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THE FALL OF THE NATURAL RATE OF INTEREST AND THE PERSPECTIVE OF SECULAR STAGNATION

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“Suppose that the short-term real interest rate that was consistent with full employment had fallen to −2% or −3% sometime in the middle of the last decade. Then, what would happen?". In this work I review the growing literature about the natural rate of interest and the specific perspective proposed by Lawrence Summers, namely the demand-side view of secular stagnation. Second, I expound in detail the model of Eggertsson and Mehrotra (2014) with three-period overlapping generations, credit constraints and downward nominal wage rigidity in order to verify the theoretical feasibility of secular stagnation in advanced economies. Third, I tweak the original model by inserting a new production function a là King and Rebelo (2000) with deterministic and stochastic productivity. This way I capture new determinants of the decline in the natural rate. I identify how the current pandemic may trigger enduring adjustments in the savings-borrowings balance, a persistently negative natural rate and macro-financial consequences intratemporally and intertemporally. Fiscal policy must take the leading role in the response.
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UNIVERSITÀ CATTOLICA DEL SACRO CUORE
SEDE DI MILANO
Facoltà di Economia
Corso di laurea in Economia – profilo in Economics
THE FALL OF THE NATURAL RATE OF INTEREST
AND
THE PERSPECTIVE OF SECULAR STAGNATION
Docente relatore: Chiar.mo Prof. Domenico DELLI GATTI
Tesi di Laurea di:
Roberto ITALIA
Matricola n. 4701777
Anno Accademico 2019/2020
2
Contents
LIST OF FIGURES .................................................................................................................... 3
LIST OF TABLES ..................................................................................................................... 4
INTRODUCTION ...................................................................................................................... 5
1 R-STAR WARS EPISODES I AND II THEORY VS EMPIRICAL EVIDENCE ........... 9
1.1 LOOKING BACK .......................................................................................................................... 11
1.2 THE NATURAL RATE OF INTEREST: THEORETICAL FOUNDATIONS AND ESTIMATES ................... 14
1.3 THE NEW SECULAR STAGNATION HYPOTHESIS: THE PHANTOM MENACE .................................. 23
2 A MODEL OF DEMAND-SIDE SECULAR STAGNATION: A FIRST APPROACH ... 34
2.1 ENDOWMENT ECONOMY ............................................................................................................ 35
2.2 PRICE LEVEL DETERMINATION ................................................................................................... 40
2.3 MODEL WITH ENDOGENOUS OUTPUT ......................................................................................... 41
2.3.1 THE STEADY EQUILIBRIA IN THE ECONOMY ......................................................................................... 46
2.4 POLICY IMPLICATIONS OF THE SECULAR STAGNATION STEADY STATE ..................................... 55
2.5 FURTHER EXTENSION WITH CAPITAL ......................................................................................... 58
3 R-STAR WARS EPISODE III - A COVID-19 MODEL OF SECULAR STAGNATION 61
3.1 BASELINE MODEL: RETURN OF PRODUCTIVITY .......................................................................... 63
3.2 “DETERMINISTIC NATURAL RATE OF INTEREST AND STEADY STATE ....................................... 68
3.3 “SHOCKED NATURAL RATE OF INTEREST AND SLOWLY TURNING ECONOMY .......................... 71
3.4 POLICY CHALLENGES OF (HARD) SECULAR STAGNATION TRAPS: A NEW HOPE? ....................... 81
CONCLUSION ........................................................................................................................ 85
BIBLIOGRAPHY .................................................................................................................... 87
SITOGRAPHY ......................................................................................................................... 91
APPENDIX .............................................................................................................................. 92
A INEQUALITY AND THE NATURAL RATE OF INTEREST ................................................................... 92
B MODEL WITH CAPITAL .................................................................................................................. 93
3
List of Figures
Figure 1.1: Central banks’ policy rates in the US, the Euro Area and Japan (percent) ............. 9
Figure 1.2: Output dynamics in the US, the Euro Area and Japan (index) .............................. 10
Figure 1.3: Nominal yields on long-term government bonds in advanced economies (10 years,
percent) ..................................................................................................................................... 11
Figure 1.4: Inflation expectations in advanced economies (annual percentage changes) ........ 12
Figure 1.5: US Treasury-Inflation Protected Securities (percent) ............................................ 13
Figure 1.6: One-sided (filtered) and two-sided (smoothed) estimates of the natural rate of
interest in the US (percent) ....................................................................................................... 18
Figure 1.7: DSGE estimates of the natural rate of interest in the Euro Area and the US (percent)
.................................................................................................................................................. 19
Figure 1.8: HLW estimates of the natural rate of interest in advanced economies (percent) .. 20
Figure 1.9: Estimates of the natural rate of interest and ex-ante short-term real rate in Japan
(percent) ................................................................................................................................... 21
Figure 1.10: Non-DSGE estimates of the natural rate of interest in the Euro Area (percent) . 21
Figure 1.11: Non-DSGE estimates of the natural rate of interest in the US (percent) ............. 22
Figure 1.12: Potential GDP estimates and actual GDP in the US, the Euro Area and Japan .. 26
Figure 1.13: Slowing population growth in advanced economies (percentage change at annual
rate) ........................................................................................................................................... 28
Figure 1.14: Slowing working-age population (aged 15-64) growth in advanced economies
(percentage change at annual rate) ........................................................................................... 29
Figure 1.15: US corporate profits as a share of GDP (billions of $/bil. of $) .......................... 30
Figure 1.16: US income and wage inequality, 1913-2014 (share of total, percent) ................. 30
Figure 1.17: Global foreign currency reserves (trillions of $) ................................................. 31
Figure 2.1: Equilibrium in the bond market ............................................................................. 40
Figure 2.2: Binding downward nominal wage rigidity (Point C) ............................................ 45
Figure 2.3: Full employment steady state (Point A) ................................................................ 51
Figure 2.4: Secular stagnation steady state (Point B) ............................................................... 52
Figure 2.5: Data versus model transition paths in advanced economies (percent) .................. 54
Figure 2.6: Effect of raising the inflation target and multiple steady states (green line) ......... 56
Figure 2.7: Expansionary fiscal policy via debt issuance in secular stagnation ...................... 58
Figure 3.1: Total factor productivity growth in advanced economies (percentage change at
annual rate, five-year moving average) .................................................................................... 61
Figure 3.2: Pre-shock full employment steady state (Point A) and secular stagnation steady state
(Point B) ................................................................................................................................... 70
Figure 3.3: Response of the “European” natural rate of interest following pandemics (percent)
.................................................................................................................................................. 74
Figure 3.4: “Shocked” natural rate of interest at time t ............................................................ 74
Figure 3.5: Disarming the shock (Point C) .............................................................................. 77
Figure 3.6: Secular stagnation trap (Point D) ........................................................................... 77
Figure 3.7: Soaring inflation (Point E) ..................................................................................... 79
Figure 3.8: Hard secular stagnation trap (Point F) ................................................................... 79
4
Figure 3.9: Effect of expansionary fiscal policy via debt issuance in a secular stagnation trap
(Point G) ................................................................................................................................... 84
Figure 3.10: Effect of expansionary fiscal policy via debt issuance in a hard secular stagnation
trap (Point H) ............................................................................................................................ 84
List of Tables
Table 1.1: Average real GDP growth rates in Japan, the US and the Euro Area from 1980 to
2024 (percent) .......................................................................................................................... 13
Table 1.2: Higher life expectancy in advanced economies ...................................................... 29
5
Introduction
Suppose that the short-term real interest rate that was consistent with full
employment had fallen to 2% or 3% sometime in the middle of the last
decade. Then, what would happen?
1
In November 2013, the American economist Lawrence H. Summers spoke these controversial
words during a conference at the International Monetary Fund. Since that thought-provoking
speech Summers and other scholars have tried to make sense of the unusually long period of
weakness in advanced economies after the Global Financial Crisis. Growth has been more
sluggish than expected and inflation has remained dormant so that the 2010s are thought of as
a lost decade like the 1930s. That has been almost paradoxical given the extraordinary, and
often solitary, interventions of major central banks and the all-time low policy rates and market
real interest rates. The ultra-accommodative monetary policy stance has slightly changed in the
last decade with no end in sight. Over time it has been clear that the United States and in
particular the Eurozone and Japan are suffering from more than the bad hangover of the Great
Recession. The malaise is even more worrying in Japan, the first country to be mired in such a
quicksand for the last twenty-five years. Reading these long-lasting episodes of
“Japanification” (read “anaemic growth”) in terms of temporary liquidity traps cannot be
satisfying. Now in 2020, to aggravate the situation, a unique large-scale shock like the
coronavirus (COVID-19) pandemics is hitting the entire world. The past, sometimes
underrated, doubts about the necessary preparation and resilience of advanced economies in the
event of a downturn are stronger.
The relevance of macroeconomic theories depends on the historical economic context and
on their ability to understand it and to formulate helpful policy recommendations. This applies
a fortiori in an age of dramatic tipping points. This thesis was born with the intention of
analysing the recent, maybe permanent, changes in the economic environment. Three questions
are at the heart of what follows. What are the reasons for this mess? Is it possible to explain it
through an economic model? How does the ongoing coronavirus crisis change the picture?
How can we explain this mess? That is the object of Chapter 1. In front of the bleak data in
advanced economies, Summers has reintroduced the old notion of secular stagnation, a Great
Depression-era idea expounded by the “American Keynes” Alvin Hansen. The so-called “new
1
Summers (2013).
6
secular stagnation hypothesis” updates and refines the insights of the late 1930s. It does so by
applying the even older concept of the “natural rate of interest” (or “r-star”), originally proposed
by the Swedish economist Knut Wicksell in 1898 and revived by one of the most prominent
New Keynesians, Michael Woodford, a century later.
2
The natural rate is the unobservable
short-term real interest rate consistent with full employment output and stable inflation. Yet, its
importance for central banks and economies is directly proportional to the uncertainty of its
value. According to Summers’ line of thinking, chronic output and inflation gaps in the current
Zero Lower Bound (ZLB) environment are due to the structural shortage of aggregate demand
and the consequent fall of the natural rate since the 1980s. In fact, several real forces have
substantially reduced the propensity to borrow and increased the propensity to save in the
developed countries. Decelerating population growth, tighter collateral constraints
(deleveraging shocks), rising intra- or inter-generational inequality, falling relative price of
investment goods have been the main culprits of these underlying changes and are unlikely to
reverse themselves. Therefore, over the years a growing literature, irrespective of the specific
approach used to measure it, has pointed out the drop of the estimated natural rate of interest
towards zero or even below in the United States, the Eurozone and Japan. What once was
considered in the academic world and in the institutions as a peaceful polar star has now become
an alarming shooting star, slightly during the Great Moderation and sharply after the financial
crisis. We are witnessing an authentic “R-Star Wars” saga. Summers asked: “What happens
then?” The key point is that, once the natural rate is permanently negative and the inflation
target is very low, the ZLB-constrained central bank cannot set the policy rate at the optimum
level balancing borrowings and savings at full employment. There is an oversupply of savings
in the economy. Not the benign decline in the real interest rate but the painful decline in output
(and income) below potential becomes the only possible adjustment. A negative natural rate
implies that episodes of high unemployment, subdued inflation and low interest rates are likely
to be more frequent and prolonged. A “new normal”, probably earlier than we think, was born
with profound implications for macroeconomic stabilisation. The Great Lockdown will have a
further heavy impact on the natural rate.
Is it possible to explain this mess through an economic model? That is the object of Chapter
2. The new secular stagnation hypothesis is still controversial in theory and in practice because
its main extreme feature is the negativity of the real interest rates.
3
At the same time, the
perspective has been very resonant in the academic world. The traditional macroeconomic
2
Wicksell (1898) and Woodford (2003).
3
Bernanke (2015b).
7
settings are inappropriate in getting demand-side secular stagnation outcomes naturally. Thus,
the theoretical deficiency has pushed two economists from Brown University, Gauti Eggertsson
and Neil Mehrotra, to fill the gap in 2014.
4
Their highly cited work, which I expound in detail,
goes beyond the previous influential literature about the ZLB.
5
In fact, the past standard
analyses are limited in their construction and in their policy prescriptions. They do not contain
enough heterogeneity to produce a permanently negative natural rate of interest. There the
variable (in gross terms) is pinned down by the consumption Euler equation, namely by the
inverse of the fixed subjective discount factor, and liquidity traps are just temporary issues.
Instead, Eggertsson and Mehrotra (2014) formalise Summers’ perspective by building a New
Keynesian framework with three-period overlapping generations, credit constraints and
downward nominal wage rigidity. With these additions, the natural rate depends on the
aforementioned fundamental forces (deleveraging cycle and secular factors), whose movement
affects the balance between borrowings and savings intratemporally and intertemporally.
Ending up in a “topsy-turvy” low growth - low inflation - low interest rates steady state is
theoretically possible because the natural rate may remain depressed for a long time. There is
no automatic recovery to full employment. A later paper of the same authors confirms
quantitatively the negativity of the natural rate in the United States and the feasibility of secular
stagnation in the United States, the Eurozone and Japan.
6
How does the ongoing coronavirus crisis change the picture? That is the object of Chapter
3. The answer to the third question cannot be fully given by the original model of Eggertsson
and Mehrotra (2014). The failure to take account of the worst recession since the Great
Depression is a drawback. The problem lies in the surprising absence of productivity slowdown
and uncertainty. Therefore, by exploiting the flexibility and the relevance of the framework in
Eggertsson and Mehrotra (2014), I prove that a tweaked version of the model can be much more
powerful and useful nowadays. To get a preview of my results, I insert a production function à
la King and Rebelo (2000) with deterministic labour-augmenting technology level and
stochastic total factor productivity. That allows for a unique distinction in the literature between
a deterministic natural rate and a “shocked” natural rate. The former prevails when the
economy performs at the trend and is in the steady state. The latter emerges when economies
shift from that steady state and are hit by unique disturbances occurring at lower-than-business-
cycle frequencies, such as pandemics or global financial crises. I determine the mechanisms
4
Eggertsson and Mehrotra (2014).
5
Krugman (1998), Eggertsson and Woodford (2003), Eggertsson and Krugman (2012).
6
Eggertsson, Mehrotra and Robbins (2017).
8
that the current COVID-19 pandemic may trigger in advanced economies, exacerbating their
pre-existing problems. The results do not change compared to the reference model. The
structure with lifecycle dynamics is adjusted so that the macro-financial effects of the adverse
event propagate for a long time on the supply and on the demand sides. The natural rate
plummets and secular stagnation may still be the painful outcome. The threat of the falling
natural rate and of “Japanification” is real. So it was before the coronavirus, let alone now.
Moreover, the analysis shows that extraordinary economic times require extraordinary and
different economic policy approaches. Major central banks have exhausted their traditional
leeway by historical standards to react to massive shocks. Their golden era is probably over,
but not their function. The leading role must be taken by the governments. In a secular
stagnation environment discretionary fiscal policy is effective to the extent that the government
exploits the relevant heterogeneity in the population. In fact, according to the financing method
there are different fiscal multipliers, which are not always positive. For sure, government
spending financed through a permanent and aggressive increase in public debt reduces excess
savings and stimulates aggregate demand. And it should be accompanied by consistent
monetary and macroprudential policies.
9
1 R-Star Wars Episodes I and II Theory vs empirical evidence
Advanced economies are living in a world characterised by low interest rates, moderate growth
and subdued inflation.
7
More than ten years ago their central banks cut sharply their policy rates
to cope with the Global Financial Crisis (GFC) and the Great Recession (Fig. 1.1). So far, they
have kept them at an all-time low. The Zero Lower Bound (ZLB) on the short-term nominal
rates has become a binding constraint for the conduct of standard monetary policy.
8
The
implementation of unconventional measures has been a solution to inject additional
accommodation but doubts about their marginal returns and macroeconomic efficacy still
remain.
Figure 1.1: Central banks’ policy rates in the US, the Euro Area and Japan (percent)
7
Henceforth, most of the data shown belong to the United States (US), the Euro Area and Japan.
8
The ZLB is the rate below which the central bank can no longer provide stimulus via interest rates because agents
would find convenient to hold cash rather than lending at a negative nominal rate. As some major central banks
have adopted slightly negative rates, it is often referred to as Effective Lower Bound” (ELB) because cash storage
and transportation costs can play a role in the trade-off. The respective authorities in the Euro Area, Switzerland,
Denmark, Sweden and Japan have all allowed rates to fall slightly below zero.
Notes: On February 16, 2016, the Bank of Japan (BOJ) began to target a deposit rate of 0.1% on one
tier of commercial-bank deposits at the central bank as its main policy rate (since September 2016
together with the yield on 10-year Japanese government bonds at around 0%). The BOJ stopped
considering its uncollateralized overnight call rate as its policy rate on April 4, 2013. Between the two
dates there are no data available because over that time the target was the monetary base.
Source:https://graphics.wsj.com/CentralBanks/#?tab=US&tab=Eurozone&tab=UK&tab=Japan
(consulted on 02/03/2020).
10
Both short-term and long-term interest rates are at historical lows. Yet, despite the ultra-
loose monetary stance, output remains below trend and inflation has barely accelerated (Fig.
1.2). From 2010 to 2019 real GDP growth has averaged 1.4% in the Asian country and in the
Euro Area, 2.3% in the US.
9
Last year was a troublesome year for real GDP growth. Japan grew
at 1%, Euro members at 1.2%, the US performed better, at 2.3%. Since 2008 underlying
inflation has been below the 2% target most of the time in Japan, the Eurozone and the United
States. In December 2019 core inflation was at 0.9% in Japan, 1.3% in the Euro Area and 1.6%
in the United States and has never been around the target in a sustained way in all the three
cases.
10
But on closer inspection the current state of advanced economies is not just the heavy
legacy of the financial crisis.
Figure 1.2: Output dynamics in the US, the Euro Area and Japan (index)
The chapter is divided in three sections. Section 1.1 gives a brief historical overview of the
decline in the real (inflation-adjusted) interest rates and the real GDP growth rates in advanced
economies. Section 1.2 introduces the concept of the natural rate of interest. Then, it provides
its derivations in the main textbook economic models and the most recent econometric
9
International Monetary Fund (2019).
10
Core inflation rate is measured by using respectively Consumer Price Index (CPI) excluding fresh food and
energy for Japan, Harmonised Index of Consumer Prices (HICP) inflation excluding unprocessed food and energy
for the Eurozone, Personal Consumption Price Expenditures (PCE) excluding food and energy for the United
States.
US
Euro Area
Notes: The vertical red lines correspond to the quarter in which the nominal interest rate was below 25
basis points for the first time in the sample. GDP (solid blue line) is indexed equal to 100 in 2007:Q1 for
the US and the Euro Area and in 1994:Q1 for Japan. GDP trend (green dotted line) is constructed using a
backward looking 10-year moving average of GDP growth.
Source: Cuba-Borda and Singh (2019).
11
estimates, which point out a downward trend. Section 1.3 presents the new secular stagnation
hypothesis, that tries to reconcile the data with the estimates.
1.1 Looking back
Falling market interest rates and weak economic growth in major industrial economies have not
been just an anomaly of the last decade due to the financial crisis. Indeed, after 2008-2009 the
dynamics has stepped up. However, since the 1980s both short-term and long-term nominal and
real rates have been experiencing a persistent decline. Japan is a case in point. There were
similar values in the 1950s and 1960s but at that time the financial world was totally different.
Monetary policy was not built upon interest rates but on monetary aggregates.
Let us look at the nominal yields on long-term government bonds (Fig. 1.3). In 1980 in the
United States the 10-year Treasury yield was around 11%. It averaged 6.7% in the 1990s and
4.5% in the first decade of the 2000s. In December 2019 it was about 1.8%.
Figure 1.3: Nominal yields on long-term government bonds in advanced economies (10
years, percent)
-2
0
2
4
6
8
10
12
14
16
18
1965-01-01
1966-06-01
1967-11-01
1969-04-01
1970-09-01
1972-02-01
1973-07-01
1974-12-01
1976-05-01
1977-10-01
1979-03-01
1980-08-01
1982-01-01
1983-06-01
1984-11-01
1986-04-01
1987-09-01
1989-02-01
1990-07-01
1991-12-01
1993-05-01
1994-10-01
1996-03-01
1997-08-01
1999-01-01
2000-06-01
2001-11-01
2003-04-01
2004-09-01
2006-02-01
2007-07-01
2008-12-01
2010-05-01
2011-10-01
2013-03-01
2014-08-01
2016-01-01
2017-06-01
2018-11-01
UK
Japan
Canada
Euro Area
US
Source: https://fred.stlouisfed.org/ (consulted on 01/15/2020).
12
As regards shorter maturities, in 1980 the yield on the 3-month Treasury securities was at
11.4%, while now it is around 1.5. The Euro Area (Germany above all), the United Kingdom,
Canada and Japan display an identical picture. The interest rates paid by businesses and
households are relatively higher but are still exceptionally low.
The pattern is only partly explained by the fall of long-term inflation expectations, which
stabilise around 2% in the late 1990s (Fig. 1.4). This is a clear manifestation of the success of
the disinflationary policies and the inflation targeting strategy.
Figure 1.4: Inflation expectations in advanced economies (annual percentage changes)
In fact, real rates, which are more relevant for economic decisions, are also low. Since the
1990s the real yields have evolved consistently down the same road as the nominal ones. For
instance, the yields on the US Treasury Inflation-Protected Securities (TIPS) have tumbled in
the wake of the crisis and have never recovered the previous levels (Fig. 1.5).
11
11
US Treasury Inflation-Protected Securities make coupon payments that are indexed to the inflation rate, so their
yields give a hint of the real interest rates. TIPS were issued for the first time in 1997, so we do not have measures
before that date.
Source: Constâncio (2016).
13
Figure 1.5: US Treasury-Inflation Protected Securities (percent)
Economic performance in terms of real GDP growth has moved in a specular way in the last
three decades (Table 1.1).
12
The sample is divided on the basis of the turning points in the path
of the real rates.
Table 1.1: Average real GDP growth rates in Japan, the US and the Euro Area from 1980
to 2024 (percent)
13
Real GDP growth
1980-1989
1990-2008
2010-2019
2010-2024
Japan
4.4%
1.4%
1.4%
1.1%
United States
3.1%
3%
2.3%
2.1%
Euro Area 14
NO DATA
2%
1.4%
1.4%
Source: International Monetary Fund (2019), https://www.imf.org/external/datamapper/datasets
(consulted in February 2020).
12
The same argument holds for real GDP per capita growth.
13
The data from 2020 to 2024 are IMF forecasts.
14
Data for 1990 and 1991 are not available.
-2
-1
0
1
2
3
4
2003-01-01
2003-08-01
2004-03-01
2004-10-01
2005-05-01
2005-12-01
2006-07-01
2007-02-01
2007-09-01
2008-04-01
2008-11-01
2009-06-01
2010-01-01
2010-08-01
2011-03-01
2011-10-01
2012-05-01
2012-12-01
2013-07-01
2014-02-01
2014-09-01
2015-04-01
2015-11-01
2016-06-01
2017-01-01
2017-08-01
2018-03-01
2018-10-01
2019-05-01
2019-12-01
5-year TIPS
10- year TIPS
Source: https://fred.stlouisfed.org/ (consulted on 01/15/2020).
14
The combination of low market interest rates and ZLB, muted inflation and sluggish growth
seems to be more than just a parenthesis of the post-crisis era in the industrialised world. Even
before the COVID-19 pandemic broke out, the forecasts for the next years were gloomy, let
alone now. Therefore, the main adverse drivers must be slow-moving secular forces.
Will the situation remain the same indefinitely? If that is the case, how does economic policy
have to be designed in order to bring economies to the so-called full employment or potential
steady state?
15
To get a full answer, we cannot simply argue that accommodative monetary
policy has been the cause of low interest rates. We need to introduce the concept of the natural
rate of interest.
1.2 The natural rate of interest: theoretical foundations and estimates
The development in advanced economies is associated with the drop in the so-called natural
rate of interest, a risk-free interest rate determined exclusively by real forces and independent
from monetary causes.
16
It is also known as r-star (denoted by ,  in the academic
literature and among professionals. We find it as the intercept term in the standard Taylor rule
that responds to inflation and output gaps:
or in gross rates:
where π* is the central bank’s inflation target,
is the output gap and the response coefficient
= 1 +> 1 (Taylor principle).
15
Despite official low unemployment rates in some countries like the United States and Japan, the behaviour of
growth and inflation signals that economic slack is not fully measured by the output/unemployment gap. For
instance, the U-6 rate, which is the unemployment rate that also includes discouraged workers no longer seeking
jobs and involuntary part-time workers, is still high. Moreover, labour force participation is persistently declining.
For this reason, it makes sense to talk about economies not being at the full employment steady state.
16
The natural rate of interest is also called neutral, equilibrium, full employment or Wicksellian interest rate.
15
The natural rate of interest is a key and at the same time controversial variable in
macroeconomics and in monetary policy. The Swedish economist Knut Wicksell (1898) was
the first to define the natural rate:
There is a certain rate of interest on loans which is neutral in respect to
commodity prices and tends neither to raise nor to lower them. This is
necessarily the same as the rate of interest which would be determined by
supply and demand if no use were made of money and all lending were
effected in the form of real capital goods. It comes to much the same thing
to describe it as the current value of the natural rate of interest on capital.
[…]
17
If it were possible to ascertain and specify the current value of the natural rate,
it would be seen that any deviation of the actual money rate from this natural
rate is connected with rising or falling prices according as the deviation is
downward or upward.
18
At the Wicksellian rate the economy is at full employment, with stable inflation and desired
saving and investment/borrowing are balanced. The extent of monetary stimulus depends on
the interest rate gap, which is the gap between the real interest rate and . Prices and economic
activity would accelerate if market rates were below the natural rate and would slow down in
the opposite case.
Therefore, the natural rate defines the current stance of monetary policy and represents a sort
of anchor for central banks because it provides the path of interest rates able to fulfil their
stability mandate. They need to steer the short-term rates looking at this blurred polar star.
Like all the other economic variables labelled as natural or potential, the natural rate of
interest is an unobservable and time-varying benchmark, that can be inferred just through the
behaviour of inflation. The abstraction of has raised several doubts in the 20th century, not
least the very existence and validity of the concept in line with the words of Wicksell.
19
The
natural rate of interest was retrieved in the context of modern macroeconomics by the New
Keynesian economist Michael Woodford in his book “Interest and Prices” (2003) and is back
to being popular amid the inflation targeting theory:
17
Wicksell (1898), p. 102.
18
Ibid., p. 107.
19
Keynes in “The General Theory of Employment, Interest and Money”, post-Keynesians and Milton Friedman
rejected the concept.
16
“The interest rate must at all time equal the Wicksellian natural rate of
interest, which may be defined as the equilibrium real rate of return in case of
fully flexible prices.
20
Given the current low rates environment and the imbalance between saving and investment,
what happens when is very low and even negative? With the 2% inflation target, central
banks cannot and will not reduce real interest rates below 2%. When the equilibrium real rate
declines towards that value, a fortiori below it, the presence of the ZLB becomes a big and
daily challenge for monetary authorities because they are not be able to replicate and reflate
the economy. Yet, economists have debated the possibility of a persistently negative natural
rate. A useful theoretical starting point to try to explain a declining natural rate is to look at the
traditional economic models. The first one is the Ramsey-Cass-Koopmans growth model with
a representative agent and effective labour.
21
The typical household intertemporal utility
maximisation problem and the resulting consumption Euler equation provide the steady state
real interest rate, which is the marginal productivity of capital net of depreciation. The natural
rate of interest can be regarded as the real interest rate in the balanced-growth equilibrium:
where σ is the intertemporal elasticity of substitution in consumption, x is the
labour-augmenting or Harrod-neutral technological change and ρ is the subjective rate of time
preference (households’ degree of impatience). The natural rate of interest varies over time
due to preference or productivity shocks. Nothing surprising. In a slowing economy (lower x),
private capital investments are likely to be less profitable, so the equilibrium interest rate is
expected to drop reflecting lower future returns. Also, an increased willingness to save
(corresponding to an increase in σ or a decline in ρ) raises the level of savings in equilibrium
and reduces the natural rate. In fact, a lower interest rate is required to stimulate higher
investments. However, given that the discount rate is positive, it is very complicated to have a
permanent negative natural rate if the economy accumulates “too much” capital stock. In the
long-run the real interest rate is such that
20
Woodford (2003), p.248
21
In the version of the model adopted the utility of each household is weighted according to its size.
17
where n is the growth rate of the population.
22
Assuming the plausibility of a persistent drop in
population and labour force and the negativity of the right hand side, the negativity of the left
hand side is very demanding. While it is low in many advanced countries, labour productivity
growth is neither sufficiently negative nor negative.
More recent analyses in the spirit of Woodford (2003) are even less helpful. In representative
agent models, such as Eggertsson and Woodford (2003), the steady state is determined by
the inverse of the fixed discount factor β of the household in the Euler equation:

where ϵ (0, 1). Eggertsson and Krugman (2012) introduce some heterogeneity in borrowing
and lending and draw the same conclusion in a model with two types of representative agents,
i.e. a patientindividual/saver and an “impatient” individual/borrower. The steady state real
interest rate is given by the inverse of the discount factor of the saver. Only temporarily the
natural rate may be negative.
The possibility of accumulating an “excessively large” amount of savings is not impossible
in overlapping generations (OLG) models. They allow for life-cycle effects of consumption and
saving decisions. In this case the economy is dynamically inefficient and a Pareto improvement
can be attained by inducing all the generations to consume more and save less when they are
young. Still it is unlikely that turns out to be below zero. Pagano and Sbracia (2014) follow
Blanchard’s OLG model in continuous time (1985) with net population change, effective labour
and age-dependent worker productivity. The balanced-growth equilibrium requires
where labour productivity, and also real wage, declines with age at the rate . If is sufficiently
big, the natural rate can be negative. But the value that the parameter needs to take is quite
implausible. The theoretical frameworks are not satisfying and cannot provide foundations to
the slump of . They do not completely get along with the evolution of the subject and the
most recent estimates of the variable.
22
According to the golden rule in the Solow model, the real interest rate is equal to the growth rate of the population
and the growth rate of labour productivity when the capital stock is at the level that maximises consumption.
18
Econometricians have split up into two main (and complementary) groups according to the
method used to identify movements in the natural rate. In the case of structural New Keynesian
dynamic stochastic general equilibrium (DSGE) models, is the real short-term rate in a
counterfactual economy that brings output into line with its potential at each point in time in
the absence of nominal adjustment frictions. This group of models has been referred to as
“Neo-Wicksellian” and is grounded on the Real Business Cycle literature and on Woodford
(2003).
23
Full price flexibility generates quite volatile estimates because of short-run
fluctuations from the steady state. At the same time, this strand models at frequencies more
relevant for monetary policy. Barsky et al. (2014) use a DSGE model and find that the natural
rate follows a highly procyclical pattern. Unlike in the past, it has remained persistently negative
in the US since 2008 (Fig. 1.6).
24
Figure 1.6: One-sided (filtered) and two-sided (smoothed) estimates of the natural rate of
interest in the US (percent)
23
Woodford (2003) characterises the natural rate of interest as the real rate that yields price stability period-by-
period.
24
Curdia et al. (2015) display the same pattern in the last three recessions in the US and so do Del Negro et al.
(2017).
Source: Barsky et al. (2014).
19
Gerali and Neri (2017) and Haavio et al. (2017), cited by Brand et al. (2018), confirm the
slump of in the Euro Area and the US (Fig. 1.7), Okazaki and Sudo (2018) for Japan.
Figure 1.7: DSGE estimates of the natural rate of interest in the Euro Area and the US
(percent)
On the other hand, there is a broad class of non-DSGE model-based methods to estimate the
real equilibrium rate. They use a medium-to-long run approach to the estimation of . The
natural rate is defined as the real short-term rate consistent with a closed output gap and inflation
gap once transitory shocks to aggregate demand or supply die out.
25
This strand of literature
does not focus on business cycle fluctuations but on highly persistent movements in the natural
rate, generating smoother estimates.
Brand et al. (2018) sort the non-DSGE models. Among the others, one approach is based on
rolling window regressions by exploiting the time series of the real interest rates. For instance,
a 5-year ahead forecast of the short-term real interest rate can be regarded as a proxy for the
natural rate.
26
In the long run the forecast reflects .
The second approach includes semi-structural models. Laubach and Williams (2003),
hereafter LW, initiated this framework for the United States and have inspired many variants.
27
In their seminal paper the natural rate is assumed to be a linear function of the estimated trend
25
Williams (2003) defines the natural rate as the “level expected to prevail in, say, the next five to ten years, after
any existing business cycle booms and busts have played out”.
26
Jarocinski (2017) for the Euro Area.
27
Hledik and Vlcek (2018), Brand and Mazelis (2018) for the Euro Area, Krustev (2018) for the US.
Source: Brand et al. (2018).
20
growth rate of potential output (g) and an unobserved component catching other determinants
such as the household’s discount rate, risk aversion, fiscal policy, and so on (z):
28
assuming σ = 1.
Holston, Laubach and Williams (2017), hereafter HLW, highlight the importance of taking
an international perspective in the analysis of the phenomenon (Fig. 1.8). They find a downward
co-movement in the natural rate of interest over the last thirty-five years in the United States,
the United Kingdom, Canada and the Euro Area.
Figure 1.8: HLW estimates of the natural rate of interest in advanced economies (percent)
LW (2003) paved the way also for Han’s semi-structural model about the Japanese case
(2019). The natural rate of interest in Japan shows the same trend as its peers and has likely
fallen into negative territory since the late 1990s (Fig. 1.9).
28
LW (2003) use the Kalman filter to jointly estimate the natural rate of interest (r*), the potential output (y*) and
its trend growth rate (g*) in the US. The technique works based on iterative updates of these unobserved variables
whenever there is a discrepancy between the model’s predictions and the actual outcomes of output gap and
inflation rate.
-1
0
1
2
3
4
5
6
7
1/1/1961
11/1/1962
9/1/1964
7/1/1966
5/1/1968
3/1/1970
1/1/1972
11/1/1973
9/1/1975
7/1/1977
5/1/1979
3/1/1981
1/1/1983
11/1/1984
9/1/1986
7/1/1988
5/1/1990
3/1/1992
1/1/1994
11/1/1995
9/1/1997
7/1/1999
5/1/2001
3/1/2003
1/1/2005
11/1/2006
9/1/2008
7/1/2010
5/1/2012
3/1/2014
1/1/2016
11/1/2017
Canada
US
Euro Area
UK
Source: https://www.newyorkfed.org/research/policy/rstar (consulted on 01/12/2020).
21
Figure 1.9: Estimates of the natural rate of interest and ex-ante short-term real rate in
Japan (percent)
In other “medium-to-long run” models the econometric estimates of for the Euro Area
(Fig.1.10) and the US (Fig. 1.11) have fallen into negative values as in DSGE cases.
Figure 1.10: Non-DSGE estimates of the natural rate of interest in the Euro Area (percent)
Source: Han (2019).
Source: Brand et al. (2018).
22
Figure 1.11: Non-DSGE estimates of the natural rate of interest in the US (percent)
The difference in estimates of the natural rate of interest across the methods hides
fundamental conceptual differences. Currently the values found range about from 2% to 1.5%.
So, we need to bear in mind the high uncertainty around the results. Mismeasurement may bring
central banks to take the wrong actions in stabilizing the economy. However, it is important
(and alarming) the fact that both the econometric groups highlight the common downward trend
in in advanced economies in the last thirty-five years (1985-2020). There have been two
periods of significant decline: a moderate one during the Great Moderation and a steeper one
after the Great Recession. This implies that the phenomenon is associated to a lethal mixture of
financial factors and slow-moving secular drivers. Moreover, the fact that the United States and
in particular the Euro Area and Japan often display negative estimates of is worrisome. It
means that episodes of ZLB are likely to be more frequent and longer.
29
Economies risk to stay
under potential, with stagnating income, deflation episodes or inflation under the 2% target.
Thus, monetary authorities would see their credibility and even their independence impaired
because they cannot fulfil their stability mandates.
Low and even negative natural rate and real rates, with their consequences on growth and
inflation, seem to be the new normal. No signs of sustained rebound are in sight. The next
step is to provide a new, deeper and more articulated theorical framework able to catch the wide
mismatch between the propensity to save and the propensity to invest. Only this way proper
29
A negative natural rate of interest does not exclude the possibility that we may see a short-term rise in the policy
rates due to temporary business cycle factors. The moves by the Federal Reserve in the last four years before the
coronavirus crisis may be interpreted this way.
Source: Brand et al. (2018).
23
policies can be built and implemented. A large amount of literature exists picking up a lot of
possible drivers. Less structural or less broad-based explanations are given by the global savings
glut idea
30
, the safe asset shortage argument
31
or the temporary liquidity trap literature.
32
On
the contrary, the new secular stagnation hypothesis provides a wider-ranging perspective.
1.3 The new secular stagnation hypothesis: the phantom menace
Weak economic growth, low inflation and low real interest rates are evident to the point that
some scholars have retrieved the old perspective of secular stagnation in the analysis of the
current environment. Like the natural rate of interest notion, a similar fate has befallen the
concept of secular stagnation. The hypothesis was coined by the President of the American
Economic Association, Alvin Hansen (1939), in the latter stages of the Great Depression:
Not until the problem of full employment of our productive resources from the
long run, secular standpoint was upon us, were we compelled to give serious
consideration to those factors and forces in our economy which tend to make
business recoveries weak and anaemic and which tend to prolong and deepen the
course of depressions. This is the essence of secular stagnation - sick recoveries
which die in their infancy and depressions which feed on themselves and leave
a hard and seemingly immovable core of unemployment.
33
“Secular” refers to the permanence of the stagnation. At that time, the US economy was still
struggling against the consequences of the crisis. Innovation, population growth, investment
spending and consumption declined with an oversupply of savings. These factors explained the
failure of the recovery to reach full employment and sowed the seed of a new era of stagnation.
However, Hansen’s gloomy prediction did not materialise because it did not anticipate the
post-war economic boom. Several decades later, the hypothesis has gained renewed traction
due to the Japanese lost decades and the post-GFC advanced world.
Within this framework, two broad visions can be distinguished: a supply-side and a
demand-side view. The former, which is mainly linked to Robert Gordon’s works, focuses on
a slowing growth rate of potential output due to a very pessimistic outlook for the evolution of
30
Bernanke (2005) and Bernanke (2015c).
31
Del Negro et al. (2017) and Caballero et al. (2017).
32
Krugman (1998).
33
Hansen (1939), p. 4.
24
future labour and total factor productivity.
34
The fact that GDP growth has been very low
despite historic lows in unemployment rates means that productivity is decelerating and labour
force is shrinking. However, lack of supply should lead to higher not lower inflation. The
demand-side position, on which I focus in this thesis, points to a chronic demand deficiency
and a negative natural rate of interest so that economies are not able to reach potential and has
been called “new secular stagnation hypothesis.
35
36
The two frameworks are not mutually
exclusive as demand-side factors can be very closely linked to supply-side factors. For instance,
a prolonged weakness in demand leads to hysteresis effects and long-term unemployment and
will reduce the productive capacity of the economy. Moreover, demographic evolution and
productivity changes are key drivers in both the positions. However, these links are often
ignored, which is the greatest flaw of secular stagnation explanations.
The American economist Lawrence H. Summers is the chief proponent of the new secular
stagnation hypothesis, even though many elements can be found in the extensive liquidity trap
literature. According to him and his followers, structural changes in the economy has generated
an excess of desired saving (supply schedule) over desired capital investment (demand
schedule) at normal rates. Secular stagnation refers to that situation in which saving can only
equal investment at full employment at a sufficiently negative natural rate. Then, potential
equilibrium with enough investment cannot be achieved in a sustainable way because of the
ZLB constraint and low inflation. The only possible adjustment takes place in the form of a
sub-optimal output and sub-target inflation, and these lower levels may continue indefinitely.
Assuming an upward-sloping demand curve at the ZLB, the result is a feedback loop: depressed
demand leads to lower inflation or deflation, lower inflation leads to real rates above the natural
one, and higher real rates lead in turn to even more depressed demand. The secular stagnation
world of economics becomes topsy-turvy: thrift, toil and flexibility are no longer virtues, but
social vices (clear example of fallacy of composition).
37
The paradox of thrift dates back to
Keynes. If everyone tries to increase savings simultaneously, then aggregate demand will fall,
which will depress output and income leading eventually to a diminishing of the aggregate
savings rate. So, although individual savings can be rational and beneficial for those who save
more, collective saving behaviour at the same time can have harmful effects on the economy.
According to the paradox of toil, a positive labour supply or technology shock that increases
34
Gordon (2012) and Gordon (2015).
35
Summers (2013) and Summers (2014), Krugman (2013), Teulings and Baldwin (2014).
36
When I mention “secular stagnation” in what follows, I always refer to “demand-side secular stagnation”.
37
Eggertsson and Mehrotra (2014).
25
productive potential of the economy is contractionary. The paradox of flexibility states that
increasing downward nominal wage flexibility exacerbates unemployment and the shortfall in
output. The last two paradoxes work in a similar way because they trigger deflationary
pressures. When the ZLB is binding, real interest rates go higher and depress demand further,
especially from high spending debtors.
At the International Monetary Fund Research Conference in November 2013, Summers
raised for the first time the argument of secular stagnation. He assumed that in the US may
have declined, reaching negative levels since at least the beginning of the century between 2%
and 3%. Central banks alone are not able to accommodate this equilibrium rate well below
zero. Full employment can be artificially achieved through easy money. Thus, economic
policy finds itself in quicksand because it cannot attain the goals of satisfactory growth with
low and stable inflation and financial soundness contemporaneously. According to Summers,
it is a long-term unhealthy phenomenon. For instance, the problem of anaemic growth in the
US is supposed to have been latent for at least twenty-five years. Only temporary unsustainable
finances, the development of bubbles, such as the dotcom bubble in the late 1990s and the
housing market bubble in 2003-2007, and lax prudential, monetary and fiscal policies allowed
good but not outstanding economic performance. There were no signs of overheating. Given a
very low at that time, bubbles worked as an alternative way to deal with excess savings when
macroeconomic policy could not absorb it.
The fact that now output gap is closing in the US, Europe and Japan may be objected to the
perspective of secular stagnation. Yet, the approach of the zero gap has to be taken with a pinch
of salt, especially looking at sleeping inflation. Moreover, it is mostly due to continuous
downward revisions of potential GDP since 2007 and not to rapid recovery in demand (Fig.
1.12). The pre-recession trend is still far. The missed economic acceleration has been surprising
and at odds with what happened in the 1930s. In fact, the countries, that started from a highly
depressed state after the management of the financial crisis, were expected to experience a
V-shaped recovery.
26
Figure 1.12: Potential GDP estimates and actual GDP in the US, the Euro Area and Japan
Source: Summers in Teulings and Baldwin (2014).
27
Summers (2014) builds a qualitative narration of the factors that have made decline and
growth inadequate in the last decades. Some of them are related to increases in saving
propensity, some others to decreases in investment propensity:
1. reduction in demand for debt-financed investment. First of all, less borrowing is the
result of a deleveraging shock and more stringent collateral requirements after the GFC
and may go on for a long time.
38
39
For example, the corporate sector in the Eurozone
went from being a net borrower of funds (11.8% of GDP in Q3 2008) to a net saver of
funds (3.4% of GDP in Q3 2014). Over the same period, the household sector raised
savings from around 7.9% of GDP to 12.5% of GDP. So, in only six years the economy
lost private sector demand (household and corporate combined) equal to 19.8% of
GDP.
40
Secondly, the lower level of investment is also a reflection of the changing character
of productive economic activity. The big tech-companies, such as Amazon, Apple,
Microsoft, Facebook and Google have a staggering market capitalisation with few
capital investments in the development of high-value added services. They swim in huge
cash hoards;
2. demographic shifts, whose development is the least uncertain.
41
People tend to have
fewer children (sub-replacement fertility). Population and working-age population
decline is already a fact in Japan since 2011 and the same transition is projected to
happen in Europe. In the US population growth has kept on decelerating (Fig. 1.13 and
1.14). Moreover, individuals live longer (decreasing mortality) (Table 1.2), which
generates a dramatic increase in the relative number of the elderly (increasing old-age
dependency ratio). The OLG literature distinguishes three sub-channels:
i. downward impact from lower labour input. Lower demand for capital
goods and investment to equip new workers, to house them or to give
them a roof over their heads as they work. It is also due to the
38
The deleveraging shock is the result of the so-called Wile E. Coyote moment or Minsky moment. It
corresponds to a sudden realization that assets were overvalued and borrowers’ collateral constraints were too lax,
so there is an abrupt downward revision of the debt limit. The reference to Wile E. Coyote is due a recurrent
episode in the famous cartoon with the Road Runner. The Coyote has the habit of running off cliffs and only when
he realises that nothing is under him, he plummets.
39
In Teulings and Baldwin (2014) Rogoff talks about a debt super-cycle coming to an end with only temporary
deleveraging and borrowing headwinds. Koo claims that the slow recovery comes from a balance-sheet recession.
40
Duprat (2015), p. 11.
41
See also Gagnon et al. (2016), Papetti (2019).
28
technological changes. Ceteris paribus, installed capital relative to the
workforce size rises, depressing the marginal product of capital and ;
ii. downward impact from higher capital supply. Ageing of the baby boom
generation and lower mortality rates imply higher savings for retirement
preparation. Moreover, assuming perfect foresight, households realise
that the growth rate of effective workers in support of the population size
is shrinking over time and “become” more patient;
iii. theoretically ageing may also have an upward impact on because in
the end relatively older individuals are dissavers. Actually, no studies
have shown the significance of this positive impact.
Figure 1.13: Slowing population growth in advanced economies (percentage change at
annual rate)
Source: https://data.worldbank.org/ (consulted on 02/02/2020).
29
Figure 1.14: Slowing working-age population (aged 15-64) growth in advanced economies
(percentage change at annual rate)
Table 1.2: Higher life expectancy in advanced economies
Share of world
GDP (%)
Life expectancy (years)
2010
1970
1990
2010
Japan
8.58
72.00
78.90
82.90
US
23.37
70.90
75.30
78.60
Germany
5.17
70.60
75.30
80.50
Source: Teulings and Baldwin (2014).
3. increasing income and wealth inequality in favour of capital income (rising corporate-
retained earnings and declining labour share) and those with more wealth and lower
propensity to spend (Fig. 1.15 and 1.16). In fact, the rich have “capitalist-spirit” type
preferences over holdings of real and financial assets due to the sense of power and
prestige they get.
42
On the other hand, the non-rich borrow from the rich subject to the
aforementioned more restrictive credit market frictions;
42
Francis (2009).
Source: Lane (2019).
30
Figure 1.15: US corporate profits as a share of GDP (billions of $/bil. of $)
Figure 1.16: US income and wage inequality, 1913-2014 (share of total, percent)
Source: https://fred.stlouisfed.org/ (consulted on 01/12/2020).
Source: https://wid.world/ (consulted on 01/12/2020).
31
4. cheaper capital goods such as those associated with information technology. A lower
level of savings can purchase the same amount of capital;
5. at any given real interest rate level, real after-tax interest rates, which matter for
investment decisions, are higher. The phenomenon is due to the past disinflationary
policies. But this reduces investment demand and also pre-tax real interest rates;
6. massive accumulation of safe and liquid assets and in particular US Treasuries by
emerging economies’ central banks and by pension funds and insurance companies to
meet regulatory requirements (Fig. 1.17). This factor is pivotal in the global savings glut
and the safe asset shortage thesis.
Figure 1.17: Global foreign currency reserves (trillions of $)
The visual inspection is consistent with some of the concerns raised by Summers. Predictions
about a secular stagnation world are hard to make. However, two things have to be underlined.
First, the market offers a simple tell-tale: the low level of the real interest rates. Secondly,
econometric estimates display the declining trend of the natural rate of interest, in some cases
into substantially negative territory. Given that the impact of the deep-seated forces, especially
demographics, on is not going to disappear in the next decades, it can be said that advanced
economies are approaching a secular stagnation equilibrium. Of course, they are not all in the
same situation. Japanese malaise has been going on since the 1990s to the point that
Source: European Central Bank (2019).
32
Japanification is used as a synonym for secular stagnation. Data for the Euro Area are more
worrisome than the American ones. Strong low inflation pressures, high old-age dependency
ratios, shrinking working age population (aged 15 to 64 years), long process of private sector
deleveraging make its members more vulnerable than the US. Current fiscal and monetary
strategies have less room to take action.
Thus, given the rich debate around the new secular stagnation hypothesis, it is fundamental
to have a theoretical framework able to collect all the pieces. As previously mentioned, past
models do not fit well and this is also true for existing liquidity trap models. In the context of
the Japanese crisis, Krugman (1998) considers a negative natural rate using an intertemporal
representative agent setting. A negative can happen when the economy’s future output is
expected to be sufficiently less than its current output. The model displays two main limits as
in Woodford (2003). First of all, the shock is temporary. It is presumed that normal economic
and policy conditions with positive real interest rates determined by the subjective discount
factor will return at some point by deus ex machina. Secondly, the factors listed by Summers
justify the need to work with heterogenous agents: the youngsters and the elderly, the borrowers
and the lenders, the rich and the not so wealthy. Krugman hints at this alternative when he adds
productive investment in the model. In fact, he sketches an overlapping generations set-up and
suggests population/labour force dynamics as an important driver of the liquidity trap in Japan.
However, the author does not get deeper and the shortcomings remain.
As the next chapter shows, one notable exception to the lack of adequate modelling is the
very influential work by Eggertsson and Mehrotra (2014), henceforth EM.
43
They formalise the
new secular stagnation within a New Keynesian overlapping generations model with credit
constraints and downward nominal wage rigidity. In the steady state the natural rate of interest
does not depend solely by the fixed discount factor of households, but also on the relative supply
of savings and demand for loans. It can be permanently below zero such that a long slump with
sub-target inflation and excessive underemployment is possible. The key condition is that
households shift from borrowing to saving over their lifecycle. Most of the structural forces
mentioned by Summers (2014) are included: a slowdown in population growth, an increase in
inequality, a tightening of borrowing limits (debt deleveraging shock) and a decline in the
relative price of investment goods. Productivity growth is completely ignored. Moreover, in the
secular stagnation equilibrium of the model the paradoxes of toil, thrift and flexibility apply
43
Eggertsson et al. (2016a) and Eggertsson et al. (2016b) analyse the secular stagnation hypothesis in the open
economy and the potential spillovers of domestic monetary and fiscal policies across countries.
33
and this has profound implications for the design of the necessary mix of macroeconomic
policies.
Eggertsson, Mehrotra and Robbins (2017), henceforth EMR, explores whether negative
natural rates and secular stagnation are quantitively realistic. EMR builds a medium-scale fifty-
six period OLG model with capital and calibrates it to match the observed behaviour of the US
economy in 2015. The results are consistent with the theoretical framework in EM. The model
is able to produce a negative natural rate of interest ranging nowadays from 1.47% to 2.20%
and a secular stagnation equilibrium with standard parameter values in the macro literature. It
is also able to fully account for and decompose the fall of over a time interval of forty-five
years (1970-2015). Demographic shifts (higher life expectancy, lower fertility rate) have
contributed the most to the trend.
44
The reduction in the labour share and in the relative price
of investment goods has had a smaller impact. Only the increase in government debt-to-GDP
ratio has avoided a worse outcome. The magnitude of the decline is surprising since inequality,
housing sector and the associated debt and firms borrowing (with their role during the financial
crisis) are not incorporated.
44
Despite being structural, the OLG framework in EMR makes their approach closer to the non-DSGE
econometric category since it captures low-frequency demographic developments. See also Gagnon et al. (2016),
Papetti (2019).
34
2 A model of demand-side secular stagnation: a first approach
Drawing from Summers (2014), Eggertsson and Mehrotra (2014) (EM) try to answer the
question about the theoretical possibility of a permanent slump. They propose a New Keynesian
OLG model in discrete time using the insights of Samuelson’s one. A model of this kind
captures both the finite-horizon and the life-cycle aspects of the behaviour of the households.
The setting is the following:
the economy is closed, thus capital flows across countries and the global savings glut
are ignored;
agents have perfect foresight: people have definite point expectations of future variables
(no uncertainty) and these expectations are correct. Thus, the expectation operator is
not present;
the economy is infinitely-lived, while households go through three stages of life and
then die. They are born in period one (the young), they become middle-aged in period
two (the middle-aged) and retire in period three (the old). In each period the three
generations/cohorts always coexist, so demographics plays a key role in the model;
there is a single consumption good;
the youngsters are credit-constrained;
the natural rate of interest rate is the equilibrium real interest rate in the bond market
that would prevail when output is at its full employment level and inflation is stable at
the target.
The structure of the economy and the interaction between the young generation and the
middle-aged may generate a persistently negative natural rate of interest, as quantified by
Eggertsson, Mehrotra and Robbins (2017) (EMR). This is in contrast with the previous
macroeconomic literature. As both the econometric groups have shown with respect to
advanced countries, the reasons for this outcome are a lethal mixture of financial factors and
long-term real drivers, in particular demographic changes. When output is endogenous, the
presence of the ZLB, a low inflation target and nominal rigidities in the aggregate supply block
make possible to have a demand-side secular stagnation equilibrium.
The chapter follows the same steps in EM. Section 2.1 starts with an endowment economy.
Section 2.2 augments it with price level. Section 2.3 adds endogenous output, nominal frictions,
government and central bank and show the steady states in the economy. Section 2.4 provides
35
some policy implications in a secular stagnation environment. Section 2.5 presents the main
insights from the model with capital.
2.1 Endowment economy
In an endowment economy there are no capital and production, that is output is just “dropped
from the sky”. Only the middle and old generations receive some income in the form of an
endowment and . The young and the middle-aged trade the unique asset, i.e. a one period
riskless bond (), with one another at a real interest rate . The gross real interest rate is
. On the one hand, the youngsters want to borrow to finance their consumption, but
they are constrained by a debt limit (exogenous time-varying collateral constraint ).
45
On the
other, the middle-aged have to repay what they borrowed in the past and need to save part of
their endowment for retirement by lending it to the young according to the consumption-
smoothing motive. The remainder is consumed. Lastly, the elderly are out of the business. They
spend all their remaining income and whatever savings they have accumulated in the previous
period. Thus, the only thing which has to be determined endogenously in the endowment
economy is the real interest rate. Within this framework there is no distinction between the real
interest rate and the natural rate of interest.
The instantaneous utility function of a generation takes the following form:

where  denotes the different type of cohort and the corresponding consumption
when households are young, middle-aged or old. Log-preferences are consistent with a
coefficient of risk aversion that goes to one. The period utility function is assumed to be twice
continuously differentiable with > 0 and  < 0. There is no altruism or bequest motive. The
lifetime (three-period) utility function of a generic household born in is time-separable:


where , is the subjective discount factor.
45
The debt limit is not deepened in EM, but the inspiration comes from Eggertsson and Krugman (2012), who
give few more details. They define the debt limit as a proxy for general views about what level of leverage on the
part of borrowers is “safe”, posing an acceptable risk either of unintentional default or of creating some kind of
moral hazard.
36
Thus, the objective function is






In each period the household faces a different budget constraint:










where and 
are respectively the real amount of borrowings by the young agent and the
real supply of savings by the middle-aged.
46
The debt limit also includes interest rate payments.
At time consumption for the young and for the old is easy to find. Substituting (1) in (4)
and assuming a binding constraint for the young generation:
47


They borrow as much as they can. The amount of borrowings depends on the real interest rate:
a drop in increases borrowing and consumption for the young.
The old consume all their remaining wealth (endowment plus the proceeds from savings
accumulated in the previous period):


The middle-aged need to solve the saving problem (. The first-order condition for a
maximum yields the consumption Euler equation:

 

46
In Eggertsson and Mehrotra (2014) the sign in front of 
in equation  is a plus, not a minus. In fact, in
terms of initial cash-flows from buying the bond, 
is negative for the middle-aged. For simplicity, in Chapter
2 and Chapter 3 I do not use the notation in the original paper, but the one in Bonchi (2017) and Ascari and Bonchi
(2019), who use the opposite sign. The setting does not change.
47
For the collateral constraint to be binding, it must be the case that 

37
where optimally the marginal utility cost of saving (opportunity cost in terms of current utility)
must be equal to the marginal utility benefit obtained by saving.
Aggregating, bond market equilibrium is found taking into account the number of the young
born in (, and of the middle-aged born in (:


where
, so is the growth rate of population.
48
Equation (5) has to be inserted into (8) to have the demand for loans by the young:


It is a decreasing function of :



Finding the supply of loans by the middle-aged requires more manipulation. We have
to use (2), (3), (7) and (5) and solve for :




 




The response of savings with respect to the interest rate is theoretically ambiguous as the
income and substitution effects work in opposite directions. In this case with the intertemporal
elasticity of substitution σ equal to one, the derivative of loan supply with respect to the real
interest rate is positive:
48
To be fair, n is the growth rate of the new cohort of young borrowers. The total population is .
Of course, a declining n generates a slower growth of the entire population.
38




The lower the endowment to the elderly is, the more rigid the curve becomes.
The asset market-clearing price () is the following equilibrium gross real interest rate:




There are clearly more elements than just the subjective discount factor: the income profile over
the life-cycle, the debt limit and population growth. The first factor can be considered as
inter-generational inequality of income. Appendix A shows how income inequality within a
given cohort, i.e. within the working-age population, enters equation . In fact, in Chapter
1.3 increased inequality is understood as a redistribution of income from those with high
propensity to consume to those that have a higher propensity to save, i.e. the wealthy or the
high-skilled households with higher endowments/wages.
It is clear that the real interest rate is not fixed but may be persistently negative, which may
entail dynamic inefficiency when endogenous output is included.
49
The determinants of the
drop are both a financial crisis and long-term drivers:
deleveraging shock (reduction in ): a Minsky moment” tighter collateral constraint
generates less borrowing;
slowdown in the population growth rate (lower ): fewer youngsters translate into less
borrowing;
more hump-shaped time-pattern of income (higher middle-aged income in 
compared to the old income 
), which increases savings for retirement. The channel
also captures the increase in life expectancy indirectly;
increase in intra-generational inequality: a distribution of more income from the poor to
the rich increases the amount of savings.
50
Figure 2.1 shows what happens at time when there is a deleveraging shock, whose impact
has been considered temporary on in the representative agent or representative saver models.
Instead, in this OLG model households are both borrowers and savers at different stages in their
lives. At time the loan demand curve shifts leftwards from 
 to 
, while there is
49
The presence of a unique and safe asset class definitely increases the chance of having dynamic inefficiency.
50
See Appendix A.
39
no change in the supply of loanable funds because the debt repayment of the middle-aged
depends upon previous commitments (). The new equilibrium is at point B: the real interest
rate needs to drop for the level of aggregate spending to remain the same. In fact, a reduction
in the real interest rate partially offsets the more restrictive borrowing constraint and fuels the
demand for loans. Moreover, having assumed an upward sloping loan supply curve, a lower
reduces desired saving by the middle-aged as the present value of future income rises.
51
Consumption today becomes more attractive to them consistently with the Euler equation.
However, and are not independent schedules. In the next period the youngsters
grow up and face a saving decision. They have less debt to repay due to the deleveraging shock,
so the disposable income (
) is higher than before. The loan supply curve shifts
rightwards and it depresses the real interest rate further. Point C is the new equilibrium.
Depending on the size of the initial shock, the natural rate of interest may plunge into negative
territory permanently, even if the reduction in D is only temporary, because of this lagged effect.
The dynamics is in contrast with the past literature, according to which the real interest rate
comes back to the previous positive value in the end (Point D).
52
The same outcome is obtained
with slower or even negative population growth and with rising inequality within the
working-age generation. As regards the former, only the demand for loans is affected by a lower
since and are normalised by the size of the middle generation . In the latter case
only the supply curve shifts and it moves to the right.
51
If the substitution effect is very weak compared to the income effect (intertemporal elasticity of substitution
lower than one) and if the endowment is concentrated in the hands of the middle generation, it may be the case
that savings increase when the real interest rate declines.
52
See Eggertsson and Krugman (2012).
40
Figure 2.1: Equilibrium in the bond market
2.2 Price level determination
So far, the fact that the natural rate of interest may be negative is not interesting as long as
output “falls from the sky” and endowment must be fully consumed. It becomes relevant when
money, price level determination and production are considered. The next step is to introduce
a one-period risk-free nominal bond (in addition to the previous risk-free real bond), the price
level, such as CPI, and the ZLB condition. The equivalence of the consumption Euler equation
for nominal bonds to its real counterpart  implies the standard Fisher equation, given perfect
foresight:


where is the nominal interest rate on the bond, controlled by the central bank, is the
aggregate price level,
 is the gross inflation rate at time . The real interest rate should
be equal to the nominal rate deflated by the growth rate of the price level. As a consequence,
middle-aged savers are indifferent between holding real debt and nominal debt.
Source: Eggertsson and Mehrotra (2014).
41
The ZLB condition is a non-negativity constraint on nominal rates because of the trade-off
with money:

Combining (14) and (15) it is clear that when the steady state real interest rate is negative the
economy needs a certain degree of inflation. A constant price level ( ) is
inconsistent with the model when the real interest rates are negative because the ZLB prevents
the nominal rate from being below zero. When prices are perfectly flexible, the only possible
equilibrium needs positive inflation so that cannot be lower than
.
53
Thus, there is a lower
bound on the steady state inflation.
54
When the natural rate of interest is positive, it is almost
impossible to violate the condition because central banks do not target deflation. Instead, when
the steady state real interest rate is negative the lower bound takes on a greater practical
significance. The more negative the steady state , the higher inflation has to be. The importance
of the statement cannot be grasped in the representative agent/saver literature, where the
condition never binds:  i.e. above 0.9.
As long as we are in the endowment economy framework, if the lower bound condition does
not hold because central banks do not tolerate high enough inflation, there is simply no
equilibrium. Yet, in a realistic sticky price world with endogenous output and low inflation
targeting, a negative natural rate may be a problem. The ZLB becomes binding and a secular
stagnation equilibrium shows up.
2.3 Model with endogenous output
The next step is to add endogenous production, government and monetary policy reaction
function. Generally speaking, the structure of the economy is the following:
firms are perfectly competitive and hire labour to maximise period-by-period profits;
labour is the only factor of production;
unlike Diamond’s two-period OLG, the middle-aged households both work and run
firms, receiving income in the form of wages and profits. The total amount of workers
and firms is the size of the middle generation ;
53
is the gross inflation rate. When inflation is mentioned henceforth, it is referred to 
 0, which
means that  The target is ().
54
Hereinafter, in the steady state the time subscript t is dropped because variables are constant.
42
taxes and public debt modify budget constraints and asset market equilibrium. No social
security system for the elderly is considered;
the central bank acts in a regime of strict inflation targeting and is guided by the natural
rate of interest;
all the aggregate variables are normalised in terms of , e.g. is output per
middle-generation household.
With respect to the endowment case, households’ budget constraints are different except (4):


 
 



 


where
is the income, the nominal wage rate, the labour supply of the
middle-aged, the profits of the firms and the lump-sum taxes levied on each generation
i.
55
Let
denote the real wage. For simplicity, it is assumed that labour endowment is
inelastic at a constant level
. The government’s budget constraint is added:




where represents the real value of public debt at time , i.e. the real value at issuance of the
quantity outstanding of one-period government bonds, is government spending.
Bond market equilibrium is 

Loan demand and loan supply are respectively


55
A more complex approach would be to consider distortionary consumption and income taxes.
43

 

Two things have to be highlighted. First of all, public sector also wants to borrow. Secondly,
loan supply is a decreasing function of the real interest rate (income effect prevails). Yet, the
shifts of the curves generate the same results as in the endowment economy assuming that loan
supply is less reactive than demand to changes in the real interest rate 

:
56




Excluding intra-generational inequality, we have a new version of equation (13):


Fiscal policy has a role in influencing the outcome in the bond market. As long as condition
(23) holds, a deleveraging shock that hits government’s borrowing or fiscal austerity lowers the
real interest rate.
Government spending, the level of public debt and taxation on the young are exogenous.
The ratio between the taxes levied on the elderly and the ones on the middle-aged is determined
by the government budget constraint and the following fiscal policy rule:

 
Therefore, changes in taxation have no effect on loan supply.
On the real economy side, the production technology of the firms is represented by


where is a total factor productivity index, which is treated as a scaling constant in this
chapter, ϵ (0, 1) measures the degree of diminishing returns to labour or the labour share. All
firms are identical.
56
EM makes calculations in the endowment economy framework and they extend them to the model with
production, where only the middle-aged agents get income. They do not highlight the change in the behaviour of
the savers probably because the further condition to be satisfied is taken for granted.
44
The profit maximisation problem under the technological constraint (26) is given by




Taking price and nominal wage as given, the optimality condition pins down demand for labour:



In a perfectly frictionless world equation (28) determines the market-clearing real wage
( 
) and constant full employment is always guaranteed (
).
The long-run aggregate supply (AS) schedule is depicted by a vertical line.
In a more realistic and less flexible setting a vertical segment of the AS curve is maintained:
there is a certain amount of inflation at which full employment holds. However, in a low
inflation/deflation environment a permanent, long-run trade-off between inflation and
unemployment exists. In this case an upward-sloping segment of the AS curve can be generated
due to imperfect functioning of labour markets even under the perfect foresight hypothesis.
Starting from Keynes, Modigliani (1944) assumes that nominal wages are perfectly flexible
upwards because workers love wage rises, but have a lower bound below which they cannot be
cut. Downward nominal wage rigidity (DNWR) could arise from several other reasons such as
government regulations and fixed-length nominal wage agreements. It has been identified in
the United States, the Euro Area and Japan.
57
The minimum acceptable wage
is the weighted
average of the past wage level and the flexible level corresponding to full employment:
 .
58
Productivity growth and indexation to the central bank’s
inflation target in the wage norm are excluded.
59
The degree of rigidity is measured by : if it
is equal to unity, wages are perfectly downwardly rigid, while if it is equal to zero, we come
back to the frictionless case. Wages behave so that

)
57
See for instance an estimate of for some peripheral Eurozone members in Schmitt-Grohé and Uribe (2016).
58
Ascari and Bonchi (2019) assume a simpler DNWR specification:
 with . When the constraint
binds, the steady state wage and price inflation are equal to and the aggregate supply curve is flat. Nonetheless,
their alternative specification does not change the results in EM.
59
See Appendix G - EMR. The steady states in the model do not change qualitatively, yet the authors do not
explore the impact that this version of the wage norm has on monetary policy effectiveness.
45
Households are unwilling to supply labour at a lower wage than
. When inflation is high, it
is not an issue because firms will offer higher nominal wages than in the previous period. The
aggregate labour demand equals the economy’s labour endowment. But when prices go below
a specific threshold, DNWR binds. Nominal wages do not adjust properly at the same pace as
prices, then firms resort to dismissal (unemployment is equal to
) and output
slumps. Recession, subdued inflation and especially deflation worsen the situation because real
wages go higher and higher and stay elevated. Negative output gap widens (Fig. 2.2).
Figure 2.2: Binding downward nominal wage rigidity (Point C)
Again the central bank sets the rate of return on the one-period risk-free bond according to
a Taylor rule (in gross rates) with a Wicksellian flavour, augmented by the ZLB:


where the term
is deleted. The Taylor principle applies. Once the gross policy rate is set
at the full-employment target , inflation is at .
60
Conventional
monetary policy is effective as long as the non-negativity constraint (15) holds. Otherwise,
60
As in the Taylor rule shown in Chapter 1.2, corresponds to . This implies two things. The
central bank does not track a steady state but a time-varying value of the natural rate of interest. Secondly, the
policymaker’s estimate of the natural rate  converges to its true value and inflation is stabilised around the
target. Perfect tracking is clearly an oversimplification so that in practice econometric mismeasurement issues
affect monetary policy stance.
46
when the shock is massive, the ZLB materialises and inflation runs below the target. By using
the Fisher equation the Taylor rule can be rewritten in real terms:

 



In conclusion, equation (24), i.e. the equilibrium real interest rate in the asset market, is adjusted
for the full employment output to get an expression for the gross natural rate of interest:

 
whose drivers are exactly the same ones of the real interest rate in the endowment economy
plus the fiscal variables. At both the output and inflation gaps are simultaneously closed and
monetary policy decisions cannot affect it. The natural rate is not time-invariant and can be
negative in the steady state.
To recap, equilibrium in the economy with nominal rigidities is a sequence of quantities

and prices 
, exogenous
processes 
and initial values of household saving, nominal interest rate,
real wage and public debt 

 that jointly satisfy:
household budget constraints , ,  and ;
consumption Euler equation ;
government budget constraint ;
bond market clearing ;
Fisher equation  and non-negativity constraint on nominal rates ;
technology constraint , profit maximisation problem  and labour demand
condition ;
wage process ;
fiscal policy rule  and Taylor rule .
2.3.1 The steady state equilibria in the economy
The steady state is determined by the intersection between the aggregate supply and the
aggregate demand (AD) curves. In contrast to textbook models, EM model has different kinds
of steady state because both the curves change their slopes and have kink points.
47
The aggregate supply schedule is defined by the equilibrium in labour market. As been
already mentioned, it depends on the double behaviour of wages. The inflation rate at which
DNWR binds has to be found. It can be done expressing nominal wages in real terms and seeing
when the real wages are above or equal the wage norm:






In the steady state  . The condition is satisfied as long as ,
which means stable prices or positive inflation. Nominal wages must rise to keep real wages
constant (). The AS curve is a vertical line at the full employment level:

 

By contrast, when there is deflation, the steady-state real wage is 
 . Real wages
are above their market-clearing level. This is substituted into the labour demand condition (28)
and then into the production function (26). Hereunder the steps to find output:


 



 

Since the inflation-output space has output on the x-axis, it is better to write (35) in terms of
inflation: 

. When DNWR binds, a positive relationship
between inflation and output shows up (long-run Phillips curve):
 







48
where




. At , . The degree of nominal rigidity puts a
lower bound on deflation. The higher is, the flatter the curve is. The upward slope indicates
that moderate rates of inflation are desirable to “grease the wheels” of the labour market (Tobin,
1972). Inflation relieves nominal wages freezing and firms hire more labour and revive
production. The limiting cases are straightforward. When is equal to unity, is equal to the
degree of rigidity. The upward sloping branch of the AS curve becomes flat à la Ascari and
Bonchi (2019): the level of employment and output is demand determined. As approaches
zero, more weight in the wage norm is put on the flexible level component and the Phillips
curve gets steeper resembling (34).
Aggregate demand also displays a double regime depending on whether the ZLB is binding
or not. In this model the components are total private and public consumption. At time


By means of the budget constraints for the young (16), the elderly (18), the government (19),
the Euler equation (7), the Fisher equation (14) and assuming loan market equilibrium (savings
expressed in terms of loan demand in (21)), aggregate demand at time is
 

Inflation enters through the “real” Taylor rule (31). A quicker way to get the steady state
demand is to immediately match the asset market-clearing real rate (24) and the two arguments
of the “real” Taylor rule (31):



Either way, at positive nominal interest rates the steady state AD is





where output and inflation are negatively related because of the Taylor principle. When
inflation accelerates and exceeds the target, the central bank raises the policy rate more than
49
proportionately so that it increases the real interest rate above its natural level and curbs
economic activity. The reverse happens when inflation is feeble.
When the central bank hits the ZLB, there is a different AD:



where output is an upward sloping linear function of inflation. The nominal rate is stuck at zero
and the gross real rate is just the inverse of gross inflation. A lower real rate oxygenates demand.
Since the inflation-output space has output on the x-axis, the slope of the  segment is
found through the inverse function theorem:
 

 

The switch from a segment of the AD curve to the other occurs at the kink point denoted by
. It is the minimum inflation rate that the central bank is able to face via the conventional
tool. The kink point is found equalising the two arguments in the max operator of equation (30):


that is an increasing function of the inflation target. This minimum inflation rate has a minimum
real interest rate counterpart: 
 
, which provides the
aforementioned lower bound condition for the steady state :



which means that if nowadays is below 2%, a full employment equilibrium does not exist.
For instance, if the natural rate is 3%, then the inflation target has to be at least equal to
3%. Otherwise the central bank is ineffective in counteracting a permanent slump because the
real interest rates remain higher than the natural rate. The lower the natural rate is, the more
likely the ZLB materialises (higher ), the more and more unattainable the real interest
rates that clear the market are (lower ). A full employment equilibrium is possible if is
50
above 2% and is unique if is above zero.
61
It is important to point out that equation 
is a steady state condition. As it will become clear in Chapter 3, it may be the case that the
natural rate of interest is pushed into negative territory because the economy is hit by a very
persistent productivity shock. In that event the economy always converges to the previous full
employment steady state with a positive natural rate. However, this slight difference is
irrelevant when it comes to monetary policy, whose action will be constrained for a long time.
The point of intersection between the aggregate demand and the aggregate supply curves
determines equilibrium output and inflation. It is contingent on the value of the natural of
interest for a given inflation target. Let us start stating that the steady state in the economy, no
matter its nature, is unique: there is either a full employment equilibrium or a secular stagnation
one. Uniqueness and determinacy are guaranteed if:
inflation target is small enough such that the kink in the AD curve is small enough;
the AS curve has a higher intercept than the segment  ;
62
linked to the previous statement, the segment must be steeper than 
around the equilibrium point: 

 

 ;
unlike EM, the central bank is assumed to target a zero steady state inflation
( rather than a positive one;
the gross natural rate is either above or below the reciprocal of the gross inflation
target ( or ). Equality is put aside.
Concavity/convexity of  and  can be disregarded and the lines can be depicted
without curving since the results do not change. Using a historical approach, in the beginning
the economy is at the full employment steady state and later winds up suffering a liquidity trap.
Full employment steady state (Fig. 2.3): If the steady state natural rate is above zero, the ZLB
does not bind because is above  and can be accommodated. The non-negativity
constraint (15) on the nominal rates in the Fisher equation holds ( and the kink in the AD
curve occurs at an inflation rate below the inflation target. Then, the central bank fulfils its
mandate (.
63
The segment  intersects  at the kink of the  curve. At
61
When the natural rate is between 2% and zero, multiple equilibria show up.
62
The point that  must have a higher intercept than  is trivial because the latter has a negative
intercept.
63
A peculiar case is when the natural rate is zero.  and the intersection occurs at the common kink
point of the curves.
51
the TR-FE equilibrium output is at potential ().
64
Even if the “usual suspects” reduce
the natural rate, the optimal outcome is the only one as long as condition (42) applies.
Figure 2.3: Full employment steady state (Point A)
Secular stagnation steady state (or deflation steady state) (Fig. 2.4): However, in the last
thirty-five years the natural rate in advanced economies has declined persistently and the crisis
has worsened this pattern. In this version of the model the focus is on the fall of population
growth and on the Minsky moment, whereas other relevant variables such as inequality and the
relative price of capital are not considered. The adverse fundamental shocks are strong enough
to have a triple impact in the model: they move leftwards the  curve, they push the natural
rate below the lower bound permanently, they raise  above the target. For instance, a debt
deleveraging shock or slowing population growth reduce aggregate spending by the youngsters
for a given level of public debt. The central bank is not able to fully offset the issue via cuts in
the policy rate in order to boost demand and absorb excess savings (. Deflation shows up
(
). In the labour market DNWR binds and steady state real wages are
unsustainable for the firms because they are above their market-clearing level. The result is
64
The uniqueness of the equilibrium can easily be proved. At 
 
 . At ,  .
52
unemployment and shrinking production (). The slump is structurally endless with no
pull towards full employment. Thus, the intersection is the one between  and 
(ZLB-DNWR equilibrium).
65
Figure 2.4: Secular stagnation steady state (Point B)
It is important to highlight three things about the unique secular stagnation steady state. First
of all, here deflation is just a by-product of the specific wage norm
assumed. A richer norm
with the inflation target makes this equilibrium consistent with inflation, that is more realistic
in the American and European context. Moreover, the deflation steady state with a negative
natural rate is locally determinate, i.e. the variables converge there in the analysis of transition
dynamics and the model does not explode.
66
This is the case when the ZLB arises due to
slow-moving fundamental factors. Instead, stability is not a property of those models where the
ZLB-DNWR equilibrium is caused by self-fulfilling deflation expectations (confidence shock)
65
The uniqueness of the equilibrium can easily be proved. At 
 
 . At ,  .
66
Determinacy is proved by log-linearizing the  and  schedules around the steady state. The
condition for the existence of such a solution is precisely that 

 

 at the intersection point.
53
and it is unclear which path leads to the steady state.
67
Lastly, EMR calibrates the model and
shows that it is able to generate dynamics consistent with the recent data in the United States,
Japan and the Eurozone (Fig. 2.5): output (per capita) below trend, interest rates at the ZLB and
long lasting fall in inflation.
When secular stagnation materialises, the paradoxes of thrift, toil and flexibility arise and
exacerbate the recession. Consistently with the old Keynesian paradox of thrift, a discount
factor shock that makes all the middle-aged more patient (higher β) cannot be absorbed through
a cut in the real interest rate because of the ZLB. A higher desire to save at the individual level
shifts the  curve leftwards. Deflation rises, real interest rates increase and aggregate income
from which to save plummets.
68
According to the paradox of toil, an increase in potential output  widens the negative
output gap. A positive labour supply (or TFP) shock that raises
(or A) moves both the 
segments rightwards but it has no impact on the  curve. Again deflation rises real interest
rates and squeezes borrowings of the young.
The paradox of flexibility is of the same kind. The DNWR parameter acts as a floor on the
fall of prices. When it is reduced, only  becomes steeper and with a lower intercept.
Therefore, the resulting wage flexibility generates deflationary pressures, higher real rates,
more unemployment and economic contraction. It is paradoxical since if all prices and wages
were flexible, then there would be no secular stagnation at all.
67
An expectations-driven liquidity trap with indeterminate ZLB-DNWR equilibrium is studied in Schmitt-Grohé
and Uribe (2013) in an infinitely-lived representative agent setting. This strand of literature has been initiated by
Benhabib et al. (2001).
68
It is improper to talk about the paradox of thrift without aggregate saving and investment. In the model with
capital, higher real interest rates imply higher rental rates of capital and lower investment and output. However,
the same general logic applies if capital is removed.
54
Figure 2.5: Data versus model transition paths in advanced economies (percent)
Notes: The absence of outright deflation in the United States and the Eurozone is captured by including
in the wage norm indexation to the inflation target. Changes in potential output in Japan and the
Eurozone are captured by including trend productivity growth and a hysteresis parameter.
Source: Eggertsson, Mehrotra and Robbins (2017).
55
2.4 Policy implications of the secular stagnation steady state
Regardless of whether advanced economies find themselves already there or not, the existence
of a secular stagnation steady state brings undeniable challenges to the policymakers. The
model offers interesting insights on the current and future policy design and its impact on the
natural rate of interest.
First of all, macroprudential policy measures are crucial for the natural rate level because
they influence the collateral constraint for the young generation. This is even truer in a
ZLB-DNWR equilibrium, where the bond market cannot clear and the demand for loans is
depressed. A relaxation of the debt limit is helpful.
What about monetary and fiscal levers? Monetary authorities are the most affected because
of the ZLB. They cannot use standard rate cuts as in the past. In the last decade central banks
have coped with the constraint by adopting unconventional tools. However, within the
framework of this model forward guidance is of limited use since policy rates are expected to
stay at zero indefinitely. Quantitative easing, as long as it is conceived as temporary asset
purchase programmes, does not play any role. In fact, in secular stagnation the ZLB is a
consequence of a permanently depressed natural rate. For a given inflation target, additional
accommodation does not make the full employment steady state materialise. Negative policy
rates are more appropriate but in practice the option to hold cash and the adverse side-effects
limit its potency. The first best to defeat the ZLB would be the removal of paper money so that
the  segment would disappear. Of course, the conversion to electronic money is not a
solution immediately on hand. Therefore, in a low environment a revision of the central bank
mandate is inevitable. Within the model, a higher inflation target  is a potential response in
order to accommodate the negative natural rate. It does not shift the entire AD curve upwards
but only its kink point (Fig. 2.6). The commitment of the central bank has to be strong enough
so that a TR-FE equilibrium exists. A modest increase in the inflation target moves the economy
nowhere and loses credibility (). Krugman has referred to small changes in the inflation
target as the “timidity trap” or the “law of the excluded middle”. In this light, for instance, we
may read the recent experience in Japan, where the estimates of the natural rate have been
gloomy. The decision of the Bank of Japan to tweak its inflation objective from  to  in
2013 has not boosted inflation and growth.
The action has to be sufficiently aggressive so that 
 . If 3%, the central
bank deals with it by setting  4%. The announcement of the new target identifies a
56
multiplicity of equilibria ().
69
The secular stagnation and the full employment steady states
are the usual ones. The latter is present because the ZLB is not binding for: the second
term in the max operator of the Taylor rule is greater than one. There is also an “unintended”
ZLB-FE equilibrium, characterised by ,
, : the second term in
the max operator of the Taylor rule is lower than one, but the Fisher equation holds for .
However, at that point inflation is locally indeterminate in transition dynamics.
In the end, the secular stagnation steady state remains and credibility is still an issue.
Households and firms can rationally ignore the new inflation commitment of the central bank
since it does not generate any pressure on the  schedule. Which of the two determinate
steady states is selected is unknown. The possible lack of impact marks a profound detachment
from the temporary liquidity trap strand. Furthermore, the natural rate is still fixed at a negative
value because the fundamental shock is not temporary. It is hard not to see the risks that
indefinitely negative real rates may spur in financial markets, e.g. asset-price bubbles. All in all
the monetary policy outlook is not shining.
Figure 2.6: Effect of raising the inflation target and multiple steady states (green line)
69
We cannot make assumptions about uniqueness as before. There is one case where the equilibria are only two:
if 
 ,   and the intersection occurs at the kink point of the AD curve.
57
Expansionary fiscal policy is theoretically the best strategy for demand deficiency issues.
For instance, it is necessary when private sector agents are repairing their balance sheets. And
it has to do much more when money has never been cheaper, i.e. when real interest rates are
negative. The fiscal policy rule (25) implies that the level and distribution of taxation between
the middle-aged and the elderly does not affect the supply curve for loanable funds. In the
steady state the reduction of savings caused by a higher is fully offset by a higher .
Instead, on the loan demand side, higher government debt plays a crucial role since it
generates more borrowings and reduces the surplus in savings. Increased public debt (as
percentage of GDP) is recognised as the main factor that has avoided a further drop in the
natural rate of interest in the developed world since the 1970s.
70
Government borrowing
decisions are not neutral. In fact, looking at the model, is positively related to and, when
the rise in government spending is debt-financed, the multiplier on output is the largest among
the different sources of funding. The positive impact depends on the durability of such a stance.
If debt issuance is viewed as delayed taxation, then it may undermine the stimulating effect
because it will lead to higher savings by the middle cohort today in order to face likely future
tax burden when old. That is fully “Ricardian”. Thus, an increase in public deficits and debt
eradicates secular stagnation and pushes the economy back to the optimum point only if it is
expected to be permanent.
71
It does so by shifting rightwards both the segments of the AD
curve. There is a minimum level of debt that makes the natural rate positive and the TR-FE
equilibrium again the unique steady state (Point B in Fig. 2.7).
72
The proposal might work in
the model, yet concerns about the feasibility and sustainability of a massive fiscal intervention
on a continuous basis are legitimate. It applies a fortiori in the United States, Japan and some
Eurozone members that already display high debt-to-GDP ratios and may not be able to issue
risk-free bonds. In their quantitative calibration using the American data, EMR acknowledges
the limits of fiscal policy since a debt-to-GDP ratio that starts at 118% needs to increase up to
215% in order to raise a natural rate of interest from a value of 1.47% to 1%.
73
Anyway, the
remarkable effectiveness of higher public debt in a secular stagnation environment does not
mean that it has to explode and taxes cannot be adjusted. Any kind of fiscal package is
successful to the extent that it absorbs excess savings in the economy.
70
Eggertsson, Mehrotra and Robbins (2017), Rachel and Summers (2019).
71
Unlike in an infinitely-lived representative agent setting, Ricardian equivalence fails in OLG models without
intergenerational altruism.
72
What affects the natural rate is public debt held by the private sector.
73
The US public debt-to-GDP ratio includes federal, state and local debt.
58
Figure 2.7: Expansionary fiscal policy via debt issuance in secular stagnation
2.5 Further extension with capital
In the analysis of secular stagnation the choice of doing without capital accumulation might
seem unexpected at first glance. Incorporating capital in the OLG model overcomplicates the
computations of the steady states and the policy implications. However, in the end the concern
about the phenomenon involves the imbalance between desired investments and desired
savings. Therefore, it is important to give the most relevant insights, while Appendix B provides
the detailed operations. These insights are helpful. The setting adds the following elements:
the middle-aged are assumed to use their savings in two ways. They either make loans
to the youngsters and the government or they accumulate capital/investment goods ,
such as business equipment, which is rented out to the firms. The productive asset does
not carry a risk premium and its relative price in terms of the consumption good is
denoted by ;
the depreciation rate is strictly positive;
74
74
Strict positivity is easy to claim since one period in an OLG model is about twenty years long.

59
in the next period, the elderly dissave and resell the undepreciated capital stock at a
value 
;
75
the production function is a constant returns to scale Cobb-Douglas;
firms pay to the middle-aged the real rental rate for capital , which is equal to its
non-negative marginal product in absence of mark-ups;
according to the no-arbitrage condition, the rental rate and the real interest rate are
positively related.
Now perfect competition and constant returns to scale allow to write aggregate income in
terms of factor costs:
The option of capital accumulation and the no-arbitrage equation between the real interest rate
and the non-negative rental rate impose a lower bound on the former in the steady state:

where  is the gross trend rate of growth of the relative price of investment goods. The
decline in the price of capital goods due to advances in information technology since the early
1980s and the strictly positive depreciation rate do not make the condition particularly binding.
According to the assumptions, investment decision has an impact only on the loan supply
schedule. In fact, the savers and not the borrowers are the ones who invest, so the amount of
funds in the bond market depends upon the value of capital purchases:

 



The equation of the gross natural rate of interest displays new drivers:



 
which means that a fall in the relative price of capital goods (lower ) or a declining trend
(lower 
) reduces persistently the resources needed for building the same capital stock and
leaves more savings available in the bond market. It may even have a negative impact on the
75
Yet in this version of the EM model the authors do not surprisingly explain what happens to the capital the
elderly sell.
60
propensity to invest since the elderly face a lower future resale value to finance consumption.
The loan supply curve shifts rightwards, the real interest rate declines and so does the natural
rate. Moreover, EM mentions the consequence of an economy built upon less durable goods,
such as non-construction assets. Even in this case a higher lowers the natural rate given that
the future resale value of capital diminishes.
Lastly, the introduction of capital connects the  segment with the AD schedule. The
former becomes downward sloping