(Endogenous) Growth Slowdowns∗
Miguel León-Ledesma
University of Kent and CEPR

Katsuyuki Shibayama
University of Kent

September 26, 2023

Abstract
We develop a model where temporary non-technology shocks can lead to permanent changes in the rate of growth of total factor productivity (TFP). The key
ingredient of the model is a matching processes between basic researchers, productdevelopers, and the stock of knowledge of the economy. In this context, search externalities generate vicious and virtuous cycles in R&D. The model has a unique equilibrium path but multiple balanced growth paths (BGPs). After a deep or long-lived
shock, the economy can transit between these BGPs, generating “super-hysteresis”
in TFP. We calibrate the model in the context of the Japanese growth slowdown
and show that, quantitatively, it can explain well the TFP growth decline after thefinancial crisis in the 1990s. The simultaneous occurrence of demographic shocks
and a persistent but temporary financial crisis gave rise to a “wretched coincidence”
resulting in the growth slowdown.
JEL Classification: O40, O49, E32.
Keywords: Growth slowdowns, permanent effects of recessions, research and development, super-hysteresis.

∗

León-Ledesma: School of Economics and Macroeconomics Growth and History Centre (MaGHiC),
University of Kent, and CEPR. Sibson Building, Canterbury, CT2 7FS, UK, email: M.A.LeonLedesma@kent.ac.uk.
Shibayama: School of Economics and Macroeconomics Growth and History Centre (MaGHiC), University of Kent. Sibson Building, Canterbury, CT2 7FS, UK, email:K.Shibayama@kent.ac.uk. We thank seminar participants at the Universities of Warwick, Portsmouth,
Kent, Exeter, Málaga, Cagliari, Alicante, Hitotsubashi, Osaka, and Keio and NIESR, Bank of England,
The Asian Development Bank, Pablo Guerrón-Quintana, Michaela Elfsbacka Schmöller, Tom Schmitz,
Thijs van Rens, Roberto Pancrazi, Diego Calderón, Julián Neira, Rish Shingania, Elisa Keller, Yuta Takahashi, and Sebastian Merkel for useful discussions.

1Introduction“One day, Sir, you may tax it.”Michael Faraday’s reply to the question about the practical worth of electricity
given by William Gladstone, Chancellor of the Exchequer (Mackay, 1977).

1

1.5

2

2.5

A fundamental question in economics concerns the potential long-term impact of temporary shocks on an economy’s growth rate. Following the Global Financial Crisis (GFC)
in 2007/2008, there has been increasing interest in the effect of business cycles on trend
output. Empirical evidence suggests that temporary financial crises can lead to permanent output level losses, a phenomenon known as “hysteresis”. Moreover, Blanchard et al.
(2015) have shown that many recessions not only result in a permanent fall in GDP levels
but also lower growth rates, which Ball (2014) refers to as “super-hysteresis”. The poor
productivity performance observed after the 1990s crisis in Japan and the GFC in many
advanced economies have raised doubts about the traditional separation of growth andbusiness cycle analysis. To illustrate this, Figure 1 displays the evolution of Total Factor
Productivity (TFP) in Japan, juxtaposed with the Nikkei Stock Market price index. A
clear TFP growth slowdown occurred around the time of the financial crisis.1

1970

1980

1990

year

2010

2020

TFP Index assuming pre-91 growth0

Stock Market Price Japan
5
10
15

TFP Index Japan

2000

1970

1980

1990

year

2000

2010

2020

Figure 1: Japan’s TFP index and stock market prices.
We propose a model of TFP growth slowdowns. In the model, permanent growth
slowdowns can happen after deep or long lasting recessions. Crucially, these permanent
growth slowdowns are the consequence of temporary shocks. The model features a uniqueequilibrium path for the economy but multiple Balanced Growth Paths (BGPs).2 In the
1

TFP growth slows down from 1.7% (1970-92) to 0.7% (1993-2019). See Betts (2021) for a detailed
account of Japan’s TFP growth slowdown from the 1990s.
2
We will use “balanced growth path”and “steady state”interchangeably throughout. In our model, theeconomy is growing in any steady state at a constant rate.

1Trend Stationary, Hysteresis and Super Hysteresis
original trend
trend stationary
hysteresis
super hysteresis

Figure 2: Three examples of different types of recoveries after a crisis.

wake of a sufficiently severe or long-lasting shock, the economy can transit from a high
TFP growth BGP to a low TFP growth BGP.3 Figure 2 shows a simplified representationof three different views of business cycles. Standard business cycle models presuppose that,after a temporary level shock, the economy returns to the initial growth rate and trendlevel of output (trend stationarity). If, for instance, there is a temporary shock to thegrowth rate of TFP, even though the economy returns back to the initial growth rate, itwill cause a permanent level loss (hysteresis). In our model, after a temporary shock there
will be both a permanent level loss and a growth rate slowdown (super-hysteresis).
Our model builds on Romer’s (1990) variety expansion model. We distinguish between
basic research and product development as separate sectors of the R&D process. Basic
research generates new ideas that add to the stock of knowledge. Product development(invention) converts existing knowledge into new products. These are produced by new
firms whose returns are shared between basic researchers and product developers. In orderto produce new research ideas, basic researchers have to be matched with the existing
corpus of knowledge. These new basic ideas increase the knowledge stock. In order todevelop a new product, developers also have to be matched with the existing stock ofknowledge.
In this environment, there are two types of externalities. First, the standard growthor knowledge externalities present in most endogenous growth models: basic researchersand developers do not internalize the spillover effects of their activities. Second, search
externalities: both types of R&D researchers do not take into account that their efforts
affect the matching efficiency between the stock of knowledge and R&D workers in the
3This is consistent with the evidence in Aikman et al. (2022) who find that deep recessions have long
lasting growth depressing effects whereas normal recessions generally do not.

2

1

16

18

R&D intensity Japan
20
22

Patent applications Japan/USA
2
3
4

24

5

26

other sector. The existence of search externalities generates multiple BGPs.
The multiple BGPs can be interpreted as an outcome of the strategic complementarity
in the R&D sector caused by the search externalities. In the high BGP, there is a virtuous
cycle of high knowledge stock leading to increased matching for basic researchers anddevelopers, resulting in faster generation of new varieties. This, in turn, supports a highlevel of knowledge stock. In contrast, in the low BGP, a vicious cycle ensues where the lower
knowledge stock reduces the probability of successful matches, which lowers the returns to
not only new product development, but also basic research. Hence, the stock of knowledge
stays low.

1990

2000
year

2010

1980

2020

1990

2000
year

2010

2020

.3

400

1980

Ratio of Japan/US STEM publications WoS
.2
.25

UK

New PhDs per million pop
200
300

Germany
S. Korea
US

France

100

.15

Japan
1980

1990

2000
Publication year

2010

2020

2000

2005

2010
year

2015

2020

Figure 3: R&D indicators for Japan. R&D intensity (top-left) is calculated as R&D flow relative
to R&D stock calculated as in Benigno and Fornaro (2018) with updated data from OECD Scienceand Technology Indicators data. Patent applications relative to the US (top-right) come from
WIPO data as reported by the World Bank. For STEM publications relative to the US (bottomleft), see footnote 5. Number of new PhDs per million population (bottom-right) are obtained
from https://www.nistep.go.jp/sti_indicator/2021/RM311_34.html.

We analyze the model in deterministic and stochastic environments. In the deterministic case, we show that the model can produce situations whereby initially capital-richeconomies may converge to the low growth BGP leading to a “middle income trap”. We
3then introduce stochastic shocks in the model. Particularly, we focus on temporary financial shocks, which we model as reduced form temporary shocks to risk premia for all
types of investment, and permanent demographic shocks which we model as shocks to thedependency ratio and the population growth rate of the economy. We analyze under which
circumstances these shocks can push the economy towards the low growth BGP.
The stochastic simulations are motivated by the experience of Japan. After the financial
crisis in the early 1990s, Japan suffers from a deterioration of R&D performance indicators
as extensively reported in Bergeaud and Verluise (2022) and Ito et al. (2023).4 This canbe seen in Figure 3, where we plot R&D intensity, Japanese patent applications relative
to the US, publications in STEM subjects relative to the US, and new PhDs per million
population.5 The coincidence of the deterioration in research indicators and R&D activities
with the financial slump motivates our focus on the role of basic research, which is often
overlooked in the growth literature.
In the stochastic analysis, the calibration aims at matching certain macroeconomic
targets for the Japanese economy in the pre-crisis period and is able to explain well the TFP
growth slowdown after the early 1990s. The stochastic simulations for the calibrated modelshow that temporary financial shocks alone can move the economy from the high to the low
BGP. In our multiple BGPs environment, the effects of exogenous shocks are discontinuous.
If the magnitude of the shocks exceeds a certain threshold their effects can become large.
Quantitatively, for financial shocks to have this effect, the crisis has to be either very deepor very long-lasting. We then introduce permanent demographic shocks as a (mild) increasein the dependency ratio and a slowdown in population growth consistent with the data. By
themselves, these permanent demographic shocks can lead to permanent but quantitatively
only minor growth slowdowns. That is, permanent demographic shocks alone do not causea change in the BGP. This does not differ from what the standard neoclassical growthmodel would predict. However, when temporary financial shocks coincide with perverse
demographic shocks, their impact is large, because they can shift the economy to the
low BGP. Our calibration shows that this happens with milder depth and duration of thefinancial shock. We term this a “wretched coincidence”. Financial and demographic shocks
combine to explain why, in the Japanese case, the financial crisis was followed by a TFP
growth slowdown.
We then analyze “big push”fiscal policies to escape from the low BGP. We study government spending, corporate taxes, and subsidies to both basic research and product development. We show that only corporate tax and basic research subsidies can move theeconomy to the high BGP for reasonable tax/subsidy values and policy support duration.
The transition towards the high BGP is slow if the economy starts at the low BGP. That
4

This decline, especially in STEM subjects, has also been highlighted by the Japanese Ministry of
Economy Trade and Industry in its “2025 Digital Cliff” report. See https://www.meti.go.jp/shingikai/
mono_info_service/digital_transformation/pdf/20180907_03.pdf.
5
Data on STEM publications from the National Science Foundation as reported by the World Bank
only go back to year 2000. To generate a series for the whole period, we extracted data from Web of Scienceby address (i.e. Japan, or USA) for papers published in the Science Citation Index Expanded, Conference
proceedings Citation Index – Science, and Current Chemical Reactions for the 1978-2022 period. The datafrom this source tracks well available data from the World Bank since year 2000.

4

is, the duration of the government support is long and the required fiscal expenditure issubstantial. However, the cost of these policies is dwarfed by the benefits in the long run,because these policies essentially trade a temporary, though long, increase in fiscal spending for a permanent growth rate increase. We also show that an increase in government
spending cannot result in a BGP change because it squeezes the R&D sectors by increasing
the returns to labor in the final goods sector. Finally, the sooner the big push policy isimplemented the lower the budgetary costs. This is particularly the case for economies
that are still in transition towards the low BGP such as Japan in the 2020s.
Related literature. There is by now a well established literature on the long-lasting
effects of temporary shocks, i.e. “hysteresis”. Both the empirical and theoretical literatureare reviewed in Cerra et al. (2023). The empirical literature has highlighted widespread
evidence of permanent output losses especially after financial crises.6 In some cases, this isalso accompanied by highly persistent or permanent growth slowdowns. Our paper presents
a theoretical mechanism generating “super-hysteresis”, i.e. a permanent growth rate decline
after a temporary shock.
The theoretical literature on the long-lasting effects of temporary shocks goes back to
the contributions of Stadler (1990), Stiglitz (1993), and Fatás (2000). Comin and Gertler(2006) observe that TFP is procyclical over frequencies longer than business cycles (themedium run). Standard business cycle models that impose exogenous persistence of TFPshocks cannot explain this phenomenon. Motivated by this fact, they build a model integrating business cycles and endogenous productivity growth, in the spirit of Romer (1990).
Endogenous technology development and adoption generate highly persistent fluctuations
in TFP in response to demand shocks. Note that these persistent growth fluctuations
also lead to permanent TFP level effects, but its growth rate converges to a unique BGP.
We have a similar approach as we use a variety expansion model of endogenous growth.
However, our model can generate multiple BGPs because of the aforementioned search
externalities in the R&D process.
Since Comin and Gertler (2006), several papers have built up on the idea that thegeneration and adoption of knowledge may be influenced by cyclical shocks. These contributions, which Fornaro and Wolf (2023) label “Keynesian Growth Theory”, introduce endogenous growth mechanisms in New Keynesian models of fluctuations. Examples of theseare Anzoategui et al. (2019), Elfsbacka Schmöller and Spitzer (2021), Guerrón-Quintana
and Jinnai (2014), Cozzi et al. (2021), Queraltó (2020), Engler and Tervala (2018), Bianchi
et al. (2019), Vinci and Licandro (2020), and Fornaro and Wolf (2023).7 These papers
highlight the role of temporary non-technology shocks in driving persistent fluctuations of
TFP leading to permanent output level losses. Our paper is similar to this literature as we
focus on the role of innovation as an engine of amplification within an endogenous growthmodel with variety expansion. Unlike them, our model, under certain conditions, generates
permanent TFP growth slowdowns because the economy can converge to a different BGP
6

For recent evidence, see Furlanetto et al. (2023).
Schmitz (2021) takes a different route and focuses on the role of firm heterogeneity within a Schumpeterian model of growth. He finds that heterogeneity amplifies the effects of a financial shocks on innovation
because small firms are both relatively more innovative than large firms and hit harder by the crisis.
7

5

after temporary non-technology shocks.8 It shows that it is possible for an economy tomake a transition from a high to a low growth regime after persistent or deep financial
crises.9
Within this tradition, our paper is closely related to Benigno and Fornaro (2018). They
also build on the idea that there is a crucial interaction between temporary shocks andproductivity through R&D efforts. They present a New Keynesian model with liquidity
traps delivering a high and a low growth steady state. There are multiple equilibrium paths
and self-fulfilling expectations and, thus, sunspots determine which equilibrium path theeconomy takes. Our paper differs in several dimensions. First, and most importantly, in
our model there are multiple steady states (BGPs) but the equilibrium path is uniquelydetermined. Hence, there is no role for self-fulfilling expectations. Secondly, our model
does not feature price and wage rigidities common in New Keynesian models. Third, we
provide a fairly realistic calibration of the model to the Japanese economy to assess thelikelihood of temporary demand shocks leading to permanent slowdowns.
Our work also relates to the idea of “strategic complementarities” in Cooper (1999).
These are static models with multiple equilibria. Which of the equilibria is chosen by
the economy cannot be determined within the model. Often, coordination is required toachieve a particular equilibrium. In our model, the multiple BGPs can be understood as acoordination game between basic researchers and product developers in which, instead ofa coordinator, history (the initial state and the exogenous shocks) determines the uniqueequilibrium path. Intuitively, because of the existence of search externalities, increased
effort in the basic research sector increases the success rate of developers and vice-versa.
This strategic complementarity in our model generates virtuous and vicious cycles of R&D
and productivity growth. See Appendix C for further discussions.
Technically, our paper is also related to the literature on “poverty traps” and “middleincome traps”. A seminal paper in this tradition is Azariadis and Drazen (1990). They
present a neoclassical overlapping generations (OLG) growth model where human capital
externalities take the form of a threshold beyond which returns to scale allow for growth“take-offs”. Their model also features multiple steady states.10 Their growth differencesmodel version, however, features also multiple equilibria. The mechanism behind our model
is similar in the sense that search and growth externalities generate regions above or below
which the economy will converge to different BGPs. As in their model, a high degree of
non-linearity is required to generate multiple BGPs, which we derive from a microfounded8

Our model also abstracts from nominal rigidities which is, however, not essential. In the class ofComin and Gertler (2006) type models, even without nominal rigidities, non-technology shocks can leadto permanent output level losses.
9
A different strand in the literature studies the growth slowdown as a consequence of slow-moving
structural changes in the economy rather than as a consequence of temporary shocks. Examples of theseare Eggertsson et al. (2019) for a model of secular stagnation caused by demographic change or exogenous
productivity slowdowns, De Ridder (2019) for the role of intangible capital, Akcigit and Ates (2021) for
declining imitation rates, Liu et al. (2022) for declining interest rates, and Aghion et al. (2023) for fallingfirm costs of expanding to multiple markets.
10
Other important contributions in this tradition include Matsuyama (2002) for learning externalities,
Acemoglu and Zilibotti (1997) for financial market incompleteness, and Doepke and Zilibotti (2005) for
child labor traps among many others. See Azariadis and Stachurski (2005) for a review.

6R&D model within a stochastic environment. Our model also emphasises the potential for“big push”policies to lift the economy from the low growth trap.
In the rest of the paper, Section 2 presents the model, Section 3 presents the quantitative
analysis and main results, Section 4 analyzes policies, and Section 5 concludes.

2

The model

We extend Romer’s (1990) variety expansion model by introducing two research matchingprocesses. As in Romer (1990), the aggregate production function uses labor and capital, while total factor productivity (TFP) is determined by the number of varieties ofintermediate goods.11
Unlike in Romer (1990), however, we explicitly separate the different roles of basic
research and product development. The proportion of labor allocated to product development determines the rate of TFP growth, while the proportion allocated to basic researchshapes the success rate of product development. Figure 4 displays the structure of the basic research and development matching process. Basic researchers match with the existing
stock of ideas to develop new basic research, and developers/inventors also match with thestock of knowledge to produce new intermediate goods. In our notation, sub/superscript
A indicates product development, while Z indicates basic research.

Figure 4: Structure of the basic research and development matching process.
11

A criticism of this class of endogenous growth model to explain a country’s growth performance isthat basic research knowledge also flows across borders and hence the knowledge stock that matters is
global. To capture this, our calibration exercises consider the exogenous component of TFP which can beinterpreted as imported technology. Importantly, Hausman (2022) argues that innovation activity relies
on local research clusters. Local developers rely on interactions with local basic research producers. These
basic researchers may either be producing new basic research or absorbing foreign produced research andadapting it to the local productive needs. The model could be reformulated in line with these arguments.

7

Because basic research activities affect the stock of research, this will impact on thelikelihood of a successful match between developers and the stock of research. However,basic researchers do not internalize this effect. Likewise, development effort will affect thesuccess rate of matching between basic researchers and the stock of basic research because
it determines the share of profits from new firms allocated to basic researchers. Thesematching externalities are key to generating vicious/virtuous cycles and hence multiple
BGPs.
To produce a new variety, developers set up new firms whose value depends on the
discounted stream of profits from their monopolistic market power. The value of the newfirms is the reward to the entire R&D sector, which is split between basic researchers andproduct developers using Nash bargaining. Because of this, the financial sector plays an important role in our model. Importantly, there is a very small mass of basic researchers thatare not pecuniary motivated. We call them Non-Pecuniary Researchers (NPRs hereafter)
and they play a crucial role when an economy “takes-off” from the low BGP.
To streamline the exposition, we briefly explain (i) labor supply, and then we focus on(ii) the two research matching functions, (iii) asset pricing equations (financial markets),and (iv) the optimization problems and their first order conditions in the basic researchand development sectors. The rest of the model is standard and hence the readers familiar
with this class of models may want to skip Sections 2.5, 2.6 and 2.7. The list of equilibrium
equations and variables are summarized in Appendix A.

2.1

Labor supply

Workers are endowed with a total number of hours Ht that can be allocated to final
production (HY,t ), innovation (HA,t ), basic research (HZ,t ), or be out of the labor force(HO,t ):
Ht = HY,t + HA,t + HZ,t + HO,t .

(1)

The proportion of worker-hours outside the labor force is given exogenously as HO,t = hO Ht ,which is introduced in order to analyze the effect of demographic changes in the quantitative
analysis of the model. Because we assume that labor is inelastially supplied, the rate of
growth of Ht is the rate of population growth and is exogenously given by γH,t .
Importantly, we assume that there is a group of researchers in the basic research sector
whose motivation is not pecuniary. Their share in the total labor supply is hZ . Thatis, there is a positive mass of researchers who engage in research because of intellectual
curiosity, intrinsic motivation, recognition or prestige. We call them “Non-Pecuniary Researchers” (NPR). Hence:
HZ,t ≥ hZ Ht .
NPRs are important to prevent the low R&D BGP from becoming an absorbing statefrom which the economy cannot exit. More specifically, it would be impossible to escapefrom the low BGP if hZ = 0. Also, the low BGP would disappear if hZ is high enough.
That is, if the proportion of NPRs is large, the economy must always converge to the high
BGP. Between extremes, as hZ becomes higher, it becomes easier to escape from the low
8BGP.
As labor is homogeneous and there is perfect mobility across sectors, the wage rate
in all three sectors must be equalized. An exception to this happens when the condition
HZ,t ≥ hZ Ht is binding. In this case, NPRs are willing to accept a lower wage than in theother two sectors.

2.2

Research and Development matching functions

The matching between aggregate research hours HZ,t and existing knowledge stock Zt generates new research outputs Ztnew . Note that individual R&D researchers take all variablesin this subsection as given. The functional form is assumed to be Cobb-Douglas:12
Ztnew = ωZ Aκ̃t Z



HZ,t
Ht

ϕ̃Z

!η̃Z
(Zt )1−ϕ̃Z.

(2)where ωZ and ϕ̃Z are scale and elasticity parameters respectively, while At is the stock ofexisting inventions. There is a growth externality given by Aκ̃t Z such that the higher thestock of inventions (e.g., Google Scholar and Github), the better the chances of a successful
match. This growth externality is common in many endogenous growth models. Unlike
in labor matching models, there is no theoretical reason for assuming constant returns toscale (CRS) so that η̃Z ⋚ 1. This subsection mainly focuses on the conditions required toprevent scale effects and some other undesirable model properties. Once these theoretical
restrictions are imposed, parameters with tilde (˜) are all eliminated.
The function contains a type of Stepping-on-Toe effect, making the model free fromscale effects (Jones and Williams, 2000). That is, successful matching leading to new
H). Although,research output depends on the share of labor hours devoted to research ( HZ,tttechnically, this formulation is simply a short-cut to prevent scale effects, it does have asimple intuition. Unlike in labor matching, many researchers can be matched with a single
research idea. With non-rivalry, most researchers will tend to access the same basic research
corpus hence leading to congestion (Stepping-on-Toe effects). E.g., most researchers tend
to read outputs published in a selective sub-set of prestigious outlets. In the extreme, allresearchers read the same set of papers. In this case, the number of academic papers thatare read by researchers is independent from the number of readers. Instead, it depends on
the share of hours that each researcher devotes to research. In our simple model setup,
HZ,tcan also be interpreted as the share of hours that the representative worker allocatesHt
to basic research.
Likewise, in order to produce a new product design Anew
, there is a matching betweent
12

In our notation, Zt and At are the stocks of basic research and invented product designs at the startof period t. Likewise, Kt denotes the capital stock for final production at the start of period t.

9

aggregate development/invention hours HA,t and existing knowledge stock Zt :Anew
= ωA Aκ̃t A
t



HA,t
Ht

ϕ̃A

!η̃A
(Zt )1−ϕ̃A,(3)where ωA and ϕ̃A are scale and elasticity parameters respectively.
With these two formulations, we need to impose a set of parameter restrictions. Firstof all, to ensure the existence of at least one balanced growth path (BGP), we need to
impose a restriction on the growth externality:
1 − 1 − ϕ̃Z η̃Z(1 − κ̃A ) ,
(4)
κ̃Z =
ϕ̃A η̃A
which implies that we must have growth externalities but these can appear either in thebasic research (κ̃Z ) or invention sectors (κ̃A ) or both as long as this restriction is satisfied.13
We also impose a set of restrictions on the degree of returns to scale (RTS) to prevent
explosive growth paths. We would have an explosive growth path if RTS in research are toolarge: η̃Z > ϕ̃1 . On the other hand, if RTS are too weak, the success rate of basic researchZgoes to infinity as the number of scientists goes to zero. That is, η̃Z < ϕ̃1 implies πtZH → ∞
Z

Z new

as H̄Z,t → 0, where πtZH = HtZ,t is the success rate of basic research. To avoid these, wemust impose η̃Z = ϕ̃1 .14 With this restriction, we regain a Romer-style formulation ofZ

knowledge production but without scale effects. Define κZ = 1− 1−κ̃Zϕ̃ η̃ . Then, we obtain:( Z) Z
HZ,t
Ztnew = ωZ AtκZ ϕZ (Zt )1−ϕZ
,
(5)
Ht


where 1 − ϕZ = 1 − ϕ̃Z η̃Z .
Following a similar reasoning, we assume that the matching function in the invention/development sector is constant returns to scale (CRS): η̃A = 1. As above, define
ϕA = ϕ̃A η̃A = ϕ̃A and κA = κ̃A η̃A = κ̃A . Then,
Anew= ωA Aκt A (Zt )1−ϕA
t



HA,t
Ht

 ϕA
,

(6)

with parameter restriction κA = 1 − (1 − ϕA ) κZ .
Finally, the laws of motion for the stock of intermediate varieties and basic research are
13

In addition, we impose the following requirement: 0 ≤ κ̃Z , κ̃A ≤ 1 because the growth externality must
be positive. In terms of the quantitative behavior of the model, where the growth externality appears has
no effect on either the BGP or transitional dynamics. In practice, we set κ̃A = 1 (κ̃Z = 0). A different
choice of κ̃Z and κ̃A only leads to a parallel shift of Zt /At both in the BGPs and in the transitional
dynamics.
14
Because ϕ̃1 > 1, the basic research matching function is increasing return to scale.
Z

10

given by:
At+1 = Anew
+ (1 − δA ) At ,t
new
Zt+1 = Zt + (1 − δZ ) Zt ,

(7)
(8)

where δA and δZ are the depreciation rates.

2.3

Financial markets

Financial markets play key role in the evolution of R&D because the asset values of productdesign and research stocks directly affect the reward to successful inventions and basic
research, respectively.
We assume that all the monopolistic profits from each product are paid out as dividends
Dt :
Yt
(9)
Dt = (1 − τD ) (1 − α̃) α ,Atwhere AYtt is output per each product design, of which (1 − α̃)α is the share belonging to
the firm owner (the successful inventor and the basic researcher who owns the successfully
matched research asset).15 Parameter composite α̃ = (θ − 1)/θ is the reciprocal of themarkup, which is determined by the elasticity of substitution among varieties.16 We allow
for a corporate tax at a rate τD which will later be used in our policy experiments.
The asset value of new firms Vt is given by the present value of dividends Dt :Λt,t+1
Vt = Dt + Et
(1 − δA ) Vt+1 ,(10)
cv
where the stochastic discount factor Λt,t+1 is derived from the households’ consumptionsavings decisions. Parameter cv captures an exogenous risk-premium reflecting the factthat innovative startups are riskier than bonds. Later, we interpret a financial crisis as anincrease in the risk-premia for all types of investment.
Once a developer successfully matches with basic research stock, Vt is split between
developers and basic researchers. That is, Nash bargaining determines the shares of productinventors (sA ) and researchers who own the existing research stock (1 − sA ). Hence sA Vt
will appear in the developers’ optimization problem as a reward while (1 − sA ) Vt appears
in the asset value of new research stock (Ztnew ).
The asset value of new research stock then is given by:
Λt,t+1ZZ
AZ(11)(1 − δZ ) Vt+1 .
Vt = (1 − sA ) Vt πt + Et
cz
The value will depend on the success rate of the matching between developers and basic
15

Output is distributed as wages (1 − α), capital cost α̃α and profit (1 − α̃)α. See also equations (20)and (21).
16
Under Romer’s (1990) parameterization, α̃ = α. Such a coincidence is not compatible with realistic
calibration. See Section 3.1.

11

Anew

research stock given by πtAZ = Z̄t t . 17 That is, the value of new research stock depends
on the value of new firms weighted by the success rate of a development match and thebargaining power of basic researchers, plus the continuation value. Equivalent to the risk
premium for new firms, cz is a risk premium for new basic research.

2.4

Optimization problems by researchers and developers

The optimization problem of basic researchers is:
maxVtZ πtZH HZ,t − (1 − τZ ) Wt HZ,t ,
HZ,t

Ztnew
,
HZ,t
HZ,t ≥ hZ Ht .

s.t. πtZH =

(12)
(13)
(14)

The success rate per unit labor input πtZH is given to basic researchers. That is, a basic
researcher successfully matches with existing research stock with rate πtZH . The wage rate
Wt and the asset value of basic research stock VtZ are also given. Parameter τZ is a government subsidy to the research sector. The choice is bounded from below by the existence of
NPRs. This condition is binding in the low BGP, in which there are no monetary-motivatedresearchers. Implicitly, we innocuously assume that the last researcher’s bargaining power
is 100%.18
The first order condition (FOC) of HZ,t is
VtZ πtZH = (1 − τZ ) Wt

if HZ,t > hZ Ht

VtZ πtZH ≤ (1 − τZ ) Wt

if HZ,t = hZ Ht(15)

The state space is split into two. In one of the two regions, which includes the high BGP,VtZ πtZH = (1 − τZ )Wt holds. In the other region, which includes the low BGP, HZ,t = hZ Htholds.19 The boundary between these two regions is determined endogenously (a freeboundary problem).
Likewise, the optimization program for developers is to choose hours worked in development to maximize:
max sA Vt πtAH HA,t − (1 − τA ) Wt HA,t ,HA,ts.t. πtAH =

Anew
t
,
HA,t

(16)
(17)

17

Note that πtAZ is not the success probability but the success rate. Because a single unit of researchstock can be matched with multiple inventors, πtAZ can exceed one.
18
That is, the share of the owner of a unit of basic research stock is 0%. The bargaining share parameterin the basic research match has no effect because the share (1 − sA ) of the value of new firms goes to basic
researchers collectively as a last researcher or as an owner of the research stock.
19
Romer (1990, pp.95-96) has already pointed out the possibility of the existence of a no-R&D steadystate, in which the non-negativity constraint of research labor is binding.

12The success rate per unit labor input πtAH is given to developers. An inventor successfully
matches with existing research stock at a rate πtAH . Again, wage rate Wt and the asset
value of basic research stock VtZ are also given. After a successful invention, they set up a
firm of which sA is their share. Parameter τA is a government subsidy to the development
sector in the same form as the subsidy to basic research activities above.
Because of the Inada property of the matching function, unlike in the basic researchsector, the negativity constraint HA,t ≥ 0 never binds. Hence, the FOC for HA,t is much
simpler:Vt πtAH = (1 − τA ) Wt
(18)

2.5

Aggregate output and TFPThe production sector follows the formulation of Romer (1990). We skip the detailed
derivation here. The resultant aggregate production function is given by
Yt = (ηKt )α (ξt Aφt A HY,t )1−α ,(19)where ξt is the exogenous component of the labor-augmenting technological progress, while
the endogenous component is Aφt A , with φA a parameter composite defined as φA =α
1
α 1−α̃ 20= 1−α.
1−α θ−1α̃The FOCs for the production sector are:Wt = (1 − α)RK,t = α̃α

Yt
Kt

Yt
HY,t

(20)
(21)

The logarithm of total factor productivity (TFP) is expressed as a weighted sum of theexogenous and endogenous labor-augmenting technology.

ln T F Pt = (1 − α) ln ξt + φA ln At(22)
Note that ξt grows at the same rate γξ,t in the entire state space. Hence, the growth rate
gap between the high and low BGPs is entirely explained by At .

2.6

Households

A representative household maximizes intertemporal utility over per-capita consumption
where we assume standard CRRA period utility with σC being the degree of relative risk
aversion. Households own the capital stock of final production firms and use their wage
income (Wt (1 − hO ) Ht ) and capital income (RK,t Kt ) to consume (Ct ) and invest in newcapital (It ). They receive lump-sum transfers (LSTt ) from the government which, under
our calibration described later, are negative (lump-sum taxes).
20

We can safely normalize η = 1.

13

We assume that the head of a household does not take into account the change ofthe family size (population) in their decisions. This assumption is quantitatively relevant,because the lower the population growth is the lower the effective discount factor is. Our
assumption is motivated by empirical findings. Some authors such as Aksoy et al. (2019)find negative effects of a higher hO on economic growth, while the evidence on the effectof population growth is mixed.21 If we assume that households take into account future
population size, the decline in the population growth rate would have larger effects. Because
this is particularly important in Section 3.5, we will come back to this point there.
The household problem is given by:"∞#
X (Ct /Ht )1−σC
βt
,
(23)
max EtCt ,It ,Kt+1
1
−
σ
C
t=0
s.t. (1 + τC )Ct + It = (1 − τH )Wt (1 − hO ) Ht + RK,t Kt + LSTt ,Kt+1 = It + (1 − δK )Kt ,

(24)
(25)

where β is the subjective discount factor (time preference parameter) and τC and τH are
consumption and labor income taxes. As long as these taxes are constant, and because
labor supply is inelastic, they have no effect on the model dynamics. They are later used
to calculate government revenues in the policy experiments.
The first order conditions yield a capital return and a stochastic discount factor conditions:
Λt,t+1(RK,t+1 + (1 − δK )) ,(26)
1 = EtcK−σCβCt+1 /Ht+1
Λt,t+1 =.
(27)
γH,t+1
Ct /HtWe add a wedge cK as an exogenous risk-premium on capital in the capital return equation.
This risk premium will be treated as stochastic later on.

2.7

Goods market clearing

We finally close the model with the goods market clearing condition:
Yt = Ct + It + Gt .

(28)

Final output equals expenditure in consumption, capital investment, and government
spending (Gt ). Government spending in the model does not affect utility or production andis hence treated as “wasteful”. It is introduced in the model to study “Big-Push” policies.
21

Pritchett (1996) finds that “There is no correlation, or a weak negative correlation, between measuresof total factor productivity growth and population growth. Nearly all of the weak correlation between
the growth of output per person and population growth is the result of shifts in participation in the labor
force, not of changes in output per worker.”

14

2.8

Equilibrium

Since several model variables are growing in steady state, we need to represent the modelin stationary form to compute the equilibrium. We first collect the equilibrium equations
and non-stationary model variables. We then divide the non-stationary variables by their
growth factors. The equilibrium equations are also converted consistently. This process is
described in detail in Appendix A. All de-trended variables will be denoted by a tilde (X̃).
Since we have discussed in detail the main building blocks of the model, we provide thefull set of (de-trended) equilibrium equations in Appendix A. We can now formally define
equilibrium.
Definition 1 An equilibrium is defined as the set of de-trended endogenous variables listed
in Table A.1 in Appendix A such that (i) the initial state (Z̃0 and K̃0 ) is given and thetransversality conditions (29) hold, (ii) exogenous variables are also given for all t ≥ 0,
and (iii) the equilibrium equations defined in Table A.3 are all satisfied for all t ≥ 0.
In this definition, we intentionally omit the dynamics of the exogenous variables because
we consider different versions in the sections below. In the deterministic version there are
only two exogenous variables: γH,t and γξ,t . In the stochastic version, the risk premium
parameters (ck , cz , and cv ), the dependency ratio (hO ) as well as population growth (γH,t )are treated as exogenous variables.
We define the transversarity conditions (TVCs) as follows:#"Λ̃0,t+1(1 − δK )t+1 = 0,(29a)lim E0t→∞cK"#
Λ̃0,t+1 γV,t+1
t+1
lim E0(1 − δA ) Ṽt+1 = 0,
(29b)
t→∞
cv
"
#
Λ̃0,t+1 γV Z,t+1
t+1 Z
lim E0(1 − δZ ) Ṽt+1 = 0,
(29c)
t→∞
cZ
where γV,t and γV Z,t are the trend growth rates of the value of firms and basic researchrespectively. Intuitively, these conditions state that the present values of the asset prices(capital stock, firm value, and the asset value of the research stock) should not explode.
The equilibrium is computed numerically using the Euler Equation Iteration (EEI)
a global solution method. Details of the computation solution method are presented inAppendix B.

2.9

Multiple balanced growth paths: intuition

To garnish intuition for the existence of multiple BGPs in the model, we focus on the labordemand of the three sectors. Assume for now that all variables are at the steady state andwe use subscript “ss” to indicate this assumption. In this exercise, we omit one equilibrium

15

R&D
Y 22
condition: Wss= Wss. That is, we omit the marginal revenue product equalization(offered wages) between the two R&D sectors and the production sectors. We then treat
R&D
Y
lines are theand Wss
HA,ss as an exogenous parameter. The crossing points of the Wss
BGPs.

3

-1

2

2.5

-2

1.5

2

-3

1

1.5

-4

0.5

1

-5
0

0.005

0.01

0.015

0.02

0

0.005

0.01

0.015

0
0.02

0.02

4

1

0.015

3

0.5

0.01

2

0

0.005

1

-0.5

0

0
0

0.005

0.01

0.015

0.02

0

0.005

0.01

0.015

-1
0.02

Figure 5: Multiple BGPs in partial equilibrium.
Figure 5 illustrates the results of this exercise under the benchmark parameter values
shown in Section 3.1. In the upper-left quadrant, we plot wages offered in the three sectors.
The right two panels show the decomposition of the offered wages.23 Among three BGPs,
the middle one is locally unstable and the outer two are stable (saddle-path stable).
To understand the intuition, note first that, for most of the range of values of HA,ss ,wages offered in the R&D sectors are downward sloping. This is “normal” in the sensethat a lower labor input is accompanied by a higher marginal revenue product of labor.24
However, due to the search externalities, wages become increasing in HA,ss around the
middle BGP. The middle BGP is a threshold point, in the sense that a vicious cycle occurs
below it while a virtuous cycle occurs above it. On the vicious cycle side, if HA,ss falls below
22

A
Z
The model implies wage equalization between the two R&D sectors WtR&D = Wss= Wss
as long as
R&DAZ
HZ,ss ≥ hZ Hss is not binding. If HZ,ss = hZ Hss , then Wss= Wss > Wss .
23
Z
ZH
Z
= constant + ln πss+ ln Vss; and
The decompositions are obtained from FOCs (15) and (18): ln Wss
AAH
ln Wss = constant + ln πss + ln Vss .
24
In most of the range, the main reason for this “normal” behavior is the downward sloping asset
values. See the bottom-right panel. The firm value is decreasing in HA,ss mainly because, as it increases,
production labor HY,ss is squeezed and hence the production size shrinks.

16the middle BGP, because the success rate of basic research matching decreases sharply (seethe top-right panel), basic researchers are discouraged, leading to a shrinkage of the stockof basic research, which, in turn, deteriorates the success rate of product development.
This further reduces incentives for basic research. On the virtuous cycle side, in contrast,
a higher HA,ss incentivizes further demand for HA,ss because of the search externalities.
This continues until the economy arrives at the high BGP.
NPRs prevent a zero R&D world from which the economy cannot escape. As NPRs are
not motivated by pecuniary rewards, they still produce basic research even when HA,ss isvery low. In the region where HZ,ss = hZ , the optimality condition in the research sector
AZ
is irrelevant and the blue and red lines decouple (i.e., Wss> Wss
in the top-left panel).
Note that, for reasonable parameter ranges, there are three BGPs even if we assume thezero NPRs (i.e., hZ = 0). But, the existence of multiple BGPs is not always guaranteed.
If, for instance, we assume an very low matching efficiency, the high BGP disappears.25

3

Quantitative analysis

3.1

Calibration

We calibrate the model to match certain aggregate features of the Japanese economy before
the 1990s financial crisis. We take 1970-1992 as the pre-crisis sample, and 1993-2019
as the post crisis sample. With this approach, we are assuming that Japan was in theneighbourhood of the high BGP in the pre-crisis period. Near the end of this period, it
is likely that Japan was close to the final stages of the standard neoclassical transitional
dynamics driven by capital accumulation. But, in the early part of the pre-crisis period,
Japan was still likely experiencing transitional dynamics. Thus, GDP growth and interest
rates in the data may exceed those produced by the model in steady state.
Table 1 presents a summary of the parameters’ notation, chosen calibration value, their
meaning, and a brief explanation of the calibration choice. These values will be usedthroughout the deterministic and stochastic analysis. For the latter, however, we later
discuss the dynamics of the exogenous shocks. The model is rich in parameters and we
do not always have a directly observable data counterpart. In some cases, this requires tomake informed guesses which we will discuss below. We have four types of parameters:
those that are calibrated to standard values in the literature, those directly taken fromthe data, those taken from previous literature and, finally, those that are estimated orcalibrated to match a particular target. Note that one period in the model is one quarter.
The first group of parameters is standard in the literature. It includes relative risk
aversion (σc ), capital share in final goods production (α), and the depreciation rate ofcapital (δK ) which corresponds to a 10% annualized rate. We have chosen the value of theelasticity of substitution among varieties θ so that the mark-up is 20%, which is within theconsensus range of New Keynesian models.
25

Appendix C presents an alternative intuitive way of viewing multiple BGPs in the spirit of coordination
games.

17

β
σC
α
δK
G
θ
ck
cv
cz
ξ
γH
hO
ωZ
ωA
δZ
δA
ϕZ
ϕA
sA
κZ
hZ
τD
τZ
τA
τH
τC

parameter
subjective discount factor
relative risk aversion
capital sharecapital depreciation
govt expenditure shareelast. of subst. varietiesrisk-premium on K
risk-premium on new firms
risk-premium on Z
exogenous TFP growth
population growth
pop share of non-labor
efficiency in research match.
efficiency in develop. match.
depreciation of Z
depreciation of A
elasticity in research match.
elasticity in develop. match.
bargaining power
knowledge externality% of NPRs
corporate tax
labor subsidy, basic researchlabor subsidy, development
labor income tax
VAT taxvalue
0.9957
0.500
0.360
0.026
0.238
6.00
1.005
1.0466
1.04661.00461.00180.380
34.47
0.03980.03980.22250.26540.80
15.0e-04
0.40
0.68
0.01
0.073
0.092

notes
Match risk-free rate
Macroeconomics literature
Capital share
Annual 10%
National Accounts
Markup of 20%
Match C/Y and I/YUsing PER data (see text)
Using PER data (see text)
World TFP growth
Population growth data
Labor market data, OECD
Normalization
Match pre-92 TFP growthHall, Maireese and Mohnen (2009)
Intellectual products deprec., SNA
Estimated matching function
Estimated matching function
Match HA,ss /HZ,ss ratio
Irrelevant for model outcomes
See text
Corporate tax data
OECD, MITI data
OECD, MITI data
National Accounts
National Accounts

Table 1: Parameter Values. See text for a detailed explanation.
The parameters directly taken from the data include the demographic parameters andfiscal policy parameters as well as the depreciation rates of R&D stocks. The growth rateof population (γH,t ) is calibrated to match the 0.73% growth in the pre-crisis period. To
calibrate the share of population outside the labor force, hO = HO,t /Ht , we use the share of
non-employed population in the pre-crisis period (38%) rather than age dependency ratio.
This parameter plays an important role in the analysis of the demographic changes later.
Because the model does not consider endogenous participation decisions, our strategy is totreat the increase in female labor force participation, which mitigates the shrinkage of labor
supply significantly in the data, as an exogenous change. If we used the age-dependency
ratio instead, it would over-estimate the rise in hO .
18

In order to study policy measures in Section 4, we need to set the tax rates and R&D
subsidies. The government spending share of output in steady state is taken from National Accounts as the average of the pre-1992 period. Since the model represents a closed
economy, government spending is the sum of government consumption, investment andnet exports.26 In addition, there are five tax rates that are taken from the data. The
corporate tax rate is set to 40% to match the typical rate before the 1990s.27 We set thelabor income tax rate to match the ratio of income tax revenue to total income. Thisis an obvious simplification as the rate is progressive in the data. In our case, however,
because labor supply is inelastic, the labor income tax is not distortionary (it is essentiallya lump-sum tax). Note that this implies that we ignore taxation of capital income. Thisis because a significant share of capital income in Japan is tax exempt and, as a result, it
is hard to measure it. The effective VAT rate is set to be 9.2% which is lower than theheadline rate of 10% because some consumption goods are VAT exempt. Implicitly, othertax revenues are included in the lump-sum transfer (LSTt ), because they do not appear togrow in proportion to the economy’s size.
For the shares of government subsidies in R&D expenditures, we take the data from
OECD and the Japanese Ministry of International Trade and Industry (MITI).28 Approximately 1% of the private sector R&D expenditure is covered by the government, while 68%
is financed publicly for the non-profit sector. Although there is no direct one-to-one mapping between our basic-research and development sectors with the non-profit and private
sectors in the data, we believe this is a reasonable approximation.
For the depreciation rates of the basic research stock we resort to Hall et al. (2010) who
state that estimates range between 10% and 36% with most researchers settling for a valueof 15% in annual terms. For the depreciation rate of At we use the “effective depreciation
rate” for R&D capital in the Japanese National Accounts.29 This is also 15%.
We turn now to parameters that are calibrated to match certain data moments. The
bargaining power between researchers and developers (sA ) is set to match the ratio of
business to non-business FTE researchers in the pre-1992 period using research personnel
data from OECD.30 The subjective discount factor (β) is calibrated to match the risk-freerate for the pre-92 period which we calculate using the uncollateralized overnight call rate(the “mutan rate”) minus the inflation rate.31 The risk-premium on capital (ck ) is set tomatch the private investment to GDP ratio of the pre-1992 SNA data. We set the riskpremia for new firms (cv ) and new basic research (cZ ) both to be 20% in annualized terms.
These numbers are within the range of those reported in the literature on the user cost of
26

With this formulation, we implicitly assume that government investment (8.8% of GDP) does not contribute to the capital formation. Likewise, government consumption (18.0%) does not enter the households’
utility function. For the pre-crisis period, the GDP share of net exports is not large (1.3%).
27
This rate changed almost every year. Subsequently, it has gradually fallen to 30% by 2018.
28
See NISTEP (2022) and https://stats.oecd.org/Index.aspx?DataSetCode=GERD_SOF
29
See
https://www.esri.cao.go.jp/jp/sna/data/reference1/h27benchmark/pdf/kaisetsu_
20211122.pdf (in Japanese).
30
David (2021) uses a value of 0.5, although those estimates are obtained in the context of firm acquisition.
31
This money market instrument was launched in July 1985 and we used the average of monthly datafor 1985-1992.

19

venture capital. For instance, Kerins et al. (2004) find a cost of capital between 16.7% for awell-diversified investor and 40% for a low diversification investor. They are also consistent
with the survival rate of the startup firm data.32
We are now left with calibrating ωZ , ωA , ϕz , ϕA , κZ , hZ , and ξss . Two of them have no
impact on either the BGP or model dynamics. We set the growth externality parameter
κZ = 1, which has no effects as long as condition (4) is satisfied. We set the scale parameterωZ to 3. Because ωZ is subsumed by the measurement units in Zt , it has no effect either. We
calibrate scale parameter ωA so that the TFP growth rate in the high steady state is close
to the TFP growth rate in the pre-1992 data. For ξss , which we interpret as exogenous orimported technology, we calibrate it to match the World TFP growth estimates in Esfahani
et al. (2020). They use World Input-Output tables to calculate the growth rate of world
TFP from 1996 to 2016. Because there is no data from the pre-crisis period, we use theirestimate of 1.28% World TFP growth for 1996-2014.
Calibrating ϕz and ϕA is a more complex task and there are no obvious literature
parallels to draw from. For this reason, we turn to a data approximation. From the two
matching functions (3) and (2) and the decomposition of the TFP (22), given the parameter
restrictions discussed in Section 2.2 and the specification for the final output production
function, we can derive two equations that would allow us to estimate ϕz and ϕA :
ϕZ κZ
ϕZ κZ
γT F P,t −γξ,t ,(30)
φA (1 − α)φAκAκAγAnew ,t − (γHA ,t − γH,t ) = (1 − ϕA )(γZ,t − (γHA ,t − γH,t )) +γT F P,t −γξ,t , (31)φA (1 − α)φAγZ new ,t − (γHZ ,t − γH,t ) = (1 − ϕZ )γZ,t +where all the γX,t ’s represent the rate of growth of any variable X. The coefficient on
the first element of the RHS of the above equations would yield an estimate of the ϕcoefficients. For Ztnew , we used the number of journal publications in all fields by Japanbased researchers. The data come from the October 2022 release of NISTEP. 33 Researchstock Zt is calculated by the perpetual inventory method with the depreciation rate δZ, we use flow of the R&D Intellectual Property Products from Japanese
above. For Anewt
SNA. We eliminate stock At using equation (22). The time-varying exogenous component
of TFP ξt , we used the rate of growth of US TFP from the Penn World Table version
10.01 taking the US as the technology frontier. Finally, for R&D labor inputs, we use thenumber of researchers in the Higher Education, Government, and Non-Profit sectors for
32

Calvino et al. (2015) provide evidence on startup survival rates for a cross-section of countries and findthat survival rates are around 40% after 7 years which, if annualized, is 88%. A naı̈ve back-of-envelope
calculation implies, if the target portfolio return is 3.7%, the required survivors’ earnings yield must be
18% (the salvage value of the exiting firms is assumed to be 0). Note that the historical PERs for the
Tokyo Stock Exchange between 1971 and 1985 (excluding the bubble periods) remained between 20 and
30 and take an average of value of 27, which implies an earnings yield of 3.7%. Our risk-premum implies
that the price earnings ratio (PER) to be 2.5 and the earnings yield to be 22%, which is slightly higher
than 18%. However, the value obtained using survival rates is likely to be downward biased because theirdata includes all firms and not just innovative startups.
33
See https://www.nistep.go.jp/research/science-and-technology-indicators-and-scientometrics/
indicators.

20

HZ,t and in the corporate sector for HA,t from the NISTEP October 2022 release (full timeequivalent). We face severe data limitations as the SNA Intellectual property statistics
only go back to 1994. Hence, our estimates are for the 1994-2017 period, which gives usvery few degrees of freedom. As a formal econometric estimation, our estimations are far
from perfect. However, we only treat these estimates as approximations to pin down thevalues for ϕA and ϕZ that are consistent with the evolution of the Japanese TFP and R&D
labor inputs. The estimates yield reasonable values of ϕZ = 0.2225 and ϕA = 0.2654.
The proportion of non-pecuniary researchers, hZ , is even more difficult to ascertain.
In the data, the proportion of researchers in the Higher Education, Government, and
Non-Profit sectors for 1994-2017 averages 0.3% of the population. We assume that nonpecuniary researchers are a small number, e.g. 0.05%, of the population which roughly
corresponds to 1 out of every 6 workers in basic research. Importantly, although qualitatively the existence of NPRs is important, quantitatively the actual value of hZ has little
impact in most simulations below. This is because the value of hZ only directly affects the
position of the low growth BGP, but it has a very minor effect on the location of the middle(unstable) and high BGPs. The exception to this is the policy experiments in Section 4
as they analyze how to escape from the low BGP. For the other simulations, the economy
starts from the high BGP and the main concern is whether or not the economy crosses
the demarcation region leading to convergence to the low BGP. Most results below arereported using hZ = 0.05%. For the policy experiments, we analyze the sensitivity of theresults to more extreme values with hZ = 0.001% and hZ = 0.15%.

γT F P
γGDP/H
Css /YssIss /Yss
Rf,ss
Rk,ss
Req,ssReq − Rf
P ERssHA,ssHZ,ssW product
W research

Model
high
unstable
1.6879 0.5407
2.6498 0.8461
0.5070 0.5270
0.2550 0.2350
3.8316 2.9153
5.9082 4.9736
21.9701 21.8504
18.1385 18.9351
2.5163 3.8323
1.5674 0.7679
0.6998 0.1643
1.9755 2.0399
1.9755 2.0399

low
0.20940.32740.53320.22882.65034.703321.808119.1578
4.53960.35080.05002.05981.4005

Data
70-92 93-19
1.690 0.690
3.065 0.564
0.501 0.548
0.264 0.203
3.866 0.165

notes
growth TFP annual (%)
growth GDP/labor (%)
consumption/output
investment/output
risk-free rate annual (%)
capital return annual (%)
equity return annual (%)
equity premium
price/earnings ratio
hours for development (%)
hours for research(%)
wage (MPL) in prod/invention
wage (MPL) in basic research

Table 2: Non-Stochastic balanced growth paths in the model and data. In bold, targeted
moments.
Table 2 reports the values of selected variables from the model in the high, middle(unstable), and low BGPs compared to the data counterpart when available. In bold, wehighlight targeted moments. As mentioned above, we chose parameter values by targeting21the rate of growth of TFP, Ct /Yt and It /Yt ratios, and the risk-free rate in the high BGP.
The model yields a rate of growth of TFP in the low BGP (a non-targeted moment) of 0.21%
against the data counterpart of 0.69%. However, it is likely that the data reflects, rather
than the new low BGP, the transitional dynamics where TFP growth is slowly converging to
the low steady state. In our stochastic simulations below, under the “wretched coincidence”
scenario that pushes the economy towards the low BGP, we can calculate the average rateof growth of TFP for the 30 years following the shock. This yields a value for TFP growth of
0.75%, which is much closer to the data. Given this, the model does a good job at explaining
the TFP growth slowdown. Regarding the Ct /Yt and It /Yt ratios in the low BGP, the model
predicts the direction of the change but cannot match the increase in consumption anddecline in investment. However, in the table above, we have kept demographics constantwhereas the data will reflect the effect of population ageing. Finally, although the model
predicts a sizeable risk-free rate decline of almost 1.2pp, it cannot account for the effective
zero risk-free rate in the post-92 Japanese data since it abstracts from nominal distortions.
Regarding other moments, the equity premia on existing capital and new firms are 2.1%
and 18.1%, respectively. The capital return should be safer because it is backed by capital,
while our equity-return is the return on the startups, which should be riskier than the listedcompanies.

3.2

Deterministic dynamics

We first look at the transitional dynamics of the model with no uncertainty. Figure 6
depicts the phase diagram.34 The isocline curves for (de-trended) ln Kt and ln Zt , thetwo state variables, represent combinations of ln Kt and ln Zt such that Kt+1 = Kt (bluehorizontal line) and Zt+1 = Zt (pink dotted curves), respectively. The isocline curve forZt is highly non-linear, reflecting the vicious/virtuous research cycles and the existence ofthe lower bound of HZ,t .35 Their crossing points are the steady states. The steady statesin the de-trended model correspond to BGPs in the original model. In the area to theleft of the green line, only NPRs engage in basic research. This green line is determined
endogenously (a free boundary). Most importantly, the black line demarcates the basinsof the low and high BGPs. That is, to the right of it, the economy converges to the highsteady state, while to the left, it converges to the low steady state. Because the middle
steady state is unstable, the demarcation line must go through it. The grey arrows show
the direction and the speed of the transition dynamics.
We can look at the transitional dynamics of the model for different initial states. Figure
7 presents the case in which two different economies start with the same level of Z0 but
they have different initial capital stocks K0 . The capital-rich economy starts above thedemarcation line, whereas the capital-poor economy starts below the demarcation line.
34

Figure 6 plots the phase diagram of the de-trended endogenous state variables. Note that, unlike inthe simple neoclassical growth model with one endogenous state, our phase diagrams omit the dynamic
jump variables such as consumption and asset values. More precisely, Figure 6 is the projection of the fullphase diagram, which is 5D, on the state space, which is 2D. In the figure, the low and high BGPs appear
as sink points in the (ln Zt , ln Kt ) plane, but they are actually saddle points in the full dimension.
35
If there are no NPRs (hZ = 0), the vertical part of the Z-isocline lies at Zt = 0.

22

Non-Stochastic Case
6

5

4

3

2

1

0

-1

-2

-3

-4
-8

-7

-6

-5

-4

-3

-2

Figure 6: Phase diagram for the deterministic (de-trended) model.
The upper left panel shows the different transitions taken by these two economies (eachdot represents a quarter). The capital-rich economy will converge to the low BGP whereas
the capital-poor will converge to the high BGP. This happens because, in the capitalrich economy, the return to working in the production sector is higher and hence morelabor is allocated to the production sector in equilibrium to the detriment of the researchsectors (lower right panel). This intuition is akin to a “resource curse”where resource richeconomies may converge to a low growth equilibrium.
The upper middle panel suggests that both economies can grow fast in the early stages
as long as their initial capital is low. The lower middle panel shows that the early stage offast growth is not driven by TFP growth. Instead, because the return on capital is high,capital accumulation is the main driver of growth (upper right panel). In later periods, thelow growth economy allocates more capital and labor to the production sector, while thehigh growth one has a lower Kt /Yt ratio and a larger R&D labor share (right two panels).36
Because any economy experiences fast growth in these early stages, there is no “poverty
trap” in our model. The model delivers a “middle income trap” as a combination of (i) fast
initial growth due to capital accumulation and (ii) failure to switch the growth engine from
capital accumulation to R&D. For an economy that is not trapped, the model exhibits thetwo growth phases, as in Galor and Weil (2000) and Galor (2005).
36
The convergence speed of the model seems to be slow. As shown in the figure, at 300 quarters, the low
initial capital economy is still on its way toward the high BGP. However, the convergence time depends
on the initial state. The higher Z0 the faster convergence will happen.

23

Low and High Initial Capital

4

GDP/pop Growth Rate

6

Capital/Output Ratio

10
8

2
4

6
0
4
2
-2

2

-4
-7

0
-6

-5

-4

Equity Return

40

0
0

100

200

300

400

TFP Growth Rate

4

0

3

61

30

2

60

25

1

59

20

0
0

100

200

300

400

200

300

400

Production Labour

62

35

100

58
0

100

200

300

400

0

100

200

300

400

Figure 7: Sample paths for two economies with different initial capital stock (K0 ).

3.3

Stochastic dynamics: business cycles and growth

The relevant question from a business cycle perspective is whether temporary shocks canmove the economy between regions along the demarcation line. To do so, we introduce
stochastic shocks in the model. More specifically, we consider a financial market shock
and a demographic shock, reflecting the events in the early 1990s and the well-known
demographic changes that Japan has experienced in the past 30 years. We pay special
attention to the depth and duration of these shocks as they will be key determinants ofgrowth regime shifts. Tables 3 and 4 describe the characteristics of these shocks which we
explain below.
The financial shock we consider is a two-point Markov process, in which the riskpremium parameters take either normal values or crisis values. Although our model
does not have a detailed financial sector, these shocks represent exogenous financial crisis
shifters.37 Relative to the baseline parameter values, we assume that all risk-premia increase by 50% during the crisis periods. We assume that the probability that an economy
in the baseline parameter values (normal times) experiences a financial crisis in the next
quarter is 1/304 (0.33%), while the probability of exit from the crisis is 1/16 (6.25%).
Hence, these shocks are temporary. These numbers imply that the average duration of thenormal times and crisis times regimes are 304 and 16 quarters (74 and 4 years), respectively. Also, the share of time of the normal periods is 95% and 5% for financial crises.
37

See, for instance, Cochrane (2011) and Di Tella and Hall (2022).

24

These numbers roughly match with the results of Tambakis (2021) based on Laeven andValencia (2020) data. See also Table 1 of Boyd et al. (2019).
For the demographic shock, as discussed in Section 3.1, to measure the change in theshare of inactive population in the labor market (hO = HO,t /Ht ) we take into accountthat female labor force participation has increased substantially throughout the periodof analysis. We set the demographic surprise consisting of a simultaneous decline in the
population growth rate (γH,t ) from 0.73% p.a. (1970-1992) to 0% p.a. (since 1993) andan increase in the share of non-active workers from 38% to 41%. The calibration based on
the data for the post-1993 period may underestimate the magnitude of the demographic
shock. We will discuss the implications of a more severe demographic shock later.
We treat this demographic shock as a permanent “MIT shock”. I.e. agents do not
anticipate it in their optimization and, once it happens, it is perceived to be permanent.
Thus, the corresponding Markov Transition Matrix for the demographic shock is the identity matrix. These demographic changes are often taken as fully anticipated trends when
explaining the Japanese growth slowdown (see Otsu and Shibayama, 2016 among others).
However, this is far from what has happened in recent Japanese history. Government
statistical offices have consistently failed to predict it and hence we treat it as a surprise.38
Table 3: Financial Shock and Demographic Shock

cK
cv
cZ

Risk-premium Shockcapitaldevelopment
basic researchnormal regime
2%
20%
20%

crisis regime
3%
30%
30%

hO
γH

Demographic Shocknormal regime aging regime
1− employment share
38%
41%
pop. growth rate
0.73%
0.00%

Combining these two types of shocks, the model contains a four-point Markov shock process.39 For convenience, we label these regimes in Table 5. In the baseline regime (regime 1),
no shocks affect the economy. In the aging regime (regime 2) the economy is only subject to(permanent) demographic shocks. In the financial shock regime (regime 3) only temporary
financial shocks affect the economy. Finally, regime 4 is labelled “wretched coincidence”
38

As Nishimura (2012) remarks: “The popular perception that demographic change is in general predictable is based on this law of large numbers. Unfortunately, this perception is not always true, or to
put it bluntly, not true in many instances. Take Japan for example. Between the 1970s and early 2000s,
the total fertility rate forecasts regularly turned out to be wrong and were consistently revised down. The
government repeatedly published its forecast in which the decline in the fertility rate was declared to be
only temporary and the birth rate expected to rise again soon. Similarly, life expectancy forecasts haveshown that the actual figures consistently exceeded the forecasts.”
39
The entire Markov transition matrix is the Kronecker product of the two transition matrices.

25

Table 4: Markov Transition Matrices
Demographic Shockfrom\to base aging
base
1
0
aging
0
1

Financial Shockfrom\to basecrisisbase
303/304 1/304
1/16
15/16
crisisas both (temporary) financial and (permanent) demographic shocks hit the economy simultaneously. Note that the baseline shock regime is very close to the non-stochastic caseabove, although it is not the exactly same because agents rationally anticipate the risk ofshocks in the future.
Table 5: Allocation of panels in the figures below
financial\demobaseline
crisisbaseline (pre 1992)
baseline (regime 1)
financial shock (regime 3)

aging
aging (regime 2)
wretched coincidence (regime 4)

Corresponding to these four shock regimes, there are four phase diagrams, each ofwhich summarizes the dynamics of the state locus for each shock regime. See Figure 8.
Importantly, even if a shock hits the economy, it does not affect the the state of an economy
immediately. The locus of the endogenous state variables (ln Zt and ln Kt ) never jumps. In
different shock regimes, the state of the economy moves gradually in different directions.
In Figure 8, both shocks expand the distance between the upper and the lower Ushaped parts of the Z-isoclines. They only have a very marginal impact on the K-isocline(blue solid line). In all four panels, to make the relative positions of the isoclines clear,
we plot three red circles denoting the BGPs in the non-stochastic case. Given our choiceof parameters, the only qualitative change in Figure 8 between the four regimes is thenumber of stationary points. In shock regimes 1 and 2 (upper panels), there are threecrossing points. Hence, as long as the economy stays to the right to the demarcation line
(solid black line), it will move toward the high BGP. On the other hand, there is only one
stationary point in regimes 3 and 4 implying that, starting from any point in the state
space, the economy moves toward the low stationary point.
In the following sub-sections, we investigate the dynamic responses of the economy toshocks under these different scenarios.

3.4

Temporary financial shocks only

We now study the dynamic effects of financial shocks. We simulate the model under thefollowing scenario: (i) the economy starts from the high BGP of the baseline regime (top left
panel of Figure 8), (ii) a financial shock hits the economy and lasts 40 quarters (bottom left
26

shock regime: 1

6
4

4

2

2

0

0

-2

-2

-4
-8

-7

-6

-5

-4

-3

-4
-8

-2

shock regime: 3

6

4

2

2

0

0

-2

-2

-7

-6

-5

-4

-7

-3

-4
-8

-2

-6

-5

-4

-3

-2

-3

-2

shock regime: 4

6

4

-4
-8

shock regime: 2

6

-7

-6

-5

-4

Figure 8: Phase diagrams for the 4 shock regimes. Regime 1: baseline. Regime 2: demographic shock only. Regime 3: financial shock only. Regime 4: financial and demographic
shocks. The red circles in all panels show the position of the BGP in the non-stochastic
case.

panel) and, (iii) it then returns back to the baseline shock regime (top left panel). Figure
9 shows the resulting impact of the shock. In all stochastic simulations throughout thissection, GDP growth and the equity market indicators have sharp spikes. They are simply
the artifact of fully flexible labor mobility. Under imperfect labor mobility, the effects of ashock should be spread out over several periods.
We have selected a length of 40 quarters, which is 2.5 times longer than the average crisis
duration in our benchmark calibration assumption, to effectively capture the characteristicsof the Japanese 1990s financial crisis. In Japan, the crisis is generally believed to have lasted
from 1991/92 to 2002. The initial stock market crash occurred at the start of 1990, but thecrisis did not fully impact the entire financial system until 1992. The crisis reached its peakin 1997/1998 with the failure of four major traditional financial institutions. To address
the crisis, the Koizumi Administration introduced the “Financial Revitalization Program”
in October 2002, aiming to halve the ratio of non-performing loans in major banks by fiscal
year 2004. However, significant corporate failures had already ceased by 2001. See Table
D.4 in Appendix D for more details.
Figure 9, under our parameter setup, illustrates that the crisis concludes before theeconomy crosses the demarcation line, resulting in a return to the high BGP (bottom
left panel). The figure also depicts the evolution of TFP and GDP growth rates, the27

4Financial Shock (40 qtrs)

TFP and GDP Growth Rates

Capital/Output Ratio

6

10

62

4

8

60

2

6

58

0

4

56

-2

2

54

3

2

1
-7

-4
-6

-5

-4

Normal Period after Shock

4

0
0

20

100

Price/Earnings Ratio

200

0

40

2

15

30

1.5

10

20

1

5

10

0.5

0
200

0

100

52
200

R&D Labour

3

2

1
-7

0
-6

-5

-4

0

100

0

100

200

Figure 9: Response to a benchmark (temporary) financial shock of 50% risk premia increase
with 40 quarters duration.

capital/output ratio, price/earnings ratio, and R&D labor allocation during the transition.
Following the financial shock, although TFP growth slows down, it eventually returns to its
high BGP level, indicating that the slowdown is only temporary. In essence, our benchmark
parameterization suggests that the temporary shock does not have any permanent growtheffects.
However, it is crucial to consider significant parameter uncertainty. For instance, theactual length of the financial crisis may exceed the assumed 40 quarters (lost decades vs.
lost decade), and the depth of the crisis may be greater than our presupposed 50% increasein the risk-premium. The latter is particularly challenging to determine accurately. Thus,
we calculated the required duration of the crisis for different depths of the risk-premium
shock, as presented in Table 6. The deeper the shock, the shorter the necessary durationof the crisis. For instance, if the shock size reaches 90% (almost doubling risk premia), a
reasonable duration of 45 quarters could lead to permanent effects on the rate of growthof TFP. Figure 10 illustrates a scenario with a duration of 60 quarters and a risk-premium
shock of 50%. The economy crosses the demarcation line and the growth slowdown becomespermanent as, even after the shock vanishes, it converges to the low growth BGP.
There are two main implications in this subsection. First, a temporary shock can
at least potentially have a permanent effect on TFP growth (super-hysteresis) even withreasonable length and depth. Second, for the case of Japan in the 1990s, although wecannot rule out a regime shift, our benchmark calibration indicates a financial shock alone28

Table 6: The Required Length of Financial Shock to Generate Permanent Effects
depth of financial shock
length of financial shock

4

Financial Shock (60 qtrs)

30%
91 qtrs

50%
60%
60 qtrs 50 qtrs

TFP and GDP Growth Rates

70%
47 qtrs90%
45 qtrs

Capital/Output Ratio

6

10

62

4

8

60

2

6

58

0

4

56

-2

2

54

3

2

1
-7

-4
-6

-5

-4

Normal Period after Shock

4

0
0

20

100

Price/Earnings Ratio

200

0

40

2

15

30

1.5

10

20

1

5

10

0.5

0
200

0

100

52
200

R&D Labour

3

2

1
-7

0
-6

-5

-4

0

100

0

100

200

Figure 10: Response to a (temporary) financial shock of 50% risk premia increase with along 60 quarters duration.

may not be enough to explain the TFP slowdown observed in the data.

3.5

Demographic shock only

We now turn to the demographic shock. Since the shock is permanent, there is no theoretical interest on whether this shock can generate super-hysteresis. Our main interest here
is whether demographic shocks alone are enough to trigger a growth slowdown consistent
with the data.
As discussed above, we assume that the population growth rate γH,t declines from 0.72%
to 0% and the the share of inactive workers hO increases from 38% to 41%. The latter isparticularly important, because (i) a smaller labor force simply squeezes R&D labor (aswell as production labor), and because (ii) the value of a firm is the reward to successful
R&D, a smaller economy size directly implies a smaller reward to R&D.
We analyze the economy starting from the high BGP in regime 1 and then moving29

to regime 2 (demographic shock only) permanently. Figure 11 shows the results. In this
scenario, crucially, the aging shock regime still retains the high BGP though its position
slightly moves to the left. To the extent that the high stationary point moves to left,
adverse demographic shocks have a permanent but quantitatively small negative impact
on R&D and TFP growth. In this experiment, R&D labor decreases but not significantly(bottom right panel). It is production labor that is most affected by the demographic
shock. This is mitigated to some extent by a higher level of capital stock (top right panel).

4

Normal Shock Regime

TFP and GDP Growth Rates

Capital/Output Ratio

6

10

62

4

8

60

2

6

58

0

4

56

-2

2

54

3

2

1
-7

-4
-6

-5

-4

Permanent Pop/Demo Shock
4

0
0

20

100

Price/Earnings Ratio

200

0

40

2

15

30

1.5

10

20

1

5

10

0.5

0
200

0

100

52
200

R&D Labour

3

2

1
-7

0
-6

-5

-4

0

100

0

100

200

Figure 11: Response to a permanent demographic shock.

However, we do not rule out the possibility that demographic shocks are strong enoughto eliminate the high BGP. There are two reasons for this. First, as discussed in Section 2.6,
there is model uncertainty. We assume that households do not internalize future population
changes. If, instead, households take the future family size into account, the shift of theZ-isocline will be larger, potentially leaving only the low BGP in the state space. Second,
there is parameter uncertainty. Our demographic shock assumptions may underestimate
the true extent of demographic surprises. If so, demographic shocks alone could be to cause
a permanent shift of the BGP.

3.6

Combining financial and demographic shocks

Having analyzed the two shocks separately, we now turn our attention to the case of the“wretched coincidence” regime. Both financial shocks and surprise demographic eventsaffected the Japanese economy from the early 1990s. We thus consider whether a combi30

Fin & Demo Shocks (40 qtrs)
4

TFP and GDP Growth Rates

Capital/Output Ratio

6

10

62

4

8

60

2

6

58

0

4

56

-2

2

54

3

2

1
-8

4

-4
-6

-4

Permanent Demo Shock

0
0

20

100

Price/Earnings Ratio

200

0

40

2

15

30

1.5

10

20

1

5

10

0.5

0
200

0

100

52
200

R&D Labour

3

2

1
-8

0
-6

-4

0

100

0

100

200

Figure 12: Temporary financial shock of 40 quarters and a 50% increase in risk-premia
combined with a permanent demographic shock.

nation of both shocks could explain the observed growth slowdown for reasonable shock
depth and duration. In particular, we consider a scenario where the economy starts in thebaseline regime (regime 1) and then moves to the “wretched coincidence” regime (regime4) for 40 quarters, and finally moves to the aging regime (regime 2). That is, it faces a temporary financial shock and a permanent demographic shock with the same characteristics
as above. As we will see, in this case, the plausibility of both shocks moving the economy
beyond the demarcation like is much less controversial.
Figure 12 shows the evolution of the key variables under consideration. The top left
panel is the phase diagram of the wretched coincidence regime. Because the low BGP
stationary point is the only attractor, starting from the high BGP, the state locus moves
towards it. In this case, the economy moves faster towards the low growth BGP. Note that,in order for the economy to reach the demarcation line, it takes less than 30 quarters (7years). This is shorter than the consensus duration of the 1990s financial crisis in Japan.
After the financial shock disappears, the demographic shock remains. Hence, the phase
diagram shifts to shock regime 2 as shown in the bottom left panel (which is the same as
regime 2 in Figure 8). Although there are two stable stationary points in the aging regime,
because the economy is in the basin of the low stationary point, it never goes back to thehigh stationary point.
In the bottom left panel, the red “+” shows the state of the economy 120 quarters(30 years) after the initial shock. That is, if the shocks hit in 1991/1992 it would display

31

where the economy is in the transition path in 2021/2022. As discussed in Section 3.1, theaverage TFP growth rate for these 30 years in this simulation is 0.75%, which is close to
the average TFP growth rate for 1992-2019 in the data.
The takeaway from this exercise is that, even though there may be uncertainty about
the shock duration and depth, the combination of temporary financial and permanentdemographic shocks is far more likely to be enough to move the economy into the low
growth trap for a wide range of parameterizations.

4“Big Push” Policies

We now study the potential for fiscal policies to push the economy from the low to the high
BGP. That is, we study which “big push” policies are effective to help the economy escapefrom the low growth regime. We will consider the following “big push” instruments:• Changes to the corporate tax rate (τD ).
• Changes to the labor subsidy in the development sector (τA ).
• Changes to the labor subsidy in the basic research sector (τZ ).
• Changes to Government spending (G).
Given the tax rates calibrated in Table 1, we look at the shares of R&D financedby the Government in the data and in the model. Table 7 shows the results. Total
R&D expenditure in Japan in 2021 was 3.2% of GDP of which 2.5% corresponds to theprivate sector and the rest to non-profit organisations including public research institutions,
Higher Education, and other non-profit institutes. The government share of total R&D is
14.8%. However, most of it is concentrated in the non-profit sector. Only 0.8% of private
corporations research is funded by the government in contrast to 67.6% for the non-profit
sector. In the model, we set 1% and 68% for the government funding shares.
Table 7: Japan R&D Spending 2021 data and model outcomesData
as % of GDP
govt funding sharegovt funds as % of GDP

Total
3.2%
14.8%
0.48%

Model (baseline)
govt funding
govt funding sharegovt funds as % of GDP

Low BGP
Develop. Basic R
1.0%
68.0%
0.004%
0.02%

32

Private
2.5%
0.8%
0.02%

Non-Profit
0.7%
67.6%
0.45%
High BGP
Develop. Basic R
1.0%
68.0%
0.02%
0.51%

Next, we introduce a shock to the tax/subsidy rates one by one with the economy
starting from the low BGP with a non-active population ratio of 41% as in the post-1992
period. The change in tax/subsidy is perceived to be unexpected and permanent. However,
if and when the economy crosses the demarcation line, the government can revert to theoriginal (i.e., current) policy, which is unanticipated by the private sector. As discussed in
Section 3.6, even though the policy change is temporary, the economy would still converge
to the high BGP. For this analysis, to make the numerical experiments meaningful, thehigh BGP must exist even after the permanent demographic shock. Recall that if thedemographic shock is stronger than what we suppose, the high BGP may disappear. If so,
after a temporal support policy, the economy converges to the low BGP. In this case, thegovernment cannot revert the tax/subsidy rates to the original level.
The results are presented in Table 8. We reduce corporate tax, increase subsidies orgovernment spending by different amounts and present the number of quarters required tocross the demarcation line. The picture that emerges is that neither government spending
nor subsidies to the product development sectors are able to push the economy towards thehigh BGP. In contrast, both corporate tax cuts or basic research subsidies can be effective.
Note, however, that the corporate tax cuts required are very large and, even when zero,
it would take more than 13 years of zero corporate tax to achieve a change in regime.
Subsidies to basic research shift the economy faster, although they would imply almost
all basic research being subsidized by the government. Government spending is the least
effective mainly because the positive effect of spending on corporate profits is outweighedby the reallocation of resources toward the final production sector and away from R&D
sectors. Corporate tax cuts and the subsidy to product developments both support R&D
but the former is better than the latter because subsidies affect directly the development
sector, not the basic research sector. Recall that the corporate profits after tax are split
between the two R&D sectors.
Table 8: Period required to escape the low BGP after the policy change
Fiscal policy instrument
Corporate tax
Periods (qtrs)
Subsidy to basic research
Periods (qtrs)
Subsidy to development
Periods (qtrs)
Govt. spending
Periods (qtrs)

baseline
40%
68%
1%
18%

Policy Change/qtrs
15% 10% 5% 0%
∞
93
68
54
78% 83% 88% 93%
∞
53
26
14
21% 41% 61% 81%∞∞∞∞
23% 28% 33% 38%∞∞∞∞

Unlike in the case when the economy starts from the high BGP, the proportion of
NPRs will now be important to determine the success of different policies. This is because
it regulates the location of the low BGP in the state space. The larger the proportion33

of NPRs, the closer to the demarcation line and hence the more likely are policies tobe successful. Given the uncertainty about this proportion in the data, we carry out asensitivity analysis for different values of hZ in Appendix E. The main conclusion fromthis exercise is that the ranking of policies remains unchanged as expected. Practically,
however, parameter uncertainty about hZ does not appear to make a large quantitative
difference.
It is perhaps more important to recognize that the Japanese economy in the early 2020s
is likely to still be in transition towards the low BGP (see Figure 12). This implies thatthe economy may be closer to the demarcation line than what has been assumed above
and the simulation results starting from the low BGP may be too pessimistic. Table E.5
in Appenxid E shows that, if the subsidy to basic research is increased to 78% in 2022,
it would only take 4.5 years for the economy to cross the demarcation line. Interestingly,
in this exercise, the corporate tax cut is as effective as the subsidy to the basic researchsector. Clearly, an important conclusion from this analysis is that the earlier the policy isimplemented, the better.
Finally, we look at the dynamic adjustment of the economy to basic research andcorporate tax shocks. We focus on the evolution of GDP and the cost of the policy interms of net government revenues given the calibrated parameters for tax policies. We
analyze two cases. First, an increase in the basic research subsidy from 68% to 83% for
60 quarters. Second, a cut in the corporate tax rate from 40% to 5% for 80 quarters. For
both scenarios, the starting point is the low BGP. In the long-run, both policies have thesame impact on government revenues minus subsidies: 0.54% of GDP per period. Because
both policies are successful in pushing the economy to the high BGP, both yield a long-run
growth rate gap of approximately 1.6%.40 Although the cost is not negligible, the returnis orders of magnitude larger.
Figure 13 presents the transition of the economy for the research subsidy shock in termsof GDP level (upper left quadrant), R&D expenditure (upper right), GDP growth (lowerleft), and government budget as a percentage of GDP (lower right). After 50 years, compared with the benchmark with no policy change, GDP is 50% higher. The time variationof the net cost in the lower right panel is almost entirely explained by the increase in R&D
subsidies. Even after the support policy ceases, the volume of subsidies remains higher
than in the baseline because there is more R&D activity in the high BGP. Importantly,
however, most of the incremental subsidy cost is offset by the increase in tax revenues dueto the higher growth rate.
Figure 14 for the 80 quarters cut in corporate taxes has similar results. That is, in thelong-run, with relatively moderate budget cost, the successful policy generates a very large
increase in GDP. The main difference is the budget cost in transition which is larger in thiscase.
In summary, there are four main lessons from these policy exercises. First, in order topush the economy back to the high growth BGP, fiscal policies need to be substantial in size
40

This includes the indirect effect of higher capital accumulation. Note also that this number is different from the gap in γGDP/H in Table 2 because, in this section, we start off with adverse demographic
conditions.

34

GDP, level

2

1
1.5

0

50

100

150

new policy
baseline

3
2
1

1
200

0
0

GDP, growth rate

6

13

50

100

4
2

150

200

Major Revenues - Subsidies
new policy
baseline
gap, RHS

new policy
baseline% of GDP

growth rate, %% of GDPbase year = 1

1.5

Resouces to R&D

4

new policy
baselineratio, RHS

3

12

2

11

1

0
-2

10
0

50

100

150

200

0

50

100

150

0
200

Figure 13: Dynamic adjustment to an increase in τZ (subsidy to basic research) to 83% for
60 quarters.
and sustained for long periods. However the actual cost is much lower than at face value
if the increase in tax revenues due to higher GDP growth is taken into account. Second,
the sooner the better. That is, as the policy response is delayed, the cost increases as
the economy moves away from the threshold above which it converges to the high growth
regime. Third, the government must carefully choose the policy instruments. Traditional
public expenditure is neutral for R&D activities. The subsidy to basic research appears tobe quantitatively most effective. Interestingly, the subsidy to product development is noteffective. Fourth and most important, growth rate effects would dwarf almost any level
effects. The cost of the big push policy that the model delivers is substantial, but its returnis orders of magnitude larger.

5

Conclusions

Standard macroeconomic models assume that, after temporary shocks, the economy either
experiences only a temporary loss of output or this loss is permanent but its rate of growth
returns back to pre-shock levels. Evidence suggests that, after prolonged or deep financial
crises, the rate of growth remains persistently low. This phenomenon has been labelled“super-hysteresis”. We develop a model where temporary financial shocks can lead topermanent changes in the rate of growth of total factor productivity (TFP).
In the model, basic research and product development are treated as separate sectors.
In order to generate basic research ideas, researchers have to match with the existing
stock of knowledge. Product developers also have to match with this knowledge stock
to produce new varieties. This creates search externalities between the two sectors thatgenerate vicious and virtuous R&D cycles. The model has a unique equilibrium path but
multiple balanced growth paths (BGPs). There is a threshold below (above) which the35

GDP, level

2

1
1.5

0

50

100

150

1
200

2

0
0

13

new policy
baseline

2

50

100

150

0
-2
50

100

150

200

7
6

11

5

new policy
baseline 4gap, RHS

10
9

3

8

2

7

1

6
0

200

Major Revenues - Subsidies

12% of GDP4

new policy
baseline

3

1GDP, growth rate

6

growth rate, %% of GDPbase year = 1

1.5

Resouces to R&D

4

new policy
baselineratio, RHS

0

50

100

150

0
200

Figure 14: Dynamic adjustment to a reduction in τD (corporate tax) to 5% for 80 quarters.
economy converges to the low (high) balance growth path (BGP). Starting from the high
BGP, if a sufficiently deep or long-lived negative shock pushes the economy beyond thisthreshold, it will converge to the low growth regime even after the shock vanishes.
We analyze the model in the context of the Japanese growth slowdown where, after
the early 1990s crisis, TFP growth declines substantially and R&D indicators deteriorate.
Quantitatively, the model can explain well the TFP growth decline in the post-1990s period.
After a financial shock, the Japanese economy can transit between these BGPs, generating“super-hysteresis” in TFP. However, for these shocks to be sufficient to tip the economy
into the low growth equilibrium, they have to either be very persistent or very large.
In the Japanese case, however, the financial crisis coincided with perverse demographic
dynamics which we term the “wretched coincidence”. When we take both demographics
and the financial crisis into account, the duration and depth properties of the financial
shock required to push the economy to the low growth regime are more realistic.
Finally, we investigate “big push” policies aimed at moving the economy back to thehigh growth regime. We show that the sooner the policy happens the better and, to be
successful, any policy needs to be sustained for a prolonged period. Government spending
shocks are unable to achieve a transition to the good regime. Corporate tax and, especially,
subsidies to the basic research sector are a effective in the sense that they trade a finite
period expenditure increase for a permanent increase in the growth rate of the economy. In
transition, the required budget cost may be non-negligible. However, because tax revenues
grow with the level of output, the long run cost of supporting basic research is low as one
day the government may tax it.

36

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39

Appendix
A

De-trending and full set of equilibrium equations

To compute the model solution, we need to express the model in stationary form. To doso, we first collect the equilibrium equations and model variables and identify which ofthese variables are not stationary. We then divide them by their corresponding growth
factors. Most quantities share the same growth factor, while most prices are unchanged,
but there are many exceptions. To maintain consistency, the equilibrium equations arealso converted consistently. Table A.3 presents all de-trended equilibrium equations, andTable A.1 presents all the definitions of the de-trended variables. De-trending is, of course,
not an arbitrary process. If we substitute the de-trended variables into the de-trended
equilibrium, we must be able to recover the original model with the raw variables. Finally,
to interpret the simulation results, we need to reverse the de-trended variables into theoriginal variables. We call this process reverse-detrending. We denote de-trended variables
with a tilde “X̃”.
In our representation of the model before de-trending (i.e., with the raw variables),excluding the definition of the stochastic discount factor, there are 6 dynamic variables
and 12 static variables (we treat the stochastic discount factor, SDF, as a short-handnotation). Among the 6 dynamic variables, three are state variables (Kt , Zt and At ). Afterde-trending, there are only 5 dynamic variables, of which the states are K̃t and Z̃t . In
the de-trended system of equations, instead of At , we use its growth rate γA,t+1 = At+1 /At(which is observable at time t). Hence, there are 13 static variables, including γA,t+1 . There
are two exogenous variables in the deterministic version of the model.

40

Table A.1: Detrended Variables
Short Name

Definition

Growth Factor

capitalresearch stock
product designK̃t = Kt /Γk,tZ̃t = Zt /Aκt Z(At )Γk,t = ξt−1 Aφt A Ht−1
unchanged (κZ = 0)(disappears)

asset value, research stock
asset value, product design
consumption

ṼtZ = VtZ /ΓV Z,tṼt = Vt /ΓV,tC̃t = Ct /ΓtΓV Z,t = ξt Aφt A −κZ Ht
ΓV,t = ξt AtφA −1 HtΓt = ξt Aφt A Ht

stochastic discount factor

Λ̃t,t+1 = Λt,t+1

unchanged

outputwage
return on capital
dividends
investment
labor, production
labor, basic researchlabor, product design
research stock, newproduct design, newsuccess rate, basic research
success rate, product designỸt = Yt /ΓtW̃t = Wt /Γw,t
R̃K,t = RK,tD̃t = Dt /ΓV,tI˜t = It /Γt
H̃Y,t = HY,t /Ht
H̃Z,t = HY,t /Ht
H̃A,t = HY,t /HtZ̃tnew = Ztnew /Aκt Z/At= Anew
Ãnew
t
t
ZH
ZH
π̃t = πt / (AκZ /Ht )π̃tAH = πtZH / (A/Ht )Γt = ξt Aφt A Ht
Γw,t = ξt Aφt A
unchanged
ΓV,t = ξt AtφA −1 HtΓt = ξt Aφt A Ht
H̄t
Ht
Ht
unchanged (κZ = 0)AtAκZ /Ht
A/Ht

growth rate of At

γA,t+1 = At+1 /At(instead of At )Table A.2: Exogenous Variables and Short-Hand Notations
Variable Name

Definition

growth rate of population Ht
growth rate of exo technology ξttγH,t = HH̄t−1ξtγξ,t = ξt−1

(short-hand notations)exogenous productivity
adj. for research stock value
adj. for prod. design asset value

41


α
ζ̃Y,t = γξ,tηγH,tγV Z,t = ΓV Z,t /ΓV Z,t+1
γV,t = ΓV,t /ΓV,t+1

Table A.3: Detrended Equilibrium Equations
Short Name

Equationlom K̃
lom Z̃
(lom Ã)K
K̃t+1 (γA,t+1 )φA = I˜t + γ1−δK̃tξ,t γH,tκZ
new
Z̃t+1 γA,t+1 = Z̃t + (1 − δZ ) Z̃t(γA,t+1 = Ãnew+ (1 − δA ) becomes static)tcapital return

1 = Etnoi
R̃K,t+1 + (1 − δK )
h
i
Λ
γ
Ãnew
Z
ṼtZ = (1 − sA ) Ṽt Z̃t + Et t,t+1cZV Z,t+1 (1 − δZ ) Ṽt+1tihΛt,t+1 γV,t+1
(1
−
δ
)
Ṽ
Ṽt = D̃t + Et
A
t+1
cv
−σ

−σ
φA
1+τC,t (γξ,t+1 γA,t+1 )C̃t+1Λ̃t,t+1 = 1+τC,t+1
βγH,t+1C̃

(26′ )

production
marginal product of labor1−αỸt = ζ̃Y,t K̃tα H̃Y,tW̃t = (1 − α) H̃Ỹt(19′ )(20′ )

marginal product of capital
dividends
goods market clearing
labor market clearing

Ỹt
RK,t = α̃α (γξ,t γH,t ) K̃t
D̃t = (1 − τD ) (1 − α̃) αỸtỸt = C̃t + I˜t + G̃t
1 − hO = H̃Y,t + H̃A,t + H̃Z,t(21′ )(9′ )(28′ )(1′ )basic research, FOC
product develop, FOC
new research output, flow
new product design, flowṼtZ π̃tZH = (1 − τZ ) W̃tsA Ṽt π̃tAH = (1 − τA ) W̃t
Z̃tnew = ωZ Z̃t1−ϕZ H̃Z,tϕA= ωA Z̃t1−ϕA H̃A,tÃnewt(15′ )(18′ )(5′ )(6′ )success rate per HZ,tπ̃tZH = H̃t(13′ )

success rate per HA,tπ̃tAH = H̃(17′ )

lom Ã (now static eqn)

γA,t+1 = Ãnew+ (1 − δA )
t

(7′ )

stock value of research ṼtZ
firm value of prod design Ṽt
SDF

B

h

(25′ )(8′ )Λ̃t,t+1cKtY,tZ̃ new
Z,t
Ãnew
t
A,t

or H̃Z,t = hZ

(11′ )(10′ )(27′ )

Solution method detailWe numerically solve the de-trended equations (i.e., express all variables as functions ofstate variables) using the Euler Equation Iteration (EEI) method, which is a projection
method (a global solution method). We update the policy functions with the following
algorithm:
0. Set an initial guess for the policy functions of dynamic jump variables (Ct , Vt and
VtZ ) as functions of state variables (Kt and Zt ).
42

Z
1. Compute Ct+1 , Vt+1 and Vt+1:

• First, compute Kt+1 and Zt+1 based on guess Ct , Vt and VtZ .
To do so, do not use their Euler equations (26′ ), (11′ ) and (10′ ).
Z• Then, compute Ct+1 , Vt+1 and Vt+1
based on Kt+1 and Zt+1 .
To do so, use the same guess policy functions. Usually, interpolation is required.

2. Update Ct , Vt and VtZ .
Z
into the Euler equations (26′ ), (11′ ) and (10′ ).
• Substitute Ct+1 , Vt+1 and Vt+1

• If the resulting Ct , Vt and VtZ are different from the guess policy functions,
update the guess policy functions.
Iterate steps 1 and 2 until the errors in the Euler equations become small enough.
Because our method is a collocation method, we can reduce the Euler equation errors
on the state grids to effectively zero. Thus, we fix the state grids and, throughout theZ, Kt+1 and Zt+1 on
iteration, we compute not only Ct , Vt and VtZ but also Ct+1 , Vt+1 , Vt+1
the same state grids.
Our solution algorithm is also a time iteration method. We could interpret step 2 (EulerZ
equations) above as a backward computation from Ct+1 , Vt+1 and Vt+1
to Ct , Vt and VtZ .
′′′Note that the TVCs (26 ), (11 ) and (10 ) are derived from the Euler equations.

C

Multiple BGPs as a coordination gameTo enhance the intuition behind the idea of vicious and virtuous cycles, we can regard thesteady states (BGPs) as equilibria of a simultaneous move game between basic researchersand product developers. In Figure C.1, the blue dashed line shows HA,ss as a best response
function (BRF) for each HZ,ss . To draw it, we assume HZ,ss is arbitrarily fixed (i.e., weignore its FOC) and assume all equilibrium equations hold except for the FOC of HZ,ss .
Likewise, the black solid line is HZ,ss as a BRF for each HA,ss . There are three equilibria, of
which the middle one is trembling-hand unstable. This is a typical outcome in coordination
games.
The arrows in the figure depend on the direction of the FOC errors. For example, if
the marginal revenue product of HA,ss is higher than the wage cost in the FOC for HA,ss
at a point on the plane, the arrow is depicted such that HA,ss is increasing. We can dothis because, unlike in the textbook coordination games in which choices are discrete, both
HA,ss and HZ,ss are continuous choices.
This figure is different from the BRFs of the standard coordination game because the
BRF of HZ,ss has an inverted Z-shape (the lines are connected to the right of the graph).
This is because there is another coordination game among basic researchers. That is, forvery high or very low HA,ss , there is only one HZ,ss that satisfies its FOC. However, for
HA,ss in the middle, there are three HZ,ss that satisfy the FOC, of which the middle one
43

-4.5
-5
-5.5
-6
-6.5
-7
-7.5
-8
-8.5
-6.5

-6

-5.5

-5

-4.5

-4

-3.5

Figure C.1: Steady State Selection as a Simultaneous Move Game(shown as the downward sloping part of the black solid line) is trembling-hand unstable.
To understand this, fix HA,ss . If all other basic researchers work at the minimum level hZ ,the resultant research stock Zss is too low, and hence it is optimal for each basic researcher
to work at the minimum level (shown by the horizontal part of the solid line). Likewise, if
everybody chooses a high HZ,ss , it is best for each researcher to choose a high HZ,ss (shownas the upward-sloping part of the solid line).

D

Japanese major bankruptcies

Table D.4 presents a short history of major Japanese corporate failures between 1989 and2019. The table displays the year, month, and name of the company as well as its main
sector. The last column displays the rank, which is based on the total debt size at thetiming of failure. The data is extracted from Teikoku Databank (2019). Note, however, thisdata does not include some major bankruptcies because several financial institutions were
rescued and, hence, they were not technically classified as bankruptcies. Thus, we added
some selected failures without ranking. The four bold face cases were the most prominent.

44

Table D.4: Major Corporate Failure in Heisei Era (1989-2019)
year mth Company Name
1996
10 Nichiei Finance
1997
4
Crown Leasing
1997
4
Nissan Life1997
11 Sanyo Securities
1997 11 Hokkaido Takushoku Bank
1997 11 Yamaichi Securities
1997
11 Tokuyo City Bank
1997
12 Toshoku Ltd.
1998
9
Japan Leasing and Sales Co.
1998 10 Long-Term Credit Bank of Japan
1998 12 Nippon Credit Bank
2000
5
Life Ltd.
2000
7
Sogo
2000
9
Kyouei Life2000
10 Chiyoda Life Insurance
2001
3
Tokyo Life
2001
9
Mycal Corp
2008
9
Lehman Brothers
20101Japan Airline
2010
9
Takefuji
20109
Incubator Bank of Japan2017
6
Takata45

Type of Business
rank
Consumers finance
10
Leasing
8Life insurance
Securities
Bank
Securities
Bank
Food wholesales
15
Leasing4Bank
Bank
Instrument sales
12
Retailer
13
Life insurance
1
Life insurance
3
Life insurance
11
Retailer
7
Investment bank
2
Airline5
Consumers finance
6
Bank
14
Auto parts manufacturing
9E

Sensitivity analysis of policies

Table E.5: Period to Escape from the low BGP for different values of the share of NonPecuniary Researchers
Fiscal Policy
corp tax
hZ = 8e−4
hZ = 5e−4
hZ = 2e−430yrs after
45yrs after

Policy Change/qtrs
15% 10% 5% 0%
52
41
34
29
∞
93
68
54
∞
∞
∞
∞
17
14
12
11
36
29
25
22

Fiscal Policy
subsidy/develop
hZ = 8e−4
hZ = 5e−4
hZ = 2e−430yrs after
45yrs after

Policy Change/qtrs
41% 61% 71% 81%
∞
∞
∞
253
∞
∞
∞
∞
∞
∞
∞
∞
∞
104
80
77
∞
∞
229 154

subsidy/researchhZ = 8e−4
hZ = 5e−4
hZ = 2e−430yrs after
45yrs after

78%
62
∞
∞
18
41

govt expend.
hZ = 8e−4
hZ = 5e−4
hZ = 2e−430yrs after
45yrs after

18%
∞
∞
∞
∞
∞

83%
28
53
∞
10
20

88%
16
26
80
6
12

93%
9
14
28
4
7

23.8%
∞
∞
∞
∞
∞

28%
∞
∞
∞
∞
∞

Notes:In the benchmark parameter setting, the share of non-pecuniary researchers (NPR) in total
population is assumed to be 0.05%: hZ = 5e−4 .
If the shock hits the economy in 1992, “30yrs after” and “45yrs after” means that the
government starts the policy measure in 2022 or 2037 respectively.

46

38%
∞
∞
∞
∞
∞