6 − 36.1) percentage points.
Taking the ratio of these numbers yields 0.758 = 0.075/0.099, so the model explains 75.8 percent of the decrease in
output growth. We implement the same calculation to obtain the number for 2017.
22
We use year-on-year inflation to remove noise in inflation.

22

dynamics become significant in the early 1990s. Once again, the model explains a substantial
fraction of the slowdown in output growth relative to the linear trend. Shown in the upper panelof Figure 3, the model explains 63.6 percent of the slowdown in output growth in 1996:Q1 (5 years
after the peak of the business cycle) and 77.5 percent of the slowdown in output growth in 2001:Q1(10 years after the peak of the business cycle). The simulated real interest rate closely keeps trackof the cubic trend of the real interest rate that declined from about 4.95 percent in 1986:Q3 to aslightly negative value of around -0.53 percent in 2019:Q4. The simulated real interest rate is 5.03
percent in 1986:Q3 and -0.28 percent in 2019:Q4 before converging to the steady-state value of -0.3
percent.23 Finally, simulated inflation also explains the observed decreases in the actual inflationin the early 1980s.
[Figure 3 about here.]5

Conclusion

Output growth in the US was persistently low after the Great Recession. In this paper, we explained this slowdown in output growth by introducing wealth preferences and DNWR into astandard monetary growth model. We theoretically showed that our model generates a slowdown
in output growth in the transition path to the stagnation steady state. Consistent with the literature on secular stagnation, the model also explains the declining real interest rate over time. Using
numerical simulations, we found that our model explains 75.8 and 95.6 percent of the slowdown in
US output growth in 2012 and 2017, respectively. We also implemented simulations for Japan andconfirmed that the model also explains the stagnation in Japan.
It is quite surprising that a simple model can account for the long-run patterns in the data.
Further research would enrich our understanding of secular stagnation and its policy prescriptions.
Many questions remain. What are the implications of growth-enhancing policy on the model
dynamics? What happens to output growth and the real interest rate if we prompt nominal wage
adjustment by removing institutional frictions in the labor market? What are the impacts offorward guidance on output growth? Exploring these questions would be important for future
research.
23

Note that we plot real interest rates from 1986:Q3 because of data availability of the nominal interest rate.

23

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A

Proofs of lemmas and propositions

A.1

Proof of Lemma 1

To prove Lemma 1, the phase diagram in the (mt , yt ) plane is convenient. We derive the loci ineach inflation regime in Figure 4.
26

[Figure 4 about here.]
We first consider the high-inflation regime. In this regime, (23) implies that ẏt > 0. We
thus focus on the ṁt = 0 locus drawn as the red solid line in Figure 4. The ṁt = 0 locus isobtained from (24) and given by µ − γ − Ω(mt , yt ) + Ω∗ = 0. Under our preference assumptions,Ω(mt , yt ) = (β + vm−ηt )yt . Therefore, the ṁt = 0 locus is:Ω∗ + µ − γ
β + vm−ηtρ + g + γ + (µ − γ)=β + vm−η
t

yt =

(35)

= fH (mt ),where the second equality comes from (22) and we define the ṁt = 0 locus as yt = fH (mt ). Heremt increases with time whenever (mt , yt ) lies to the right of the ṁt = 0 locus. Together with ẏt > 0
under the high-inflation regime, the directions of the changes are indicated by red arrows in the
figure.
Next, consider the low-inflation regime. Under the assumption of µ > γ (see (17)), (28) implies
that ṁt > 0. We thus focus on the ẏt = 0 locus drawn as the blue solid line in Figure 4. Theẏt = 0 locus is obtained from (27) and given by g − Ω∗ + Ω(mt , yt ) = 0. Again, noting that∗Ω(mt , yt ) = (β + vm−ηt )yt and Ω = ρ + g + γ, we have the ẏt = 0 locus as follows:Ω∗ − gβ + vm−ηt
ρ+γ=β + vm−η
t

yt =

(36)

= fL (mt ),where we define the ẏt = 0 locus as yt = fL (mt ). Note that fL (mt ) < fH (mt ) holds for any mt > 0
because µ − γ > 0 and g > 0. Here yt increases with time whenever (mt , yt ) lies to the left of theẏt = 0 locus. Together with ṁt > 0 under the low-inflation regime, the directions of the changes
are indicated by blue arrows in the figure.
Let us introduce another locus that determines the regime change. This locus is drawn as theblack solid line in Figure 4 and defined as a set of (mt , yt ) in which output growth is g and inflation27

is γ. In other words, the locus specifies a set of (mt , yt ) that satisfies Ω(mt , yt ) = Ω∗ . Using∗Ω(mt , yt ) = (β + vm−ηt )yt , we rewrite Ω(mt , yt ) = Ω as:Ω∗
β + vm−ηt
ρ+g+γ=
β + vm−η
t

yt =

(37)

= fT (mt ),where we define the locus as yt = fT (mt ). As shown in Figure 4, we have fL (mt ) < fT (mt ) <fH (mt ) given mt because of µ > γ and g > 0. It is easy to show that the economy is in thehigh-inflation regime if (mt , yt ) lies above the locus.24 When (mt , yt ) lies below the locus, DNWR
is binding and the economy is in the low-inflation regime.
Figure 4 also draws the optimal transition path (curve with arrows) starting from the initialstate of the economy. In the figure, π0 > γ is assumed. Thus, the economy is in the high-inflation
regime, and (m0 , y0 ) is located above the locus of fT (mt ). Once (mt , yt ) moves to the right of thefT (mt ) locus, the regime in the economy turns to the low-inflation regime. Note that, as t → ∞,mt → ∞ from (31) and ẏt /yt → 0 from (34). Thus, (mt , yt ) asymptotically converges to the dottedline located at the bottom where yt = y ss = (ρ + γ)/β. In the transition, ẏt /yt is always positive
because (mt , yt ) is located above the blue solid line (see Figure 4).
We prove two claims to show the uniqueness of the transition path to the stagnation steadystate. First, we show that the transition path is saddle-path stable around the stagnation steadystate. Second, we prove that the regime change from the high-inflation to the low-inflation regime
occurs at a unique pair of (mt , yt ) = (mt∗ , yt∗ ). The first claim ensures that the transition path in
the low-inflation regime is unique. The second claim implies that the transition path in the highinflation regime is unique because (mt , yt ) is smoothly and uniquely connected to the transition
path in the low-inflation regime. With these claims, we prove that the transition path to thestagnation steady state is unique as a whole. It is necessary to prove the second claim because, if(mt , yt ) moves along the curve represented by yt = fT (mt ) in which ẏt /yt = g and πt = γ, multiple
24

We prove this claim by contradiction. Suppose that πt = γ and ẏt /yt < g. Then, (21) implies that Ω(mt , yt ) =ρ + ẏt /yt + πt < ρ + g + γ = Ω∗ . That is, Ω(mt , yt ) < Ω∗ . It immediately follows from Ω(mt , yt ) = (β + vm−η
t )yt
−η
∗
that yt = Ω(mt , yt )/(β + vm−ηt ) < Ω /(β + vmt ) = fT (mt ), which contradicts the supposition. Note also that we
do not need to consider πt < γ and ẏt /yt > g because they are infeasible.

28

transition paths connected to the curve may exist in the high-inflation regime.
Let us prove the first claim. Define zt = 1/mt and consider the system of the equations under
the low-inflation regime:
ẏt
yt
żt
zt

= g − [Ω∗ − Ω(1/zt , yt )] ,= −(µ − γ).

By linearizing the above two equations around the stagnation steady state (zt , yt ) = (0, y ss ), we
have the eigenvalues ζ such that:
ρ+γ−ζ

0

0

−(µ − γ) − ζ= 0,

where we use (33) to replace βy ss by ρ + γ. As mentioned in the main text, we assume thatρ + γ > 0. The condition ρ + γ > 0 then ensures that one eigenvalue is positive. In addition,(17) implies that the other eigenvalue is negative. Because the system of the equations under the
low-inflation regime includes one jump variable yt and one predetermined variable zt , the transition
path is saddle-path stable and unique under this regime. Along any path located above the saddle
path, mt eventually becomes negative so that it is infeasible. Along any path located below thesaddle path, yt eventually becomes zero so that the transversality condition does not hold.
Next, we prove the second claim by contradiction. Suppose that there exist other pairs of (mt , yt )in which output grows at a rate of g and inflation is equal to γ. In this case, the transition pathof (mt , yt ) must be along the curve represented by yt = fT (mt ) in Figure 4. Note also that mt andyt cannot jump between the high- and the low-inflation regimes and (mt , yt ) must be connected tothe unique transition path in the low-inflation regime that goes through (mt∗ , yt∗ ). In other words,
if (mt , yt ) moves along the curve without a jump, then there exists a pair of (mt , yt ) = (mt̃ , yt̃ ) thatis not only along the curve but is also close to (mt∗ , yt∗ ) in time. Because (mt̃ , yt̃ ) exists in theneighborhood of (mt∗ , yt∗ ), t̃ = t∗ + ∆ where ∆ is an infinitesimally small increment of time. In
this case, the growth rate of Ωt̃ must be zero around t = t∗ because Ω(mt∗ , yt∗ ) = Ω∗ = ρ + g + γ
along the curve.

29Let us denote Ω(mt , yt ) by Ωt . The growth rate of Ωt is given by:
Ω̇t̃
ẏ
v
ṁt̃
= t̃ − ηηΩt̃yt̃v + βmt̃ mt̃
=g−η

v(µ − γ)v + βmηt̃

(38)

= 0.
The first equality is derived from the total derivative of Ωt = Ω(mt , yt ) = (β + vm−ηt )yt . In thesecond equality, ẏt /yt = g and ṁt /mt = µ−γ because Ω(mt̃ , yt̃ ) = Ω∗ . The third equality shows thatthe growth rate of Ωt̃ must be zero. The above equation suggests that mt̃ = {v[η(µ − γ) − g]/β}1/η
is constant. However, this contradicts the assumption that ṁt̃ /mt̃ = µ − γ > 0 if Ω(mt , yt ) = Ω∗holds.
We conclude that the transition path to the stagnation steady state is unique as a whole. The
regime change occurs only at t = t∗ and the transition path under the high-inflation regime issmoothly and uniquely connected to the unique transition path under the low-inflation regime.

A.2

Proof of Lemma 2

We first prove that Ω(mt , yt ) > Ω∗ for 0 ≤ t < t∗ . The economy is in the high-inflation regime
in 0 ≤ t < t∗ . As shown in Figure 4, yt is located above the locus of fT (mt ). Thus, we haveyt > fT (mt ) for 0 ≤ t < t∗ . Then, the definition of fT (mt ) shown in (37) leads to Ω(mt , yt ) > Ω∗for 0 ≤ t < t∗ .
We also prove that Ω(mt , yt ) < Ω∗ for t > t∗ from Figure 4. The figure indicates that yt <fT (mt ) for t > t∗ because the economy is in the low-inflation regime. Again, the definition offT (mt ) shown in (37) leads to Ω(mt , yt ) < Ω∗ for t > t∗ .

A.3

Proof of Proposition 3

To prove the proposition, we focus on the growth rate of Ωt = Ω(mt , yt ) = (β + vm−ηt )yt . Whenβ = 0, the growth rate of Ω(mt , yt ) is written as:
Ω̇t
ẏt
ṁt
=
−η .
Ωt
yt
mt
30

(39)

In the general case of β 6= 0, Ω̇t /Ωt = ẏt /yt −ηv/(v+βmηt )ṁt /mt . In the case of β = 0, however, thecoefficient on ṁt /mt is a constant η. We will show below that Ωt is constant in equilibrium whenβ = 0. The constant Ωt rules out the possibility of an endogenous slowdown in output growth.
Output growth under the high-inflation regime

Suppose that the economy is initially inthe high-inflation regime. Substituting (23) and (24) into (39) yields the differential equation forΩt under the high-inflation regime:Ω̇t= g − η(µ − γ − Ωt + Ω∗ )
Ωt

(40)

= η(Ωt − ΩH ),wheregΩH = Ω∗ + (µ − γ) − .
η(41)

Equation (40) is the differential equation for Ωt with a positive coefficient on Ωt . If there is adeviation of Ωt from ΩH , Ωt would explode to either ∞ or −∞. Therefore, when β = 0, onlyΩ̇t = 0 is feasible in equilibrium. Therefore, Ωt in equilibrium is constant at Ωt = ΩH .
The high-inflation regime under β = 0 is feasible only when g < η(µ − γ). To see this,
suppose that g ≥ η(µ − γ) in the high-inflation regime. In this case, (41) implies that ΩH ≤ Ω∗ .
Given ẏt /yt = g in the high-inflation regime, (21) and (22) imply that ΩH ≤ Ω∗ is rewritten asΩH = ρ + ẏt /yt + πt = ρ + g + πt < Ω∗ = ρ + g + γ and thus πt < γ. However, it violates theassumption of DNWR, πt > γ.
If Ωt is constant at ΩH , ẏt /yt , ṁt /mt , πt , and rt are all constant over time. In particular,(23)–(26) reduce to ẏt /yt = g, ṁt /mt = g/η, πt = µ − g/η, and rt = ρ + g. The economy achieves
the first-best allocation on the balanced growth path.
Output growth under the low-inflation regime

Substituting (27) and (28) into (39) yields:Ω̇t= g + (Ωt − Ω∗ ) − η(µ − γ)Ωt
= (Ωt − ΩL ),

31

(42)

whereΩL = Ω∗ + η(µ − γ) − g.

(43)

As in (40), (42) is the differential equation for Ωt with a positive coefficient on Ωt . Once again,
only Ω̇t = 0 is feasible in equilibrium. Therefore, Ωt in equilibrium is constant at Ωt = ΩL .
The low-inflation regime under β = 0 is feasible only when g ≥ η(µ − γ). To prove this, supposethat g < η(µ − γ) in the low-inflation regime. In this case, (43) implies that ΩL > Ω∗ . Given
πt = γ in the low-inflation regime, (21) and (22) imply that the condition ΩL > Ω∗ is rewritten asΩL = ρ + ẏt /yt + πt = ρ + ẏt /yt + γ > Ω∗ = ρ + g + γ and thus ẏt /yt > g. However, it is infeasible
because output growth cannot exceed g.
If Ωt is constant at ΩL , then ẏt /yt , ṁt /mt , πt , and rt are all constant over time. In particular,(27)–(30) reduce to ẏt /yt = η(µ − γ), ṁt /mt = µ − γ, πt = γ, and rt = ρ + η(µ − γ). The economy
achieves the inefficient allocation on the balanced growth path because output growth is η(µ − γ),
which is lower than g.
To summarize, there is no endogenous slowdown in output growth when β = 0. The parametersin the model fully determine the regime. If g < η(µ − γ), the growth rate of output is g and theeconomy remains in the high-inflation regime. Alternatively, if g ≥ η(µ − γ), the growth rate ofoutput is η(µ − γ) and the economy is always in the low-inflation regime starting from the initialperiod.

32

Figure 1: Real GDP per capita and trend

Notes: Each panel of the figure plots the estimated output trend for log real GDP per capita, along withthe actual data. The upper panel plots US data for 1990:Q1–2019:Q4, while the lower panel shows Japanesedata for 1980:Q1–2019:Q4. In each panel, the dotted line represents actual GDP. The dashed line is the
cubic trend of output. The dot-dashed line is the linear trend estimated from the subsample. We use theperiod over 1990:Q1–2007:Q1 for the US and the period over 1980:Q1–1991:Q1 for Japan as the subsample
for their linear trend. The linear trend after the last period of the subsample represents the projected values.

33

Figure 2: Simulation results for the US: Output, real interest rate, and inflation

Notes: Each panel of the figure compares the simulated data, estimated cubic trend, and actual data in the
US. The solid line represents the simulated data, the dashed line represents the estimated cubic trend, andthe dotted line represents the actual data. The upper panel is log real GDP, the middle panel is the realinterest rate, and the lower panel is (year-on-year) inflation. The dot-dashed line in the upper panel is thelinear trend estimated from the data over 1990:Q1–2007:Q1. For details of the data, see the main text.

34

Figure 3: Simulation results for Japan: Output, real interest rate, and inflation

Notes: Each panel of the figure compares the simulated data, estimated cubic trend, and actual data in Japan.
The dot-dashed line in the upper panel is the linear trend estimated from the data over 1980:Q1–1991:Q1.
For other details, see the notes in Figure 2.

35

Figure 4: Phase diagramߩ + ݃ + ߛ + (ߤ െ ߛ)ߚ‫ݕ‬௧ 

‫ݕ‬௧ = ݂ு (݉௧ ) ֞ ݉ሶ ௧ = 0
ߩ+݃+ߛ

ߚ

‫ݕ‬௧ = ்݂ (݉௧ )˳ߨ௧ = ߛ             ‫ݕ‬ሶ ௧ /‫ݕ‬௧ = ݃

ߩ+ߛ

ߚ

‫ݕ‬௧ = ݂௅ (݉௧ )˳‫ݕ‬ሶ ௧ = 0





݉௧ 

Notes: The red solid line denoted by yt = fH (mt ) represents the locus that achieves ṁt = 0 when theeconomy is in the high-inflation regime. This ṁt = 0 locus determines the direction of change in mt in thehigh-inflation regime. In this regime, ẏt > 0 always holds. The blue solid line denoted by yt = fL (mt ) is thelocus that achieves ẏt = 0 when the economy is in the low-inflation regime. This ẏt = 0 locus determines
the direction of change in yt in the low-inflation regime. In this regime, ṁt > 0 always holds. The blacksolid line denoted by yt = fT (mt ) points to the locus that achieves ẏt /yt = g at the lowest level of inflation
πt = γ. If (mt , yt ) is located above (below) the locus, the economy is in the high- (low-)inflation regime.

36