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

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§
May 10, 2022

Abstract
Previous studies have argued that output growth in advanced economies declined during
the Great Recession and remained low afterward. This paper proposes a model to explain thisslowdown in output growth. We incorporate wealth preferences and downward nominal wage
rigidity into a standard monetary growth model. Our model demonstrates that output initiallygrows at the same rate as productivity and slows endogenously in the transition path to thestagnation steady state. This stagnation is persistent even if productivity continues to grow ata steady rate. Applying our model to US data, we show that it successfully explains the declines
observed in the real interest rate, inflation, and the velocity of money, along with the slowdown
in output growth.
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 Matthias Schlegl for his helpful comments and Shuhei Kishimoto for providing excellent research assistance. The authors also gratefully acknowledge the financial support of the Joint Usage/ResearchCenters at the Institute of Social and Economic Research (ISER) and Kyoto Institute of Economic Research (KIER),
and Grants-in-Aid for Scientific Research (20H05631, 20H05633, 21H00700, and 21H04397). This paper is an extended version of Ono’s 2015 working paper entitled “Growth, secular stagnation, and wealth 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 advanced economies 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 prior to theGreat Recession, the actual output after the Great Recession failed to catch up with the expected
output trend.1 Figure 1 plots two estimated linear trends for log real GDP per capita using thedata from 1990:Q1 to 2019:Q4. The dashed line is the linear output trend under the assumptionthat there was a structural break starting in 2007:Q1 in the growth rate of real GDP per capita.
The dot-dashed line is the counterfactual linear output trend that removes the structural break.2
The two estimated output trends grow at the same rate of 1.93 percent up until 2007:Q1. However,after 2007:Q1, the growth rate of the output trend shown in the dashed line declines to 0.77 percent.
That is, there was a slowdown in output growth.3[Figure 1 about here.]This paper accounts for this slowdown in output growth using wealth preferences. What motivates our approach is the data on the velocity of money. Figure 2 shows that while the velocity of
M2 increased in the 1990s, it quickly decreased after the 2000s. In the standard monetary growthmodel (e.g., Sidrauski (1967)), velocity is often constant over time and thus difficult to reconcile
with the data. If we introduce wealth preferences into the standard monetary growth model, thestrong demand for money as part of wealth may slow the velocity of money. Furthermore, the literature on wealth preferences has found that strong wealth preferences could lead to an inherently
stagnant economy in the steady state.4 As such, we consider that the dynamics of velocity are notirrelevant to the observed slowdown in output growth.
[Figure 2 about here.]
1

Similar observations have been made for the Eurozone countries after the Great Recession and Japan after the
1997 financial crisis.
2
We regress the log of real GDP per capita (ln yt ) on a constant, time (t), and dt × (t − t∗ ), where dt is a dummy
variable that takes a value of one for t after 2007:Q1 and t∗ is the 2007:Q1 period. The estimation equation isln yt = a0 + a1 t + b1 dt (t − t∗ ) + εt , where εt is the error term. In plotting the dot-dashed line, we remove thestructural break by setting b1 at zero. The regressor of dt × (t − t∗ ) ensures that the two output trends are connected
at t =2007:Q1 with the adjustment of a constant term. The coefficient on this regressor b1 is estimated to be -0.0029
with a standard error of 0.0003 and statistically different from zero.
3
We confirmed a very similar pattern using quadratic, cubic, and quartic detrending. We also confirmed a similar
pattern when using the potential GDP series estimated by the Congressional Budget Office.
4
The earliest examples of these studies are Ono (1994) and Ono (2001), among others.

2

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 andSaez (2021), Hashimoto, Ono, and Schlegl (2021)), we introduce wealth preferences with a strictly
positive marginal utility in equilibrium. This preference assumption leads to a strong desire forsaving compared with the case without wealth preferences. The second component is downward
nominal wage rigidity (DNWR), which is widely discussed in recent studies.5 Together with wealthprefrences, this DNWR plays an important role for 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 stagnation steady state. In particular, we theoretically show that output
initially follows productivity that represents potential output in our model, but output later falls
below productivity. In our model, the household with wealth preferences accumulates wealth rather
than consuming enough to reach the potential output. The aggregate demand weakened by wealthpreferences leads to disinflation while the DNWR is not binding. However, once the DNWR
binds, inflation no longer decreases, the aggregate demand determines output, and output growthis endogenously determined. Then, the weakened aggregate demand falls short of the potential
output, the aggregate demand shortage occurs, and the growth rate of output is lower than that ofproductivity.
We conduct numerical simulations assuming that the slowdown in output growth takes place in2007:Q1. These numerical simulations show that the simulated output tracks the estimated output
trend with a structural break. More specifically, our model explains 88.8 percent of the decline
in output in 2015 and almost 100 percent of the decline in output in 2017. While our model isextremely simplified to derive our main results theoretically, it predicts the slowdown in US output
growth after the Great Recession remarkably well.
Our model can also predict the permanent decline in the real interest rate, which is closely
linked to the secular stagnation hypothesis.6 In the standard consumption Euler equation, thereal interest rate is determined by the household’s subjective discount rate and the growth rate of5

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.
6
Characterizations of secular stagnation appear in Baldwin and Teulings (2014) and Krugman (2014).

3

consumption. However, as pointed out by Michaillat and Saez (2021), wealth preferences in the
consumption Euler equation create discounting in the real interest rate. This discounting leads toa persistently low real interest rate, consistent with the data under secular stagnation.
In our model, inflation also declines until the output trend deviates from the productivity trend.
This observation is roughly consistent with inflation in the US before and after the Great Recession.
Hall (2011) points out that US inflation declined during the 1990s, but became stable even in thepresence of long-lasting slack in the economy from the Great Recession. Our model interprets themissing deflation as being a consequence of the binding DNWR, where inflation stops declining
even if aggregate demand falls short of aggregate supply.
Our model can also replicate the decline in the velocity of money in the 2000s shown in Figure
2. In our model, households have strong wealth preference so that they accumulate money rather
than increasing consumption. These strong preferences for money generate nominal money holdingsthat grow faster than does nominal aggregate demand. Consequently, velocity declines over time.
This prediction sharply contrasts with the standard monetary growth model, which predicts theconstant velocity of money along the balanced growth path.
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)). While this group considers the decline in productivity as the main driver ofsecular stagnation, our paper assumes a constant productivity growth rate to highlight the degree towhich wealth preferences alone can explain the observed slowdown in output growth. The second
group emphasizes the impact of demographic changes on saving in explaining the declining realinterest rate. (e.g., Carvalho, Ferrero, and Nechio (2016), Gagnon, Johannsen, and Lopez-Salido(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 saving. 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) integrate these three factors to replicate secularstagnation and discuss policy evaluation in their model.7
7

Ikeda and Kurozumi (2019) discuss monetary policy rules to prevent secular stagnation in a model with financial
frictions and endogenous total factor productivity growth.

4Our paper is categorized into the fourth group, which introduces wealth preferences into astandard macroeconomic model. This group assumes a strong desire for liquidity or wealth (e.g.,Michau (2018), Illing, Ono, and Schlegl (2018)).8 The study closest to ours is Michau (2018). He
incorporates wealth preferences and the DNWR into the standard neoclassical growth model andshows the existence of both the neoclassical and stagnation steady states in his model. By contrast,
our model has a unique steady state, and endogenously generates a regime change from efficient
allocation in the neoclassical economy to inefficient allocation in the stagnant economy.9
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 model

2.1

Setup

The representative household solves the following maximization problem:Z ∞e−ρt [ln ct + v(mt ) + β(at )] dt,(1)ȧt = rt (at − mt ) − πt mt + wt nt − ct + τt ,(2)

nt ≤ 1,

(3)

max
0

s.t.

where a0 is given. The notation is standard: ct is consumption, mt is real money balances (mt =Mt /Pt , where Mt is nominal money balances and Pt is the price level), and at is total real asset
holdings. In the budget constraint, rt is the real interest rate (at − mt = bt represents the illiquid
asset holdings of the household), πt is inflation or the opportunity cost of holding money, wt isreal wages, nt is labor supply, and τt are lump-sum transfers from government. The parameterρ > 0 is the household’s subjective discount rate. The budget constraint (2) suggests that the8
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)).
9
In this sense, our analysis also differs from Benigno and Fornaro (2018), who develop an endogenous growthmodel with downward nominal wage rigidities. They show that weak growth depresses aggregate demand and thatthe resulting aggregate demand shortage may lead to the stagnation steady state. In contrast to our study, stagnationarises as a self-fulfilling equilibrium.

5sources of consumption and saving (ȧt ) equal income from asset holdings (rt (at − mt ) − πt mt ),labor income (wt nt ), and the lump-sum transfers from government (τt ). In this maximization
problem, we simplify the labor supply and assume that nt ≤ 1. We also assume no capital in theeconomy, so at = mt holds for all t (i.e., bt = 0 in equilibrium).
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 ′ (at ) > 0, β ′′ (at ) ≤ 0,
and β ′ (at ) is strictly positive and constant in equilibrium. Except for β(at ), the utility functions
are standard and take a constant-relative-risk-aversion form. The utility from consumption is
given by ln ct , and the utility from real money balances satisfies v ′ (mt ) > 0, v ′′ (mt ) < 0, andlimmt →∞ v ′ (mt ) = 0.
The simplest specification that satisfies the above conditions for β(at ) is a linear function β(at ) =β × at , where β is a positive constant. The linearity assumption follows Michau (2018) and extends
Ono (1994) and Ono (2001), in which the household has an insatiable desire for liquidity. We
assume the linearity of β(at ) for simplicity, not for the necessity of our main results.
A necessary condition for our main results is that the marginal utility from wealth is strictly
positive and constant in the stagnation steady state. As argued by Michau (2018) and Michaillat
and Saez (2021), there are a variety of alternative specifications for the utility from wealth thatgenerates positive constant marginal utility. In these studies, while the concavity of the utility
function is ensured, the marginal utility from the wealth is constant in equilibrium.10 Because
these specifications lead to the same results, we employ linear utility for simplicity.11
10Michau (2018) and Hashimoto, Ono, and Schlegl (2021) 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 ′ (at − mst ) > 0,β ′′ (at − mst ) < 0, and limat →∞ β ′ (at − mst ) = 0, but β ′ (0) > 0. Thus, the marginal utility from the wealth is constantin equilibrium, where bt = at −mst = 0. Michaillat and Saez (2021) allow for the utility from relative wealth at (i)− ãt .
Here, at (i) denotes the wealth at the individual household level and ãt is the average wealth in the economy, and thehousehold takes ãt as given. They assume that β ′ (at (i) − ãt ) > 0, β ′′ (at (i) − ãt ) < 0 and limat (i)→∞ β ′ (at (i) − ãt ) = 0,
but again β ′ (0) > 0 where at (i) = ãt .
11
The constant marginal utility results in equilibrium money holdings beyond the amount the consumers use for
their transactions. Nevertheless, constant marginal utility is not necessarily inconsistent with the neuroscientificevidence. 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)

6

The first-order conditions arec−1= λt ,tv ′ (mt ) = (rt + πt )λt ,λ̇t = (ρ − rt )λt − β ′ (at ),where λ is the Lagrange multiplier for the budget constraint. The transversality condition is given
by limt→∞ e−ρt λt at = 0.
We rewrite the above equations asv ′ (mt )ct = rt + πt ,

(4)

ċt
= rt − ρ + β ′ (at )ct .
ct(5)

In (4), the household pays the opportunity cost of holding money, rt + πt , to receive the marginal
benefits v ′ (mt ) (or v ′ (mt )ct when measured by the unit of consumption goods). In (5), the householdpays the marginal cost of saving, ρ + ċt /ct . This is the household’s consumption discount rate thatallows for the household’s risk aversion.12 Regarding the marginal benefits of saving, the household
receives market returns on illiquid assets rt and the marginal utility from wealth β ′ (at ) (or β ′ (at )ct
when measured by the unit of consumption goods). When the wealth preferences are absent, (5)reduces to the standard Euler equation ċt /ct = rt − ρ and saving only yields market returns rt .
When the wealth preferences are present, however, saving generates additional benefits of β ′ (at )ct .
Thus, the household would give up more consumption and accept a lower interest to enjoy holding
more wealth.
Eliminating rt from (4) and (5) and allowing for bt = 0 in equilibrium yield the condition forthe substitution between consumption and liquidity.

Ω(mt , ct ) = ρ +
where

ċt
+ πt ,ct(6)Ω(mt , ct ) ≡ [v ′ (mt ) + β ′ (mt )]ct .

12

The consumption discount rate is, in general, represented by the sum of the steady-state discount rate and thegrowth rate of marginal utility. In equation, it is given by ρ − [du′ (ct )/dt][1/u′ (ct )], where u(ct ) is the (instantaneous)utility from consumption. Under our assumption that u(ct ) = ln ct , we obtain ρ − [du′ (ct )/dt][1/u′ (ct )] = ρ + ċt /ct .

7

Here Ω(mt , ct ) denotes the marginal benefits of holding real money balances (measured by theunit of consumption goods). In Ω(mt , ct ), v ′ (mt ) is benefits from increasing money and β ′ (mt ) is
an additional benefit of increasing wealth as a whole. The right-hand side of (6) represents theopportunity cost of holding money. To hold an additional unit of the liquid asset, the household
has to give up consumption goods by an amount equal to the household’s consumption discountrate (ρ + ċ/ct ) and the inflation rate (πt ).
There is a representative firm in the competitive market in the economy. In our model, thefirm’s technology is linear:yt = θt n t ,(7)where yt is output and θt is productivity. With this production function, the firm’s labor demandcondition is
wt = θ t .

(8)Critical assumptions in our model are that nominal wage inflation Ẇt /Wt cannot be lower thanthe lower bound γ (i.e., the DNWR), and that θt increases at g > 0.13 Real wage inflation isg from (8) and thus nominal wage inflation is πt + g. As a result, the DNWR is expressed as
Ẇt /Wt = πt + g ≥ γ, or πt ≥ γ − g, where γ − g is the lowest level of price inflation.
The government has a budget constraint τt = µmst , where mst = Mts /Pt and Mts is the nominal
money supply. Throughout the paper, we assume that the money growth rate is strictly positive(µ > 0) and sufficiently high:µ > γ − g,(9)which means that the money growth rate always exceeds the lowest level of inflation.
We rewrite the assumption for the DNWR as the complementarity slackness condition in thelabor market:(πt + g − γ)(1 − nt ) = 0.

(10)

13
We assume that θt grows at an exogenous rate of g, but with a sufficiently large natural limit θ̄. We require thisassumption for a technical reason. Later, we show that aggregate demand converges to a constant level. Unless we
impose a natural limit on θt , labor demand decreases with productivity and will eventually disappear. To circumvent
this, we assume that θ̄ is at least as large as (ρ + γ)/β. We further assume that the household does not know when
productivity hits θ̄ in the future. In our model, these assumptions ensure that the representative household makes
all decisions under the prediction that θt eternally grows at g.

8

The economy has two regimes depending on the complementary slackness condition. If nt = 1,
the economy is in the state of full employment and πt > γ − g. We refer to this regime as thehigh-inflation regime. Alternatively, if nt < 1, there is unemployment in the labor market with thebinding DNWR and πt = γ − g. We refer to this regime as the low-inflation regime.
The market-clearing conditions are
1. Goods market ct = yt = θt nt ,2. Labor market (πt + g − γ)(1 − nt ) ≥ 0,
3. Money market mt = mst ,4. Bond market bt = 0.
A competitive equilibrium of the model is the set of allocations {ct , yt , nt , 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 andmoney supply are specified as above; and iv) All markets clear except for labor market. The labormarket-clearing condition depends on the complementary slackness condition (10).

2.2High-inflation regime

The high-inflation regime is characterized by full employment nt = 1 and high inflation πt > γ − gwhere the DNWR is not binding. We summarize the equilibrium conditions as follows:
ẏt
yt
ṁt
mt

= g,

(11)

= µ − Ω(mt , yt ) + ρ + g,

(12)πt = Ω(mt , yt ) − (ρ + g),(13)

rt = ρ + g − β ′ (mt )yt .

(14)

To derive (11), recall that the production function (7) and goods market-clearing condition imply
ct = yt = θt . Here, yt = θt means that the growth rate of output is g. Next, (5) leads to (14),where we use ẏt /yt = g and at = mt in equilibrium. It is straightforward to show (13) from (6).
By assumption, inflation is higher than γ − g in this regime. Finally, (12) is obtained from the9definition of mt = Mt /Pt , ṁt /mt = µ − πt . Given the initial value θ0 , we obtain the output levelyt = θt . Moreover, we can numerically solve (12) for mt given m0 .
It is convenient to define the threshold value of Ω(mt , yt ) in which output grows at the rate ofg but inflation is as low as γ − g. Substituting ẏt /yt = g and πt = γ − g into (6) gives the thresholdvalue of Ω(mt , yt ):Ω∗ = ρ + γ.

(15)

We use the threshold Ω∗ to evaluate the allocation in the low-inflation regime.

2.3

Low-inflation regime

The low-inflation regime is characterized by unemployment nt < 1 and the lower bound of inflation
πt = γ − g where the DNWR is binding. We summarize the equilibrium conditions as follows:
ẏt
yt
ṁt
mt

= g − [(ρ + γ) − Ω(mt , yt )] ,(16)

= µ − (γ − g),(17)πt = γ − g,(18)

rt = v ′ (mt )yt − (γ − g).

(19)

Our assumption corresponds to (18). Equation (17) immediately follows from (18) because ṁt /mt =µ − πt . To obtain the expression for output growth (16), we use (6) and the equilibrium conditionct = yt . The real interest rate rt can be obtained from (4) together with πt = γ − g.
To compare the system of equations between the high- and low-inflation regimes, we rewrite(16) – (19) as
ẏt
yt
ṁt
mt

= g − [Ω∗ − Ω(mt , yt )] ,(20)

= µ − Ω∗ + ρ + g,(21)

πt = Ω∗ − (ρ + g),(22)

rt = ρ + g − β ′ (mt )yt − [Ω∗ − Ω(mt , yt )] ,(23)

where we use the definitions of Ω(mt , yt ) and Ω∗ shown in (6) and (15), respectively.
10Comparisons between the two regimes reveal that there is additional downward pressure on
output growth and the real interest rate when the DNWR binds. Equation (20) indicates that
output growth can be lower than g depending on the sign of Ω∗ − Ω(mt , yt ). In particular, it
indicates that output growth is lower than or equal to g if Ω(mt , yt ) ≤ Ω∗ . Similar interpretations
can apply to the real interest rate from a comparison between (14) and (23).

3

The model dynamics

In this section, we specify the functional forms of v(mt ) and β(at ) and investigate the model
dynamics. We first show that the stagnation steady state exists in the presence of the wealthpreferences. We then demonstrate that output initially grows at the same rate as productivity
but slows in the transition path to this stagnation steady state. This slowdown in output growthendogenously occurs even if productivity continues to grow at a steady rate. Finally, we also showthat the model without wealth preferences fails to generate the slowdown in output growth.
In what follows, we assume that the functions v(mt ) and β(at ) are given byv(mt ) = v

mt1−η
,
1−η

β(at ) = βat ,v > 0,β ≥ 0,

η > 0,(24)(25)

where, with a slight abuse of notations, v > 0 and β ≥ 0 represent a parameter for the functionsv(mt ) and β(at ).

3.1

The stagnation steady state

We characterize the steady state as a preparatory step for analyzing the model dynamics. The
steady state in our model is the “stagnation” steady state in which the DNWR binds and output
converges to a constant value. We show that the stagnation steady state can arise under our
preference assumptions (24) and (25).
Suppose that nominal wage inflation hits the lower bound γ at t = t∗ . Equation (17) implies

11that mt goes to ∞ as t → ∞ because of (9). In particular,
mt = mt∗ exp[(µ − γ + g)(t − t∗ )] ⇒ lim mt = ∞,
t→∞

(26)

where mt∗ is the real money balances in the period from which the DNWR starts binding. Then,in the stagnation steady state where yt converges to a constant value, we havelim Ω(mt , yt ) = lim [β ′ (mt ) + v ′ (mt )]yt = βy ss ,

t→∞

t→∞

where a superscript ss on a variable denotes the stagnation steady-state value.
To prove the existence of the stagnation steady state, we use the transversality condition:
limt→∞ λt at exp(−ρt) = limt→∞ λt mt exp(−ρt) = 0. Taking the time derivative of this conditiontranslates the condition into λ̇t /λt + ṁt /mt − ρ < 0. In the stagnation steady state, λ̇t /λt =−ċt /ct = 0 and ṁt /mt = µ − (γ − g). Thus, µ − (γ − g) < ρ ensures the transversality condition.
Note that as long as the assumption of µ > 0 holds, the transversality condition implies thatg < ρ + γ.
Now, the allocations in the steady state are characterized as follows. Regarding output, (6)becomes βy ss = ρ + γ − g because limt→∞ Ω(mt , yt ) = βy ss , ẏt /yt = 0, and πt = γ − g. The
steady-state output is given byy ss =

ρ+γ−g
,
β

(27)

which is strictly positive because g < ρ+γ. Because of the DNWR, we have π ss = γ −g. The nominal interest rate in this steady state is zero because of (4): limt→∞ (rt + πt ) = limt→∞ [v ′ (mt )yt ] = 0.
From this equation, the real interest rate is given by rss = −π ss = g − γ. Clearly, ẏt /yt = 0 andṁt /mt = µ − (γ − g) > 0 from (9).

3.2Transition dynamics to the stagnation steady state

We are now ready to discuss the transition path to the stagnation steady state. The following
proposition summarizes our main results.
Proposition 1. Suppose that productivity grows at a strictly positive growth rate (g > 0) and thatthe money growth rate exceeds the lower bound of inflation (µ > γ − g) and is strictly positive12

(µ > 0). Then, under the preference assumptions specified by (24) and (25) with a strictly positiveβ, we have a unique dynamic path of output toward the stagnation steady state. Output growth isendogenously determined. In particular, let t∗ be the period of time from which the DNWR binds.
Then,• For 0 < t < t∗ , the growth rate of output is equal to that of productivity.
• For t = t∗ , the growth rate of output starts declining.
• For t > t∗ , the growth rate of output is lower than that of productivity.
• As t → ∞, the growth rate of output converges to zero.
Here, we assume that the initial productivity θ0 is such that π0 = Ω(m0 , θ0 ) − (ρ + g) > Ω∗ − (ρ + g).
To ensure the transversality condition, we assume that µ − (γ − g) < ρ.
Proof. See Appendix A.1.
The model endogenously generates the slowdown in output growth. The economy experiencesa regime change from the high- to the low-inflation regime. Given that π0 > Ω∗ − (ρ + g), output
initially equals productivity, or equivalently, the potential level (i.e., yt = θt ). Thus, output grows
at the same rate as productivity. A steady growth of output persists when the DNWR is notbinding (i.e., πt > γ − g). However, once the DNWR binds at t ≥ t∗ , the growth rate becomes
lower than g. Output growth then declines over time and converges to zero in the stagnation steadystate.
To understand the model dynamics, we focus on Ω(mt , yt ) given by (6). Using the marketclearing conditions ct = yt in equilibrium, we haveΩ(mt , yt ) = ρ +ẏt+ πt .
yt(28)

Equation (28) is the condition for substitution between consumption and saving, where the saving
is in the form of money. Under our specification, Ω(mt , yt ) = (β + vm−ηt )yt is decreasing in mt andincreasing in yt . That is, there are two offsetting effects on Ω(mt , yt ). In the growing economy, both
mt and yt grow over time so that the dynamics of Ω(mt , yt ) are not clear analytically. Numerically,13

however, the marginal benefits of holding real money balances Ω(mt , yt ) decreases over time because
the effect of mt on Ω(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 consumption.
Now, given the declining marginal benefits of holding money, the marginal cost represented
by the right-hand side of (28) must decline in equilibrium. As (28) shows, given a constant ρ,either output growth (ẏt /yt ) or inflation (πt ) must decline. If the economy is in the high-inflation
regime, output growth can be kept at ẏt /yt = g. This is because decreased inflation can lower themarginal cost of holding money. By contrast, inflation can no longer decrease if the economy isin a low-inflation regime. Only through the slowdown in output growth can the marginal cost ofholding money decrease.
We emphasize that the aggregate demand shortage drives this slowdown in output growth. In
our model, the household’s wealth preferences weaken the aggregate demand growth by substituting
wealth for consumption goods. Initially, the weakened aggregate demand leads to disinflation since
the DNWR is not binding. Because prices are flexible, the economy can achieve the efficient level of
output. However, when the DNWR makes price adjustment rigid, the weakened aggregate demand
determines equilibrium output. Moreover, the equilibrium output is inefficient in terms of welfare.
Therefore, the growth rate of output is g for t < t∗ and becomes lower than g for t ≥ t∗ .
Some remarks on the transition path are in order. First, all variables including ẏt /yt andṁt /mt smoothly switch from a path described from (11)–(14) to a path specified by (20)–(23)without a jump. We can confirm this by evaluating Ω(mt , yt ) at Ω∗ . When the marginal benefitof holding money is at the threshold level, (20)–(23) at t = t∗ all reduce to (11)–(14), suggestingsmooth transitions from the high- to the low-inflation regime. Second, output growth also exhibits
a smooth transition to the stagnation steady state. Since Ω(mt , yt ) in (16) smoothly converges to
its limit of βy ss , output growth smoothly converges to zero as t → ∞:

lim

ẏt

t→∞ yt= lim [g + Ω(mt , yt ) − (ρ + γ)] = g + βy ss − ρ − γ = 0t→∞where the last equality comes from (27).

14

(29)

3.3

The role of wealth preferences

The results of Proposition 1 critically depend on the household’s wealth preferences with a strictly
positive β. In this subsection, we show that if there are no wealth preferences (i.e., β = 0), themodel fails to generate the endogenous slowdown in output growth, in contrast to Proposition 1.
The following proposition summarizes our results under β = 0.
Proposition 2. Suppose that productivity grows at a strictly positive growth rate (g > 0) and thatthe money growth rate exceeds the lower bound of inflation (µ > γ − g). Then, under the preference
assumptions specified by (24) and (25) but with β = 0, we have a unique dynamic path of output
growth such that
• For any t > 0, the growth rate of output is constant.
• The growth rate of output is equal to min{g, η[µ − (γ − g)]}.
The growth rate of output is g when the DNWR does not bind or η[µ − (γ − g)] when the DNWR
binds. The transversality condition is given by (1 − η) min{g, η[µ − (γ − g)]} < ρ.
Proof. See the Appendix A.2.
The proposition suggests that the model without wealth preferences (β = 0) fails to generate an
endogenous slowdown in output growth. Instead, parameters in the model predetermine the regime
in equilibrium. For example, if the economy is initially in a high-inflation regime, the growth rate
moves in tandem with productivity. Output always equals its potential level and the allocation isefficient. If the economy instead starts from a low-inflation regime, the economy experiences low
economic growth given the rigid wage adjustment arising from the DNWR. In this case, output isalways below its potential level and the allocation is inefficient. The aggregate demand shortage
make economic growth slower than g. In either case, however, we do not observe a slowdown in
output growth, in contrast to the case of β > 0. Moreover, the growth rate of output with thebinding DNWR does not converge to zero. Even if the DNWR is binding, there is no stagnation
steady state under β = 0. Therefore, the essential ingredient for generating slowdown in output
growth and stagnation steady state is a strictly positive β in the wealth preferences.14
14

A natural question is what happens if nominal wages are fully flexible, but β is strictly positive. As (28) suggests,
πt may continue to decline without the slowdown in output growth. However, it can be shown that there is nomonetary equilibrium with a strictly positive β. For details, see Ono (2001).

15

Our analysis suggests that the model with wealth preferences is qualitatively consistent withthe observations in advanced economies. However, it is unclear whether the model numericallyexplains the data. Moreover, it is also worth assessing macroeconomic variables other than output
growth. The following section assesses the model with wealth preferences based on output growth,
the real interest rate, inflation, and the velocity of money.

4

Simulating the model

4.1

Calibrations

This section simulates the US output trend and sees whether our model can replicate the slowdown
in US output growth after the Great Recession. In the debate on secular stagnation, it was expectedthat output growth would remain high after the Great Recession. Given these expectations, wecalibrate g at the growth rate of output before 2007:Q1, g = 0.019.15 We set µ = 0.036 based on
the mean growth rate of M2 stock (per capita) over 1990:Q1–2007:Q1. Regarding the subjective
discount rate ρ, we parameterize ρ = 0.04. As for the lower bound of nominal wage inflation, we set
γ at a slightly larger value than that in previous studies (e.g., Schmitt-Grohé and Uribe (2016)).
We set γ at 0.019.16
To pin down v, η, and β in the utility functions v(mt ) and β(at ), we target the moments of thereal interest rate and the velocity of money between 1990:Q1 and 2007:Q1. For the real interest
rate, we use the 10-year real Treasury yields from the database of the Cleveland Federal Reserve
Bank and estimated by Haubrich, Pennacchi, and Ritchken (2012).17 We set our target for the realinterest rate at 2.80 percent, which is the mean over the period 1990:Q1–2007:Q1. For the velocity
of money, which is yt /mt in our model, we target the mean velocity of M2 (2.03) and the mean
growth rate (0.45 percent) over 1990:Q1–2007:Q1. The resulting calibrated values are v = 0.45,
15

We estimated g from a regression of log output discussed in footnote 2. Using the regression, we set g at 4 × â1
where â1 is the estimated coefficient on a linear time trend.
16
The value of γ = 0.019 appears large because the actual nominal wage inflation temporarily falls short of 0.019.
In the data, the index of compensation per hour in the nonfarm business sector declined by 10.5 percent in 2009:Q1.
Nevertheless, we calibrate γ at 0.019 for the following reasons. First, we are calibrating the lower bound of the longrun nominal wage inflation that excludes the impact of temporary shocks. For the above compensation per hour, themean wage inflation rate over 2007:Q2–2019:Q4 is higher than two percent, which exceeds our calibration value ofthe lower bound. Second, the parameter value of γ = 0.019 satisfies the parameter restriction, 0 < µ − (γ − g) < ρ,
where the latter points to the transversality condition. If γ = −0.105, it violates the transversality condition.
17
The most recent data are available at https://www.clevelandfed.org/our-research/indicators-and-data/
inflation-expectations/background-and-resources.aspx#research.

16η = 5.84, and β = 0.01, respectively.
While the data are discrete in time, our simulated data are continuous. We assume that theregime change takes place after 2007:Q1, that is, t∗ = 2007:Q1. This timing is matched with ourestimated output trend with a structural break because the deviation between the two estimatedoutput trends starts at 2007:Q1.
We perform the numerical simulation as follows. First, we guess at the output level in the
period of regime change (denoted by yt∗ ). Second, given the guess of output at t = t∗ , we computethe transition path backward from t∗ = 2007:Q1 to t = 1990:Q1. In this transition path for t < t∗ ,output growth is equal to productivity growth. Third, we calculate the transition path forwardfrom t∗ = 2007:Q1 to t = t̄, where t̄ is a sufficiently large to approximate t = ∞. In this transition
path, the economy moves toward the stagnation steady state. We compute the output at t = t̄(denoted by yt̄ ) and compare it with the steady state output y ss . Finally, 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 .

4.2

Simulation results

Figure 3 plots the simulated output trend predicted by the model (solid line) along with the two
estimated output trends that replicate those in Figure 1. The initial value of the simulated output
trend is the value in 1990:Q1 and is normalized by the value of the estimated output trend datain the same period. Because we directly calibrate g at 0.019 from the estimated output trend, thisnormalization makes the simulated output trend match perfectly with the estimated output trend
until 2007:Q1. Thus, the figure particularly depicts the simulated and estimated output trends
from 2000:Q1 to focus on the output trends after 2007:Q1.
The simulated output trend in Figure 3 indicates the slowdown in output growth after the GreatRecession, as the theory predicts. Comparing the solid and dashed lines, we see that the model canaccount for a substantial fraction of the slowdown in output growth. For example, the model can
explain 88.8 percent of the total decline in output in 2015.18 The dashed and solid lines intersect
18

Based on Figure 3, the estimated counterfactual output trend that removes a structural break is 48.3 percent
higher in 2015:Q1 than in 1990:Q1. The estimated output trend with the structural break is 39.4 percent higher
in 2015:Q1 than in 1990:Q1. Thus, the output declines are 8.9 percentage points between the trend lines with andwithout the structural break. Next, the model’s simulation results suggest that the simulated output trend is 40.4
percent higher in 2015:Q1 than in 1990:Q1, so output declines 7.9 percentage points. Thus, the model can explain17

in 2017. Thus, the model can almost entirely explain the total output decline in 2017.
[Figure 3 about here.]
The model can also successfully replicate the real interest rate that has decreased over timesince the 1990s. Figure 4 shows the real interest rate (dotted line), the cubic trend componentof the real interest rate (dashed line), and the simulated real interest rate predicted by the model(solid line). As suggested by the trend (dashed) line, the trend real interest rate declines over time.
In particular, the decline in the trend real interest rate is 3.16 percentage points lower, from 3.82
percent in 1990:Q1 to 0.66 percent in 2019:Q1. The simulated real interest rate also decreases
over time. As discussed in the previous section, there is additional downward pressure on the realinterest rate represented by Ω∗ − Ω(mt , yt ) when the DNWR binds (See (23)). This downward
pressure arises from the household’s insatiable strong desire for wealth. As a result, the simulated
real interest rate was 3.31 percent in 1990:Q1 and 0.19 percent in 2019:Q1. The magnitude ofthe decline in the simulated real interest rate is 3.12 percentage points, which is close to the 3.16
percentage point decline in the trend real interest rate.
[Figure 4 about here.]
Figure 5 plots inflation. As before, the solid, dotted, and dashed lines are the simulated,
actual, and cubic trend inflation, respectively. Here, we calculate inflation using the price index
of Personal Consumption Expenditure, excluding foods and energy.19 The figure indicates that
estimated trend inflation decreased steadily by 2003:Q2 and did not exhibit large declines after
2003:Q2. In particular, the dashed line shows that the trend inflation after 2003:Q2 is stable
at around 1.5–2 percent. The simulated inflation stops decreasing after 2007:Q1 and is equal
to a constant value γ − g (See (22)). Overall, the simulated trend inflation is lower than theestimated trend inflation, especially after 2007:Q1, and the timing in which the simulated trend
inflation becomes stable is slightly delayed. Nevertheless, the model replicates the magnitude of
disinflation well. In particular, while the estimated trend inflation decreases by 2.35 percent (from3.84 percent in 1990:Q1 to 1.49 percent in 2019:Q1), the simulated inflation decreases by 3.20
percent (from 3.20 percent in 1990:Q1 to zero percent in 2019:Q1).20 Clearly, the decline in the88.8 percent of the total decline (0.079/0.089 = 0.888).
19
We use year-on-year inflation to remove the noise in inflation.
20
The simulated inflation after 2007:Q1 is πt = γ − g and becomes zero because we assume γ = g.

18

simulated inflation results from the household’s wealth preferences and the resulting weak aggregate
demand for consumption goods.
[Figure 5 about here.]
We finally assess the velocity of money. We have seen that the trend velocity yt /mt eventually
decreases with time because wealth preferences make real money balances mt grow faster than
output yt . Here, we evaluate how fast the velocity of money decreases. Figure 6 compares thesimulated velocity (solid line) with the actual velocity (dotted line) and its trend (dashed line).
Recall that our calibration target was the level and growth rate of trend velocity between 1990:Q1
and 2007:Q1. We thus assess the performance of the model using trend velocity after 2007:Q1.
The estimated trend velocity decreases by 27.6 percent from 2007:Q1 to 2019:Q4 and the simulated
trend velocity declines by 35.8 percent during the same period.21 Therefore, our model explains
the trend velocity quantitatively well.
[Figure 6 about here.]
Before closing this section, we provide some numerical examples of the model without wealthpreferences. Proposition 2 implies that a high-inflation regime is chosen as long as g ≤ η[µ−(γ −g)].
Using the parameter values in the previous section, we translate the condition as η ≥ g/[µ−(γ−g)] =0.019/0.036 = 0.528. We thus reassign two parameter values to η: η = 1 and η = 1/2. The former
achieves the high-inflation regime, and the latter leads to the low-inflation regime.
If η = 1, the economy is in the high-inflation regime. It immediately follows from Proposition 2and (14) that ẏt /yt = 0.019 and rt = ρ + g = 0.059. These results indicate that both output growth
and the real interest rate are constant. Furthermore, Ω(mt , yt ) = v ′ (mt )yt = vyt /mt under β = 0and η = 1. Appendix A.2 proves that Ω(mt , yt ) is constant in equilibrium when β = 0. Therefore,ṁt /mt = ẏt /yt = 0.019 and πt = µ − g/η = 0.017. The model fails to explain the disinflation
observed in the 1990s and early 2000s.
If η = 1/2, the low-inflation regime is chosen. In this case, the DNWR binds, and output growth
is lower than g. More specifically, Proposition 2 implies that ẏt /yt = η[µ − (γ − g)] = 0.5 × 0.036 =21
The estimated trend velocity is 0.017 in 2007:Q1 and -0.259 in 2019:Q1 in logarithms, respectively. Thus, thepercentage change in trend velocity is −27.6(= −25.9 − 1.7) percent. Likewise, the simulated trend velocity in
2007:Q1 and 2019:Q1 is 0.092 and -0.266 in logarithms, respectively. Thus, the percentage change in the simulated
trend velocity is −35.8(= −26.6 − 9.2) percent.

19

0.018. From (17) and (18), we have ṁt /mt = µ − (γ − g) = 0.036 and πt = γ − g = 0, respectively.
Finally, the Euler equation (5) reduces to its standard form under β = 0: rt = ρ + ẏt /yt , so
rt = ρ + η[µ − (γ − g)] = 0.058. While inflation is quite low, output growth, the real interest rate,and inflation do not decrease over time.
The velocity of money declines over time if η < 1. This is because of the constancy of Ω(mt , yt ) =vyt m−ηt under β = 0 (See Appendix A.2). Given that Ω(mt , yt ) is constant in equilibrium, we haveẏt /yt = η ṁt /mt . For the velocity of money to decrease, real money balances grow faster than does
output. This is possible only if η < 1.

5

Conclusion

Output growth in advanced economies was persistently low after the Great Recession. In this paper,
we explained this slowdown in output growth by introducing wealth preferences into a standard
monetary growth model. We theoretically showed that our model generates a slowdown in output
growth in the transition path to the stagnation steady state. Using numerical simulations, we also
reported that our model explains 88.8 percent of the decline in output trend in 2015 and nearly 100
percent of the decline in output trend in 2017. In addition to output growth, the model successfully
reproduces the magnitude of the decreases in the real interest rate. Our model also explains thedeclines in inflation and the velocity of money in the sample period.
It is quite surprising that a simple model can account for the long-run patterns in the data.
Further work would enrich our understanding of secular stagnation and its policy prescriptions.
Many questions are left open. 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 of theforward guidance on output growth? Exploring these directions would be important for future
research.

References
Baldwin, Richard and Coen Teulings. 2014. “Introduction.” In Secular Stagnation: Facts, Causes,
and Cures, edited by Coen Teulings and Richard Baldwin, chap. 1. CEPR Press, 1–23.
20

Barattieri, Alessandro, Susanto Basu, and Peter Gottschalk. 2014. “Some evidence on the importance of sticky wages.” American Economic Journal: Macroeconomics 6 (1):70–101.
Benigno, Gianluca and Luca Fornaro. 2018. “Stagnation traps.” The Review of Economic Studies
85 (3):1425–1470.
Camerer, Colin, George Loewenstein, and Drazen Prelec. 2005. “Neuroeconomics: How neuroscience can inform economics.” Journal of Economic Literature 43 (1):9–64.
Carvalho, Carlos, Andrea Ferrero, and Fernanda Nechio. 2016. “Demographics and real interest
rates: Inspecting the mechanism.” European Economic Review 88:208–226.
Eggertsson, Gauti B and Paul Krugman. 2012. “Debt, deleveraging, and the liquidity trap: A
Fisher-Minsky-Koo approach.” The Quarterly Journal of Economics 127 (3):1469–1513.
Eggertsson, Gauti B, Neil R Mehrotra, and Jacob A Robbins. 2019. “A model of secular stagnation:
Theory and quantitative evaluation.” American Economic Journal: Macroeconomics 11 (1):1–48.
Fallick, Bruce, Michael Lettau, and William Wascher. 2016. “Downward nominal wage rigidity in the United States during and after the Great Recession.”

Available at SSRN:

https://ssrn.com/abstract=2725019.
Fernald, John G. 2015. “Productivity and potential output before, during, and after the GreatRecession.” NBER Macroeconomics Annual 29 (1):1–51.
Gagnon, Etienne, Benjamin K Johannsen, and David Lopez-Salido. 2021. “Understanding the newnormal: The role of demographics.” IMF Economic Review 69 (2):357–390.
Gordon, Robert J. 2015. “Secular stagnation: A supply-side view.” AEA Paper & Proceedings
105 (5):54–59.
Grigsby, John, Erik Hurst, and Ahu Yildirmaz. 2021. “Aggregate nominal wage adjustments: New
evidence from administrative payroll data.” American Economic Review 111 (2):428–71.
Guerrieri, Veronica and Guido Lorenzoni. 2017. “Credit crises, precautionary savings, and theliquidity trap.” The Quarterly Journal of Economics 132 (3):1427–1467.

21

Hall, Robert E. 2011. “The long slump.” American Economic Review 101 (2):431–469.
Hashimoto, Ken-ichi, Yoshiyasu Ono, and Matthias Schlegl. 2021. “Structural unemployment,
underemployment, and secular stagnation.” ISER Discussion Papers No. 1088.
Haubrich, Joseph, George Pennacchi, and Peter Ritchken. 2012. “Inflation expectations, real rates,
and risk premia: Evidence from inflation swaps.” The Review of Financial Studies 25 (5):1588–
1629.
Hazell, Jonathon and Bledi Taska. 2020. “Downward rigidity in the wage for new hires.” Available
at SSRN: https://ssrn.com/abstract=3728939.
Ikeda, Daisuke and Takushi Kurozumi. 2019. “Slow post-financial crisis recovery and monetary
policy.” American Economic Journal: Macroeconomics 11 (4):82–112.
Illing, Gerhard, Yoshiyasu Ono, and Matthias Schlegl. 2018. “Credit booms, debt overhang andsecular stagnation.” European Economic Review 108:78–104.
Jones, Callum. 2022. “Aging, secular stagnation and the business cycle.” The Review of Economics
and Statistics forthcoming.
Krugman, Paul. 2014. “Four observations on secular stagnation.” In Secular Stagnation: Facts,
Causes, and Cures, edited by Coen Teulings and Richard Baldwin, chap. 1. CEPR Press, 61–68.
Mian, Atif and Amir Sufi. 2014. “What explains the 2007–2009 drop in employment?” Econometrica 82 (6):2197–2223.
Michaillat, Pascal and Emmanuel Saez. 2021. “Resolving New Keynesian anomalies with wealth in
the utility function.” The Review of Economics and Statistics 103 (2):197–215.
———. 2022. “An economical business-cycle model.” Oxford Economic Papers 74 (2):382–411.
Michau, Jean-Baptiste. 2018. “Secular stagnation: Theory and remedies.” Journal of Economic
Theory 176:552–618.
Ono, Yoshiyasu. 1994. Money, Interest, and Stagnation: Dynamic Theory and Keynes’s Economics.
Oxford University Press.
22

———. 2001. “A reinterpretation of chapter 17 of Keynes’s general theory: Effective demand
shortage under dynamic optimization.” International Economic Review 42 (1):207–236.
Schmitt-Grohé, Stephanie and Martin Uribe. 2016. “Downward nominal wage rigidity, currency
pegs, and involuntary unemployment.” Journal of Political Economy 124 (5):1466–1514.
Sidrauski, Miguel. 1967. “Rational choice and patterns of growth in a monetary economy.” American Economic Review 57 (2):534–544.
Sigurdsson, Jósef and Rannveig Sigurdardottir. 2016. “Time-dependent or state-dependent wagesetting? Evidence from periods of macroeconomic instability.” Journal of Monetary Economics
78:50–66.
Summers, Lawrence H. 2014. “U.S. economic prospects: Secular stagnation, hysteresis, and thezero lower bound.” Business Economics 49 (2):65–73.
Takahashi, Yuta and Naoki Takayama. 2022. “Tech-driven secular stagnation: Cross-country evidence.” Unpublished manuscript.

A

Proofs of Propositions

A.1

Proof of Proposition 1

To prove Proposition 1, the phase diagram in the (mt , yt ) plane is convenient. We derive the lociin each inflation regime in Figure 7.
[Figure 7 about here.]
We first consider the high-inflation regime. Equation (11) implies that ẏt > 0 for any t < t∗ .
We thus focus on the ṁt = 0 locus. The ṁt = 0 locus is obtained from (12) and given byµ − Ω(mt , yt ) + ρ + g = 0. Under our preference assumptions, Ω(mt , yt ) = (β + vm−ηt )yt . Therefore,

23the ṁt = 0 locus can be rewritten as
ρ+g+µ
β + vm−ηtρ + γ + [µ − (γ − g)]=β + vm−η
t

yt =

(30)

= fH (mt ),where we define the ṁt = 0 locus by yt = fH (mt ). In Figure 7, the locus is drawn as the red solidline. Here mt 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 (9), (17) implies that ṁt > 0
for any t > t∗ . We thus focus on the ẏt = 0 locus. The ẏt = 0 locus is obtained from (16) and given
by g − (ρ + γ) + Ω(mt , yt ) = 0. Again, noting that Ω(mt , yt ) = (β + vm−ηt )yt , the ẏt = 0 locus canbe rewritten as
yt =

ρ+γ−g
β + vm−ηt(31)= fL (mt ),where we define the ẏt = 0 locus by yt = fL (mt ). Note that fL (mt ) < fH (mt ) holds becauseµ − (γ − g) > 0 from (9) and g > 0. In Figure 7, the locus is drawn as the blue solid line. Hereyt increases with time whenever (mt , yt ) lies to the left of the ẏt = 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 defined as a setof (mt , yt ) in which output growth is g and inflation is γ − g. Substitute ẏt /yt = g and πt = γ − ginto (28) to get Ω(mt , yt ) = ρ + γ. Again, using Ω(mt , yt ) = (β + vm−ηt )yt , we rewrite this conditionat the threshold as
yt =

ρ+γ
β + vm−ηt= fT (mt ),

24

(32)

where we define the locus as yt = fT (mt ). As shown in Figure 7, we have fL (mt ) < fT (mt ) <fH (mt ) given mt because of µ > γ − g and g > 0. It is easy to show that the economy is in thehigh-inflation regime when (mt , yt ) lies above the locus.22 When (mt , yt ) lies below the locus, the
DNWR is binding and the economy is in the low-inflation regime.
Figure 7 also draws the optimal time path (the curve with arrows) starting from the initialstate of the economy. By assumption, at the initial state, π0 > γ − g. Thus, the economy is in thehigh-inflation regime, and (m0 , y0 ) is located above the locus of fT (mt ). Once (mt , yt ) goes to theright of the fT (mt ) locus, the economy moves to the low-inflation regime. Recall that (20) and (21)indicate that (mt , yt ) switches from the high-inflation regime to the low-inflation regime without ajump. Also, as t → ∞, mt → ∞ from (26) and ẏt /yt → 0 from (29). Thus, (mt , yt ) asymptoticallyconverges to the dotted line located at the bottom in which yt = y ss . The optimal time path neverexceeds the dotted line because ẏt /yt is always positive as long as (mt , yt ) is located above the blue
solid line (See Figure 7).
To complete the proof, we confirm that the time path shown in the figure is saddle-path stable.
Let us redefine zt = 1/mt to obtain the system of the equation under the low-inflation regime:
ẏt
yt
żt
zt

= g − [(ρ + γ) − Ω(1/zt , yt )] ,= −(µ − γ + g)

By linearizing the above two equations around the stagnation steady state (zt , yt ) = (0, y ss ), we
have the eigenvalues ζ such thatρ+γ−g−ζ00−(µ − γ + g) − ζ= 0,

where we use (27) to replace βy ss = ρ + γ − g. As mentioned in the main text, combining thecondition µ − (γ − g) < ρ derived from the transversality condition with the condition µ > 0 implies
that g < ρ+γ. The condition g < ρ+γ then ensures that one eigenvalue is positive. In addition, the22

We can prove this claim by contradiction. Suppose that πt = γ − g and ẏt /yt < g. We do not need to consider
πt < γ −g and ẏt /yt > g because they are infeasible. Then, (28) implies that Ω(mt , yt ) = ρ+ ẏt /yt +πt < ρ+g+γ −g =
ρ + γ = Ω∗ . That is, Ω(mt , yt ) < Ω∗ . It immediately follows from Ω(mt , yt ) = (β + vm−ηt )yt that yt < fT (mt ), which
contradicts the supposition.

25

assumption µ − (γ − g) > 0 in (9) implies that the other eigenvalue is negative. Because the systemof equations under the low-inflation regime includes one jump variable yt and one predeterminedvariable zt , the time path is saddle-path stable and unique.
Along any path located above the saddle path mt eventually becomes negative so that it isinfeasible. Along any path located below the saddle path ct eventually becomes zero so that thetransversality condition does not hold. Therefore, the saddle path is the unique equilibrium path.

A.2

Proof of Proposition 2

The proposition discusses the case of β = 0 to clarify the role of wealth preferences. To prove theproposition, we focus on the elasticity of Ω(mt , yt ) with respect to mt and study the dynamics ofΩ(mt , yt ) = [v ′ (mt ) + β ′ (mt )]yt . We then discuss the equilibrium under the high- and low-inflation
regimes.
Let εm be the elasticity of Ω(mt , yt ) with respect to mt .

εm (mt ) =∂Ω(mt , yt ) mt
vm−η
t
.
= −η
∂mt
Ωt
β + vm−ηt(33)

The growth rate of Ωt = Ω(mt , yt ) is
Ω̇t
ẏt
ṁt
=
+ εm (mt ) .
Ωtytmt
In general, the dynamics of Ωt are described by a nonlinear differential equation because theelasticity of Ωt depends on mt . However, when β = 0, the above equation simplifies to
ẏt
ṁt
Ω̇t
=
−η .
Ωt
yt
mt

(34)

Next, we study the dynamics of Ωt under the high- and low-inflation regimes. We will show
below that Ωt is constant in equilibrium when β = 0. This constancy of Ω(mt , yt ) along with (34)
implies that ẏt /yt = η ṁt /mt holds in equilibrium.

26

Output growth under the high-inflation regime

Substituting (11) and (17) into (34) yields

the differential equation for Ωt under the high-inflation regime:Ω̇t= g − η(µ − Ωt + ρ + g)
Ωt

(35)
∗

= g − η[µ − (γ − g)] + η(Ωt − Ω ),where we use (15) for the second equality. Equation (35) is the differential equation for Ωt with apositive coefficient on Ωt . Thus, if there is a deviation of Ωt from its steady-state value, Ωt would
explode to ∞. Therefore, when β = 0, only Ω̇t = 0 is feasible in equilibrium. Imposing Ω̇t = 0 on(35), we have the steady-state value of Ωt under the high-inflation regime with β = 0.
g∗Ωss
H = Ω + [µ − (γ − g)] − .
η(36)The high-inflation regime under β = 0 is feasible only when g < η[µ − (γ − g)]. To see this,
suppose that g ≥ η[µ − (γ − g)] under the high-inflation regime. In this case, (36) implies that
∗
ss
∗
Ωss
H ≤ Ω . Given ẏt /yt = g in the high-inflation regime, (15) and (28) imply that ΩH ≤ Ω is∗rewritten as Ωss
H = ρ + ẏt /yt + πt = ρ + g + πt < Ω = ρ + γ and thus πt < γ − g. However, itviolates the assumption of the DNWR, π > γ − g.

Output growth under the low-inflation regime Substituting (17) and (20) into (34) yieldsΩ̇t= g − Ω∗ + Ωt − η[µ − (γ − g)].
Ωt

(37)

As in (35), (37) is the differential equation for Ωt with a positive coefficient on Ωt . Once again,
only Ω̇t = 0 is feasible in equilibrium. Imposing Ω̇t = 0 on (37), we have the steady-state value
under the low-inflation regime with β = 0:∗
Ωss
L = Ω + η[µ − (γ − g)] − g(38)The low-inflation regime under β = 0 is feasible only when g ≥ η[µ − (γ − g)]. To prove this,∗
suppose that g < η[µ − (γ − g)] in the low-inflation regime. In this case, (38) implies that Ωss
L >Ω .

27

∗Given πt = γ − g in the low-inflation regime, (15) and (28) imply that the condition Ωss
L > Ω is∗rewritten as Ωss
L = ρ + ẏt /yt + πt = ρ + ẏt /yt + γ − g > Ω = ρ + γ and thus ẏt /yt > g. However, itviolates the feasibility because yt = θt nt where nt ≤ 1.
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 − g)], the growth rate of output is g andthe economy remains in the high-inflation regime. Alternatively, if g ≥ η[µ − (γ − g)], the growthrate of output is η[µ − (γ − g)] and the economy is always in the low-inflation regime from theinitial period.

28

Figure 1: Real GDP per capita and trend

Notes: The figure plots log real GDP per capita from 1990:Q1 to 2019:Q4 and an estimated linear trend withand without a structural break in 2007:Q1. The dotted line represents actual GDP. The dashed (dotteddashed) line is the linear output trend with (without) a structural break in trend in 2007:Q1. In the figure,actual GDP is normalized to its 1990:Q1 value and the linear trend lines are normalized accordingly.

29

Figure 2: Velocity of M2 and trend

Notes: The figure plots the log velocity of M2 from 1990:Q1 to 2019:Q4. The dotted line represents theactual data for the log velocity of M2, and the dashed line corresponds to the estimated cubic trend. The
actual velocity is normalized to its 1990:Q1 value, and the trend line is normalized accordingly.

30

Figure 3: Simulated output trend

Notes: The figure compares the estimated output trend with the simulated output trend. The solid line
represents the simulated output trend, where a regime change occurs in 2007:Q1. The estimation andsimulation of trends cover the period from 1990:Q1 to 2019:Q4. For exposition, the figure depicts the trendlines from 2000. See Figure 1 for other details.

31

Figure 4: Simulated real interest rate

Notes: The solid line represents the simulated real interest rate. The dotted line is the real interest rate estimated using the approach in Haubrich, Pennacchi, and Ritchken (2012). The real interest rate corresponds
to 10-year real Treasury yields. The dashed line is the estimated cubic trend, based on data from 1990:Q1
to 2019:Q4.

32

Figure 5: Simulated inflation

Notes: The solid line represents simulated inflation. The dotted line is the actual (year-on-year) core inflation
rate. The dashed line represents the estimated cubic trend, based on data from 1990:Q1 to 2019:Q4.

33

Figure 6: Simulated trend of velocity

Notes: The solid line represents the simulated trend of velocity. The dotted line is the log of M2 velocity.
The data are normalized to the value in 1990:Q1. The dashed line represents the cubic trend of velocity,
based on data from 1990:Q1 to 2019:Q4.

34

Figure 7: Phase diagramߩ + ߛ + [ߤ െ (ߛ െ ݃)]ߚ‫ݕ‬௧ ݂ு (݉௧ ) ֞ ݉ሶ ௧ = 0ߩ+ߛߚ்݂ (݉௧ )˳ߨ௧ = ߛ െ ݃

ߩ+ߛെ݃

ߚ

݂௅ (݉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 − g. If (mt , yt ) is located above (below) the locus, the economy is in the high- (low-)inflation regime.

35