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REVIEW
FEDERAL RESERVE BANK OF ST. LOUIS
THIRD QUARTER 2020
VOLUME 102 | NUMBER 3
A Short Tour of Global Risks
Carmen M. Reinhart
Offshoring to a Developing Nation with a Dual Labor Market
Subhayu Bandyopadhyay, Arnab Basu, Nancy Chau, and Devashish Mitra
Asset Pricing Through the Lens of the Hansen-Jagannathan Bound
Christopher Otrok and B. Ravikumar
Reconstructing the Great Recession
Michele Boldrin, Carlos Garriga, Adrian Peralta-Alva, and Juan M. Sánchez
The Case of the Reappearing Phillips Curve: A Discussion of Recent Findings
Asha Bharadwaj and Maximiliano Dvorkin
REVIEW
Volume 102 • Number 3
President and CEO
James Bullard
Director of Research
Christopher J. Waller
Chief of Staff
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Deputy Directors of Research
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Review Editor-in-Chief
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Research Economists
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Serdar Birinci
YiLi Chien
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Maximiliano Dvorkin
Miguel Faria-e-Castro
Victoria Gregory
Sungki Hong
Kevin L. Kliesen
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Fernando Leibovici
Oksana Leukhina
Fernando M. Martin
Michael W. McCracken
Amanda M. Michaud
Alexander Monge-Naranjo
Christopher J. Neely
Michael T. Owyang
Paulina Restrepo-Echavarría
Hannah Rubinton
Juan M. Sánchez
Ana Maria Santacreu
Guillaume Vandenbroucke
Yi Wen
Christian M. Zimmermann
221
A Short Tour of Global Risks
Carmen M. Reinhart
237
Offshoring to a Developing Nation
with a Dual Labor Market
Subhayu Bandyopadhyay, Arnab Basu, Nancy Chau, and Devashish Mitra
255
Asset Pricing Through the Lens of the
Hansen-Jagannathan Bound
Christopher Otrok and B. Ravikumar
271
Reconstructing the Great Recession
Michele Boldrin, Carlos Garriga, Adrian Peralta-Alva, and Juan M. Sánchez
313
The Case of the Reappearing Phillips Curve:
A Discussion of Recent Findings
Asha Bharadwaj and Maximiliano Dvorkin
Managing Editor
George E. Fortier
Editors
Jennifer M. Ives
Lydia H. Johnson
Designer
Donna M. Stiller
Federal Reserve Bank of St. Louis REVIEW
Third Quarter 2020
i
Review
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© 2020, Federal Reserve Bank of St. Louis.
ISSN 0014-9187
ii
Third Quarter 2020
Federal Reserve Bank of St. Louis REVIEW
A Short Tour of Global Risks
Carmen M. Reinhart
This article is based on the author’s Homer Jones Memorial Lecture delivered at the Federal Reserve
Bank of St. Louis, Wednesday, June 25, 2019. (JEL E51, F3, G15, G28)
Federal Reserve Bank of St. Louis Review, Third Quarter 2020, 102(3), pp. 221-35.
https://doi.org/10.20955/r.102.221-35
I
t’s a pleasure and an honor to deliver the 2019 Homer Jones Lecture. What I’d like to do
is examine global risks and connect those risks to the literature and work that I’ve done.
The tour begins with some of the risks. This is not meant to be encyclopedic; but I will
try to be brief so we can cover a lot of ground—some of the risks in the advanced economies
and then, truly global in nature, move on to risks in emerging markets. I would note that,
in the past, interest in emerging markets was really limited to traders that bought emerging market bonds and occasionally equity. But while in the early 1980s emerging markets
accounted for about a third of global gross domestic product (GDP), now they account for
about two-thirds of global GDP. So, it’s really not possible to talk about the global economy
without a full, rounded view of advanced and emerging economies.
I’ll first focus on what I see as more short-term concerns and then talk about a long-term
issue that keeps cropping up on my radar screen—something that I’ve been working on for a
long time: What’s going to happen with the U.S. dollar as a reserve currency? And I’ll conclude
there.
So, on to global risks in the advanced economies. I am going to start where much of the
discussion has been recently, which is, of course, on issues relating to trade and globalization
but also on issues relating to how much ammunition the advanced economies have in the
event of a downturn. Let me start there.
Figure 1—not surprisingly, given the kind of work that I’ve done in the past on debt—
basically shows the level of public debt from 1900 through the present for over 20 advanced
economies. I want you to have three takeaways from this graph. Number one is pretty self-
evident. Advanced economies as a whole have the highest levels of debt since World War II
Carmen M. Reinhart is the Minos A. Zombanakis Professor of the International Financial System at Harvard Kennedy School.
© 2020, Federal Reserve Bank of St. Louis. The views expressed in this article are those of the author(s) and do not necessarily reflect the views of
the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Articles may be reprinted, reproduced, published,
distributed, displayed, and transmitted in their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses,
and other derivative works may be made only with prior written permission of the Federal Reserve Bank of St. Louis.
Federal Reserve Bank of St. Louis REVIEW
Third Quarter 2020
221
Reinhart
Figure 1
Advanced Economies: Government Debt as a Percent of GDP, 1901-2019
Percent
How were WWII
debts reduced?
Default, financial
repression/inflation,
fiscal tightening,
and growth.
140
120
100
80
How were WWI and
depression debts
reduced?
Default, restructuring,
and conversions—
a few hyperinflations.
How will GFC debts
be reduced?
U.S.
Unweighted average for
22 advanced economies
60
40
20
0
1901
1911
1921
1931
1941
1951
1961
1971
1981
1991
2001
2011
NOTE: GFC, global financial crisis.
collectively (blue shaded area). Number two, the United States (black line), which had more
fiscal space than a number of advanced economies on the eve of the global financial crisis,
has since had a bigger surge in debt and now has more sustained debt than what we’ve seen
in other advanced economies.
Number three is tricky because you have to use your imagination: It’s not what you see
that should worry you; it’s what you don’t see that should worry you. This is strictly on-budget
public debt, and therefore any off-balance-sheet items are not included. And two points on
off-balance sheet items. At the end of World War II, public debt was the whole story. Private
debt had been unwound through the Great Depression and the war, so the private sector was
lean and mean. That applied to households. That applied to corporations. The whole story was
what you see in this graph. In addition, advanced economies had much younger populations
and very limited pension liabilities at that time, which you also don’t see here.
The point that I am making is that, if Figure 1 already highlights that public indebtedness
is a limiting factor (a limiting factor in how much fiscal space advanced economies have to
cope with a downturn), I would add that limiting that fiscal space are further considerations
in that private debt. In the context of the United States, I think there are some concerns on
the corporate side. I will talk about that later. But generally, private debt levels are quite high
in the advanced economies, certainly very high by postwar measures. And pension liabilities
are understandably an issue unlike ever before, because of the aging structure of the popula222
Third Quarter 2020
Federal Reserve Bank of St. Louis REVIEW
Reinhart
Figure 2
Twin Deficits
A. Twin balances: General government budget and current
accounts relative to nominal GDP, average 2018-20 (percent)
12
Twin surplus quadrant
Current account/GDP (percent)
8
4
–6
–4
–2
0
0
2
4
–4
U.S.
–8
Twin deficits quadrant
–12
General government budget balance/GDP (percent)
6
B. Gross government debt relative to GDP (percent)
Japan
Greece
Italy
Portugal
United States
Belgium
France
Spain
Cyprus
United Kingdom
Canada
Austria
Slovenia
Ireland
Israel
Finland
Germany
Netherlands
Slovak Republic
Malta
Switzerland
Australia
Korea
Norway
Iceland
Denmark
Sweden
Latvia
Lithuania
Czech Republic
New Zealand
Luxembourg
Estonia
Hong Kong SAR
0
50
100
150
200
250
NOTE: Red dots and red bars indicate countries with twin deficits.
SOURCE: IMF and World Economic Outlook.
tion and the pension liabilities we’ve accumulated. So bottom line: Fiscal space is a lot more
limited, notwithstanding many arguments out there that debts don’t matter and deficits don’t
matter.
Note again the solid line in Figure 1. As I’ve mentioned, the United States has less fiscal
space than the other advance economies because of the growing U.S. debt. The Congressional
Business Office has done studies that basically show that in about a decade’s time, even with
interest rates remaining where they are or even with interest rates moving lower by about 50
basis points, the United States still has a debt sustainability issue arising—so bear that in mind.
I would also like to point out for the United States that, if we were an outlier in Figure 1
in our accumulation of debt, we’re also an outlier in how quickly we are adding both public
debt and external debt to our balance sheets. The bottom-left quadrant of Panel A of Figure 2
shows countries with twin deficits. Basically, a twin deficit means you have a current account
deficit—you’re borrowing from the rest of the world—and a fiscal deficit.
Former Federal Reserve Chairman Bernanke for many, many years talked about the saving glut. The saving glut is basically what you see here on the left side—the deficits. The saving
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Reinhart
Figure 3
Three-Month LIBOR Interest Rates
Percent
8
7
6
5
4
3
2
1
0
–1
1/1999
Japan
7/2002
1/2006
7/2009
1/2013
Euro
U.S.
7/2016
SOURCE: FRED®, Federal Reserve Bank of St. Louis.
Table 1
Monetary Policy “Space”?
Starting federal funds
rate (percent)
Lowest federal funds
rate (percent)
Cumulative cut
(percentage points)
1990
8.25
3.0
5.25
2001
6.50
1.0
5.50
2007
5.25
0
Recession
~5.25
SOURCE: Federal Reserve.
glut basically amounts to China saving a lot. We don’t save as much. Surpluses in Asia and
Germany also are offset by our deficit. That’s an old story. Certainly, it’s a story that carried
weight in the 1980s. It began in the 1990s, it continued in the 2000s, and it has continued to
the present.
What is relatively new is that in addition to the old flow problem, we now have more of a
stock problem, meaning we are adding debt when our relative standing in terms of global
indebtedness has notched up considerably. I will return to this issue later when I talk about
what we can expect over the medium term for the U.S. dollar as a reserve currency.
Again, fiscal space is much more limited now for the advanced economies than it was at
the time of the global financial crisis. I would argue that monetary policy space is also much
more limited for obvious reasons. As shown in Figure 3, Japan has had negative interest rates
for some time. Europe has had negative interest rates for some time. And for the United States,
which is the outlier with positive interest rates, in the past, the average decline in the federal
funds rate to combat recession had been 600 basis points (Table 1). This is something that we
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Reinhart
Figure 4
Global Annual Export Growth, 1928-2009
Annual change (percent)
45
+/– One standard deviation
35
25
15
5
–5
Average (1928-2008) = 7.8
–15
2009 estimate
–25
The Great Depression
–35
1928 1933 1938 1943 1948 1953 1958 1963 1968 1973 1978 1983 1988 1993 1998 2003 2008
SOURCE: Reinhart and Rogoff (2009).
are not capable of delivering at the moment. So I think a real risk—and I will later conclude
on this point—is that the advanced economies, collectively, are seriously constrained in terms
of policy tools to deal with bad shocks. And that is any bad shock.
Let’s turn to a shock that has been very much in the press—the trade wars. Figure 4 shows
global annual export growth from 1928 to 2009. The figure ends in 2009 because it is taken
from my book with Ken Rogoff. What are people worried about? Well, people are worried
about a replay. In one potentially bad scenario, they’re worried about a replay of the aftermath
of the Smoot-Hawley tariffs and the trade war of the 1930s, which produced that record contraction in global trade.
Are we there? Are we close to there? What’s going on with globalization? Well, let me
make a couple of points about globalization. Actually, I wrote about this in Project Syndicate
years ago. The peak in globalization was in the year before the crisis. Since the crisis, we’ve been
moving toward a lower level of global growth in terms of trade (Figure 5). If you look at the
decade before the global financial crisis, average trade growth volume was about 6 percent.
In the decade after the crisis, it was less than half of that. This is not unique to the post-global
financial crisis experience. This is not the first era of globalization we’ve had.
I think people don’t realize that in the late 1800s to early 1900s, before World War I, we
had a very globally integrated capital and goods and service system, albeit limited by the technology at the time. But that globalization was shot to pieces, first by World War I, then by the
Depression, and certainly by World War II. Although the financial crisis did not have the
extent of drama that the two world wars and major depression produced, it did put a big dent
in global trade: It made countries running a current account deficit—such as Spain, Greece,
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Reinhart
Figure 5
Global Trade Now in Negative Territory: Will This Time Be Different?
Volume of world trade 12-month change (percent)
25
Pre-GFC average growth 5.9%;
20
Post-GFC average growth 2.4%
15
10
5
0
–5
–10
–15
–20
–25
1/2001
7/2003
1/2006
7/2008
1/2011
7/2013
1/2016
7/2018
NOTE: GFC, global financial crises.
SOURCE: CPB Word Trade Monitor, February 2019.
Italy, Ireland, and others—realize that you can’t finance a current account deficit from the
rest of the world. So you have to watch. You have to import less. You have to look more to
home. I think the issue of the rise of home bias dates back to the global financial crisis. I would
note that Brexit was another major blow, and more recently, of course, what we’re seeing in
the trade wars is yet another.
What, in a nutshell, do I take away from the trade wars? Well, I found that my first assessment was completely wrong. If you had asked me in 2018 what I thought, I would have said
the trade wars would be resolved a lot quicker, that it would be more of a NAFTA-type situation with a swifter resolution. It wasn’t. It isn’t. And it’s my view now that it’s not likely to be
because what I’ve seen is that, over the course of this period, it’s become not just about trade,
but about geopolitical issues, about security issues and all kinds of issues that are unlikely to
be resolved entirely with a handshake. So I think because of the electoral cycle in the United
States, we are going to get some news, some deliverables on trade, but not a resolution. And
certainly, I’m not looking for a return to the pre-crisis globalization era.
Let’s continue on our global tour of advanced economy risks. I just returned from Europe.
I was in Paris giving a talk there last week, and always the question is, what do you think are
the weak points? Well, I think one can’t talk about the next crisis in Europe without really
making the point that the previous crisis hasn’t been resolved for all of Europe. If you look at
Figure 6, two things stand out: Financial crises produce a recession, a drop in per capita GDP,
but ultimately countries recover. The recovery is pretty dramatic for Korea, but for Italy and
Greece, it’s nonexistent. If you take the level of Greek GDP or Italian per capita GDP today,
it is below what it was in 2007. And if you take the International Monetary Fund (IMF) projections out to 2024, even by 2024, per capita income in Greece and in Italy will still be below
what it was in 2007. That is a pocket of weakness that I think will continue to be a source of
both tension and recurring bouts of global uncertainty when we talk about Europe.
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Reinhart
Figure 6
Post-Crisis Recovery (or Lack Thereof in Southern Europe): Unresolved Debt Overhangs
Index of real GDP per capita
190
Average breakeven for the
worst 100 crises ≈ 7 years
170
150
Korea Asian crisis comparison
1997 = 100
130
U.S. GFC comparison
2007 = 100
110
90
70
Italy 2007 = 100
t
Greece 2007 = 100
t+1 t+2 t+3 t+4 t+5 t+6 t+7 t+8 t+9 t+10 t+11 t+12 t+13 t+14 t+15 t+16 t+17
Number of years after start of GFC (2019-2024, IMF estimates)
NOTE: GFC, global financial crisis.
Figure 7
Federal Funds Rate and Financial Conditions
Percent
9
8
7
6
5
4
3
2
1
0
2/1990
Index
106
Federal funds rate (left axis)
Financial conditions (right axis)
Neutral (right axis)
104
102
100
98
96
2/1994
2/1998
2/2002
2/2006
2/2010
2/2014
2/2018
94
SOURCE: Federal Reserve and Goldman Sachs, accessed via Bloomberg, May 13, 2019.
I say “global uncertainty” because if you go back to 2018, every time the new Italian government made an announcement about possibly leaving the euro or contemplating not servicing
debt, those comments translated into an appreciation in the dollar. A depreciation in the euro
and an appreciation in the dollar is bad news for emerging markets that have dollar debts. It
is a global shock. Let’s now move on to yet another but very different type of risk.
U.S. financial conditions remained fairly accommodative through 2018 (Figure 7, black
line). Look, for instance, at the financial conditions index that Goldman Sachs publishes
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Reinhart
Figure 8
Yield Ratios: Risky Yields Relative to Corporate Debt Yields
21
16
High-yield corporate bonds
EMBI+
11
6
1
2/1/2000
2/1/2003
2/1/2006
2/1/2009
2/1/2012
2/1/2015
2/1/2018
NOTE: EMBI+, J.P. Morgan Emerging Bonds Index.
SOURCE: Bloomberg and author’s calculations.
(Figure 7, green line). I do not suggest that this is, by any means, the be-all end-all liquidity
measurement of financial conditions. But by and large, the point I’m making here is that financial conditions, on the whole, up until the end of 2018 were relatively accommodated. And
that, in turn, also contributed—in a world of low yields—to the eternal search for yield. I have
done work going back to 1815 on the search for yield (Reinhart, Reinhart, and Trebesch, 2016):
The search for yield is eternal. And it drove investors into high-yield corporate debt and,
notably, comparatively newer instruments like collateralized loan obligations (CLOs).
I think we should pause a minute and think about what kinds of risk the honeymoon that
we’re seeing or have seen in these episodes is bringing to the table. Concretely, CLOs have
some similarities to the mortgage-backed security problem in that they’re not only attracting
local interest but also global interest. In other words, foreign banks: Japanese banks and
European banks are coming into the CLO market, which also potentially means that if that
market sours, there will be global spillovers into other markets. Let’s be clear: I’m not suggesting comparable magnitudes, but magnitudes in some dimension reminiscent of 2008-09.
I always remind people that German banks did not get into trouble in 2009 because they
had a real estate bubble in Germany. They did not have a real estate bubble. They got into
trouble because they had bought U.S mortgage-backed paper. So there’s some scope there
for an international contagion dimension.
There is also a worrisome trend: The quality of the borrowers and the quality of the covenants of these loans have been deteriorating. And that is also reminiscent of the run-up to
the global financial crisis in which the earlier tranches of the mortgage pools were better quality
than the later tranches. Just to be clear, what I’m saying is that, historically, emerging market
yields—emerging markets, high-risk debt—have moved together with corporate high-yield
debt (Figure 8). What we have seen in the last year and a half is that emerging market yields
on the whole went higher while—at the same time—corporate yields went lower.
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Reinhart
Figure 9
Foreign Exchange Reserve Ratios and the Currency
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
1/1999
8.5
Reserves/M2
(left scale)
8.0
7.5
7.0
Renminbi/USD
(right scale)
6.5
6.0
7/2001
1/2004
7/2006
1/2009
7/2011
1/2014
7/2016
5.5
1/2019
NOTE: USD, U.S. dollar.
SOURCE: FRED®, Federal Reserve Bank of St. Louis.
So the question to you is this: What does this mean? Are we overpricing—overestimating—
the risks in emerging markets, or are we underestimating the risks in the corporate sector?
The arguments I’ve made thus far is the latter, and that’s the convergence that I’m showing in
Figure 8.
Let’s turn to emerging markets, specifically China, which by almost any metric is the size
of the U.S. in the global economy. So we’re talking about the world’s second-largest economy
and, depending how you measure it, in some instances the same size as the U.S. economy.
The same points that I made about advanced economies having more-limited ammunition, I
am going to make about China—and not about fiscal policy but about monetary policy.
Let me clarify. One of the concerns we’ve had when discussing the potential for a global
slowdown is not just that the United States appears to be slowing or that Europe appears to
be slowing, but that China also appears to be slowing—and slowing big-time. We should be
concerned, again, because China is the world’s second-largest economy. And a level of comfort is usually drawn from the idea that it can provide stimulus. And indeed, China is providing fiscal stimulus to the tune of about 1.75 percent of GDP. However, I don‘t think it’s
reasonable to expect China to provide fiscal stimulus and monetary stimulus, which for them
is credit creation—meaning providing accessible credit to the corporate sector, to exporters,
to banks, and so on.
Let me explain. In 2008-09, China really did record—by almost any metric—fiscal stimulus and monetary stimulus. At that time, however, China was growing double digits. They
had large capital inflows. They were accumulating U.S. Treasuries. They were accumulating
reserves. They were trying to lean against the wind to avoid a renminbi appreciation. That is
not where they are now.
Figure 9 shows international reserves converted into renminbi, divided by M2 (the blue
line). What this indicator shows is that China has gone through the full phase of the capital
flow cycle. (This indicator goes back to work that Graciela Kaminsky and I [Kaminsky and
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Reinhart
Figure 10
General Government Debt: Emerging Market and Middle-Income Economies, 1880-2023
SOURCE: Bredenkamp et al. (2019); https://www.imf.org/en/News/Seminars/Conferences/2018/05/24/sovereign-debta-guide-for-economists-and-practitioners.
Reinhart, 1999] did on indicators of the capital flow cycle and indicators of financial crises
many, many years ago.) China had a surge in inflows associated with the boom, and it is in
the outflow phase.
In other words, a country facing capital outflows and trying to maintain a more-or-less
stable currency can only do three things. One, they can try to stabilize the exchange rate by
losing reserves (i.e., selling their dollar holdings), which China’s been doing, intervening to
stabilize their renminbi. Two, they can tighten controls, which it’s also been doing. This is
related to the turmoil in Hong Kong. And three, they can keep tight money, which basically
goes to the point that I was making. I don’t think China has the ability right now to really
engage in very stimulative monetary policy, at least nothing like what it’s done in the past.
This is a global risk because China’s footprint, as we shall also see, among emerging markets is nothing less than major. We hear a lot about the impact of trade, but China also has a
huge impact through finance. And I end my commentary on China by saying that if you do
backward exercises, meaning you look at China’s trading partners, whether they’re commodity
producers or other Asian economies that export intermediate goods to China, and you look
at how much they’ve slowed, you would not infer that the remnibi-to-U.S. dollar slowdown
(Figure 9, green line) has only been to 6 or 6.5. You would infer that the slowdown is even
greater. So I think that that is also a serious—bigger, more protracted—Chinese slowdown
that is also a serious headwind to the global economy.
Briefly, I’ll also mention two other types of risk now moving out of the big countries and
into emerging markets. Figure 10, which is taken from recent work (Bredenkamp et al., 2019)
that I did for an IMF conference and volume, shows the indebtedness for emerging markets.
Since we are comparing previous episodes, let’s choose a relevant previous episode we all are
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Reinhart
Figure 11
The Rise of China as a Global Official Creditor, 1998-2018
NOTE: FDI, foreign direct investment.
SOURCE: Horn, Reinhart, and Trebesch (2019).
familiar with: the Asian crisis or the crisis of the 1980s. I would note emerging markets are in
more vulnerable territory. Not just Turkey. Not just Argentina. But emerging markets as a
whole have slowed down dramatically, largely as a consequence of the slowdown in China and
partly, also, as a consequence of dollar strength. These countries tend to have a high share of
either corporate or public debt, or both, in U.S. dollars. So a dollar appreciation means
higher debt servicing costs, which means more problems. But emerging markets are in more
vulnerable territory than they’ve been in a while.
Finally, before talking about the long-term issue, I mentioned that China’s role in emerging markets is not just the vast expansion in trade. It’s been an expansion in finance. (Figure 11)
Right now, Chinese lending to the emerging world is bigger than all such lending from the
Paris Club and the IMF combined—the official creditors, all the major advanced economies,
that lend bilaterally to emerging markets. China’s loans are bigger than all of those and the
IMF and the World Bank combined.
Well, what’s the problem? To say that the problem is that China’s lending is opaque would
be an understatement. It is not recorded by the Bank for International Settlement. It is only
partially recorded in the World Bank database. It is a thorn in the side of the IMF. The IMF’s
program with Pakistan—a big to-do—was actually trying to find out how much debt Pakistan
owed China. You can’t do debt sustainability exercises that are meaningful if you don’t know
what the outstanding level of debt is.
Hidden debts are a big problem for countries that have borrowed from China. And if you’re
an investor, you also worry about hidden debts: If you’re buying an Ecuadoran or Angolan
bond and you’re pricing them, thinking that the external debt of that country is, let’s say, 40
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Reinhart
Figure 12
Total External Debt: Officially Reported (World Bank) and “Hidden Debts” to China, 2000-18
NOTE: Median ext. PPG debt (IDS), median external, public and publicly guaranteed debt to GDP according to the
World Bank’s International Debt Statistics.
SOURCE: Horn, Reinhart, and Trebesch (2019).
Table 2
Countries Restructuring External Chinese Debt Since 2011
Tanzania
Cuba
Sudan
Ecuador
Bangladesh
Seychelles
Venezuela
Zambia
Cote D’Ivoire
Togo
Ukraine
Sri Lanka
percent, and it’s really 60 percent, you have a problem. And if also you don’t know who the
senior creditors are, you have a problem.
The issue that I’m raising here is based on ongoing work with Christoph Trebesch and
Sebastian Horn (Horn, Reinhart, and Trebesch, 2019) (Figure 12). The issue is that these are
economies that are not systemic. On the whole, they tend to be small. Collectively, they’re
not trivial either. And there is a problem of seriously underreported debts. The World Bank
database captures only about 50 percent of China’s loans to these countries. So with that uplifting note, let me turn to one final point. By the way, it is not a hypothetical that these debts
cause problems. If you look at sovereign restructurings, we’ve had about a dozen sovereign
restructurings already of Chinese debt that we know of (Table 2.) There may be more.
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Figure 13
Role of the Dollar and the Global Footprint of the U.S. Economy, 1950-2016
Percent
70
60
Percent
30
U.S. GDP as a share of world GDP
(right scale)
28
26
24
50
22
40
20
18
30
16
Share of countries where the U.S. dollar
is the principal anchor currency (left scale)
20
14
12
10
1950:M1
1960:M1
1970:M1
1980:M1
1990:M1
2000:M1
10
2010:M1
SOURCE: Ilzetzki, Reinhart, and Rogoff (2019).
Figure 14
Role of the French Franc and Deutsche Mark 1950-1998 and Euro 1999-2016
Percent
30
28
France and Germany GDP as a share of world GDP
(right scale)
Percent
12
10
26
24
8
22
6
20
18
16
Share of countries where the euro
(and previously the French franc and Deutsche
mark) is the principal anchor currency (left scale)
14
12
10
1950:M1
1960:M1
1970:M1
1980:M1
1990:M1
2000:M1
2010:M1
4
2
0
NOTE: Notice the difference in scales in comparison with those for the U.S./U.S. dollar in Figure 13.
SOURCE: Ilzetzki, Reinhart, and Rogoff (2019).
I now turn to my last topic: What about the long horizon? I talked about U.S. debt rising.
I talked about the twin deficit problem. I talked about the COB projecting that, even with rates
roughly where they are, there is still a debt sustainability problem. Well, let me bring up an old
topic—the Triffin dilemma. It arose in the late 1960s when the United States was borrowing
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Reinhart
heavily to finance the Vietnam War. And at the time, which was still under the Bretton Woods
system, countries held dollar reserves and stabilized against the dollar. There was demand for
U.S. dollar debt: Remember, countries and central banks do not buy greenbacks. They buy
debt. The essence of the Triffin dilemma is that for domestic considerations, you would like
to be more circumspect about your debt levels. The external dimension is that if you’re the
world’s reserve currency, you have a lot of rope to hang yourself with: The rest of the world is
willing to buy a lot of debt to sustain what may appear like a strictly domestic unsustainable
situation.
How did the Triffin dilemma resolve itself last time? It resolved itself with the breakdown
of the Bretton Woods system and dollar depreciation. The dollar depreciated versus the
Deutsche mark by about 55 percent. Figure 13 is from Ilzetzki, Reinhart, and Rogoff (2019).
The solid line shows one of our measures of the demand for dollars, or dollar debts, from the
rest of the world; this is the share of countries where the U.S. dollar is the main anchor currency.
The dashed line shows U.S. GDP as a share of global GDP. The modern Triffin dilemma, if
you will, is that the U.S. share in the global economy is getting smaller while the demand for
U.S. assets is getting bigger. (Also see Figure 14.)
How do you reconcile the two? Last time, the reconciliation was, we can say, a devaluation
because it was in an era of fixed exchange rates. The question now is, will this mean that once
again the equilibrating factor to impose a tax, if you will, on foreign bond holders is a secular
depreciation of the dollar? Now, every time you say “secular depreciation of the dollar” at a
time of uncertainty, you know you’re going to be wrong. You just know you’re going to be
wrong because every time you say “dollar depreciation” and there’s uncertainty, you have to
face the flight to quality, the flight to the dollar. Why are we in a situation where the long-term
secular trends tell you one thing and in the short run something else happens? I would have
to say that at the moment, it’s a lack of alternatives.
The fact is that the euro hasn’t delivered what everyone hoped it would deliver. There is
no liquid euro debt market. You have Italian debt. You have Greek debt. You have a more
fragmented system. The renminbi is not a convertible currency. And given the trends that I
described earlier for China, it’s clear that China has been scaling back on its ambitions to
make it an international currency relative to what their ambitions were six years ago.
Is the dollar going to depreciate on a secular basis, or are we going to continue to have
dollar appreciation every time global uncertainty pops up? Because if you look at every moment
of turmoil, it’s usually characterized by a flight into U.S. assets and an appreciating dollar. n
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REFERENCES
Bredenkamp, Hugh; Hausmann, Ricardo; Pienkowski, Alex and Reinhart, Carmen M. “Challenges Ahead,” in Ali Abbas,
Alex Pienkowski, and Kenneth Rogoff, eds., Sovereign Debt: A Guide for Economists and Practitioners. Chapter 9.
London: Oxford University Press, 2019; https://doi.org/10.1093/oso/9780198850823.003.0010.
Kaminsky, Graciela and Reinhart, Carmen. “The Twin Crises of Banking and Balance-of-Payments Problems.”
American Economic Review, June 1999, 89(3), pp. 473-500; https://doi.org/10.1257/aer.89.3.473.
Horn Sebastian; Reinhart, Carmen M. and Trebesch, Christoph. “China’s Overseas Lending.” Unpublished manuscript,
2019.
Ilzetzki, Ethan; Reinhart, Carmen M. and Rogoff, Kenneth. “Exchange Rate Arrangements in the Twenty-First Century:
Which Anchor Will Hold?” Quarterly Journal of Economics, May 2019, 134(2), pp. 599-646;
https://doi.org/10.1093/qje/qjy033.
Reinhart, Carmen M.; Reinhart, Vincent and Trebesch, Christoph. “Global Cycles: Capital Flows, Commodities, and
Sovereign Defaults, 1815-2015.” American Economic Review, May 2016, 106(5), pp. 574-80;
https://doi.org/10.1257/aer.p20161014.
Reinhart, Carmen M. and Rogoff, Kenneth S. The Time Is Different: Eight Centuries of Financial Folly. Princeton University
Press, 2009; https://doi.org/10.2307/j.ctvcm4gqx.
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Offshoring to a Developing Nation
with a Dual Labor Market
Subhayu Bandyopadhyay, Arnab Basu, Nancy Chau, and Devashish Mitra
We present a model of offshoring of tasks to a developing nation characterized by a minimum-wage
formal sector and a flexible-wage informal sector. Some offshored tasks are outsourced by the formal
sector to the lower-wage informal sector. Productivity improvements in performing offshored tasks
in the developing nation increase offshoring, but not necessarily formal-to-informal sector outsourcing, which can cause the developed nation’s wage to fall. Productivity improvements in the developing nation’s informal sector expand both offshoring and outsourcing, causing the developed nation’s
wage to rise. When the minimum wage is reduced in the developing nation, the developed nation’s
wage falls when most of the efficiency gains accrue to the informal sector. (JEL F1)
Federal Reserve Bank of St. Louis Review, Third Quarter 2020, 102(3), pp. 237-53.
https://doi.org/10.20955/r.102.237-53
1. INTRODUCTION
This article analyzes developed-to-developing nation offshoring in the presence of a dual
labor-market structure in the developing nation. While the developed nation’s labor market
is assumed to feature flexible wages and full employment, the developing nation is characterized by a dual labor market where a formal and an informal sector coexist. While the formal
sector is subject to a minimum-wage regulation, the informal sector is assumed to be able to
circumvent that law or the law does not apply to it and pay a lower market-clearing wage. It
is also possible that the formal sector circumvents the law by outsourcing to the informal
sector or hiring informal or casual workers to perform certain tasks. Consideration of labor-
market duality leads to some important departures from the existing literature on trade in tasks,
which was pioneered by Grossman and Rossi-Hansberg (2008, GRH hereafter) among others.
As described in Bhagwati and Panagariya (2013), India has over 200 labor regulations
that apply to firms in the formal sector. These regulations make labor costs higher than what
they otherwise would be and adversely affect the flexibility of firms in responding to shocks.
Subhayu Bandyopadhyay is a research officer and economist at the Federal Reserve Bank of St. Louis. Arnab Basu and Nancy Chau are professors at
the Charles H. Dyson School of Applied Economics and Management at Cornell University. Devashish Mitra is a professor at The Maxwell School
of Citizenship and Public Affairs at Syracuse University. The authors thank two anonymous referees for their very helpful comments.
© 2020, Federal Reserve Bank of St. Louis. The views expressed in this article are those of the author(s) and do not necessarily reflect the views of
the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Articles may be reprinted, reproduced, published,
distributed, displayed, and transmitted in their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses,
and other derivative works may be made only with prior written permission of the Federal Reserve Bank of St. Louis.
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In practice, firms find ways of getting around these labor regulations by incurring some costs.
For example, Ramaswamy (2003) documents that formal sector manufacturing firms in India
are able to circumvent labor regulations by hiring temporary (casual) or contract workers to
whom those regulations do not apply. Hasan and Jandoc (2013) show that even in large Indian
manufacturing firms with employment over 200 workers, casual or contract workers constitute about 30 percent of total employment. Harris-White and Sinha (2007) provide anecdotal
evidence supporting outsourcing of certain activities from formal sector to informal sector
firms in India. Sundaram (2015) also provides evidence indicative of outsourcing of relatively
labor-intensive activities from formal sector to informal sector firms in India. And, finally,
Sundaram, Ahsan, and Mitra (2012, p. 79) provide evidence of “linkages between the formal
and informal manufacturing sectors through outsourcing.” In addition to the evidence for
India, there is also evidence for Mexico showing that about 25 percent of employees of formal
firms are informal workers; thus, formal firms are able to avoid many labor regulations
(Samaniega de la Parra, 2016).
The paper by GRH is one of the first to model trade in tasks in the context of a developed
nation offshoring tasks to a lower-wage nation. The paper’s structure is similar to neoclassical
competitive models of trade. Accordingly, as in the Heckscher-Ohlin type framework, a reduction in the cost of offshoring has a positive wage effect similar to a productivity increase. This
leads to the somewhat counterintuitive result that technological improvements in offshoring
that lead to more tasks being offshored can actually lead to a higher wage for labor in the developed nation. This is possible because technological improvements lead to cost savings and
scale expansion, and these are reflected in a higher domestic wage at full employment. The
paper by GRH focuses on the developed nation, and the nation that performs the offshored
tasks is modeled simply as a nation with a fixed wage.
Bandyopadhyay et al. (2020) provide a model for the joint determination of wages in the
developed (source) nation that offshores tasks and a developing (recipient) nation that completes the tasks. Within the context of this model, they derive several results that show that
while a developed nation may gain from technological improvements in offshoring, the developing nation could lose if the labor-saving effect of technological improvements outweighs
the scale-expansion effect. One major issue not considered by Bandyopadhyay et al. (2020) is
the importance of the informal sector in developing nations. Indeed, while the formal sector
can feasibly be monitored by the government, the informal sector is often out of reach of government regulations. This means that labor standards or minimum-wage laws are hard to enforce
in the informal sector, which creates an incentive for firms to outsource some of their tasks
to the informal sector. Keeping this duality between the formal and informal sectors in mind,
we analyze how technological improvements may impact wages and employment in a simultaneous labor market equilibrium in three markets: the developed nation’s labor market, the
developing nation’s formal sector labor market, and finally the developing nation’s informal
sector labor market.
We build a model where two nations, which are small in the output market, have a bilateral
offshoring relationship in the production of a manufacturing good.1 As in GRH, competitive
firms based in the developed nation produce this good by completing a range of tasks. Some
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of these tasks are relatively complex and require more labor to be completed in the developing
nation, so they are completed in the developed nation, where it is cheaper, while the rest of the
tasks are offshored. Among the offshored tasks, intermediate-complexity tasks are completed
in the developing nation’s minimum-wage formal sector, while the least-complex tasks are
completed in its lower-wage informal sector where it is cheaper.2 This second layer of task
allocation is commonly referred to as “domestic outsourcing,” which allows formal sector
firms to circumvent the minimum wage.3
The focus of our general equilibrium model is simultaneous labor market clearing in the
developed and the developing nations, where each nation has two sectors, a manufacturing
sector and a numeraire agricultural (food) sector.4 In the developing nation, there is a dual
labor market characterized by a rigid-wage formal manufacturing sector and a common
flexible wage in the informal manufacturing sector and the agricultural sector. Flexible wages
characterize the developed nation’s labor market. The residual labor supplies of the manufacturing sectors are absorbed by the respective agricultural sectors of the two nations. We primarily analyze how the flexible wages in the two nations are affected by changes in offshoring
technology and outsourcing technology. We also analyze how these factors and parametric
changes affect other endogenous variables of interest, such as the levels of offshoring and
outsourcing and the share of the informal sector in the developing nation’s economy.
The comparative static analysis yields some results that depart from the existing literature.
For example, while a rise in offshoring productivity raises offshoring, it may reduce the developed nation’s wage. This can happen because the developing nation’s informal sector wage
may rise through offshoring demand effects and also because of the accompanying shift of
marginal tasks from the low-wage informal sector to the higher-wage formal sector. As a result,
the degree of informality, given by the ratio of informal-to-formal sector manufacturing
employment, may fall. Some other results are counterintuitive at first glance. For example,
although increased informal sector productivity in the developing nation will raise formalto-informal sector (formal-informal) outsourcing, it may reduce both the informal sector’s
wage and the degree of informality in the nation’s manufacturing sector. Similarly, while a
minimum-wage cut reduces informality, it may actually increase the informal wage.
Section 2 presents the model and the description of the equilibrium. Section 3 presents
comparative static analyses. Section 4 concludes.
2. THE MODEL AND EQUILIBRIUM
2.1 The Basic Structure
Consider two nations, a developed nation F and a developing nation H. There are two
homogeneous goods, a numeraire manufactured good and food. We assume that the two
nations are small in the output market, so the prices of both goods can be set at unity, without
loss of generality. The output levels of the manufactured good and food in nation F are denoted
by x* and y*, respectively. Nation H also produces food, for which the output level is denoted
by y, and workers in nation H may also perform tasks offshored by nation F’s manufacturing
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sector. For simplicity, we assume that all of the manufacturing sector’s activity in H is completion of the tasks offshored by F.
Following GRH, we assume that production of a unit of x* requires a continuum of labor
tasks i [0,1] to be performed either in H or in F. Labor is the only input used to perform the
required tasks. While each task i requires a unit of labor in F, the same task requires βt(i) > 1
units of labor in H, where β is a general technology parameter and t(i) is the part of technology
specific to task i in nation H. Tasks that are more complex and require more labor to complete
in the developing nation are indexed by higher values of i. Therefore, by construction tʹ(i) > 0.
Developing nations are often characterized by a dual labor-market environment, where
a formal manufacturing sector coexists with (i) an informal manufacturing sector and (ii) the
food (agricultural) sector. The formal manufacturing sector features large and well-organized
firms bound by laws and regulations: They are required to pay corporate income taxes, get
import licenses, have labor unions, etc. The informal manufacturing sector and the agricultural sector are usually characterized by small firms or farmers in rural settings, respectively,
where labor laws and regulations do not apply or are not enforceable (because of prohibitive
monitoring costs). Accordingly, we first assume that there is a minimum wage in the formal
manufacturing sector and a flexible wage in the informal manufacturing sector, where the
latter conducts the simplest of manufacturing tasks and is characterized by perfect labor mobility with the agricultural sector. Second, we assume that completion of tasks in the informal
sector involves some additional costs. These costs may arise because of a lack of infrastructure
that allows the simplest tasks to be transported to the informal sector or the inferior production technology that characterizes the informal sector. Furthermore, to the extent that the
informal sector has more infrastructure constraints, such as unreliable electricity, worker productivity in the sector suffers. Since higher values of i represent more-complex tasks, the labor
required to outsource from the formal to the informal sector is assumed to be an increasing
markup over the labor required to complete the task in the formal sector. This markup is
β̃τ(i), where β̃ is a general informal sector technology parameter and τʹ(i) > 0 captures that
the informal sector is less technologically advanced and thus has increasing difficulty in completing more-complex tasks. The labor required to complete task i in the informal sector is
then β̃τ(i)βt(i).
Denoting land and labor by T and L, respectively, the constant-returns-to-scale (CRS)
production function for food in nation H is y = G(Ly ,T), where Ly is labor used in H’s agricultural sector. Similarly, y* = G*(L*y ,T *) represents nation F’s production function for food. Since
land is specific to food production and its endowment in each nation is fixed, the CRS production functions for food in the two nations are characterized, respectively, by diminishing
returns to labor:
240
(
)
(
)
(
)
(1)
y = G L y ,T , GLy L y ,T > 0, GLy Ly L y ,T < 0 and
(2)
y* = G* L*y ,T * , G*L* L*y ,T * > 0, G*L* L* L*y ,T * < 0.
Third Quarter 2020
(
)
y
(
)
y y
(
)
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Bandyopadhyay, Basu, Chau, Mitra
2.2 The Labor Supply in the Manufacturing Sector
Let us denote the developed nation’s wage by w* and the developing nation’s wage in the
agricultural sector as w. Recalling that output prices are fixed at unity, competitive profit-
maximization conditions in the agricultural sectors in nations H and F are w = GLy(Ly ,T) and
w* = G*L*y(L*y ,T* ), respectively. Inverting these functions and suppressing T and T* from the
functional forms, we obtain the respective labor demand functions in H and F as
(3)
Ldy = Ldy ( w ), Ldy ′ ( w ) < 0 and
(4)
L*yd = L*yd w * , L*yd ′ w * < 0.
( )
( )
–
–
Given the respective labor endowments L and L * of nations H and F, the labor supply
functions for the manufacturing sectors of nations H and F are respectively given by5
(5)
L ( w ) = L − Ldy ( w ), L′( w ) > 0 and
(6)
L* w * = L* − L*yd w * , L* ′ w * > 0.
( )
( )
( )
2.3 Offshoring to the Developing Nation: Formal-Informal Task Allocation
We assume that technology in the agricultural sectors and endowments in the two nations
are such that the developed nation’s wage w* exceeds the developing nation’s minimum wage
w– in its formal manufacturing sector.6 Labor mobility between H’s informal sector and agricultural sector equalizes the wage between these sectors at w. Although w* and w are endogenous, labor-allocation decisions are best explained for a given vector of wage rates (w*,w–,w).
Any task i can be completed by a unit of labor in F at a cost of w*. This same task can be completed in nation H’s formal sector at a lower wage rate w–, albeit with a greater labor requirement βt(i) > 1. The cost of completing this task in H’s formal sector is w–βt(i). As i goes to zero,
we have tasks that are less complex and the labor cost of completing these tasks in the developing nation are small enough such that w–βt(i) < w* and hence the developed nation offshores
these tasks. On the other hand, as i goes to 1, we assume that the tasks require sufficiently
more labor to be completed in the developing nation such that w–βt(i) > w*, so the tasks are
completed in the developed nation. Given continuity and monotonicity of the underlying
functions, the marginal offshored task is denoted by I, where
(7)
wβt ( I ) = w * ⇔ t ( I ) = ρ I ⇒ I = I ( ρ I ) , I ′ ( ρ I ) =
1
> 0,
t ′( I )
where ρI = w*/(βw–) is the effective relative factor price of completing a task in the developed
nation. Thus, tasks in the range i [0,I] are offshored, while the remaining tasks i [I,1] are
completed in the developed nation. Next, notice that for the minimum wage to be binding,
the informal sector of the developing nation must have a lower wage w. The least-complex
offshored tasks (i.e., as i goes to zero) can be performed in the informal sector at a lower cost
wβ̃τ(i)βt(i) than in the formal sector, where the cost is w–βt(i). This is true for all tasks where
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wβ̃τ(i) < w–. On the other hand, the most-complex offshored task (i.e., i = I) is such that the
high labor requirement dominates the wage advantage of the informal sector, such that
wβ̃τ(I) > w–. Thus, task I is completed in the developing nation’s formal sector. The marginal
task outsourced from the formal to the informal sector is J, where
( ) ( )
% (J )= w ⇔τ (J )= ρ ⇒ J = J ρ , J′ ρ =
wβτ
J
J
J
(8)
1
> 0,
τ ′( J )
where ρJ = w–/(β̃w) is the effective relative factor price of completing an offshored task in the
developing nation’s formal sector (relative to the informal sector). Given the assumed continuity and monotonicity of the τ(i) function, (8) implies that out of the offshored tasks, i [0,J]
are completed in the informal sector and the remainder i [J,I] are completed in the formal
sector.
2.4 Equilibrium
Given that the manufacturing good is produced through a CRS production technology
where each task requires a unit of labor, (1–I) tasks that remain in the developed nation require
x*(1–I) units of labor when output is x*. Thus, in the presence of offshoring, the labor demand
in the manufacturing sector in the developed nation is x*(1–I). Labor demand in the agricultural sector y* of the developed nation is L*yd(w*). Thus, using equation (6), the developed
nation’s labor market-clearing condition where the aggregate demand for labor from the two
–
sectors equals the labor endowment L * is
( )
( )
x * (1− I ) + L*yd w * = L* ⇔ x * (1− I ) = L* w * .
(9)
Let us now consider labor required to complete the offshored tasks i [0,I]. Notice that
an offshored task i performed in the developing nation’s formal sector requires βt(i) labor units.
Furthermore, only tasks i [J,I] are completed in the formal sector. Since the labor required
to complete these tasks vary in the developing nation, the total labor used for completion of
I
these tasks per unit of output is β ∫ t (i )di in that nation. Therefore, to produce x* units of output,
J
I
the labor required in the developing nation’s formal sector is x* β ∫ t (i )di . Similarly, ββ̃t(i)τ(i)
J
is the labor requirement to complete a task i in the informal sector, where tasks in the range
i [0,J] are completed. Thus, production of x* units of output leads to an informal sector labor
J
demand of x * ββ% ∫ t (i )τ (i )di. Labor demand in the developing nation comes from three sources:
0
the formal manufacturing sector, the informal manufacturing sector, and the agricultural
sector. The labor demand in the agricultural sector is Ldy (w). The developing nation’s labor
market clears when the aggregate labor demand of these three sectors equals the developing
nation’s labor endowment such that using equation (5) we have
(10)
242
I
J
I
⎡ J
⎤
x * β ∫ t (i )di + x * ββ% ∫ t (i )τ (i )di + Ldy ( w ) = L ⇔ x * β ⎢ β% ∫ t (i )τ (i )di + ∫ t (i )di ⎥ = L ( w ).
⎢⎣ 0
⎥⎦
J
0
J
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The flexible-wage rates of the two nations (w,w*) adjust to clear their respective labor markets
simultaneously.
It is convenient to analyze equations (9) and (10) in the form of relative demand and supply between the two nations. If we take the ratio of the left-hand sides of the second equalities
in equations (9) and (10), we get the relative demand for labor in the manufacturing sectors
of the two nations. Similarly, the ratio of the right-hand sides of the same equations yields the
relative supply of labor in the manufacturing sectors of the two nations. The relative demand-
supply equality is7
% ( J,I ) + γ ( J,I )⎤ = L ( w ) ,
β ⎡⎣ βµ
⎦ L* w *
(11)
( )
⎛I
⎞
⎞
⎛J
where µ ( J,I ) = ⎜ ∫ t (i )τ (i ) di ⎟ / (1− I ) and γ ( J,I ) = ⎜ ∫ t (i ) di ⎟ / (1− I ) . Notice that μ(.) can be
⎠
⎝0
⎝J
⎠
⎡⎛ J
⎞ ⎤
written as µ ( J,I ) = ⎢⎜ ∫ t (i )τ (i ) di ⎟ x * ⎥ / ⎡⎣(1− I ) x * ⎤⎦ , which is the parameter-adjusted labor
⎠ ⎦
⎣⎝ 0
demand of the developing nation’s informal sector (i.e., informal sector’s labor demand
divided by ββ̃) relative to labor demand in the developed nation’s manufacturing sector.
Similarly, γ(.) is the parameter-adjusted labor demand in the developing nation’s formal sector
(i.e., formal sector’s labor demand divided by β) relative to the labor demand in the developed
nation’s manufacturing sector. When offshoring is high, I is larger, and given J, relative demand
μ(.) has to be higher because the unit labor demand in the developed nation’s manufacturing
sector (i.e., 1–I) is lower. Similarly, given J, γ(.) is increasing in I because of two effects: (i) the
aforementioned effect of a reduction in unit labor demand in the developed nation’s manufacturing sector and (ii) the expansion of the upper limit of the range [J,I] of tasks performed
in the developing nation’s formal sector. Similarly, one can explain the effects of changes in J
given I. In reality, both (I,J) change in response to changes in relative prices (ρI,ρJ), as described
in equations (7) and (8).
The cost of producing a unit of x* is the sum of the costs of completing all tasks necessary
to produce that unit. The cost of completing (1–I) tasks in the developed nation is w*(1–I),
while the cost of completing the offshored tasks in the developing nation’s formal and informal
I
J
J
0
sectors are wβ ∫ t (i )di and wββ% ∫ t (i )τ (i ) di, respectively. Noting that the price of good x* is
unity, the zero profit condition for the good is
(12)
I
⎤
⎡ J
w * (1− I ) + β ⎢wβ% ∫ t (i )τ (i )di + w ∫ t (i )di ⎥ = 1.
⎢⎣ 0
⎥⎦
J
Equations (11) and (12) jointly determine the international equilibrium (w,w* ) at a given
minimum wage w– and for given technology parameters β and β̃.
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3. COMPARATIVE STATICS
The offshoring equilibrium is affected by various parameters underlying the model
–. This section explores how
described in the previous section and also by the minimum wage w
the equilibrium is affected by (i) a change in offshoring technology parameterized by β, (ii) a
change in β̃ reflecting changes in outsourcing technology related to formal-informal outsourcing within the developing nation, and (iii) a change in the minimum wage in the developing nation’s manufacturing sector. In particular, we focus on how changes in these parameters
or policy variables affect offshoring (from the developed nation to the developing nation) and
outsourcing (from the developing nation’s formal to informal sector), the wages in the two
nations, and the share of the informal sector in the total manufacturing employment of the
developing nation. We first derive some equations that apply to all of the aforementioned
parameter and policy changes. After that, we analyze offshoring technology and outsourcing
technology changes in Sections 3.1 and 3.2, respectively, and finally the effects of changes in
the minimum wage in Section 3.3. Propositions 1, 2, and 3 correspond to Sections 3.1, 3.2,
and 3.3 and summarize the findings of each of these subsections.
Let us define the share of the informal sector employment in total manufacturing sector
employment in the developing nation as δ =
⎞ *
⎛ %J
⎜⎝ β ∫ t (i )τ (i )di ⎟⎠ x
0
I
⎛ %J
⎞ *
⎜ β ∫ t (i )τ (i )di + ∫ t (i )di ⎟ x
⎝ 0
⎠
J
. Using the definitions
%
βµ
of μ and γ above, this reduces to δ =
. Next, consider the elasticity of the relative demand
% +γ
βµ
for labor (see equation (11)) with respect to change in the relative factor price ρI , given (β,β̃,ρJ).
% +γ
d lnβ βµ
∂ lnµ
∂ lnγ
This elasticity is ξ I =
=δ
+ (1− δ )
and strictly positive for the fold lnρ I
∂ lnρ I
∂ lnρ I
lowing reasons: Given ρJ , J is fixed and the rise in ρI raises I. On inspection of the respective
expressions for μ(.) and γ(.) provided below equation (11), it is clear that both these functions are strictly increasing in I. Thus, a rise in ρI must raise μ(.) and γ(.), which means that
∂ lnµ
∂ lnγ
ξI =δ
+ (1− δ )
> 0.. This elasticity is critical to understanding how the margin for
∂ lnρ I
∂ lnρ I
offshoring shifts in response to parametric changes. For example, when β falls, the first-round
–), which reflects the fact that at a lower β,
effect is an increase in the relative price ρI = w*/(βw
an offshored task can be performed at a lower wage cost in the developing nation’s formal
sector. The marginal offshored task increases until the difficulty of transporting the new marginal task offsets the cost savings from technological improvements. As more tasks are offshored,
the relative labor demand of the developing nation’s formal sector rises, with this effect mea∂ lnγ
sured by the term
. Similarly, the relative labor demand of the developing nation’s
∂ ln ρ I
(
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Bandyopadhyay, Basu, Chau, Mitra
informal sector rises as the unit labor demand of the developed nation’s manufacturing sector
∂ ln µ
falls, with this effect measured by the term
.
∂ ln ρ I
Aggregating these effects by weighting them by the shares δ and 1–δ of the developing
nation’s informal and formal sectors, respectively, we get the effect on relative labor demand
from a rise in the relative factor price ρI . This effect is captured by ξ I above. Similarly, given ρI ,
I is fixed by equation (7) and we can explore the effect of a change in ρJ on relative labor
demand through changes in the outsourcing range [0,J]. The effects of changes in ρJ on relative
demand can be expressed by another elasticity represented as a weighted average:
% +γ
d lnβ βµ
∂ ln µ
∂ lnγ
.
ξJ =
=δ
+ (1− δ )
d lnρ J
∂ ln ρ J
∂ ln ρ J
(
)
This elasticity must also be positive because given ρI , I is fixed and the rise in ρJ raises J, which
must in turn raise β̃μ + γ.8 Differentiating (11), we get
(13)
(η
*
)
(
) (
) (
) (
)
+ ξ I ŵ * − η + ξ J ŵ = ξ I −1 β̂ + ξ J − δ β%̂ + ξ I − ξ J ŵ.
Equation (13) yields an upward-sloping locus in (w,w*) space because a higher w* increases
offshoring, raising labor demand in nation H so that labor markets of the two nations clear
after a suitable increase in the wage rate w. Differentiating (12) we get
(14)
(
)
(
) (
)
θ * ŵ * + 1− θ − θ * ŵ = − 1− θ * β̂ − 1− θ − θ * β%̂ − θ ŵ,
where θ* is F’s cost share in the production of x*, θ is H’s corresponding cost share of its formal
sector, and the remainder (1–θ–θ*) is H’s cost share of its informal sector. This relationship
yields a familiar negative relationship corresponding to the zero-profit condition in the factor
price space (w,w*). Given the output price, a higher wage for labor in nation F can be consistent
with zero profit only if the wage for nation H’s labor is lower.
3.1 Technological Improvements in Offshoring (Fall in β)
Using equations (13) and (14,) we consider the effects of a change in β (i.e., inverse of labor
–):
productivity of offshoring) on w and w* for a given vector (β̃,w
ŵ
β̂
(15)
ŵ *
β̂
) ( )(
) and
θ (η + ξ ) + (1− θ − θ )(η + ξ )
(1− θ − θ )(ξ −1) − (1− θ )(η + ξ ) .
=
θ (η + ξ ) + (1− θ − θ )(η + ξ )
=
(
−θ * ξ I −1 − 1− θ * η* + ξ I
*
J
*
*
*
*
I
J
I
*
*
J
*
I
Proposition 1.9 A reduction in β leads to
(i) an increase in the range of offshoring [0,I],
(ii) an increase in w and a decrease in the range of formal-informal outsourcing [0,J] if and
only if ξ I > θ* – η*(1– θ*),
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Bandyopadhyay, Basu, Chau, Mitra
(iii) an increase in w* if and only if ξ I <1+
(iv) a decrease in δ if ξ I ≥ θ* – η*(1– θ*).
(1− θ )(η + ξ ) , and
(1− θ − θ )
*
J
*
Comment. A reduction in β reflects improved offshoring technology that spurs offshoring
of tasks and hence an expansion of the range [0,I]. If an increase in the demand for offshoring raises the informal wage, then there is a greater incentive to complete some tasks in the
developing nation’s formal sector and the range of outsourcing [0,J] decreases. Finally, if
technological improvements do not spur a lot of offshoring, then cost savings that raise scale
must raise the demand for labor in the developed nation’s manufacturing sector, pushing up
w*. This is similar to the productivity effect on the developed nation’s wage noted in both GRH
and Bandyopadhyay et al. (2020). The effect of β on the equilibrium share δ is more complicated and is discussed below.
The specific results in Proposition 1 are better understood by digging deeper. The decrease
⎛ w* ⎞
in β has the following effects. First, it raises the relative factor price ρ I ⎜ =
of completing
⎝ wβ ⎟⎠
a task in the developed nation, leading to more tasks being offshored
to the formal sector of the developing nation. Second, notice that β is an offshoring cost parameter that reflects the labor required to move a task out of the developed nation and applies to
both the formal and informal sectors. Therefore, a fall in β at a given I tends to reduce labor
demand in both of these sectors. Finally, lower labor costs tend to drive down unit costs and
a competitive equilibrium is restored through the expansion of scale. This last effect tends to
raise the developing nation’s labor demand. The sum of these three effects determines whether
the developing nation’s labor demand as a whole rises or falls in response to a fall in β. When
ξ I > θ* – η*(1– θ*), the demand for offshore labor responds strongly to a change in the effective
factor price ρI . In this case, the expansionary effect on labor demand dominates and the developing nation’s labor market clears when more labor flows into the nation’s manufacturing
sector from its agricultural sector (y) through a rise in the wage w (recall that the labor supply
of the manufacturing sector L(w) is positively sloped). When w rises, the effective relative
–/(β̃w) must fall. This reduces formal-informal outsourcfactor price for the formal sector ρJ = w
ing J. Turning to the effect on the developed nation’s wage, notice that if ξ I is relatively small,
the effect of a shift in labor demand is relatively small (toward the developing nation) and
dominated by the scale expansion effect and hence w* rises. Finally, the relative size of the
informal sector δ must fall when ξ I ≥ θ* – η*(1– θ*) because the relative size is independent of
scale, and hence all that matters is the range of tasks that are outsourced from the formal to
the informal sector. Since J falls when ξ I > θ* – η*(1– θ*), a smaller range of tasks are completed
in the informal sector. Even when ξ I = θ* – η*(1– θ*), the relative size of the informal sector must
fall because J remains unchanged but I rises, which means that a higher fraction of offshored
tasks are now completed in the formal sector. n
3.2 Technological Improvements in the Informal Sector (Fall In β̃):
The effect of a rise in informal sector productivity (i.e., fall in β̃) can be obtained by using
equations (12) and (14):
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Bandyopadhyay, Basu, Chau, Mitra
(
(
) (
) (
)(
)(
)
)
*
J
*
*
I
ŵ −θ ξ − δ − 1− θ − θ η + ξ
= *
and
J
*
*
I
β%̂ θ η + ξ + 1− θ − θ η + ξ
(16)
(
)
− 1− θ * − θ (δ + η )
ŵ *
= *
.
J
*
*
I
β%̂ θ η + ξ + 1− θ − θ η + ξ
(
) (
)(
)
Proposition 2. A reduction in β̃ leads to
(i) an increase in the range of offshoring [0,I] and also an increase in the range of formalinformal outsourcing [0,J],
1− θ * − θ η* + ξ I
(ii) an increase in w if and only if ξ J > δ −
,
θ*
*
(iii) an increase in w , and
(iv) a decrease in δ if the τ(i) schedule is relatively steep at i = J.
(
)(
)
Comment. A reduction in β̃ reflects an improved technology of outsourcing from the formal
to the informal sector. The first-round effect should be an increase in the range [0,J] of tasks
performed in the informal sector. However, the cost reduction for firms spurs scale expansion
and raises demand for labor in the developed nation, raising w* and spurring more offshoring.
Thus, the range of offshoring [0,I] rises. If τ(i) is steep, then there is not much scope for increasing outsourcing, hence share δ falls. The effect on w is more nuanced and explained in more
detail in the discussion that follows.
–/(β̃w) of completing the
At the initial w, the fall in β̃ raises the effective factor price ρJ = w
tasks in the formal sector compared with the informal sector, which shifts more tasks to the
informal sector (i.e., J rises). However, because of the decline in β̃, each informal sector task
requires less labor, which creates a cost reduction at the initial equilibrium that leads to reallocations that increase scale. The scale expansion drives up labor demand in the developed
–). Thus, more tasks are offshored. There are different
nation, raising w* and hence ρI = w*/(βw
opposing effects on demand in the informal sector. First, labor demand in the informal sector
decreases due to greater efficiency from a lower β̃. On the other hand, increased offshoring,
increased outsourcing of tasks to the informal sector, and scale expansion all suggest an increase
in the labor demand of the manufacturing sector of the developing nation. When the offshoring and outsourcing elasticities (ξ I,ξ J ) are relatively large, the inequality in part (ii) of Proposi
tion 2 is more likely to be satisfied, and the expansionary effects dominate the contractionary
effect of labor-saving technological improvements. In this case, aggregate labor demand of
the manufacturing sector of the developing nation rises. The labor market clears at a higher
wage w, where more labor moves from the agricultural sector to the manufacturing sector in
the developing nation. Finally, consider the ratio of formal-to-informal sector labor employment. Suppose τ(i) is very steep at the initial equilibrium. As β̃ falls, there is not much of a
change in J because τ rises rapidly to equal the new factor price ρJ . Without much change in J,
there are two effects of a fall in β̃, both of which reduce the ratio δ. First, each informal sector
task requires less labor, which shrinks this sector’s relative employment through the labor-
saving effect. Second, as I rises in response to a higher w*, a rigid J means a greater range of
tasks [J,I] are completed in the formal sector. This effect also shrinks δ. In other words, unless
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Bandyopadhyay, Basu, Chau, Mitra
the τ(i) schedule is sufficiently flat to allow for an elastic response of J to a rise in ρJ , the share
of informal sector employment is inversely related to informal sector productivity. n
3.3 The Effects of a Change in the Minimum Wage
If the developing nation’s government decides to change the minimum wage, the effects
can be analyzed using equations (12) and (14) as follows:
(
) (
)
−θ * ξ I − ξ J − θ η* + ξ I
ŵ
= *
,
ŵ θ η + ξ J + 1− θ * − θ η* + ξ I
(
(17)
(
(
) (
)(
) (
)(
) (
)(
)
)
1− θ * ξ I − ξ J − θ η + ξ I
ŵ *
=
.
ŵ θ * η + ξ J + 1− θ * − θ η* + ξ I
)
– leads to
Proposition 3. A reduction in w
(i) an increase in the range of offshoring [0,I] and a decrease in the range of formalinformal outsourcing [0,J],
θ * ξ J − θη*
(ii) an increase in w if and only if ξ I >
,
θ* +θ
(iii) an increase in w* if and only if ξ I <
(iv) a decrease in δ.
(1− θ )ξ
*
*
J
+ ηθ
1− θ − θ
, and
Comment. A minimum-wage cut reduces the effective wage of completing the marginal
offshored task in the formal sector of the developing nation, which must raise the range of
offshoring [0,I]. In addition, it also reduces the effective cost of completing the marginal outsourced task in the formal sector compared with the informal sector. Thus, fewer tasks are
done in the informal sector, reducing the outsourcing range [0,J] and also the relative size of
the informal sector δ. The wage effects are more nuanced and are better understood in the
detailed discussion below.
– raises the relative price of completing tasks in the developed nation (i.e., ρ )
A cut in w
I
and reduces the relative price ρJ of completing tasks in the developing nation’s formal sector
(vis-á-vis the informal sector). This expands the offshoring margin I and shrinks outsourcing
margin J and has three effects on the informal wage w. First, as the marginal offshored task I
rises, demand shifts from the developed to the developing nation, tightening the latter’s labor
market and exerting upward pressure on w. Second, at a lower minimum wage, more tasks
are completed in the developing nation’s formal sector, reducing the demand for labor in the
informal sector, which has a negative impact on the informal wage. Finally, lowering the unit
cost at the initial equilibrium leads to scale expansion, which raises demand in all of the labor
markets, exerting upward pressure on all the flexible factor prices. If ξ I is large relative to ξ J,
the formal-informal reallocation effect (i.e., the second effect) is small and the expansionary
effects dominate. The net increase in the manufacturing sector’s demand for labor in the developing nation drives up the informal wage w. Finally, the comparative statics effect on w* is best
–,w) in the unit cost function may
understood by focusing on how the factor rewards (w*,w
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– falls, given the technology and output price, zero profit
change vis-á-vis each other. When w
requires that at least one of the factor rewards (w*,w) rises. When ξ J is large relative to ξ I,
demand shifts disproportionately from the informal to the formal sector in response to the
minimum-wage cut. In this situation, there may be a net reduction in the developing nation’s
manufacturing sector’s labor demand, which requires w to fall to clear the market. When w
falls, the only possible outcome consistent with a zero-profit equilibrium is a higher w*. Put
differently, if ξ J is relatively small, then it is possible that the informal wage w rises (as explained
– zero profit can be reestablished only through a
above)—to such an extent that even at a lower w
(
)
*
J
θ * ξ J − θη* 1− θ ξ + ηθ
<
fall in w . It is easy to check that
. Using this fact and part (iii) of
θ* +θ
1− θ * − θ
1− θ * ξ J + ηθ
θ * ξ J − θη*
I
I
*
Proposition 3, we have that if w falls, it must be that ξ >
.
⇒
ξ
>
1− θ * − θ
θ* +θ
In turn, using part (ii) of the proposition and the last inequality in the previous sentence, it
must be that a necessary (but not sufficient) condition for w* to fall is a rise in w. In other
words, a rising factor reward in the informal sector is what allows the developed nation’s
wage to fall in spite of the fall in the developing nation’s minimum wage. Finally, notice that
a larger I and a smaller J in response to a minimum-wage cut imply that fewer tasks [0,J] are
completed in the informal sector and a greater range of tasks [J,I] are completed in the formal
sector. Thus, the ratio of informal sector employment δ must decline. n
*
(
)
4. CONCLUSION
This article argues that given the overwhelming importance of the informal sector in many
developing nations, it is important to consider the dual labor-market structure that characterizes these nations. It is important not only because the structure is closer to reality, but also
because it leads to important differences in the comparative statics responses. While GRH
and Bandyopadhyay et al. (2020) both point to developed-nation wage increases due to the
productivity effect, we find that the dual labor-market feature can overturn this effect. We
see this in Proposition 1 of this article, which notes the possibility of a reduction in the developed nation’s wage, in contrast to Proposition 1 of Bandyopadhyay et al. (2020). The reduction is possible because when offshoring elasticity is large, the increase in labor demand in the
developing nation can push up costs on two fronts: a higher informal wage and a greater share
of the work performed in the relatively higher-cost formal sector. These reallocation effects
and factor price changes allow a fall in the developed nation’s wage in spite of technological
improvements.
Other important factors that are missed by models that do not consider the informal sector is the possibility of purely domestic factors that can raise the productivity of the informal
sector. While the direct effect of such changes is a boost in the informal sector, the indirect
effect encompasses the offshoring decision as well as the developed nation’s wage. With more
work being done more efficiently by the informal sector and with the factor price of the formal
sector being held constant by the minimum wage, the developed nation’s wage s must rise to
reflect this efficiency. Such an international effect of a purely domestic technological change
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is missed by models that ignore the dual labor-market structure. Finally, without a dual labor-
market structure, one cannot fruitfully talk about the impact of changes in minimum-wage
laws on the vast majority of urban informal workers. Our work shows that while a cut in the
minimum wage will shrink the relative size of the informal sector, informal workers can
actually be better off because of a rise in the informal wage through the expansionary effects
of a minimum-wage cut.
Our agenda for future work on the topic of dual labor markets includes the analysis of
the effects of different types of labor standards (including the minimum wage) after allowing
for imperfect monitoring of these standards in the formal sector. It is also relevant to look at
competing offshoring destinations and how labor standards or the degree of informality in
one nation affects other offshoring recipients and possibly their labor standards. Finally, we
have abstracted in this article from considerations arising out of terms-of-trade changes in the
output market. Interactions between output-market terms of trade and factor-market terms
of trade in the presence of informality is another possible avenue for our future work. n
APPENDIX
A1. PROOF OF PROPOSITION 1
–, the definition of ρ in (7), and also (15), we get
Given w
I
(
(
)(
) (
) (
)(
)
− 1− θ * − θ 1+ η* − η + ξ J
ρ̂ I ŵ *
=
−1 = *
< 0.
θ η + ξ J + 1− θ * − θ η* + ξ I
β̂ β̂
(A1)
)
Equation (A1) implies that a fall in β must raise ρI. Therefore, using (7), we have that I must
⎛ dI
⎞
rise when β falls ⎜
< 0⎟ . The first relationship in (15) establishes that
⎝ dβ
⎠
ŵ
–), ρ̂ = –ŵ. Thus,
< 0 ⇔ ξ I > θ * − η* 1− θ * . Notice from the definition of ρJ that given (β̃,w
J
β̂
(
ρ̂ J
β̂
=−
ŵ
β̂
)
(
)
> 0 ⇔ ξ I > θ * − η* 1− θ * . In turn, from (8) we get
the second relationship in (15) shows that
ŵ *
β̂
<0⇔ξ
I
Ĵ
β̂
(
)
> 0 ⇔ ξ I > θ * − η* 1− θ * . Now,
(1− θ )(η + ξ ) . Finally, notice
<1+
(1− θ − θ )
*
J
*
I
%
βµ
β%
γ
that δ = %
= %
, where λ ( J ,I ) ≡ =
µ
βµ + γ β + λ ( J ,I )
∫ t (i )di
J
J
∫ t (i )τ (i )di
. As shown above, when
0
ξ I ≥ θ* – η*(1– θ*), w will either rise or be constant when β falls. Thus, the marginal task J will
either fall or remain constant. The increase in I without any increase in J means that the
numerator of the expression for λ(J,I) rises, but the denominator remains constant or falls.
Thus, λ(J,I) must rise, implying that δ must fall when ξ I ≥ θ* – η*(1– θ*). n
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A2. PROOF OF PROPOSITION 2
The second relationship in (16) shows that the developed nation’s wage w* must always
–) must rise, which implies that I must
rise when β̃ falls. In turn, this means that ρI = w*/(βw
rise.
Using the first relationship in (16) we get
d wβ%
θ * (η + δ )
ŵ
–
+1 = *
>
0
⇔
J
*
*
I
% > 0. Thus, ρJ = w/(β̃w) must rise when
%̂
θ
η
+
ξ
+
1−
θ
−
θ
η
+
ξ
d
β
β
(
) (
)(
( )
)
β̃ falls, which implies that J must rise. Using the first relationship in (16) we find that
ξ
J
(1− θ
>δ −
*
)(
− θ η* + ξ I
*
) is a necessary and sufficient condition for the informal wage w
θ
to rise when β̃ falls. Turning to the relative size of the informal sector, recall that
I
∫ t (i )di
γ
β%
, where = J J
≡ λ ( J ,I ). A fall in β̃ for a given λ reduces δ. However,
δ= %
µ
β + λ ( J ,I )
∫ t (i )τ (i )di
0
since I and J both rise, the direction of the change in λ is, in general, ambiguous. If the τ(i)
schedule is steep at i = J, the comparative static change in J will be small. In this event, the
denominator for the expression for λ does not change much, but the numerator rises because
of a rise in I. Thus, λ rises (assuming that t(i) is not too steep at i = I). Therefore, in this case, a
reduction in β̃ and an increase in λ both reduce δ. If both schedules t(i) and τ(i) are steep, the
offshoring and outsourcing margins do not change much and λ does not change much. How
ever, the fall in β̃ reduces δ. Therefore, as long as τ(i) is sufficiently steep at i = J, the informal
share δ must fall with a fall in β̃. n
A3. PROOF OF PROPOSITION 3
ρ̂ ŵ *
Using equations (7), (8), and (17) for a given β and β̃, we get I =
−1 < 0 and
ŵ ŵ
ρ̂ J
ŵ
= 1− > 0. These imply that a minimum-wage cut must raise ρI and reduce ρJ . In turn,
ŵ
ŵ
equations (7) and (8) show that I must rise and J must fall. The two inequalities in (17) yield
(
)
1− θ * ξ J + ηθ
ŵ
θ * ξ J − θη*
ŵ *
I
> 0 if and only if ξ I <
and
if
and
only
if
ξ
<
. Finally,
<0
θ* +θ
1− θ * − θ
ŵ
ŵ
I
β%
recall that δ = %
, where λ ( J ,I ) =
β + λ ( J ,I )
∫ t (i )di
J
J
∫ t (i )τ (i )di
. As I rises and J falls, the numerator of
0
the expression for λ rises and the denominator shrinks. Thus, λ rises as the minimum wage
falls, meaning that δ must fall. n
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NOTES
1
Bandyopadhyay et al. (2020) also present their main results in the context of two nations that are small in the output market. The appendix of their paper considers how the analysis may be extended to large countries and how
that can modify their findings.
2
The more-complex tasks will also be more costly to domestically outsource, since greater skills might be required
to perform them and skills cannot be fully transferred from the formal to the informal sector. Also, supervision by
formal-firm managers of informal sector firms or of casual workers is more difficult. Informal workers, due to the
temporary nature of their jobs, have little incentive to acquire skills on the job. For the same reason, their employers
have virtually no incentive to invest in their human capital or productivity. Despite the low productivity of informal
workers, formal firms transfer some of the relatively simple tasks to them because of the lower informal sector wage.
3
See, for example, Goldschmidt and Schmieder (2017), where “domestic outsourcing” in Germany is analyzed.
4
The small-nation assumption in the output market considerably simplifies the analysis of factor markets by ensuring that any excess supply or excess demand in the output market is absorbed by the world market at fixed international prices. Utility of each nation is entirely determined by national income because output prices are fixed in
the indirect utility function of each nation. Of course, national income is endogenous and determined by factor
allocation between the two sectors—manufacturing and agriculture. Excess labor supply from agriculture in each
nation is absorbed under labor market clearing (modeled) in the offshoring manufacturing sector. Technological
change affects allocation of labor both between the sectors and across the nations, and all of this is considered
in our analysis. Dropping the small-nation assumption is possible, and following Bandyopadhyay et al. (2020,
pp. 222-23) we may pursue this line of inquiry in our future work. The analysis is done in the context of a representative North-South model of offshoring. It may be possible to extend the analysis to several such nations to characterize the global economy. However, such a model is much more complex and beyond the scope of this article.
– –*
5 We assume that the technology in sectors (y,y*) and the endowments (L
,L ) are such that there is an excess supply
of labor in the agricultural sector for the relevant range of wages within which a sensible interior offshoring equilibrium (described in the next section) obtains.
6
In our competitive model, each firm hires similarly and the excess labor supply is absorbed at a flexible wage that
prevails in the informal sector and in the agricultural sector. In a model of heterogeneous firms (that will require
some sort of imperfect competition, such as monopolistic competition with product differentiation) with firm-
specific wage negotiations, although the minimum wage may not be binding for all firms, it could be binding for
the marginal firm. Our competitive model captures this behavior in a simple and tractable way without invoking
a monopolistically competitive framework and the associated complexity.
– ) and J by ρ = w
– /(β̃w). If we
Note that equations (7) and (8) above show that I is entirely determined by ρI = w*/(βw
J
–)
explicitly note these relationships in equation (11), then we get a relationship between the factor prices (w,w*,w
and the technology parameters (β,β̃). For analytical convenience, we take a slightly different approach, although
this aforementioned relationship is at the heart of equation (13) later in the text. Equation (13) captures the labor-
market equilibrium in the two nations in the presence of offshoring, outsourcing, and formal-informal duality. On
the other hand, equation (14) is derived from the zero-profit condition of competitive firms and also represents a
relationship between these factor prices and technology parameters. Equation (14) ensures that a representative
firm’s scale has to adjust to ensure price-unit cost equality. Equations (13) and (14) together characterize the comparative static effects of parametric changes.
% +γ
% ( J ) −1⎤
t ( J ) ⎡⎣ βτ
∂ βµ
⎦ > 0, because using equation (8), we have βτ
% ( J ) = w >1.
8 Notice that
=
w
∂J
1− I
9 Proofs of all propositions are provided in an appendix.
7
(
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REFERENCES
Bandyopadhyay, S.; Basu, A.K.; Chau, N.H. and Mitra, D. “Consequences of Offshoring to Developing Nations:
Labor-Market Outcomes, Welfare, and Corrective Interventions.” Economic Inquiry, 2020, 58(1), pp. 209-24;
https://doi.org/10.1111/ecin.12833.
Bhagwati, J. and Panagariya, A. Why Growth Matters? How Economic Growth in India Reduced Poverty and the Lessons
for Other Developing Countries. New York: Public Affairs, 2013.
Goldschmidt, D. and Schmieder, J.F. “The Rise of Domestic Outsourcing and the Evolution of the German Wage
Structure.” Quarterly Journal of Economics, 2017, 132(3), pp. 1165-217.
Grossman, G.M. and Rossi-Hansberg, E. “Trading Tasks: A Simple Theory of Offshoring.” American Economic Review,
2008, 98(5), pp. 1978-97; https://doi.org/10.1257/aer.98.5.1978.
Harris-White, B. and Sinha, A. Trade Liberalization and India’s Informal Economy. Oxford University Press, 2007.
Hasan, R. and Jandoc, K.R.L. “Labor Regulations and Firm-Size Distribution in Indian Manufacturing,” in J. Bhagwati
and A. Panagariya, eds., Reforms and Economic Transformation in India. Oxford University Press, 2013, pp. 15-48.
Ramaswamy, K.V. “Liberalization, Outsourcing and Industrial Labor Markets in India: Some Preliminary Results,” in
S. Uchikawa, ed., Labor Market and Institution in India: 1990s and Beyond. Manohar, 2003.
Samaniega de la Parra, B. “Formal Firms, Informal Workers and Household Labor Supply in Mexico.” Unpublished
manuscript, Department of Economics, University of Chicago, 2016.
Sundaram, A. “The Impact of Trade Liberalization on Micro Enterprises: Do Banks Matter? Evidence from Indian
Manufacturing.” Oxford Bulletin of Economics and Statistics, 2015, 77(6), pp. 832-853;
https://doi.org/10.1111/obes.12082.
Sundaram, A.; Ahsan, R.N. and Mitra, D. “Complementarity between Formal and Informal Manufacturing in India:
The Role of Policies and Institutions,” in J. Bhagwati and A. Panagariya eds., Reforms and Economic Transformation
in India. Oxford University Press, 2012, pp. 49-85.
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Asset Pricing Through the Lens of the
Hansen-Jagannathan Bound
Christopher Otrok and B. Ravikumar
Stochastic discount factor (SDF) models are the dominant framework for modern asset pricing. The
Hansen-Jagannathan bound is a characterization of the admissible set of SDFs, given a vector of asset
returns. The admissible set provides (i) a test of the asset-pricing model and (ii) information on how
to modify the SDF to be consistent with asset returns, neither of which requires solving the model.
In this article we use the Hansen-Jagannathan bound to examine asset-pricing implications and to
test specific asset-pricing models using bootstrap experiments. (JEL G1, C15, E44)
Federal Reserve Bank of St. Louis Review, Third Quarter 2020, 102(3), pp. 255-69.
https://doi.org/10.20955/r.102.255-69
1 INTRODUCTION
An asset-pricing model is typically defined by its stochastic discount factor (SDF). For
instance, Mehra and Prescott (1985) used constant-relative-risk-aversion (CRRA) preferences
and the SDF in their model was a function of consumption growth. The validity of an SDF is
determined by its ability to match the observed asset returns. An early test of an asset-pricing
model with CRRA preferences was the Hansen and Singleton (1982) J-test. For U.S. stock and
bond returns data, this test typically rejects the model. The J-test tells us whether or not an
asset-pricing model has statistically significant pricing errors. It does not provide information
on how to modify the SDF to improve the fit. Hansen and Jagannathan (1991) derive a volatility bound (HJ bound) that is based on necessary conditions that an asset-pricing model
must satisfy. The HJ bound characterizes the admissible set of SDFs that is consistent with
the observed asset returns.
The HJ bound exploits two conditions: (i) the intertemporal Euler equation that connects
the price of an asset to the covariance of the asset’s payoff with the SDF and (ii) the implication
from linear pricing that the SDF be a linear function of payoffs. The asset-pricing model is
said to be consistent with the data if the volatility of the proposed SDF (evaluated at the mean
SDF) is greater than the volatility implied by the HJ bound. The HJ bound is a lower bound
Christopher Otrok is the Sam B. Cook Professor of Economics at the University of Missouri–Columbia and a research fellow at the Federal Reserve
Bank of St. Louis. B. Ravikumar is an economist, a senior vice president, and the deputy director of research at the Federal Reserve Bank of St. Louis.
© 2020, Federal Reserve Bank of St. Louis. The views expressed in this article are those of the author(s) and do not necessarily reflect the views of
the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Articles may be reprinted, reproduced, published,
distributed, displayed, and transmitted in their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses,
and other derivative works may be made only with prior written permission of the Federal Reserve Bank of St. Louis.
Federal Reserve Bank of St. Louis REVIEW
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Otrok and Ravikumar
and, hence, is a necessary but not sufficient condition that an asset-pricing model must satisfy.
In other words, the HJ bound provides a “test” of an asset-pricing model based solely on
necessary conditions implied by the model.
The HJ bound approach in a sense works backward: Instead of writing down a model,
solving it, and then testing it, the HJ bound asks what a valid SDF should look like in the mean-
variance space. The HJ bound approach has several advantages. First, the bound is model-free;
that is, it is constructed using only observed asset returns. Second, one does not need to solve
the nonlinear asset-pricing model. Specifically, there is no need to find a partial equilibrium
or a general equilibrium solution to the model. Third, there is no limit on the number of assets
used in the construction of the bound. Fourth, the bound is informative on how to modify
the SDF in order to be consistent with the data.
In this article we provide a derivation of the HJ bound and then apply the bound to examine
a few popular SDFs. The results provide an illustration of the equity premium puzzle. We then
check the robustness of the resolutions of the puzzle with a bootstrap experiment. Our bootstrap results indicate that minor variations in asset return moments and consumption moments
can yield large variations in the distance between an SDF’s volatility and the HJ bound. We
conclude with some implications for business cycle models.
2 THE HANSEN-JAGANNATHAN BOUND
For frictionless asset-pricing models, Hansen and Jagannathan (1991) showed that the
volatility of the SDF that satisfies the representative consumer’s Euler equation must exceed
a lower bound that is a function of only asset returns. The derivation of the HJ bound is presented here purely for completeness. (In the appendix, we derive the Sharpe-ratio version of
the HJ Bound; see also Ljungqvist and Sargent, 2018.)
Let R denote the n×1 (gross) return vector of risky assets. Consider an SDF m that prices
the n assets according to
Et ( Rt+1mt+1 ) = ι ,
where Et is the expectation operator conditional on information in period t and ι is an n×1
vector of 1s. This is the standard Euler condition, which equates the expected marginal cost
and marginal benefit of delaying consumption one period. For example, in the case of time-
separable preferences, mt+1 is the ratio of the marginal utility of future consumption to the marginal utility of current consumption. The unconditional version of the Euler equation is then
(1)
E ( Rm ) = ι .
1
. In the absence of a risk-free
E (m )
asset, we cannot pin down the mean of the SDF using return data.
Suppose we compute the least-squares projection of the SDF onto the linear space spanned
by a constant and contemporaneous returns. The projection is of the form
Note that if there is a risk-free asset, then its gross return is
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m = mv + ε ,
(2)
where
mv = v + ( R − E ( R ))′ β ,
(3)
β n, v = E(m) = E(mv), and ε is orthogonal to the constant as well as contemporaneous
returns. This implies E(ε) = 0 and E(Rε) = 0. Together with the Euler equation (1), this implies
E(Rm) = E(Rmv) = ι. Then
var (m ) = var (mv ) + var ( ε ) + 2cov (mv ,ε ) .
By construction of mv , the projection error ε is orthogonal to mv , so E(mv ε) = 0. Thus,
var (m ) = var (mv ) + var ( ε ) ≥ var (mv ) ,
meaning that a lower bound on the variance of a model’s SDF m is the variance of mv . To
find this lower bound, we need to know var(mv ).
From (3) it is easy to see that var (mv ) = βʹΩβ, where Ω is the variance-covariance matrix
of asset returns. Since (2) and (3) describe a linear least-squares projection, we can estimate
the projection coefficient β via OLS as β = Ω –1cov(R,m). Rewriting cov(R,m), we have
β = Ω –1(E(Rm) – E(m)E(R)). Since the model implies E(Rm) = ι, we can solve for β with
β = Ω−1 (ι − E (m ) E ( R )) .
Thus, we can write var(mv) = (ι – E(m)E(R))ʹΩ–1(ι – E(m)E(R)). In terms of standard deviations,
we can write the lower bound as
(4)
{
}
std (m ) ≥ (ι − E (m ) E ( R ))′ Ω−1 (ι − E (m ) E ( R ))
1
2
.
The right-hand side is the HJ bound. Note that the lower bound on the standard deviation
of a model’s SDF is a function of the mean of the model’s SDF; so, it would seem like the lower
bound depends on the model. However, we can generate a lower-bound frontier by picking
different means. It is easy to see that the bound is a quadratic function of the mean SDF. A
necessary condition for an SDF with mean E(m) to be consistent with asset-return data is that
it satisfies the inequality (4).
Computing the HJ bound frontier is straightforward. First, we calculate the sample mean
of gross returns to use as a proxy for E(R). Second, we calculate the variance-covariance matrix
of the gross returns. Third, we choose a set of values for E(m). For each value we compute the
right-hand side of (4) to trace out a bound frontier.
Figure 1 illustrates the HJ bound using two asset returns from 1959:Q2 to 2019:Q2: the
return on a 3-month Treasury bill and the return on the S&P 500. Both returns are transformed
into real returns using the price deflator for personal consumption expenditures. (We use
this deflator because when we conduct model evaluations later, we will be using personal
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Figure 1
HJ Bound Frontier
Standard deviation SDF
0.74
0.72
0.70
0.68
0.66
0.64
0.62
0.60
HJ bound
0.58
0.994
0.995
0.996
0.997
0.998
0.999
1.000
Mean SDF
NOTE: The figure depicts the set of admissible SDFs in mean-standard deviation space implied by stock and bond
returns from 1959:Q2-2019:Q2.
consumption data.) The horizontal axis is E(m), and the vertical axis is the HJ bound. The
frontier is U-shaped, implying that SDFs with means far from the one associated with the least
volatility will need to have higher volatility to satisfy the bound.1
3 EQUITY PREMIUM PUZZLE
In this section, we use the HJ bound to illustrate the equity premium puzzle. To “test” a
model using the HJ bound, we need the SDF implied by the model. As an example, suppose
we want to check whether the Mehra and Prescott (1985) model is consistent with asset-return
data. The preferences in their model are described by
∞
E0 ∑ β t
t=0
ct1−σ
, σ > 0,
1− σ
where E0 is the conditional expectation given information at time 0, ct is the representative
agent’s consumption at time t, β (0,1) is the subjective discount factor, and σ is the coefficient of relative risk aversion. (The preferences are assumed to be logarithmic when σ = 1.)
The SDF for these preferences is given by
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Figure 2
Mehra and Prescott (1985)
Standard deviation SDF
14
12
10
8
6
4
2
0
0.85
HJ bound
Mehra-Prescott
0.90
0.95
1.00
1.05
1.10
Mean SDF
NOTE: Sample: 1959:Q2-2019:Q2. The HJ bound frontier in this figure is the same as in Figure 1. It also has the means
and standard deviations of the Mehra-Prescott SDF for the range of the risk-aversion parameter (σ), from 1 to 10.
⎛c ⎞
mt+1 = β ⎜ t+1 ⎟
⎝ c ⎠
−σ
.
t
We can compute the time series of mt+1 using consumption data and parameter values for β
and σ. Different values of β and σ imply different means, E(m), and different volatilities, std(m).
The question is whether there are any empirically plausible β and σ such that the pair (E(m),
std(m)) is inside the frontier. That is, given the mean of the model’s SDF, the test is whether
std(m) satisfies the bound in (4).
Figure 2 plots the same bound as in Figure 1 for the stock and bonds returns data. Figure 2
also plots the pairs (E(m), std(m)) using quarterly nondurables and service consumption data
from 1959:Q2 to 2019:Q2, for β = 0.99 and values of σ from 1 to 10. These values of β and σ
are in the range investigated by Mehra and Prescott. The volatility for σ = 1 is the right most
“x.” As risk aversion is increased, the x’s move to the left, but the increases in volatility are
small.
For no value of σ is the bound satisfied. In fact, the volatilities of the SDF are far below
the bound. We conclude that the model with this parameterization is rejected. A natural question is whether or not there exists a parameterization of the model that satisfies the bound.
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Figure 3
High Risk Aversion in Mehra and Prescott (1985)
Standard deviation SDF
14
HJ bound
Mehra-Prescott
12
10
8
6
4
2
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Mean SDF
NOTE: Sample: 1959:Q2-2019:Q2. The figure illustrates the means and standard deviations of the Mehra-Prescott SDF
for a wider range of the risk-aversion parameter (σ), relative to Figure 2, from 1 to 460.
To do so we increase risk aversion and find that the Mehra-Prescott model generates enough
volatility to satisfy the bound when σ = 460; see Figure 3.
The high value of σ is unreasonable for two reasons. First, it implies an extreme aversion
to risk. Second, it implies a high risk-free rate of 36 percent annually.2 Figure 3 thus demonstrates the risk-free rate puzzle as well: The level of risk aversion that matches the observed
equity premium comes at the cost of unreasonable values for the risk-free rate.
In sum, the Mehra-Prescott model of asset pricing is rejected for reasonable parameterizations of risk aversion. One needs implausibly high values of risk aversion to generate sufficient volatility to satisfy the bound. Moving forward, we need to find an SDF that generates
higher volatility without high risk aversion.
Note that the above evaluation of the Mehra-Prescott model did not require us to solve
the model or compute equilibrium asset returns. The test involved merely checking whether
a necessary implication of the model was satisfied. We learned the same lessons that emerge
from a full solution of the model. Another alternative to testing models using just the first-order
conditions would be to estimate the Euler equation via GMM (generalized method of moments)
as in Hansen and Singleton (1982) and then apply a J-test to the overidentifying restrictions.
As is well known, this would lead to a statistical rejection of the Mehra-Prescott model. It
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would not, however, provide any guidance as to why the model was rejected and what to do
to fix the model.
4 RESOLUTIONS
A similar procedure can be applied to two other popular asset-pricing models. Both are
based on relaxing separability in the utility function: in one case state separability and in the
other case time separability. Both add just one parameter to the Mehra-Prescott model, and
both increase the volatility of the SDF.
Epstein and Zin (1991) and Weil (1989) generalize the time-separable preferences to allow
for an independent parameterization of attitudes toward risk and intertemporal substitution.
Following Weil (1989), these state-nonseparable preferences have a recursive representation:
Vt = U [ ct ,EtVt+1 ],
where V is a von-Neumann-Morgenstern utility index and
⎛ 1−σ ⎞
⎛ 1− ρ ⎞ ⎫⎜⎝ 1− ρ ⎟⎠
⎧
⎟⎠
⎜⎝
1− ρ
1−
β
c
+
β
1+
1−
β
1−
σ
V
−1
(
)
⎡
⎤
(
)
(
)
⎨
⎣
⎦ 1−σ ⎬
⎩
⎭
U [c,V ] =
.
(1− β )(1− σ )
The elasticity of intertemporal substitution is 1/ρ, and σ is the coefficient of relative risk aversion. As shown by Weil (1989), the SDF for these preferences simplifies to
⎛ 1−σ ⎞
⎛ 1−σ ⎞
⎡ ⎛ c ⎞ −σ ⎤⎜⎝ 1− ρ ⎟⎠
−1
⎢ β ⎜ t+1 ⎟ ⎥
Rt+1 ]⎜⎝ 1− ρ ⎟⎠ ,
[
⎢⎣ ⎝ ct ⎠ ⎥⎦
where Rt+1 is the return on the market portfolio.
Constantinides (1990) models consumers as habitual, in that levels of consumption in
adjacent periods are complementary. That is, the time-nonseparable preferences of consumers
(in a discrete-time, one-lag version of Constantinides, 1990) are given by
1−σ
ct − δ ct−1 ]
[
U 0 = E0 ∑ n β
,
1− σ
∞
t
t=0
where δ > 0. The representative agent’s SDF is given by
−σ
−σ
ct+1 − δ ct ) + βδ Et+1 ( ct+2 − δ ct+1 )
(
mt+1 = β
.
(ct − δ ct−1 )−σ + βδ Et (ct+1 − δ ct )−σ
Figure 4 plots the bound and the SDF volatilities for the two models. Again, the HJ bound
is the solid curve, as in Figure 1; the “x” represents the habit SDF, while the “o” the EpsteinZin SDF. For state-nonseparable preferences, the parameters are β = 0.99, ρ = 0.9, and σ = 1.7.
For time-nonseparable preferences, the parameters are β = 0.99, δ = 0.8, and σ = 1.61.3 Both
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Figure 4
Epstein-Zin and Habit-Formation Models
Standard deviation SDF
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
HJ bound
Epstein-Zin
Habit formation
0.60
0.55
0.990
0.992
0.994
0.996
0.998
1.000
1.002
Mean SDF
NOTE: Sample: 1959:Q2-2019:Q2. The figure illustrates the means and standard deviations of the SDFs for the EpsteinZin and habit-formation preferences that satisfy the HJ bound. The HJ bound is the same as in Figure 1.
models satisfy the HJ bound by increasing the volatility of the SDF. In the case of the EpsteinZin model, wealth, which is volatile, is part of the SDF. In the case of the time-nonseparable
models, consumption growth is operated on by a difference operator, which increases volatility when raised to moderate powers.
4.1 Are the Resolutions Robust?
Our model evaluation shows that both Epstein-Zin and habit-formation models satisfy
the HJ bound for apparently reasonable parameter values.4 The evaluation was simple: It compared just two points—the volatility of the SDF and the HJ bound at the mean of the same
SDF. The evaluation does not account for sampling variability in (i) asset-return data and (ii)
consumption data. The sampling variability in (i) and (ii) might affect our inference on the
model since the HJ bound is affected by (i) and the SDF is affected by (ii).
We now conduct a bootstrap experiment to take into account the two sampling variabilities
and check whether the resolutions are robust to changes in the data sample. We adopt a variant
of the bootstrap procedure in Otrok, Ravikumar, and Whiteman (2004): They first find parameters of the asset-pricing models that satisfy the HJ bound for the whole post-WWII sample.
Using these parameters, they then show that the models do not satisfy the bound for subsamples.
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Our bootstrap experiment here, however, investigates how the HJ bound and the volatility of the SDF vary across artificial samples drawn from the full data set (1959:Q2-2019:Q2).
To do this we compute the time series for the representative agent’s SDF for the two successful
asset-pricing models in Section 3 using consumption growth data. We then use a bootstrap
procedure to sample a vector of asset returns and the SDF. We bootstrap the entire vector—
consumption, equity return, and bond return—so that the observed correlation properties
between the two returns and the SDF are maintained in our experiment.
The bootstrap procedure is as follows:
(i) Use the parameters from Section 3 and observed consumption growth data to get
time series for the SDFs of the models. (The parameters are β = 0.99, ρ = 0.9, and σ = 1.7
for the Epstein-Zin SDF and β = 0.99, δ = 0.8, and σ = 1.61 for the habit SDF.)
(ii) Draw (with replacement) a time series of length 241 from the joint “empirical”
distribution of the SDFs, equity returns, and T-bill returns. That is, for each period
we draw a 3-tuple (SDF, Requity, RT-bill).
(iii) Calculate the mean and volatility of the SDF.
(iv) Calculate the HJ bound using the time series for equity and T-bill returns at the
mean SDF.
(v) Repeat steps ii-iv 1,000 times.
Figure 5 is a scatter plot of the distance between the HJ bound and SDF volatility, calculated as the SDF volatility minus the HJ bound, for each of the 1,000 bootstrap simulations.
Panel A plots the habit-formation model, while Panel B plots the Epstein-Zin model. The
striking feature of these figures is that the distance is almost always negative, implying that
the models miss the bound in most simulations. In fact, the habit model misses in 96 percent
of the simulations and the Epstein-Zin model misses in 95 percent of the simulations.
Burnside (1994) casts the distance to the bound in a GMM framework. He studies statistical measures of the distance between the HJ bound and SDF volatility in the time-separable
model and argues that the over rejection is partly due to variations in the mean of the SDF.
Cecchetti, Lam, and Mark (1994) also show that in the context of models with time-separable
preferences and habit formation preferences, much of the variability in the distance is due to
the uncertainty in estimating the mean of the SDF. To their point, even if we consider only
the lowest possible bound from the bootstrap simulations, we will reject the models most of
the time. The reason is that the mean of the SDF varies greatly across bootstrap samples. Since
the bound itself rises rapidly for E(m) different from 0.99, the distance to the bound becomes
large and leads to a rejection.5
Statistically speaking, for the HJ bound to be a useful evaluation device, the test should
not reject a true model. Specifically, suppose one uses observed consumption data to solve an
asset-pricing model, that is, compute the equilibrium asset returns implied by the model. Then
the test based on the distance between the HJ bound associated with the equilibrium returns
and the volatility of the model SDF should not reject the true model. One can judge the test
by simulating the true model many times and counting the number of times the the HJ bound
is violated. Gregory and Smith (1992) conduct this exercise for time-separable preferences
and conclude that the true model is rejected frequently.
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Figure 5
Bootstrap Simulations
A. Habit-formation model
Standard deviation SDF
2
Habit formation
0
–2
–4
–6
–8
–10
–12
–14
–16
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.02
1.04
Mean SDF
B. Epstein-Zin model
Standard deviation SDF
5
Epstein-Zin
0
–5
–10
–15
–20
–25
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Mean SDF
NOTE: Sample: 1959:Q2-2019:Q2. The figure plots the distance between the standard deviation of the SDF and the HJ
bound evaluated at the mean of the SDF for each draw of the bootstrap. The parameters for the Epstein-Zin SDF are
β = 0.99, ρ = 0.9, and σ = 1.7; the parameters for the habit SDF are β = 0.99, δ = 0.8, and σ = 1.61.
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A formal statistical evaluation involves calculating rejection rates based on critical values
of the test statistic, as in Burnside (1994) and Cecchetti, Lam, and Mark (1994). However,
Otrok, Ravikumar, and Whiteman (2002) show that tests based on the distance to the HJ
bound are non-pivotal in finite samples: The finite-sample critical values depend upon the
SDF parameters: risk aversion and the discount factor. Therefore, one has to calculate parameter-
specific critical values for each point in the null hypothesis of interest. Nevertheless, for the
case of time-separable preferences, Otrok, Ravikumar, and Whiteman (2002) show that the
finite-sample distribution of the test statistic associated with the risk-neutral case is extreme.
The critical values for the risk-neutral case deliver type-I errors no larger than intended, regardless of risk aversion or the discount factor. They also show that the maximal type-I error critical values for time-separable preferences are appropriate for habit formation as well as state
nonseparable preferences. Their conclusion is that the HJ bound is indeed a useful statistical
evaluation device, in that type-I errors can be controlled, while type-II error rates are acceptably small. Using their finite-sample critical values, they report evidence against time-separable
preferences and mixed evidence for Epstein-Zin and habit preferences.
5 ASSET-PRICING IMPLICATIONS OF BUSINESS CYCLE MODELS
Our focus so far in this article has been on asset-pricing models and financial returns
data typically used in the asset-pricing literature. The HJ bound is also useful for analyzing
the asset-pricing implications of business cycle models. Such an approach is useful since a
business cycle model is typically solved with a first-order approximation, which eliminates
risk premia. Higher-order solutions are possible but costly for moderate-sized models. An
early approach to using the HJ bound in the context of a business cycle model was Tallarini
(2000). That paper first showed that risk-aversion per se did not affect the business cycle
behavior of standard macroeconomic aggregates. It then showed that the SDF from that model
did satisfy the HJ bound with sufficiently high risk aversion.
Typically, in business cycle models that study asset-pricing implications, asset return is
measured by the S&P 500. Gomme, Ravikumar, and Rupert (2011) argue that business cycle
theory does not necessarily imply using financial return to measure the return to capital. They
construct the return to capital in the United States using NIPA statistics on capital income
and capital stock. They show that while the mean return is roughly the same for equity returns
and the return to capital, the NIPA return to capital is less volatile.
In Figure 6 we construct the HJ bound using the return to capital as in Gomme, Ravikumar,
and Rupert (2011) and the return to the 3-month Treasury bill. We also plot the HJ bound
from Figure 1 for comparison. The sample period here is 1959:Q2-2008:Q4. The bound for
the return to capital is significantly higher. This follows from the fact that the return to capital
is less volatile than the return to equity, which leads to a sharper set of restrictions on the set
of admissible SDFs.
The implication for business cycle models is that the asset-pricing puzzle is in fact more
challenging than the one we see with financial return data. As in Figure 2, the time-separable
model does not generate enough volatility to satisfy the HJ bound with equity returns, so it
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Figure 6
Return to Capital: Gomme, Ravikumar, and Rupert (2011)
Standard deviation SDF
2.5
HJ bound with equity returns
HJ bound with return to capital
2.0
1.5
1.0
0.5
0
0.985
0.990
0.995
1.000
Mean SDF
NOTE: Sample: 1959:Q2-2008:Q4. The figure depicts the set of admissible SDFs in the mean-standard deviation space
implied by stock and bond returns as in Figure 1. It also depicts the set of admissible SDFs implied by return to capital
from Gomme, Ravikumar, and Rupert (2011) and bond return.
certainly will not satisfy the bound with return to capital. While it may be possible to find timeor state-nonseparable preferences that satisfy this bound, they will still suffer from the stability
problem we documented earlier with these resolutions. In addition, in the case of time-nonseparable preferences, Otrok (2001) shows that the data prefer only moderate amounts of
habit formation, which will not generate much volatility in the SDF.
6 CONCLUSION
The Hansen-Jagannathan bound is a helpful tool for understanding asset-pricing implications. By characterizing the admissible set of SDFs, we can use the bound to test proposed
SDFs. Further, we can understand what types of asset-return data will pose greater difficulty
for an asset-pricing model. Lastly, the bound can be constructed with only the first and second
moments of asset-return data, hence implementation of the bound requires no computing
power beyond a spreadsheet program.
The HJ bound uses the means, variances, and contemporaneous correlations of assetreturn data to construct the lower bound that an SDF must satisfy. Otrok, Ravikumar, and
Whiteman (2007) develop a volatility bound that uses serial correlation properties of the
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return data as well. This generalization allows for an evaluation of whether models fail to
match the data in the long run, at business cycle frequencies, etc. That is, the generalized
bound can help identify the frequencies at which a model violates the necessary conditions.
A business cycle model that violates the bound at business cycle frequencies might be unacceptable, but violations at other frequencies might not be a cause for concern. The generalization involves projecting the SDF onto the space of current, past, and future returns. Because
the projection is onto a larger space than that for the HJ bound, the generalized bound is tighter
than the HJ bound. They find that the state-nonseparable SDF satisfies the bound at business
cycle frequencies, while the time-nonseparable SDF does poorly at those frequencies. Open
questions for future research are whether the resolutions at those frequencies are stable and
whether these SDFs can satisfy the generalized bound at business cycle frequencies when the
bound is constructed with Gomme, Ravikumar, and Rupert (2011) capital-return data. n
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APPENDIX: HJ BOUND AND THE SHARPE RATIO
The Sharpe ratio is the mean excess return on an asset (relative to the risk-free rate)
divided by the standard deviation of that asset’s return. The Sharpe ratio measures how the
market views risk: A higher Sharpe ratio implies that the market demands a higher return for
a given level of risk. The connection between the HJ bound and the Sharpe ratio presented
here follows Cochrane (2001).
Consider the unconditional Euler equation used to price assets:
(
)
ι = E R equitym .
For the sake of exposition we will assume that the only asset is equity, with return Requity,
1
and there is a risk-free rate such that R f =
. We can write the right-hand side of the
E (m )
equation as
(
)
(
)
E R equitym = E (m ) E R equity + ρ Requity, mσ Requity σ m ,
where ρRequity,m is the correlation of equity returns with m and σs represent standard deviations.
Next, divide through by E(m) to get
ρ equity σ equity σ m
1
= E R equity + R , m R
.
E (m )
E (m )
(
Replacing
)
1
with R f, dividing by σRequity, and rearranging terms yields
E (m )
(
)
E R equity − R f
σ Requity
Since –1 ≤ ρRequity,m ≤ 1, we have the inequality
(
=−
)
E R equity − r f
σ Requity
ρ Requity, mσ m
≤
E (m )
.
σm
.
E (m )
The left-hand side is the Sharpe ratio. For given E(m), a higher Sharpe ratio implies that
the lower bound on SDF volatility is higher.
NOTES
268
1
Matlab code and data used for this and all subsequent examples in this article can be found on Christopher Otrok’s
REPEC webpage: https://ideas.repec.org/e/pot2.html.
2
For the value of σ that satisfies the bound, Em = 0.9258, or a quarterly return of 1/Em = 1.0801.
3
Note that σ is not the coefficient of risk aversion in the habit model, though it is proportional to various measures
of risk aversion. See Boldrin, Christiano, and Fisher (1997).
4
Since we can dismiss the Mehra-Prescott model for achieving the bound with only unreasonable amounts of risk
aversion, we will focus on only these two models in this section.
5
The value 0.99 is in the lower part of the bound in Figure 1.
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REFERENCES
Boldrin, M.; Christiano, L.J. and Fisher, J.D.M. “Habit Persistence and Asset Returns in an Exchange Economy.”
Macroeconomic Dynamics, 1997, 1, pp. 312-32; https://doi.org/10.1017/S1365100597003027.
Burnside, C. “Hansen-Jagannathan Bounds as Classical Tests of Asset Pricing Models.” Journal of Business and
Economic Statistics, 1994, 12, pp. 57-79; https://doi.org/10.1080/07350015.1994.10509991.
Cecchetti, S.G.; Lam, P-S. and Mark, N.C. “Testing Volatility Restrictions on Intertemporal Marginal Rates of
Substitution Implied by Euler Equations and Asset Returns.” Journal of Finance, 1994, 49, pp. 123-52;
https://doi.org/10.1111/j.1540-6261.1994.tb04423.x.
Cochrane, J.H. Asset Pricing. Princeton University Press, 2001.
Constantinides, G.C. “Habit Formation: A Resolution of the Equity Premium Puzzle.” Journal of Political Economy,
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Epstein, L.G. and Zin, S.E. “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset
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Gomme, P.; Ravikumar, B. and Rupert, P. “The Return to Capital and the Business Cycle.” Review of Economic
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Gregory, A.W. and Smith, G.W. “Sampling Variability in Hansen-Jagannathan Bounds.” Economics Letters, 38,
pp. 263-67; https://doi.org/10.1016/0165-1765(92)90068-A.
Hansen, L.P. and Jagannathan, R. “Implications of Security Market Data for Models of Dynamic Economies.” Journal
of Political Economy, 1991, 99, pp. 225-62; https://doi.org/10.1086/261749.
Hansen, L.P. and Singleton, K.J. “Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations
Models.” Econometrica, 1982, 50(5), pp. 1269-86; https://doi.org/10.2307/1911873.
Ljungqvist, L. and Sargent, T.J. Recursive Macroeconomic Theory. MIT Press, 2018.
Mehra, R. and Prescott, E.C. “The Equity Premium: A Puzzle.” Journal of Monetary Economics, 1985, 15, pp. 145-61;
https://doi.org/10.1016/0304-3932(85)90061-3.
Otrok, C.; Ravikumar, B. and Whiteman, C.H. “Evaluating Asset-Pricing Models Using the Hansen-Jagannathan
Bound: A Monte Carlo Investigation.” Journal of Applied Econometrics, 2002, 17, pp. 149-174;
https://doi.org/10.1002/jae.640.
Otrok, C.; Ravikumar, B. and Whiteman, C.H. “Stochastic Discount Factor Models and the Equity Premium Puzzle.”
Unpublished manuscript, 2004; https://ideas.repec.org/p/pra/mprapa/22938.html.
Otrok, C.; Ravikumar, B. and Whiteman, C.H. “A Generalized Volatility Bound for Dynamic Economies.” Journal of
Monetary Economics, 2007, 54(8), pp. 2269-90; https://doi.org/10.1016/j.jmoneco.2007.06.028.
Otrok, C. “On Measuring the Welfare Cost of Business Cycles.” Journal of Monetary Economics, 2001, 47(1), pp. 61-92;
https://doi.org/10.1016/S0304-3932(00)00052-0.
Tallarini, T. “Risk-Sensitive Real Business Cycles.” Journal of Monetary Economics, 2000, 45(3), pp. 507-32;
https://doi.org/10.1016/S0304-3932(00)00012-X.
Weil, P. “The Equity Premium Puzzle and the Risk-Free Rate Puzzle.” Journal of Monetary Economics, 1989, 24,
pp. 401-21; https://doi.org/10.1016/0304-3932(89)90028-7.
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Reconstructing the Great Recession
Michele Boldrin, Carlos Garriga, Adrian Peralta-Alva, and Juan M. Sánchez
This article uses dynamic equilibrium input-output models to evaluate the contribution of the construction sector to the Great Recession and the expansion preceeding it. Through production interlinkages and demand complementarities, shifts in housing demand can propagate to other economic
sectors and generate a large and sustained aggregate cycle. According to our model, the housing boom
(2002-07) fueled more than 60 percent and 25 percent of employment and GDP growth, respectively.
The decline in the construction sector (2007-10) generates a drop in total employment and output
about half of that observed in the data. In sharp contrast, ignoring interlinkages or demand complementarities eliminates the contribution of the construction sector. (JEL E22, E32, O41)
Federal Reserve Bank of St. Louis Review, Third Quarter 2020, 102(3), pp. 271-311.
https://doi.org/10.20955/r.102.271-311
1 INTRODUCTION
With the onset of the Great Recession, U.S. employment and gross domestic product
(GDP) fell dramatically and then took a long time to return to their historical trends. There is
still no consensus about what exactly made the recession so deep and the subsequent recovery
so slow. In this article we evaluate the role played by the construction sector in driving the
boom and bust of the U.S. economy during 2001-13.
The construction sector represents around 5 percent of total employment, and its share
of GDP is about 4.5 percent. Mechanically, the macroeconomic impact of a shock to the construction sector should be limited by these figures; we claim it is not. Rather, we claim one
reason why the Great Recession was particularly deep and persistent is that the construction
sector and housing consumption are strongly interconnected to the rest of the economy.
Michele Boldrin is professor of economics at Washington University in St. Louis. Carlos Garriga is a vice president and economist at the Federal
Reserve Bank of St. Louis. Adrian Peralta-Alva is a senior economist at the International Monetary Fund. Juan M. Sánchez is an assistant vice president and economist at the Federal Reserve Bank of St. Louis.
The authors are grateful for the stimulating discussions with Morris Davis; Bill Dupor; Ayse Imrohoroglu; Rody Manuelli; Alex Monge-Naranjo;
Alp Simsek; Jorge Roldos; Paul Willen; and the seminar participants at the Bank of Canada, Southern Methodist University, International Monetary
Fund, 2012 Wien Macroeconomic Workshop, 2012 ITAM Summer Camp in Macroeconomics, 2012 SED, 2015 GBUREES, Stony Brook, and Queens
College CUNY. Constanza Liborio, James D. Eubanks, and You Chien provided excellent research assistance.
© 2020, Federal Reserve Bank of St. Louis. The views expressed in this article are those of the author(s) and do not necessarily reflect the views of
the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Articles may be reprinted, reproduced, published,
distributed, displayed, and transmitted in their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses,
and other derivative works may be made only with prior written permission of the Federal Reserve Bank of St. Louis.
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Such linkages are important at the production stage (purchases of intermediate goods) but
also at the final consumption stage (broad demand complementarities).
Our vision of how a shock to the demand for housing travels to the rest of the economy
is the following. In response to a demand-driven housing boom, the construction sector leads
the rest of the economy by fueling an expansion through its purchases of inputs. The expansion generates an increase in consumption (housing and nonhousing) as well as investment
(residential and capital). This continues either until a new steady state is reached or, as in the
historical case we study, a sudden drop in housing demand generates a decline in construction
output. This translates into a general reduction of demand (for intermediate inputs and complementary consumption goods), thereby propagating and magnifying, again, the negative
sectoral shock. Further, a sudden drop in the demand for housing also generates a slow recovery because the excessive inventory of housing units takes a long time to be absorbed, hence
the particularly long delay in the aggregate recovery.
In the empirical analysis, we construct measures of sectoral interlinkages (multipliers)
and show that the construction sector is one of the most interconnected in the economy. We
use this to quantify the contribution of the construction sector during the period 2002-13 and
estimate it to have been unusually large. Construction is capable of accounting for about 52
percent of the decline in total employment and 35 percent of the decline in aggregate gross
output. Eliminating the production multipliers weakens the impact on total employment to
20.8 percent and on gross output to 19.3 percent.
In a simplified version of the model, we illustrate the importance of production interlinkages and demand complementarities. This exercise provides a set of sufficient conditions
under which the presence of interlinkages generates larger effects in aggregate employment
and output than in their absence. The algebra indicates that, for the amplification effect to exist,
the sectoral interlinkages must be asymmetric, with the construction sector buying relatively
more inputs from the rest of the economy than vice versa. This condition is supported—by
more than two orders of magnitude—by estimates from the U.S. input-output table from the
Bureau of Economic Analysis (BEA). To generate a multiplier effect via interlinkages, it is
also necessary to have an elasticity of substitution between housing and consumption goods
lower than 1.1 With an elasticity of substitution larger than 1, a decline in housing demand
generates the reallocation of productive inputs away from the construction sector and a boom
in the nonhousing sectors, which may (more than) compensate for the decline due to the
supply-side interlinkages. With a unitary elasticity, these two effects cancel out.
One of the limitations of the static model is that it does not allow study of the dynamic
adjustment of consumption, residential investment, and productive capital. It also ignores
the process of the adjustment of relative prices, and it is not ideal for quantitative purposes.
To overcome these limitations, we solve the full dynamic general-equilibrium model numerically. We use that model to answer the following question: If demand for housing shifts exogenously over time to match the observed dynamics of employment in the construction setor,
what will happen to the remaining macro quantities and prices?2
Our simulations indicate that, in the presence of sectoral interlinkages and consumption
complementarities, the size of the boom-bust cycle in total employment and output is sub272
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stantially larger and more coherent with historical observations than otherwise. During the
demand-driven housing boom, both sectors in our model expand and contribute to the growth
of output and employment, by 2 percent and 2.5 percent, respectively. During the housing
bust, the decline in output is 3.3 percent and in total employment is 3.8 percent. The model
also captures the leading role of construction during booms and busts (see Leamer, 2007) and
the comovement with nonhousing expenditures and investment. Further, the separation of
productive capital and residential structures, together with the irreversibility constraints, introduces an asymmetry between expansions and recessions similar to that observed by many previous researchers and that is traditionally hard to obtain in most real business cycle models.3
As in the theoretical exercise, the dynamic quantitative simulations show that reducing the
importance of either the sectoral interlinkages or the demand complementarities weakens
the transmission mechanism. Moreover, the model without linkages also fails to capture the
lead-lag pattern of housing and consumption expenditures observed in the data. These results
indicate that modeling production linkages provides a quantitatively relevant transmission
channel.
The burst of the real estate “bubble” might have substantially lowered potential output
and created a substantial “displacement effect,” for both labor and capital, which took quite
some time to absorb. Some researchers have referred to this displacement effect as a worsening
of labor frictions. For example, Arellano, Bai, and Kehoe (2019) and Ohanian and Raffo (2012)
attribute the Great Recession primarily to this factor. Since our model captures significant
declines in employment and output in the absence of such frictions, we also perform a business cycle accounting exercise on simulated data from the model. Through the lens of the
one-sector neoclassical growth model, the presence of intersectoral linkages, movements in
relative prices, and shifts in housing demand can be interpreted as “distortions.” Business cycle
accounting would attribute the recession generated in the model to the labor wedge. In our
model, the magnitude of the worsening of the labor wedge is about 62 percent of the total
change observed in the data. Importantly, in both our model and the data, the worsening is
due to the consumer side of the labor wedge and not to differences between wages and the
marginal product of labor.
Obviously, the fluctuations of the construction sector cannot fully account for the dynamics of employment and output since 2002. Other relevant factors not incorporated into the
analysis are important. Many suggest (Black, 1995; Hall, 2011; and Kocherlakota, 2012) that
high interest rates could be responsible for the slow recovery. These authors argue that even
in models with perfect competition and price flexibility (i.e., lacking the typical frictions of
New-Keynesian business cycle models), too-high interest rates may result in substantially
lower levels of output and employment. Since some interest rates appear to be currently constrained by the zero lower bound, such analyses appear particularly pertinent. Others argue
that the level of uncertainty (Bloom, 2009, and Arellano, Bai, and Kehoe, 2019), government
policies (Herkenhoff and Ohanian, 2011), and excessive debt overhang in the economy
(Garriga, Manuelli, and Peralta-Alva, 2019; Herkenhoff and Ohanian, 2012; and Kehoe, Ruhl,
and Steinberg, 2013) may be responsible for the lackluster recovery. Our exercise is silent with
respect to these factors.
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The remainder of the article is organized as follows. Section 2 connects our work with
the related literature. Section 3 evaluates the importance of production multipliers first in
the data, using the input-output tables of the U.S. economy, and then theoretically, using a
stylized two-sector model. Section 4 presents the quantitative multisector model, illustrates
the key mechanism at work in the analysis, and quantifies the importance of production linkages and demand complementarities. Section 6 compares the implications of the model in terms
of business cycle accounting methodology, and Section 7 offers some concluding comments.
2 RELATED LITERATURE
There is a large literature studying the connection between housing and the macroeconomy,4 for example, Gervais (2002), Campbell and Hercowitz (2005), Leamer (2007), Fisher
(2007), and Davis and Van Nieuwerburgh (2015). Most of these papers explore the effects of
housing at the traditional business cycle frequency, ignoring large swings in growth rates as
in the case of the Great Recession and the precedent boom. For example, Davis and Heathcote
(2005) study the comovement of residential and nonresidential investment in a dynamic multisector model with interlinkages and unitary elasticity of substitution between housing and
other goods. Our results show that intersectorial production linkages have large propagation
and magnification effects on most macroeconomic variables and that these effects are larger
in the presence of demand complementarities.
A small strand of the growth-theoretical literature has long argued that asymmetries in
the input-output structure of multisectoral neoclassical models amplify the effect of sectoral
shocks and may even generate endogenous cycles for appropriate configurations of the parameters, for example, Benhabib and Nishimura (1979), Boldrin and Deneckere (1990), and Long
and Plosser (1983). This theoretical theme has recently seen a revival in a growing literature,
both theoretical and applied, stressing that the intersectoral composition of the production
sector is an important source of propagation of idiosyncratic sectoral shocks (i.e., Horvath,
1998 and 2000; Carvalho, 2010; Foerster, Sarte, and Watson, 2011; Gabaix, 2011; Acemoglu
et al., 2012; Carvalho and Gabaix, 2013; Caliendo et al., 2014; Acemoglu, Ozdaglar, and Tahbaz-
Salehi, 2015; and Atalay, Dratzburg, and Wan, 2018).5 This literature focuses on idiosyncratic
technological shocks affecting the shape of the production possibility frontier and thereby
generating movements of aggregate output. These sectoral supply shocks have been proved
to be an important channel through which local perturbations may lead to aggregate fluctuations, absent any change in the composition of aggregate demand. For our part, we study how,
absent any technological variation, sectoral demand shocks may also cause aggregate fluctuations. In the case of the Great Recession, it is hard to identify a specific “sectoral production
possibility shifter,” while there clearly was a dramatic drop in the demand for housing. Inter
estingly, our analysis shows that technological asymmetries—for example, that a one-sector
model is a poor representation of the underlying production possibility set—play a crucial
role also in the case of demand shocks. It also shows that demand complementarities, largely
ignored in the business cycle literature, are in fact quantitatively relevant and should not be
ignored.
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Further along the sectoral-aggregate divide, Li and Martin (2014) study the transmission
of shocks using dynamic factor methods and explicitly look at input-output linkages. They
find that a significant part of traditionally defined aggregate fluctuations are driven by “sector-
specific shocks.” In the case of the Great Recession, more than half of aggregate volatility is
accounted for by an additional aggregate shock—which they label the “wedge factor”—emerging
only during this period. Most crucially, and consistently with our bottom line, they find that
shocks originating in the construction sector generate the largest spillover effects over time,
dominating those of all the other sectors. In our model, the driving force is an exogenous
shifter in housing demand that acts like a wedge factor.
There is also an extensive literature that explores the role of financial conditions as drivers
of the Great Recession and of the delayed recovery (i.e., Black, 1995, and Bloom, 2009; Christiano
Motto, and Rostagno, 2010; Arellano, Bai and Kehoe, 2019; Gertler and Karadi, 2011; Hall,
2011; Jones, Midrigan, Philippon, 2018; Kocherlakota, 2012; Jermann and Quadrini, 2012;
Brunnermeier and Sannikov, 2013; He and Krishnamurthy, 2019; and Mitman, Kaplan, and
Violante, 2020). Most of the literature abstracts from the role of housing during this episode,
with a few exceptions. Among them is Garriga, Manuelli, and Peralta-Alva (2019). In their
model, an increase in the cost of housing financing generates a collapse of house prices, inducing a recession through deleveraging. Similarly, Martinez, Hatchondo, and Sánchez (2015)
use heterogeneous agent models to analyze the aggregate effects of a house price decline and
of its propagation to the rest of the economy through household balance sheets and housing
defaults. Iacoviello and Pavan (2013) argue that a tightening of household budgets, due to
the drop in real estate wealth, induced a sharp decline in aggregate consumption. Rognlie,
Shleifer, and Simsek (2018) explore the aggregate effect of insufficient housing demand resulting from a period of overbuilding. Our article is complementary to this literature because,
once again, we take as granted the drop in housing demand and then study its supply-side
propagation due to sectoral interlinkages.
3 CONSTRUCTION IN AN INPUT-OUTPUT ECONOMY
This section first provides empirical evidence and then a simple theoretical evaluation
of the importance of interlinkages. The data analysis places special attention on the Great
Recession but also uses detailed U.S. sectoral input-output data for the period 1990-2013.
The theoretical framework provides a set of sufficient conditions for the amplification mechanism to work.6
3.1 Construction and Aggregate Fluctuations: 1990-2013
For the analysis of economic fluctuations, it is common to use aggregate data for the
whole postwar period. Unfortunately, the current availability of uniform input-output data
is limited to the years 1990-2014. According to the BEA, the U.S. economy has experienced
three recessions (1990-91, 2000-01, and 2007-09) during that interval of time. To evaluate
the direct contribution of the construction sector to each episode, Table 1 summarizes the
changes in employment and real income for the construction and private sectors. We measure
the direct contribution of construction as the ratio of the change in construction to each total
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Table 1
The Role of Construction in the Past Three Recessions
Employment (millions)
Real income (billions of dollars)
Construction
sector
Private
sector
Construction
sector
Private
sector
Peak, 1990:Q3
5.24
109.6
346.0
7,455
Trough, 1991:Q1
4.93
108.7
320.6
7,400
Difference
–0.31
–0.87
–25.4
–54.9
Recessions
1990-91
Percent accounted for by construction
35.4
46.3
2000-01
Peak, 2000:Q1
6.84
132.6
Trough, 2001:Q4
6.79
Difference
–0.05
Percent accounted for by construction
610.4
10,870
131.0
598.7
10,629
–1.55
–11.70
–241.1
3.3
4.9
2007-09
Peak, 2007:Q4
7.53
137.9
682.3
12,586
Trough, 2009:Q2
6.09
131.0
540.5
11,852
Difference
–1.4
–6.9
–141.8
–734.0
Percent accounted for by construction
20.3
19.3
SOURCE: BEA and BLS.
change. The top panel of Table 1 reminds us that the 1990-91 recession was mild: Between the
peak of 1990:Q3 and the trough of 1991:Q1, for the private sector, employment and income
each declined by less than 1 percent. In relative terms, the declines in the construction sector
were sizeable: slightly less than 6 percent for employment and more than 7 percent for income.
The middle panel of Table 1 shows the recession that started in 2000:Q1 and ended in 2001:Q4.
This recession was slightly more severe: for the private sector, employment fell by more than
1 percent and income by more than 2 percent. However, the declines in the construction sector
were almost negligible and the shares of the aggregate declines they accounted for were small.
The Great Recession started in 2007:Q4 and lasted until 2009:Q2. During this period, total
employment decreased by roughly 7 million jobs. Table 1 shows that the direct contributions
of the drops in construction employment and real income were 20.3 percent and 19.3 percent,
respectively. This recession was dramatically bigger than the previous two, and the size of the
drop in employment was 18.6 percent, not 5.9 percent as in 1990-91 or 0.7 percent as in 2000-01.
These calculations ignore the fact that construction leads the cycle and that during the
Great Recession this industry went into recession 18 months before the overall economy.
Measuring the decline from the perspective of the construction cycle shows that employment
fell from 7.7 million (2006:Q3) to 5.5 million (2011:Q1) and recovered little thereafter. Figure 1
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Figure 1
The Construction Sector During the Great Recession
Index, 2006 = 100
105
Employment, construction
Gross output, construction
GDP, construction
100
95
90
85
80
75
70
65
2005
2006
2007
2008
2009
2010
2011
2012
2013
SOURCE: BEA.
shows that employment, gross output, and GDP in the construction sector each dropped about
30 percent, whereas Table 1 infers employment dropped by 18.6 percent.
3.2 Evidence and Implications of Production Interlinkages
This subsection uses input-output data to provide a rough estimate of the role of interlinkages in the aggregate amplification of sectoral demand shocks during the cycle leading
up to the Great Recession. Despite its relatively small size, the contribution of the construction
sector to the Great Recession was a combination of two factors: the large size of the shocks
affecting the sector and the sector’s strong interlinkages with suppliers. One way of measuring sectoral interlinkages is through the purchases of a sector from other sectors, expressed
as a percentage of the total output of the latter. For the period 1990-2013, these calculations
are reported in Figure 2.
These numbers show by how much the total demand for the (gross) output of these sectors
would immediately decrease if construction demand vanished. For instance, the total demand
for the manufacturing sector would immediately decrease by 7 percent if the construction
sector vanished. This measures only the direct effect: Because each sector purchases goods and
services from other sectors as inputs, the process continues—virtually for an infinite number
of steps—until it converges, thereby inducing a “production multiplier” effect. This multiplier
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Figure 2
Purchases from Other Sectors
Other Services
Arts, Entertainment, and Recreation
Education and Health Services
Professional and Business Services
Financial Services
Information
Transportation and Warehousing
Retail Trade
Wholesale Trade
Manufacturing
Construction
Utilities
Mining
0
1
2
3
4
5
6
7
8
Percent of industry’s gross output
SOURCE: Use matrix from the 2006 BEA input-output tables.
Figure 3
Sectors’ Multipliers
Employment
Gross output
Wholesale Trade
Utilities
Mining
Retail Trade
Financial Services
Other Services
Information
Professional and Business Services
Manufacturing
Financial Services
Wholesale Trade
Government
Government
Education and Health Services
Transportation and Warehousing
Mining
Agriculture
Utilities
Construction
Leisure and Hospitality
Professional and Business Services
Information
Leisure and Hospitality
Construction
Retail Trade
Transportation and Warehousing
Other Services
Agriculture
Education and Health Services
Manufacturing
5
10
15
20
Multiplier
25
1
1.5
2
2.5
Multiplier
SOURCE: Use matrix from the 2006 BEA and BLS input-output tables.
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Figure 4
The Construction Sector’s Contribution to the Dynamics of Employment and Gross Output
Index, 2002 = 100
110
Employment
Index, 2002 = 100
120
108
115
106
110
104
105
102
100
Without construction, total effect
With construction
Without construction, direct effect
95
100
98
1998
Gross output
2000
2002
2004
2006
2008
2010
2012
2014
90
1998
2000
2002
2004
2006
2008
2010
2012
2014
SOURCE: Authors’ calculations using BEA and BLS requirements tables.
can be used to calculate the total effect of a sectoral shock on the rest of the economy. Figure 3
ranks sectors according to the size of their multipliers in terms of gross output and employment.
In terms of gross output, the two sectors with the largest multipliers are manufacturing
(2.4) and agriculture (2.3). Construction has the fourth largest production multiplier: A $1
decline in the output of the construction sector generates (absent changes in relative prices
and in the composition of final demand) a slightly below $2 decline in gross output of all other
sectors combined. Recall that the construction sector is larger than the agriculture sector, its
final demand much more volatile than those of both the manufacturing and agriculture sectors,
and its output composition more homogeneous. In terms of employment, the construction
sector also has a relatively large multiplier. It is worth noting that with respect to employment,
the multipliers of the manufacturing and agriculture sectors are not as significant as the multiplier for the construction sector. This highlights that the construction sector is important
overall because of its employment and gross output multipliers.
Our goal is to understand how significant these multipliers are when it comes to aggregate
fluctuations. To this end, we add up the direct and multiplier effects for the construction sector
to compute the total effect of construction on the rest of the economy. We use the requirement
matrices and compare the actual evolution of U.S. employment and gross output with a counterfactual economy without the construction sector. Figure 4 displays the paths of employment
and gross output for the three cases: the actual values and the values without the direct and
without the total effects of construction. The difference between these paths is a rough estimate
of the construction sector’s (direct and total) impact on the aggregate dynamics. When construction is included, total employment increases about 6 percent between 2002 and 2006,
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Figure 5
Sectoral Changes During the Great Recession: The Data and Input-Output Simulations
Employment, percent change
(BLS requirement matrix)
Gross output, percent change
(BEA requirement matrix)
Data, 2006-2009
Data, 2007-2009
Simulation
Other Services
Leisure and Hospitality
Education and Health Services
Professional and Business Services
Financial Services
Information
Transportation and Warehousing
Retail Trade
Wholesale Trade
Manufacturing
Construction
Utilities
Mining
Total nonfarm
–40
–30
–20
–10
Percent change
0
10
–25
–20
–15
–10
–5
0
5
10
Percent change
SOURCE: Authors’ calculations using BEA data.
which is then entirely lost. In contrast, the economy without a construction sector and with
the total effect has a slower recovery from the 2000-01 recession; employment growth picks
up only in 2005 and employment destruction starts in 2009. The magnitude of the subsequent
decline is half that actually experienced. Unlike in the actual economy, employment starts
recovering already in 2010 and in 2012 surpasses the previous peak. This exercise shows that
the construction sector contributed greatly to employment growth between 2002 and 2005
and to employment destruction during the Great Recession. A similar, if somewhat weaker,
conclusion can be drawn by analyzing the panel for gross output. This simple decomposition
reveals that during the Great Recession construction alone may have accounted for 52 percent
of the decline in employment and 35 percent of the decline in gross output.
At a more micro level, construction interacts differently with the various sectors in the
economy. Therefore, a decline in the activity of the construction sector will have a larger impact
on those sectors that sell to it directly as opposed to those that do not. To show this, Figure 5
reports for each sector the actual sectoral declines in employment and gross output, respectively, between 2006-09 and 2007-09 and those estimated using the input-output matrix for
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Figure 6
Sectoral Changes During the Recovery: The Data and Input-Output Simulations
Employment, percent change
(BLS requirement matrix)
Other Services
Gross output, percent change
(BEA requirement matrix)
Data, 2009-2011
Simulation
Leisure and Hospitality
Education and Health Services
Professional and Business Services
Financial Services
Information
Transportation and Warehousing
Retail Trade
Wholesale Trade
Manufacturing
Construction
Utilities
Mining
Total nonfarm
–10
–5
0
5
10
Percent change
15
20
–15
–10
–5
0
5
10
15
20
Percent change
SOURCE: Authors’ calculations using BEA data.
2007-09, as a consequence of the observed decline in construction.7 The blue (2006-09) and
green bars (2007-09) in Figure 5 represent the historical percent changes in gross output and
employment for 13 industries and for the total economy (total nonfarm). In 2006-09, gross
output in the construction sector and in the aggregate (total nonfarm) declined close to 25
percent and 6.2 percent, respectively. Employment in the construction sector decreased by
roughly the same amount as gross output, 21.5 percent, while aggregate employment declined
by 4.4 percent. The aggregate numbers are slightly larger when considering the period 2007-09.
The yellow bars represent the declines attributable to the construction sector on the basis of
the input-output multipliers (for gross output and employment, respectively). For example,
according to this methodology, the drop in the construction sector accounts for a significant
part of the gross output decline in the mining sector, about 68 percent of it, while it accounts
for little of the decline in the retail trade sector.
According to this methodology, construction is capable of accounting for about 35 percent
of the decline in aggregate gross output and for about 52 percent of the decline in aggregate
employment. These numbers contrast with the direct impact estimates that account for 20.3
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percent of the decline in employment and 19.3 percent of the decline income, as shown in
Table 1. The difference between the direct and the total effects of construction is due to the magnifying role of the production interlinkages, and this is what we label the production multiplier.
The construction sector played an important role not only in the Great Recession but also
in the subsequent slow recovery. Its contribution can be measured by performing a similar
counterfactual for the period 2009-11. Figure 6 shows the simulated growth rates for the 13
sectors and the total economy (nonfarm economy) under the assumption that construction
grows from 2009 forward at pre-recession rates. The blue bars display the actual percent changes
of gross output and employment, respectively. Between 2009 and 2011, gross output increased
by 5 percent and employment increased by roughly 1 percent. The yellow bars represent the
counterfactual simulation and show that if construction had grown at its pre-recession levels,
total gross output and employment would have increased by 6 percent and 2 percent, respectively. In this scenario, the sectors that would have grown the most in terms of gross output
are wholesale trade (20 percent), retail trade (10 percent), mining (13 percent), and transportation and warehousing (11 percent). The differences between the above growth rates and the
actual growth rates (blue bars) indicate that the contribution of the construction sector to the
dynamics of aggregate employment and output is nontrivial. The next section proposes a
simple model of interlinkages that explains the nature of these effects.
4 THEORETICAL MODEL OF INPUT-OUTPUT LINKAGES
This section presents a stylized two-sector model with housing and production interlinkages. The model also incorporates the durable consumption goods nature of housing, the
presence of a fixed factor in the production of housing services (i.e., land), and the complementarity between consumption of housing services and of all other goods.
4.1 Households
Total population size, Nt, is normalized to 1. Household preferences are defined by a
time-separable utility function, U(ct , θt ht , nt ), where ct represents consumption goods, ht
represents housing services, and nt represents labor supplied in the market. Housing provides
utility, and it is a complement to aggregate consumption. The shifts in housing consumption
are driven by adjustments in the parameter θt . The utility function U satisfies the usual properties of differentiability and concavity. The sequence of utilities is discounted by the term
β (0,1). Housing services are obtained by combining physical structures, st , and land, lt ,
according to H(st , lt ). The latter is homogeneous of degree 1 and satisfies Hʹi > 0, Hʹʹi > 0 and
Hʹʹij > 0. Housing structures depreciate at a constant rate, δs. In each period, the numeraire is
the spot price of the nonconstruction good. Formally, the representative consumer chooses
∞
to maximize
{ct , ht ,nt , kt+1 , st+1 , lt+1 }t=0
max ∑t=0 β tU ( ct ,θt ht ,nt ) ,
∞
s.t.
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ct + xtk + pts xts = wt nt + rtk kt + ptl ( lt − lt+1 ) + π t ,
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Boldrin, Garriga, Peralta-Alva, Sánchez
(1)
ht = H ( st , lt ) ,
(2)
xtk = kt+1 − (1− δ k ) kt ≥ 0,
(3)
xts = st+1 − (1− δ s ) st ≥ 0.
The maximization is also subject to transversality and no-Ponzi-game conditions. Prices are
defined as follows: pts is the price of infrastructure, ptl is the price of land, wt is the wage rate,
and rt is the gross return on capital. To facilitate computing the rental rate for housing services,
our specification allows land trading, lt , even if in equilibrium there is no trading of land,
which is owned by the representative household and inelastically supplied. The term πt represents profits from the construction sector. All investment decisions are subject to an irreversibility constraint and have different depreciation rates in the two sectors, construction
and nonconstruction.
The relevant first-order conditions of the consumer problem are
U n ( ct ,θt ht ,nt )
= wt ,
U c ( ct ,θt ht ,nt )
∀t,
U c ( ct ,θt ht ,nt )
k
= 1+ rt+1
−δk ,
β U c ( ct+1 ,θt+1ht+1 ,nt+1 )
∀t,
when the irreversibility constraints do not bind, xtk > 0. The relevant conditions for housing
decisions for the case with positive housing investment (xts > 0) satisfy
U h ( ct ,θt ht ,nt )
= Rt ,
U c ( ct ,θt ht ,nt )
pts =
∀t,
1
s
⎡
⎤
k ⎣ Rt+1 H s ( st ,lt ) + pt+1 (1− δ s ) ⎦ ,
1+ rt+1
ptl =
1
l
⎡ Rt+1H l ( st ,lt ) + pt+1
⎤⎦ ,
1+ rt+1 ⎣
where Rt represents the implicit rental price for housing services measured in terms of consumption units. Notice that a no-arbitrage condition holds between investment in land and
housing. The last two expressions state that the current cost of purchasing a unit of housing
structures (land) equals the future return of housing services derived from the housing capital
(land) valued at market prices, plus its capitalization.
4.2 Nonconstruction Sector
The model uses a 2 × 2 input-output structure: To operate, each sector requires, among
other things, that the output of a sector uses intermediate inputs from other sectors as well as
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its own. To capture this fact, we deviate from common practice and write all production
functions in terms of gross (as opposed to net, i.e., value-added) output. Capital goods, which
are produced in the nonconstruction sector, must be distinguished from the intermediate
inputs from the same sector since they last more than one period. In the baseline model, capital
goods are used only in the nonconstruction sector for simplicity. Both investments satisfy the
putty-clay assumption, which is sector specific.
Formally, let mi,j be the intermediate input produced by sector i and used by sector j.
The nonconstruction sector operates in a competitive market and uses the technology
Aty F kt ,nty ,mty ,y ,mts ,y to produce its gross output:
(
)
Yt = ct + xtk + mty,y + mty,s .
The production function F has constant returns to scale. The firm’s optimization problem is
π ty =
max
kt ,nty ,mty, y ,mts,y
s.t.
Yt − wt nty − rtk kt − mty,y − ptsmts,y ,
(
)
Yt = Aty F kt ,nty ,mty,y ,mts,y ,
∀t,
∀t,
where the price of the nonconstruction sector’s output is normalized to 1. The constant-returnsto-scale assumption implies zero equilibrium profits, πty = 0, and marginal cost pricing for
each input
(
w = A F ( k ,n ,m
1 = A F ( k ,n ,m
p = A F ( k ,n ,m
)
),
),
).
rtk = Aty F1 kt ,nty ,mty,y ,mts,y ,
t
s
t
y
t 2
t
y
t
y,y
s,y
t ,mt
y
t 3
t
y
t
y,y
s,y
t ,mt
y
t 4
t
y
t
y,y
s,y
t ,mt
4.3 Construction Sector
The construction sector is also competitive. Its net output consists of residential structures,
purchased by the households, while its gross output also includes structures used as intermediate inputs in both sectors. In the baseline case, purely for simplicity, we assume this sector
has a fixed stock of capital; hence its value added is split between the wages of labor and the rent
accruing to the owner of the fixed capital stock (the representative household). Implicit in this
formulation is a somewhat extreme assumption about the mobility of factors from one sector
to another: While labor can move freely, the stock of capital invested in the construction sector
is completely immobile (either way), and variations in investment activity have an impact only
on the nonconstruction sector. The technology for gross output is represented by
(
(
XtS = xts + mts,s + mts,y = AtsG nts ,mts mts,s ,mty,s
))
and exhibits decreasing returns to scale in labor and the intermediate input mix. The function
G(.) has a constant elasticity of substitution, and the aggregator of intermediate inputs is
homogeneous of degree 1. The optimization problem of the representative firm is now
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π ts = s max
pts XtS − wt nts − ptsmts,s − ptymty,s ,
y ,s
s ,s
∀t,
nt ,mt ,mt
(
))
(
XtS = AtsG nts ,mts mts ,s ,mty ,s ,
s.t.
∀t.
The first-order conditions are similar to those of the representative firm in the nonconstruction sector and are not repeated here. Note that because of the presence of a fixed stock of
capital, firm profits are not zero in equilibrium in this sector. It is worth emphasizing that pts
reflects the cost of producing new structures. The equilibrium price of a house differs from
this value since it depends on the relative value of the structures and land.
4.4 Competitive Equilibrium
The competitive equilibrium of this economy is defined as follows:
y
s
Competitive Equilibrium: Given a sequence of values { At ,At ,θt }t =0, a competitive equilib∞
rium consists of allocations {ct ,xtk ,xts ,lt ,nty ,nts ,mts ,s ,mty ,s ,mty ,y ,mts ,y }t=0 and prices
∞
{wt ,rtk , ptl , pts ,rt ,Rt }t=0 that satisfy the following:
∞
(i) the household’s optimization problem;
(ii) profit maximization in the construction and nonconstruction sectors; and
(iii) the clearing of markets:
(a) the labor market (wt ):
nt = nty + nts ,
∀t ;
lt = lt−1 = l ,
∀t ;
(b) the land market (ptl ):
(c) the capital market (rtk ):
(
)
rtk = Aty F1 kt ,nty ,mty ,y ,mts ,y ,
∀t ;
(d) the nonconstruction output market (ptc = 1):
(
)
∀t ; and
))
∀t.
ct + xtk + mty ,y + mty ,s = Aty F kt ,nty ,mty ,y ,mts ,y ,
(e) the construction output market (pts ):
(
(
xts + mts ,s + mts ,y = AtsG nts ,mts mts ,s ,mty ,s ,
For a given sequence of housing demand shifters {θt } the model endogenously generates
time series for all macroeconomic quantities and prices. Before comparing the predictions of
the model with the data, it is useful to understand how to characterize the amplification process. This is described in the next subsection.
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4.5 The Macroeconomic Effects of Production Interlinkages and Demand
Complementarities
To proceed analytically it is necessary to make a number of simplifying assumptions relative to the baseline model: (i) the economy lasts one period, (ii) labor and intermediate goods
are the only inputs, and (iii) the share of land in the housing aggregator is zero. The utility
index is defined as U(c, θ, h, n), and the budget constraint of the representative household is
c + ph = wn, where w is the wage rate and p the price of housing, both measured in units of
the nonconstruction good.
Part of the output of the nonconstruction sector, my , is used as an input to produce construction, and part of the output of the construction sector is used to produce nonconstruction goods, mh. The gross output flows of the two sectors are c + my = Y = Ay f(ny , εy mh ) and
h + mh = H = Ah g(nh, εhmy), respectively, where Aj represents the productivity of sector j = y,h.8
The εj(j = y,h) terms capture the relative importance in sector j of the intermediate inputs from
the other sector. Aggregate labor satisfies the restriction ny + nh = n. Free mobility implies that
the wage rate is the same across sectors.
A competitive equilibrium in this economy is an allocation {c, h, ny, nh, my, mh} and prices
{w, p} that solve (i) the optimization problem of the household, (ii) the optimization problem
of the firms in each sector, and ( iii) the market-clearing conditions.
As a function of the preferences (θ) and technology (ε) parameters, value added in this
economy is defined as
VA(θ , ε ) = c (θ , ε ) + p (θ , ε )h (θ , ε ) .
The goal is to identify conditions under which a shift Δθ = θʹ – θ in the demand for housing
has a larger impact on total employment and value added when there are interlinkages (ε > 0)
rather than when there are not (ε = 0), that is, the conditions under which we have
∂VA(θ , ε ) ∂VA(θ , 0 )
≥
.
∂θ
∂θ
There are three interacting channels through which a change in the demand for housing may
affect value added: (i) a direct change in the desired quantities of c and h, (ii) a change in their
relative prices (and the consequent second-order changes in the quantities demanded), and
(iii) a change in the supply of labor due to wealth and price effects. To highlight the difference
of this transmission mechanism relative to the recent literature discussed in Section 2 above,
the next examples abstract from movements in relative prices due to sectoral shocks. The full
quantitative model will also consider changes in relative prices and the related adjustments
in quantities.
4.5.1 Example: Leontief Production. A simple way to eliminate the price effects is to consider
an economy in which both production functions have fixed coefficients
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⎧⎪ m ⎫⎪
c + m y = Y = Ay min ⎨n y , h ⎬ ,
⎪⎩ ε y ⎪⎭
⎧ my ⎫
h + mh = H = Ah min ⎨nh , ⎬ .
⎩ εh ⎭
The parameters εy ≥ 0 and εh ≥ 0 capture the intensity of the sectoral interlinkages.9 Using the
nonsubstitutability of inputs and the constraint on total employment yields a linear production
possibility frontier:
⎛ Ay + ε h ⎞
⎛ Ay Ah − ε y ε h ⎞
c+⎜
⎟h=⎜
⎟ n.
⎝ Ah + ε y ⎠
⎝ Ay + ε h ⎠
To satisfy feasibility, it must be the case that AyAh > εyεh. If the intermediate input requirements
are too high relative to the productivity of each sector, it would not be feasible to produce
positive amounts of both goods.10 The linearity of the production possibility frontier implies
that the relative price of new houses depends only on the technical coefficients,
p=
Ay + ε h
Ah + ε y
.
In the model without interlinkages εj → 0, the price is given by the ratio of productivities
p = Ay /Ah. Similarly, wages are determined by
⎛ Ay Ah − ε y ε h ⎞
w=⎜
⎟.
⎝ Ay + ε h ⎠
In the absence of sectoral shocks on (Ay,Ah), exogenous changes in housing demand (Δθ)
have no effect on prices and wages. All the macroeconomic effects of value added are driven
by changes in the production of each sector:
⎛ Ay + ε h ⎞
VA(θ , ε ) = c (θ , ε ) + ⎜
⎟ ⋅h (θ , ε ) .
⎝ Ah + ε y ⎠
From this expression, it is direct to derive specifications for which a shift in housing
demand is not amplified in the aggregate. The first one ignores sectoral interlinkages, εy = εh = 0
as the model collapses to the standard two-sector model where relative prices are determined
by factor productivities. The second one assumes perfectly symmetric sectors, εy = εh = ε and
Ay = Ah = A, implying steady-state prices and wages equal to p = 1 and w = (A – ε). One can
easily add a third trivial case where the nonconstruction good is completely independent of θ;
that is, c(θ,ε) = c(ε).
Case 1: Preferences with Perfect Complementarity. This specification allows the housing-
demand shifter Δθ to change directly the consumption demand of both goods. The utility
index is given by U(θc, h, n) = min{θc, h} – an1+γ/(1 + γ). This corresponds to the extreme case
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of perfect complementarity; but only some degree of complementarity (less than unitary
elasticity of substitution between c and h) is sufficient for the mechanism to operate. With
this utility function consumption is given by
⎡
Ay Ah − ε y ε h
ĉ = ⎢
⎢ Ah + ε y + θ Ay + ε h
⎣
(
) (
)
⎤
⎥ n,
⎥
⎦
and the demand for housing is just ĥ + θĉ. In the model without interlinkages (εj = 0),
employment is allocated in the two sectors according to n̂hnolink = Ay / Ay + θ Ah n.
Solving for the aggregate level of employment yields
( (
⎡θ ⎛
Ay Ah − ε y ε h
n̂link = ⎢ ⎜
⎢ a ⎜⎝ Ah + ε y + θ Ay + ε h
⎣
(
) (
))
1
)
⎞ ⎤γ
⎟⎥ ,
⎟⎠ ⎥
⎦
whereas in the absence of interlinkages the employment level is
1
⎡ θ ⎛ Ay Ah ⎞ ⎤ γ
n̂nolink = ⎢ ⎜
⎟⎥ .
⎢⎣ a ⎝ Ah + θ Ay ⎠ ⎥⎦
Measured economic activity is given by
⎡ Ay Ah − ε y ε h ⎤ i
VA = c + ph = ⎢
⎥ n̂ (θ ).
⎢⎣ Ay + ε h ⎥⎦
Notice that value added is proportional to total employment, and the scaling factor does not
depend on the parameter θ. The change in value added due to a change in housing demand
driven by Δθ is
∂VA ⎡ Ay Ah − ε y ε h ⎤ ∂n̂(θ )
=⎢
.
⎥
∂θ ⎢⎣ Ay + ε h ⎥⎦ ∂θ
We ask next, how do changes in the preference parameter θ affect aggregate employment,
n, and value added, VA? Notice first, from the formulas above, that the economy with interlinkages and the one without them have different levels of aggregate employment. Hence, we
will compute the two elasticities of employment with respect to variations in θ.
In the economy with linkages, this elasticity is
⎡
Ay + ε h
link 1
⎢
e n,
=
θ
γ ⎢ Ah + ε y + θ Ay + ε h
⎣
(
) (
)
⎤
⎥ > 0,
⎥
⎦
and in the economy without linkages it is
1 ⎡ Ay ⎤
e nolink
⎢
⎥ > 0.
n,θ =
γ ⎢⎣ Ah + θ Ay ⎥⎦
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The presence of interlinkages amplifies the effect of any given preference shock when
link
nolink
11
e n,
>
θ e n,θ , which reduces to Ahεh > Ayεy after a bit of algebra. This condition is clearly satisfied when the construction sector purchases intermediate inputs from the rest of the economy
(εh > 0), but not the other way around (εy = 0). Hence, in general, the condition holds (for given
levels of sectoral productivities) when the construction sector absorbs lots of inputs from the
other sector but the other sector does not use housing as an intermediate input, which does
not sound so unrealistic. Notice that when there is symmetry between both sectors the condition fails. This is consistent with the earlier theoretical results of Horvath (1998) and Dupor
(1999) cited in Section 2.
Is there some empirical evidence that supports this asymmetry? We have used the direct-
input requirement matrices to carry out a back-of-the-envelope test of these conditions. Our
procedure was simple: We aggregated the matrices into a 2 × 2 format: construction and everything else. Next, we eliminated the “own intermediate inputs,” which the model assumes away
for simplicity, and collapsed all the value added of the two sectors into labor income. The
sectoral price indices were used to compute the relative price p during the available sample
period and then to compute, by simple algebra, the four parameters of our model. We found
that Ahεh /Ayεy equals 636, after rounding up. The inequality is amply satisfied, thereby suggesting, on the basis of this admittedly simplified model, that in the real world the magnification
effects of asymmetries are likely to be present.
Case 2: Preferences with Unitary Elasticity. To highlight the importance of the complementarity
between c and h, we consider the case of Cobb-Douglas preferences, u(c,h) = log c + θ log h.
This specification is very common in macro housing models (i.e., Davis and Heathcote, 2005,
and Iacoviello, 2005) and in the recent literature on production networks (i.e., Acemoglu et al.,
2012). It implies a total employment level
1
n̂ = ω i ((1+ θ ) / a )1+γ ,
where ωnolink = 1 in the economy without linkages and ωlink > 1 in the economy with them,
independently of θ. The level of aggregate employment is larger in the economy with linkages
than in the one without them; but in response to changes in housing (Δθ), the two economies
share the same aggregate labor elasticity,
link
nolink
e n,
θ = e n,θ =
1
1+ γ
⎛ θ ⎞
⎜⎝
⎟.
1+ θ ⎠
Even with Leontief technologies, the unitary elasticity eliminates the contribution of the
input-output multipliers. For this reason, in the quantitative exercise in Section 5, we explore
the importance of demand complementarities by adopting a more general class of preferences
that includes the Cobb-Douglas as a special case.
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5 QUANTITATIVE ANALYSIS
In this section we use the general model developed in the previous section to carry out
various quantitative exercises.
5.1 Parameterization
To proceed we need to specify functional forms and then assign parameter values. The
choice of functional forms is relatively general with the exception of the housing demand shifter,
⎡ηc − ρ + (1− η )ht− ρ ⎤⎦
u ( ct ,ht ,N t ) = ⎣ t
1− σ
−
1−σ
ρ
+ θt ht
+log(1 – Nt ),
where the parameter ρ controls the degree of demand complementarity between consumption,
c, and housing services, h. The parameter σ represents the intertemporal elasticity of substitution, and the parameter η represents the relative importance of consumption. The utility from
leisure is logarithmic, as is standard in the real business cycle literature with a representative
agent.12 Housing services enter as complementary to consumption goods but also as a linear
term scaled by the parameter θt that shifts housing demand to generate a construction boom
and bust. This is interpreted as a “reduced-form housing market wedge” given by θ̃t that, when
measured relative to consumption goods, implies the housing price equation
ph
pth = Rt + θ%t + t+1 ,
1+ rt+1
where the notion of rents comes from owner-equivalent rent given by
Rt
(1− η ) ⎛ ct ⎞
=
η
1+ ρ
.
⎜⎝ h ⎟⎠
t
Housing services are obtained from housing structures and land according to a CobbDouglas mixture,
()
h = H ( s,l ) = zh ( s ) l
e
1−e
,
where zh represents a transformation factor between stock and flow. The production of the
nonconstruction goods also uses a Cobb-Douglas technology:
(
)
α1
F k,n y ,m y ,y ,m s ,y = A y ( k )
(n y )α (m y ,y )α (ms ,y )1−α −α −α ,
2
3
1
2
3
where αi represents the share in production for input i. Notice that the specification allows for
substitutability between intermediate goods. The technology used in the construction sector,
instead, is a constant elasticity of substitution with diminishing returns to scale:
(
s
s ,s
G n ,m ,m
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y ,s
)
⎡
= A ⎢γ 2 n s
⎣
s
( )
− γ 1γ 4
((
+ (1− γ 2 ) m
)
−γ 4
y ,s 1−γ 3
) (m )
s ,s γ 3
⎤
⎥
⎦
−1/γ 4
.
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Table 2
Parameter Values
Parameter
Value
α1
α2
α3
γ1
γ2
γ3
γ4
Ay
As
zh
ε
η
σ
ρ
0.18
0.50
0.035
0.62
0.40
0.04
1.5
2.4
1.74
0.175
0.28
0.435
1
5
The model parameters are set to match long-run averages of their data counterparts
between 1952 and 2000. The implied parameter values are relatively robust to the choice of
the sample period; however, during the housing boom some of the ratios and long-run averages departed significantly from their historical trends. Hence, to avoid stacking the cards in
favor of our model, we used data from only the period before the housing boom and bust to
calibrate our model.
The time unit is a year, as input-output tables are yearly at best. The discount factor is
β = 0.96. The depreciation rates of residential structures and nonresidential capital are
δ s = 0.015 and δ y = 0.115, respectively. The weight on leisure, = 0.33, is such that total hours
worked equal one-third of the time endowment in the steady state. The preference parameters
are set to match the consumption-to-output and housing-to-output ratios. The parameters
of the production functions are set to satisfy the following:
the ratio of gross output in the two sectors, Y s/Y y = 0.08;
the average labor share in the construction sector = 0.7;
the average labor share in the nonconstruction sector = 0.65;
the ratio of consumption to nonconstruction gross output = 0.35;
the observed shares of intermediates in gross output of own sector (myy and mss)
= 0.4, 0.007;
(vi) time allocated to market activities, ny + ns = 1/3; and
(vii) the ratio of employment in the two sectors, ny/ns = 16.
(i)
(ii)
(iii)
(iv)
(v)
The values of the parameters not listed here are displayed in Table 2. The intratemporal
elasticity of substitution between consumption and housing services is determined by the
parameter εch = 1/(1 + ρ). Quantitatively, the value of ρ is an important determinant of the
spillover effects from housing into the rest of the economy. If consumption services are close
substitutes, a decline in the demand for housing services can generate an increase in the demand
for the consumption good; whereas if they are close complements, a decline in housing demand
can translate into a decline also in the demand for consumption. Various recent papers, part
of an extensive literature on the topic, estimate this elasticity to be less than 1. For example,
Flavin and Nakagawa (2008) use a model of housing demand and estimate an elasticity of less
than 0.2. Others (i.e., Song, 2010, and Landvoigt, 2017) use alternative model specifications
and also estimate values for the elasticity to be less than 1. The simulations consider elasticities
εch {0.17, 0.25}.
To generate dynamics in housing demand and construction, households face unanticipated shocks to the demand shifter θt. Households have some initial expectations about their
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Figure 7
Dynamics of the Construction Sector: The Model and Data
Construction employment
Construction value added
Percent
40
Percent
25
Data
Model
20
30
15
20
10
10
5
0
0
−5
−10
−20
−30
1998
−10
Data: Employment
Data: Hours
Model
2000
2002
2004
−15
2006
2008
2010
2012
2014
−20
2000
2005
2010
2015
SOURCE: BEA and authors’ calculations.
housing demand set by the initial values θt = θ2000 for all t ≥ 2000. Looking forward, they assume
this parameter will remain unchanged in the future. In 2001, households are surprised by an
initial increase in θ perceived as permanent going forward, θt = θ2001 > θ2000 for all t ≥ 2001. In
each subsequent period, this parameter is adjusted. In 2007, there is a demand reversal that
generates a decline in housing demand until 2010; thereafter it remains constant forever.13
The dynamics of this parameter are calculated to generate equilibrium paths of valued added
and employment and hours in the construction sector that are in line with the data, as shown
in Figure 7.
5.2 Role of Residential Investment in Growth and Employment
The goal of this exercise is to measure the sectoral contribution of the construction sector
to the macroeconomy. The baseline case considers a boom and a bust in the construction
sector, which generates the total employment and aggregate value-added series summarized
in Figure 8. The shocks to the construction sector have nontrivial effects on total employment
and aggregate value added. To measure the ability of the model to mimic the data, we measure
the fraction of the changes in employment and GDP during the expansion period (2000-07)
and during the recession (2007-10) it generates. During the boom, the exogenous changes in
the demand for housing explain 60 percent of the change in total employment and 25 percent
of value added.14
Similarly, the housing crash, started by a sudden decrease in housing demand, generates
declines in the employment of the construction sector and the demand for intermediate inputs
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Figure 8
The Aggregate Implications of a Shock to Construction
Total employment
Aggregate value added (GDP)
Normalized = 1 (2007)
Percent
5
1.02
0
1.00
−3.3%
−3.8%
0.98
−5
0.96
0.94
−7.0%
−6.0%
Data: Employment
Data: Hours
Model: Predicted
−10
−8.2%
Data
Model
0.92
2004
2006
2008
2010
2012
2014
−15
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
SOURCE: BEA and authors’ calculations.
from suppliers. The input-output structure of the model once again lowers the demand for
the nonconstruction sector as demand for intermediates from the construction sector falls.
In the short run, the decline in the demand for housing generates a very small and short-lived
increase in nonhousing consumption that is consistent with the empirical evidence, as shown
in Figure 9. This temporary consumption increase is not sufficient to compensate for the decline
in other key macroeconomic aggregates. In the model, the collapse of the construction sector
(starting in 2007) generates a 3.8 percent decline in total employment and a 3.3 percent decline
in aggregate value added. Comparing the numbers with the data, the model can rationalize
44 percent of the decline in employment and 56 percent of the decline in total value added.
An important feature of construction in the business cycle is its leading role during both
booms and busts (see Leamer, 2007). Similarly, the data suggest that, during a boom, purchases
of housing and durable goods increase faster than purchases of food and services; during a
bust, purchases of housing and durable goods decline very sharply, while nonhousing related
purchases continue to increase for a few more quarters. Our model captures these lead-lag
patterns almost perfectly, as illustrated by the two panels of Figure 9.
In terms of prices, the model generates an 11 percent increase in the house price-to-rent
ratio (price-rent ratio) during the housing boom and a 15 percent decline during the bust, as
shown in the left panel of Figure 10. According to the OECD data, the price-rent ratio increased
40 percent during the boom and declined 25 percent during the bust. Relative to the data, the
model captures around 30 percent of the boom and 60 percent of the bust. The model performs
remarkably well given that this is a nontargeted moment and that there is only one force driving
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Figure 9
Lead-Lag Responses of Consumption and Housing Spending
Model
Data
Percent
Percent
1.0
Goods
Housing
0.8
4
0.6
2
0.4
0
0.2
0
−2
−0.2
−4
−0.4
−0.6
−6
−0.8
−1.0
−5
−4
−3
−2
−1
0
1
2
3
4
5
−8
−5
−4
−3
−2
−1
0
1
2
3
4
5
Year
Year
SOURCE: BEA and authors’ calculations.
Figure 10
Model-Implied Prices
Price-rent ratio
Wages
Percent
Percent
15
2.0
1.5
10
1.0
0.5
5
0
0
−0.5
−1.0
−5
−1.5
−10
2000
2005
2010
2015
−2.0
2000
2005
2010
2015
SOURCE: Authors’ calculations.
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Figure 11
Lead-Lag Responses of Consumption and Housing Spending
Model (ρ = 5)
Model (ρ = 3)
Percent
Percent
1.0
1.0
Goods
Housing
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1.0
−5
−4
−3
−2
−1
0
1
2
3
4
5
Year
−1.0
−5
−4
−3
−2
−1
0
1
2
3
4
5
Year
SOURCE: Authors’ calculations.
the aggregate dynamics. For a detailed discussion on other drivers for house prices, see Garriga,
Manuelli, and Peralta-Alva (2019).
The model can also reconcile large movements in employment and hours with very modest
movements in real wages, as shown in the right panel of Figure 10. There is abundant research
arguing that a variety of different frictions affecting the labor market are important to generate significant movements in aggregate employment (see, for example, Boldrin and Horvath,
1995; Christiano, Motto, and Rostagno, 2010; Arellano, Bai, and Kehoe, 2019; Gertler and
Karadi, 2011; Hall, 2011; and Jones, Midrigan, and Philippon, 2018). Our model abstracts
from such features, but modeling the use of intermediates allows us to reduce the response of
wages to changes in the labor supply. Section 6 explores the connection of these findings with
the existing literature.
5.3 The Role of Demand Complementarities
In our model, the complementarity between consumption and housing is an important
driver of the employment dynamics. To compare the model implications for different levels
of complementarity, we calculate, in each case, a sequence of demand shifters {θt} that match
the dynamics of employment in the construction sector. The qualitative implications are the
same, but the calibration with stronger complementarity generates a more pronounced boom-
bust pattern. With a lower degree of complementarity, GDP falls 2 percent instead of 3.3
percent and total employment declines by 2.4 percent instead of 3.8 percent. The degree of
complementarity also has implications for other variables. Figure 11 emphasizes the different
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lead-lag responses of consumption and housing spending. When the elasticity of substitution
between consumption and housing is increased from 0.16 to 0.25, consumption declines significantly less during a bust. With low elasticity, expenditures on goods respond with a lag
relative to housing but have similar dynamics. As the elasticity increases, the dynamics of
goods expenditures diverge from the declining path of housing. Increasing the elasticity to
higher numbers (ρ < 2) would generate a boom of the nonconstruction sector during the
collapse of the construction sector. A low elasticity of substitution magnifies the aggregate
responses of employment and capital investment.
5.4 The Role of Interlinkages and Supply Complementarities
In our view the interlinkages have been an important driver of aggregate output and
employment during the housing boom and bust. To isolate the effects of the interlinkages
from those derived purely from consumer demand for housing, we study two alternative
specifications. The first uses the same parametric calibration and shuts off interlinkages by
holding the sectoral demand of intermediates fixed at the level of the initial steady state in
1998 mts ,s = m0s ,s ,mty ,s = m0y ,s ,mty ,y = m0y ,y ,mts ,y = m0s ,y . This case is referred to as the “no interlinkages specification.” The second formulation completely ignores the role of intermediate
goods mts ,s = mty ,s = mty ,y = mts ,y = 0 , and the production functions are specified for value added
and not gross output.15 In this “value-added specification,” the relevant technologies ignore
the use of intermediate inputs from other sectors in the economy. The nonconstruction sector satisfies
(
)
(
)
(
)
ct + xtk = Aty F kt ,nty ,
and the construction sector satisfies
( )
xth = AtsG nts .
For both specifications, we carry out the same simulation experiment under the assumption that ρ = 5. The housing demand shifter is adjusted to generate movements in construction
employment consistent with the data. Figure 12 compares the key macroeconomic aggregates
in the three cases: the baseline model with interlinkages, a specification with no interlinkages,
and the value-added specification.
Consider first the case labeled “no interlinkages,” which is simpler because the intermediates are fixed to the initial steady-state levels. Both sectors are committed to producing the
same amount of intermediates every period. During the housing boom, the only way to produce more structures is to use more capital and labor. Since the quantity of intermediates
cannot adjust, prices adjust more relative to the baseline level. Qualitatively speaking, the
equilibrium dynamics of this version of the model are similar to those of the baseline one.
However, the quantitative implications are very different. Since intermediates are constant,
the marginal product of labor in the construction sector does not increase as much as in the
baseline experiment and employment also does not increase as much either. The construction
sector expands during the boom, but because the links to the other sector have been severed,
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Figure 12
The Impact of Construction: Role of Interlinkages (ρ = 5)
Aggregate value added (GDP)
Normalized = 1 (1999)
1.04
Employment
Normalized = 0 (2006)
Interlinkages
No interlinkages
Value added
1.03
1.02
Data: Employment
Data: Hours
Interlinkages
No interlinkages
Value added
5
0
1.01
−3.3%
1.00
−5
0.99
0.98
2000 2002 2004 2006 2008 2010 2012 2014 2016 2018 2020
−10
2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
Price-rent ratio
Percent
35
Wages
Percent
Interlinkages
No interlinkages
Value added
30
25
2.0
1.5
1.0
20
0.5
15
0
10
−0.5
5
0
−1.0
−5
−1.5
−10
2000
Interlinkages
No interlinkages
Value added
2005
2010
2015
−2.0
2000
2005
2010
2015
SOURCE: BEA and authors’ calculations.
the latter barely moves in spite of the consumption complementarity: All movements are less
than 0.5 percent. Consequently, the changes in GDP and employment are an order of magnitude smaller than in the economy with intersectoral links. The input-output links operate,
de facto, as total factor productivity changes in the manufacturing sector, turning the variations in the demand for houses into a variation in the marginal value of output for the second
sector.
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Figure 13
Lead-Lag Responses of Consumption, Housing Spending, and Investment
No interlinkages specification
Value-added specification
Percent
1.0
Percent
1.0
Goods
Housing
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1.0
−5
−4
−3
−2
−1
0
1
2
3
4
5
−1.0
−5
−4
−3
−2
−1
Year
0
1
2
3
4
5
Year
Capital investment
Percent
5
Residential investment (structures)
Interlinkages
No Interlinkages
Value Added
4
3
2
Percent
5
0
−5
−10
1
−15
0
−20
−1
−25
−2
−3
−30
−4
−35
−5
2000
2005
2010
2015
−40
2000
2005
2010
2015
SOURCE: BEA and authors’ calculations.
In the value-added model, the change in the demand for housing also generates very small
booms and busts in output and employment. Here the propagation from housing to the rest
of the economy travels only on the demand side (via consumption complementarity), and
the effect is consequently small. For the value-added case, the adjustment via prices is more
severe because the only way to prevent individuals from purchasing more housing is to increase
house prices. In this case, the effects of a positive demand shock are a sizeable appreciation of
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Table 3
Quantitative Implications of Alternative Models
Share of changes accounted for by the construction sector (percent)
Expansion 2000-07
Experiment
Recession 2007-10
Employment
GDP
Employment
GDP
Baseline (ρ = 5)
60.2
25.3
43.9
56.2
Lower complementarity (ρ = 3)
28.7
14.1
28.7
40.6
Value-added specific (ρ = 5)
14.9
3.2
14.5
8.5
No interlinkages specific (ρ = 5)
14.5
2.5
10.6
4.8
SOURCE: Authors’ calculations.
house values but small macroeconomic spillovers on the production side, as the construction
sector is not directly interconnected with the rest of the economy. The adjustment in relative
prices leads wages to remain relatively constant.
As shown in Figure 13, the responses of consumption, house spending, and investment
are very different in each specification. In the model with fixed interlinkages, the dynamics
are similar to those in the baseline model presented in Figure 9. However, the magnitudes are
significantly smaller. The dynamics of the value-added specification are very different and
resemble the case of high elasticity of substitution.
The study of these three alternative specifications illustrates an important point. The
presence of interlinkages is necessary to generate large aggregate changes from fluctuations
in construction. In fact, both alternative models generate very small changes in output and
employment (for given shifts in the demand for housing) even though both maintain the complementarity between consumption and housing. Complementarity between housing and
consumption, alone, delivers only very small aggregate fluctuations, which instead appear when
the input-output structure of the economy is accounted for. Interlinkages are also crucial for
the behavior of investment. In response to demand shocks, the model with interlinkages generates a simultaneous increase in consumption (housing and nonhousing) as well as investment
(residential and nonresidential), whereas the value-added specification fails to account for
such strong comovements. No asymmetric input-output structure, no business cycle action.
5.5 Quantitative Implications of Alternative Models
The different specifications studied lead to a general conclusion: The aggregate importance
of the construction sector is significant despite its relatively small share in terms of employment
and value added. Table 3 presents a summary of all the results discussed above. The table shows,
for each of the model specifications considered, the fractions of the changes in employment
and GDP accounted for by shocks to the construction sector during the expansion (2000-07)
and the recession (2007-10). In light of the previous discussion, the numerical values should
be easy to interpret at this point. The left side of Table 3 considers the role of the construction
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sector in the expansion. Regardless of the complementarity between housing and consumption goods, the model with interlinkages reveals that the construction sector accounts for a
very significant share of the growth in total employment: between 29 and 60 percent. It also
reveals that the contribution of construction to GDP is also much larger than its share: between
14 and 25 percent, which is somewhat smaller than its contribution to employment.
According to our model, the contribution of construction to employment and output
was arguably even more important during the Great Recession. Depending on the specification, the decline in employment generated in the models with interlinkages is between 28 and
44 percent of the actual decline during the recession. In the case of GDP, the model generates
between 41 and 56 percent of the observed changes. The model suggests that construction has
been an important macroeconomic driver during the housing boom and bust and also highlights the asymmetry of its contribution between the expansion and the recession. During
expansions the spillover on employment is larger than on output, but during recessions it is
the opposite.
6 INTERLINKAGES AND BUSINESS CYCLE ACCOUNTING
An alternative methodology to identify the sources of economic fluctuations, within the
context of a one-sector growth model, is “business cycle accounting,” based on Chari, Kehoe,
and McGrattan (2007). Recent works, including Arellano, Bai, and Kehoe (2019) and Ohanian
and Raffo (2012), document that the Great Recession can be accounted for, mostly, by a worsening of labor market distortions. Both of these studies find that the labor wedge worsened by
about 12 percent during the 2007-09 recession. Different explanations have been proposed
to rationalize the measured increase in distortions in the labor market. For instance, Arellano,
Bai, and Kehoe (2019) propose a model of imperfect financial markets and firm-level volatility.
Such a model captures about half of the worsening in the labor wedge.
The wedge can be computed using data on employment, consumption, and wages generated by any model. It is defined as
Xt = −
U Nt
U Ct
/ wt ,
where UNt is the marginal disutility measured at the aggregate level of employment, UCt is the
marginal utility of consumption measured at the aggregate level of consumption, and wt is
the aggregate wage rate. Assuming wages are flexible and considering an aggregate CobbDouglas production function with capital share α, the wage can be replaced with
wt =
Yt
(1− α ).
N
Furthermore, using a log utility function for consumption and the following function for the
disutility of employment,
U (N ) = B
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N 1+υ
,
1+ υ
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Figure 14
Business Cycle Accounting of the Model with Interlinkages
Labor wedge
Percent
2
Implied total factor productivity
Labor wedge (ν = 1)
Labor wedge (ν = 0.5)
Labor wedge (ν = 2)
0
−6
−0.5
Δ07–10 = –5.6
−1.0
Δ07–10 = –7.4
−1.5
−8
−10
0.5
0
−2
−4
Percent
1.0
−2.0
Δ07–10 = –10.8
−2.5
−12
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
−3.0
2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
SOURCE: BEA and authors’ calculations.
the wedge can be written as
Γt = −
B Ct 1+υ
N .
(1− α ) Yt t
Notice that the parameters B and α are not important to understand fluctuations in the
labor wedge; only the time series for aggregate consumption, output, and employment, and a
value for υ, are required. We consider three values of υ = {0.5, 1, 2} and compute the labor
wedge implied by our model using simulated data for consumption, output, and employment.
Since our model has multiple sectors, several adjustments in the data are necessary. Consump
tion of goods and housing services are aggregated using relative prices Ct = ct + Rtht. Aggregate
output is Yt = Ct + Xtk, and total employment is Nt = nty + nts.
In the context of our model, any action in terms of implied distortions must be derived
from the input-output structure and changes in relative prices. Figure 14 displays the changes
in the labor wedge for our benchmark simulation. Its behavior is consistent with the data:
The labor wedge worsens during the recession and does not recover quickly. For the case
computed with υ = 1, which is consistent with the value used by Ohanian and Raffo (2012),
the labor wedge worsens by 7.4 percent; this is about 62 percent of the total change in the labor
wedge during this period. Notice that our computation of the labor wedge assumes that wages
are perfectly flexible. If this condition does not hold, the labor wedge has another component,
referred to as the “firm-side” labor wedge in Arellano, Bai, and Kehoe (2019).16 This wedge is
basically the difference between the marginal product of labor and the wage. These authors
refer to the other component of the labor wedge as the “consumer-side” labor wage, which is
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basically our Γt. Arellano, Bai, and Kehoe (2019) find that (i) the firm-side labor wedge has
been fairly flat since 2006 and (ii) a worsening of the consumer-side labor wedge accounts
for most of the Great Recession. Recall that there are no frictions in our model, so wages equal
the marginal product of labor in every period. Thus, not only the behavior of the labor wedge
during the Great Recession but also its decomposition in the data are consistent with what
our model predicts. It can also be shown that our model would be consistent with a large and
fairly persistent negative shock to total factor productivity. The combination of these two
would rationalize the model-predicted Great Recession.
7 CONCLUSIONS
This article analyzes the contribution of the construction sector to U.S. economic growth,
particularly during the Great Recession, using a two-sector input-output model. Historically,
the construction sector has been relatively small in terms of its contributions to employment
and GDP, but it is highly interconnected with other sectors in the economy and highly volatile.
Our empirical analysis reveals how these sectoral interlinkages propagate changes in housing
demand, greatly amplifying their effect on the overall economy. In our model, construction
accounts for 52 percent of the decline in employment and 35 percent of the decline in output
during the Great Recession and for similar, albeit slightly smaller, shares during the preceding
boom.
The importance of the sectoral interlinkages is illustrated first using a simple static multisector model. We prove that, in our model, changes in housing demand have a much larger
effect on aggregate activity when the sectors are asymmetrically interconnected. Also, the
presence of irreversibility constraints on investment introduces an asymmetry between the
expansion and the recession in the dynamic model. The simulation exercise is calibrated to
reproduce the boom-bust dynamics of construction employment in the period 2002-10. In
the model, during the housing boom all sectors expand and contribute 2 percent and 2.5 percent to the growth of output and employment, respectively. During the housing bust, the
irreversibility constraint binds, amplifying the asymmetric response: The declines in output
and employment are 3.3 percent and 3.8 percent, respectively. With a lower degree of complementarity, the asymmetric effect is not as large but still significant. These numbers can be
used to calculate the contribution of construction in the data. The model suggests that during
the expansion (2002-07), the construction sector accounted for a significant share of the growth
in employment (between 29 and 60 percent) and GDP (between 14 and 25 percent). The construction sector’s contribution was more important during the Great Recession (2007-10):
Our calibrated model suggests that movements in housing demand—propagating through
the economy—accounted for 29 to 44 percent of the variation in aggregate employment and
41 to 56 percent of the variation in GDP.
The presence of intersectoral linkages substantially amplifies the impact of changes in
housing demand. In the model specifications without this mechanism, changes in housing
demand consistent with the dynamics in construction employment have only a small effect
on macroeconomic quantities. This is true even when the complementarity between consump302
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Boldrin, Garriga, Peralta-Alva, Sánchez
tion goods and housing services is high. A direct implication of this result is that the presence
of interlinkages is necessary to generate large aggregate variations from changes in construction, and a high degree of complementarity is not sufficient to obtain the propagation of adjustments in housing demand to the rest of the economy we obtained in our model. To capture
the intricacies of this mechanism it is necessary to formalize the aggregate economy with a
multisector model with asymmetric interlinkages.
Since in our model the equilibrium is efficient, the behavior of output is also the behavior
of potential output. Taking into account that both output and potential output were affected
during the Great Recession, we perform a business cycle accounting exercise on simulated data
from the model using the now common “wedge” approach. Despite the lack of any friction
or distortion, the data our model generates attribute the recession to a worsening in the labor
wedge. The magnitude generated by the model accounts for 62 percent of the total change
observed in the data. Clearly, in our model, the metrics of the wedge measures are explained
by the fact that we account for sectoral linkages, irreversibilities, and the movements of relative prices between sectors; they are not explained by frictions. This approach shows how
multi-sector models of the business cycle can improve, or at least challenge, our understanding of the factors driving aggregate fluctuations.
A direct policy implication of our findings is that the output gap could be lower than historical estimates suggest. The historical anomalies in the events that took place between 2007
and 2013 can be accounted for by the equally anomalous evolution of housing demand in the
six years before 2007 and in those following it. As far as policy is concerned, the basic implication of our research is simple: Estimations of output gaps using pre-2007 trends, and aggregate one-sector models, may lead to misleading policy prescriptions. n
APPENDIX
A1 MICROFOUNDATIONS FOR THE HOUSING DEMAND SHIFTERS
The modeling strategy used in the article uses changes in “effective” housing demand as
the driver of the amplification mechanism that affects sectoral interlinkages. There are many
potential drivers of the changes in housing demand during the period we study. This particular
episode witnessed sizeable changes in home ownership and significant innovations in housing
finance at the household level (i.e., new mortgage products) and the industry level (i.e., the
use of mortgage-backed securities as a liquid asset). This section provides a microfoundation
of housing demand shocks using two different specifications. The first one uses credit constraints in housing finance, where changes in collateral requirements (i.e., loan-to-value ratios)
are isomorphic to variations in the relative weight that housing has on preferences (intensive
margin). The second one considers the case where a large number of new households enter
into the owner-occupied housing market. At the aggregate level this is also captured as an
increase in the aggregate demand for housing (extensive margin). Either specification would
be consistent with the approach used by Chari, Kehoe, and McGrattan (2007) that reduces all
the frictions in the model to distortions/wedges in the equilibrium conditions.
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A1.1 “Effective” Housing Demand and Credit Constraints
The first specification relates housing demand to the presence of credit constraints. Con
sider a simple two-period version of the household optimization problem to allow for borrowing and collateral constraints of the form b ≤ λph. This is the standard constraint that restricts
the amount of housing finance to be proportional to the value of the house, λ [0,1]. The cost
of borrowing, R > 1, is paid in the second period and can differ from the return of other
assets,17 r. For ease of exposition, consider the case where housing fully depreciates at the
end of the second period and labor is inelastically supplied. The optimization problem of the
representative consumer is
max u ( c1 ) + βu ( c2 ) + θ%v ( h ) ,
c1,c2 ,h,s ,b
s.t.
c1 + s + ph = w1 + b,
b ≤ λ ph,
c2 = w 2 + (1+ r )s + bR.
The optimality condition for housing measured in terms of t = 1 consumption goods can
be written as18
θ
v′(h)
= p ⎡⎣1− λ (1+ φ ) ⎤⎦ .
u′ ( c1 )
When the solution is interior, increases in the value of λ reduce the cost of housing relative
to the cost of consumption goods. This is observationally equivalent to an exogenous increase
in θ. Similarly, a tightening of credit conditions reduces housing demand. From this perspective, the relevant value is θ% = θ / ⎣⎡1− λ (1+ φ ) ⎦⎤ . When housing finance is not present, λ = 0,
the expression for housing demand is the same as in the previous model and θ% = θ .
A1.2 “Effective” Housing Demand and Home Ownership
Part of the housing boom was fueled by an increase in the home ownership rate.19 The
second specification relates the aggregate change in housing demand to increasing participation in the housing market, using a model based on Garriga, Manuelli, and Peralta-Alva (2019).
Consider an economy where households are ex-ante heterogeneous in their labor ability
ε [ ε , ε ] and the ability distribution is uniform ε ~U ( ε ,ε ) ≡ f ( ε ). Preferences are represented by a utility function u(c,h) = c(θ + h), where consumption goods are perfectly divisible,
–
c R+, and housing is a discrete good with only one size of home available, h {0,h }. The
implicit assumption is that renters consume zero housing and homeowners consume a fixed
positive amount; we could allow for the purchase of different size homes at the cost of introducing unnecessary notation. The parameter θ > 0 can be interpreted as a reservation value
for rental housing, and as θ → 0 owner-occupied housing becomes more desirable.
The optimization problem for the consumer is
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{ ( ) ( )
v ( ε ) = max ur c r ,0 ,uo c o ,h ,
h
s.t.
(
)
c o = wε − ph + φ ,
r
c = wε ,
where w is the wage, p is the house price, and the price of consumption goods is 1. The term ϕ
represents an exogenous transaction cost associated with buying a house, measured in terms
of consumption goods. The optimal decision rule determines a cut-off level of ability necessary
to purchase a house. For the specified preferences and under the necessary assumptions for
an interior solution, the threshold for homeownership, ε*, is characterized by
(
)
ε * θ ,h ,φ , p,w ≥
p
φ
θ +h +
.
wh
w
(
)
In the model, the determinants of ownership are the cost of housing relative to income,
–
p/w; the house size, h ; transaction costs, ϕ; and the reservation value of rental housing,20 θ.
The comparative statics are straightforward. Increases in the house price, minimum house
size, and transaction costs increase the income threshold required for home ownership, whereas
an increase in wage income decreases it. Notice that the demand shifter changes the number
of individuals buying houses, as the size of the latter is fixed.
Given this threshold, aggregate housing demand and the home ownership rate are in fact
proportional:
ε
H ( p ) = h ∫U ( ε ,ε )dε =
ε*
h ⎡
φ ⎤
p
.
ε − θ +h −
⎢
wh ⎥⎦
(ε − ε ) ⎣ w
(
)
Despite its simplicity, the expression shows the connection between housing demand and
the key individual variables. A reduction in the rental threshold, θ, affects the total quantity
demanded from the construction sector but also reduces the transaction costs, ϕ, affecting
housing demand.
A2 ALTERNATIVE SPECIFICATIONS QUANTITATIVE MODEL:
FIXED INTERLINKAGES AND VALUE-ADDED ECONOMIES
In our quantitative analysis, we made an effort to disentangle the role of interlinkages. It
is always challenging to compare different models, but the quantitative analysis suggests similar results from the various alternatives. The first alternative considers an economy calibrated
to the same initial steady state (parameters and targets) and compares the economy with interlinked production with an economy where the amount of intermediates is fixed at the initial
steady-state level. The second alternative compares the value-added economy with the linkage
economy. Both are calibrated to the same target values for the baseline year, but the underlying parameters are different.
∞
For a given sequence of land lt
, there is an optimization problem that solves for the
t=0
equilibrium in each case. In the baseline case with interlinkages, the social planner chooses a
∞
sequence of quantities {ct ,xtk ,xts ,nty ,nts ,mts ,s ,mty ,s ,mty ,y ,mts ,y }t=0 to maximize
{}
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(
)
∞
max ∑t=0 β t ⎡⎣u ( ct ,θt ,ht ) + γ v 1− nty − nts ⎤⎦ ,
s.t.
(
= A G (n ,m (m
)
ct + xtk + mty ,y + mty ,s = Aty F kt ,nty ,mty ,y ,mts ,y ,
xts + mts ,s + mts ,y
s
t
s
t
s
t
y ,s
s ,s
t ,mt
xtk = kt+1 − (1− δ k ) kt ≥ 0,
( )
∀t ,
∀t ,
xts = st+1 − (1− δ s ) st ≥ 0,
ht = H st ,lt ,
)) ,
∀t ,
∀t ,
∀t ,
s0 ,k0 ≥ 0.
In the model with no interlinkages, the production of intermediate goods is fixed at the
steady-state level before the boom. In this case, the social planner is forced to produce the
same quantity of intermediates each period mts ,s = m0s ,s ,mty ,s = m0y ,s ,mty ,y = m0y ,y ,mts ,y = m0s ,y .
(
To satisfy this constraint, the social planner picks a vector {
}
∞
ct ,xtk ,xts ,nty ,nts t=0
(
) − (m
to maximize
)
)
∞
max ∑t=0 β t ⎡⎣u ( ct ,θt ht ) + γ v 1− nty − nts ⎤⎦ ,
s.t.
(
ct + xtk = Aty F kt ,nty ,m0y ,y ,m0s ,y
(
)) (
(
y ,y
0
)
+ m0y ,s ,
)
xts = AtsG nts ,m0s m0s ,s ,m0y ,s − m0s ,s + m0s ,y ,
xtk = kt+1 − (1− δ k ) kt ≥ 0,
∀t,
xts
∀t,
= st+1 − (1− δ s ) st ≥ 0,
( )
ht = H st ,lt ,
∀t,
∀t,
∀t,
s0 ,k0 ,m0s ,s ,m0y ,s ,m0y ,y ,m0s ,y ≥ 0.
The last case studied is that of a value-added economy, where intermediate goods are∞
y s
k s
completely eliminated. The social planner chooses a vector of quantities {ct ,xt ,xt ,nt ,nt }t=0
to maximize
(
)
∞
max ∑t=0 β t ⎡⎣u ( ct ,θt ht ) + γ v 1− nty − nts ⎤⎦ ,
s.t.
(
= A G (n ) ,
)
ct + xtk = Aty F kt ,nty ,
xts
s
t
s
t
∀t,
xtk = kt+1 − (1− δ k ) kt ≥ 0,
xts = st+1 − (1− δ s ) st ≥ 0,
( )
ht = H st , lt ,
∀t,
∀t,
∀t,
∀t,
s0 ,k0 ≥ 0.
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A3 CALIBRATION OF INTERLINKAGES
Interlinkages are calibrated using input-output data. In particular, the information shown
in Table A1 is used to calibrate the parameters of the two production functions and is calculated from the BEA’s 2010 use input-output table. The use table shows the uses of commodities by intermediate and final users; rows present the commodities or products, and columns
display the industries and the final users. The sum of the entries in a row is the output of that
commodity. The columns show the products consumed by each industry and the three components of value added—compensation of employees, taxes on production and imports less
subsidies, and the gross operating surplus. Value added is the difference between an industry’s
output and the cost of its intermediate inputs; total value added is equal to GDP.
Table A1 displays input-output values (which are originally in millions of dollars) as a
fraction of each industry’s output. Construction receives most of its inputs from other industries (48.3 percent of its gross output) and less than 1 percent from itself. The reverse is true
for the other industries, as they receive most of their inputs from themselves (43.0 percent of
their total gross output).
Table A1
Coefficients with Respect to Column Industries
Commodities/industries
Construction
Other industries
Construction
0.0009
0.0058
Other industries
0.4828
0.4301
Compensation of employees
0.3625
0.2802
Taxes on production and imports, less subsidies
0.0072
0.0471
Gross operating surplus
0.1466
0.2368
Total
1.0000
1.0000
SOURCE: BEA 2010 use input-output table.
NOTES
1
Davis and Heathcote (2005) construct a real business cycle with housing and interlinkages. In the baseline economy with Cobb-Douglas preferences, the presence of interlinkages generates a relatively small contribution to
aggregate fluctuations in response to productivity shocks. Our theoretical model shows that Cobb-Douglas preferences completely eliminate the role of interlinkages because of insufficient complementarity. This is true even
when the sectoral linkages are asymmetric. Iacoviello (2005) generates house price fluctuations using shocks to
Cobb-Douglas preferences, but the productive structure of the economy does not have interlinkages, so most of
the action is driven by the presence of binding collateral constraints and price rigidities.
2
The appendix provides two microfoundations for the drivers of housing demand. We show that this shock is isomorphic to a model that relaxes collateral constraints or a model with housing demand at the extensive margin.
3
The analysis abstracts from both the increase in the burden of debt brought about by the decline in home prices
(which is the focus of Garriga, Manuelli, and Peralta-Alva, 2019) and the reduction in credit activity it implied, two
factors that are likely to have played a major role in the overall process. Although these factors could interact with
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the sectoral interlinkages, abstracting from them captures the contribution of the real side of the economy in the
recession.
4
It is often argued that the housing sector is of great relevance to the aggregate economy because housing wealth
is a major determinant of consumption demand (see Carroll, Otsuka, and Slacalek, 2011; Case, Quigley, and Shiller,
2005; and Mishkin, 2007, which are among the most cited articles). An ample and somewhat more recent literature
(e.g., Calomiris, Longhofer, and Miles, 2009, and Iacoviello, 2011, and the references therein) has cast serious doubts
on the quantitative relevance of this channel for business cycles analysis. While the housing sector is certainly very
cyclical, this is most likely not due to a causal chain going from housing wealth to consumption and aggregate
demand to output, but to a host of other common factors driving such comovements. Further, the same literature
also reveals that, when empirical evidence of a causal link is found, the latter is not only quantitatively weak but its
magnitude is also dependent on demographic and financial variables.
5
The traditional view of the business cycle literature is that idiosyncratic sectoral shocks are likely to average out
and have no aggregate effects as the number of sectors in the economy gets larger (i.e., Lucas, 1981; Kydland and
Prescott, 1982; Long and Plosser, 1983; and Dupor, 1999).
6
In the analysis hereafter, the definition of the construction sector does not include the real estate and leasing sector,
because the two sectors are quite different. Including it in the definition of the construction sector would substantially increase the construction sector’s significance in accounting for the Great Recession.
7
For employment, we use the employment requirements matrix from the Bureau of Labor Statistics (BLS).
8
For expositional purposes, in this section we assume that the diagonal coefficients of the requirement matrix A in
the Leontief model are zero. The general formulation would be c + m yy + m yh = Y = Ay f n y ,ε yym yy ,ε yhmhy and
h + mhy + mhh = H = Ah g nh ,ε hym yh ,ε hhmhh , where in each mij, the first subscript denotes the origin (i) and the second
denotes the destination (j).
(
9
(
)
)
As εj → 0, the required quantity of the intermediate good converges to zero, mj → 0. When both coefficients converge to zero, the technologies become c = Ayny and h = Ahnh , respectively. In this case, the interlinkages disappear.
10 This condition is the well-known “all or nothing” property of Leontief input-output models. When it is met, the
economy is productive and any nonnegative value added is reachable if enough labor input is available.
11 This condition is related to the irrelevance result in Dupor (1999).
12 This specification implies a Frisch elasticity of labor equal to 2. Keane and Rogerson (2012) argue that this elasticity
can be reconciled with lower elasticity estimates at the micro level.
13 The long-run value of θ has small quantitative implications for the short-run dynamics discussed in this article.
14 The magnitudes of these numbers vary with the time interval considered, but the overall magnitudes are within
reasonable bounds.
15 See the appendix for model details.
16 They follow Galí, Gertler, and López-Salido (2007) in this decomposition.
17 In this class of model, the consumer usually has an incentive to borrow to purchase the house when the cost of
borrowing is lower than the return of other assets, R/1 + r = ϕ < 1 (additional conditions are discussed later in the
article). In many countries, interest payments are tax deductible, reducing the effective cost of borrowing relative
to other assets. Under this assumption, the Lagrange multiplier of the collateral constraint binds, and housing
demand directly determines the amount of borrowing.
18 For an interior solution with borrowing it suffices that λ < 1/(1 + ϕ).
19 See Chambers, Garriga, and Schlagenhauf (2009a,b) for a detailed discussion on the home ownership rate boom
between 1994 and 2007.
20 When the transaction cost is proportional to the value of the house, the budget constraint of the buyer is slightly
different, c o = wε − ( p + φ )h , and the homeownership threshold is ε * ≥
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w
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).
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The Case of the Reappearing Phillips Curve:
A Discussion of Recent Findings
Asha Bharadwaj and Maximiliano Dvorkin
The Phillips curve seems to have flattened over time. In this article, we use a simple New Keynesian
model to analyze potential pitfalls in the estimation of the slope of the structural Phillips curve.
Changes in the conduct of monetary policy or in the relative importance of supply and demand
shocks may bias simple estimations of the slope of the Phillips curve. Recent proposals have favored
estimations using regional or city data in an effort to overcome these issues. We use a simple model
of a monetary union with a continuum of economies and find that some of the drawbacks of the
aggregate model are still present in a cross-section of many regions in a monetary union. The relative
importance of the demand and supply shocks largely determines the empirical relation between
unemployment and inflation in both the aggregate and the cross-section of regions. Our analysis
shows potential pitfalls in estimating the slope of the Phillips curve, even if using regional data.
(JEL E12, E31, E58, R13)
Federal Reserve Bank of St. Louis Review, Third Quarter 2020, 102(3), pp. 313-37.
https://doi.org/10.20955/r.102.313-37
1 INTRODUCTION
Central banks around the world intervene in financial markets by setting the short-term
nominal interest rates to stimulate economic activity and control inflation. There are many
theories that suggest that changes in interest rates due to monetary policy have effects on real
activity: An increase in interest rates is associated with a tightening of the economy and a
decrease in the real output, while a decrease in interest rates is associated with an increase in
real output. Interest rates can affect output through several channels. For instance, an increase
in interest rates implies an increase in the return on savings. In this case, individuals have an
incentive to save more and forego current consumption, which translates to a decrease in
current real output, all else equal. Clearly, the opposite is true in the case of a decrease in
interest rates. Other possible mechanisms through which movements in interest rates affect
real activity include borrowing costs and investment. As interest rates decline it becomes
cheaper to borrow and invest, so business investment goes up, increasing total output.
Asha Bharadwaj is a research associate and Maximiliano Dvorkin is a senior economist at the Federal Reserve Bank of St. Louis.
© 2020, Federal Reserve Bank of St. Louis. The views expressed in this article are those of the author(s) and do not necessarily reflect the views of
the Federal Reserve System, the Board of Governors, or the regional Federal Reserve Banks. Articles may be reprinted, reproduced, published,
distributed, displayed, and transmitted in their entirety if copyright notice, author name(s), and full citation are included. Abstracts, synopses,
and other derivative works may be made only with prior written permission of the Federal Reserve Bank of St. Louis.
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While these theories shed light on the effects of monetary policy on the real economy,
there is no consensus on the link between interest rates and inflation. A widespread view
among policymakers is that there is a trade-off between real activity and inflation and that
monetary policy decisions on interest rates, by affecting the real economy, ultimately affect
inflation. This trade-off is known as the Phillips curve.
The Phillips curve was popularized by A.W. Phillips in 1958, when he showed a statistically significant negative relation between the unemployment rate and the growth rate of
nominal wages—that is, wage inflation. Based on this empirical relationship, Samuelson and
Solow (1960) argued that a looser monetary policy could reduce the unemployment rate by
allowing inflation to rise. This then implied that monetary authorities could exploit this tradeoff. Since it was first discovered empirically, the Phillips curve has guided discussions of monetary policy and has shaped our understanding of the transmission of monetary policy to prices.
More recently, several theories on price setting by firms can rationalize the existence of a
Phillips curve in an economic model.
Over the years, the Phillips curve has received several criticisms. Recent articles have
argued that inflation can be approximated by statistical processes unrelated to the amount of
slack in the economy (Atkeson and Ohanian, 2001; Cecchetti et al., 2017; and Stock and Watson,
2007). Moreover, the lack of a stable relationship between inflation and various measures of
slack has led several articles to conclude that the Phillips curve has weakened over the years
(Blanchard, Cerutti, and Summers, 2015; and Coibion and Gorodnichenko, 2015).
Several articles have pushed back on this criticism and have attempted to “recover” the
Phillips curve. Fitzgerald and Nicolini (2014) argue that aggregate data are uninformative
about the true structural relationship between unemployment and inflation, and that in fact,
under a specific definition of inflation targeting, the evolution of equilibrium inflation is a
random walk. They then show that regional data can be used to identify the structural relationship between unemployment and inflation. The main intuition is that monetary policy typically reacts to the aggregate state of the economy, but not to regional conditions. Thus, it is
possible to use the deviations of regional economic activity relative to the aggregate and the
deviation of inflation relative to the aggregate to recover the relationship between unemployment and prices.
A recent article by McLeay and Tenreyro (2019) also supports this view. The authors argue
that it is difficult to identify the slope of the Phillips curve empirically, even if a negative relationship does hold true in the underlying model. This is because monetary policy will react to
economic shocks in order to stimulate output when it is below potential and reduce inflation
when it is above target. The actions of the monetary authority will typically affect the empirical
slope of the Phillips curve. McLeay and Tenreyro (2019) then use a simple New Keynesian
model to highlight this estimation bias due to the endogeneity of monetary policy. They propose several solutions including using regional Phillips curves to circumvent this identification
problem.
In our article, we follow the approach suggested by McLeay and Tenreyro (2019) closely
and use a simple New Keynesian model to highlight the issues that arise with the identification
of the empirical slope of the aggregate Phillips curve. We then use a simple New Keynesian
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model of a monetary union, as given by Gali and Monacelli (2008), to attempt to recover the
Phillips curve at the regional level. However, we find that this approach is not sufficient to
overcome the identification issues highlighted by McLeay and Tenreryo (2019), since even
at the regional level our model fails to recover the slope of the Phillips curve. We argue that
several factors, including the relative importance of the demand and cost-push shocks, affect
the estimation of the slope of the Phillips curve, at both the aggregate and regional levels.
The structure of this article is as follows. Section 2 briefly reviews existing literature on
the Phillips curve. Section 3 presents empirical evidence on the aggregate relationship between
unemployment and inflation. Section 4 introduces a simple New Keynesian model of optimal
policy with the Phillips curve and describes the empirical relationships we obtain when the
model is used as a data-generating process. Section 5 discusses empirical challenges with
regional Phillips curves. Section 6 presents a simple model of a monetary union with a continuum of regions and discusses the results we obtain when we use the model to simulate
data. Section 7 concludes.
2 LITERATURE REVIEW AND BACKGROUND ON THE PHILLIPS CURVE
Phillips (1958) showed, for the United Kingdom and for the years 1861-1913, a statistically
significant negative relationship between the unemployment rate and the growth rate of nominal wages held in the data. This result led to an outpouring of work on this topic, including
the work by Samuelson and Solow (1960), who argued that policy could exploit the trade-off
between inflation and the unemployment rate: Looser monetary policy would lead to an increase
in inflation and a decrease in the unemployment rate. Figure 1 portrays the relationship
between inflation and unemployment for the United States and United Kingdom, between
1900 and 1940, and we observe a clear negative trend between the level of unemployment
and the growth rate of prices.
However, in the 1960s and 1970s, the relationship between unemployment and inflation
started to change and was no longer a robust negative relationship. Figure 2 represents the
relationship between unemployment and inflation in the 1960s for three countries—the United
Kingdom, the United States, and France—and we observe that the negative relationship we
would expect is no longer consistent across countries. It is negative for the United States, while
it is positive for France and the United Kingdom. Lucas (1972) argued that the statistical
relationship between unemployment and inflation depends on the parameters governing
monetary policy and, therefore, is not “structural.” Lucas provided a theoretical model consistent with the existence of the observed statistical relationship between inflation and unemployment, but in which systematic attempts by policy to exploit the trade-off are not successful
and change the statistical relation of these two variables.
Gali and Gertler (1999) developed a structural model of the Phillips curve with microfoundations, based on initial work by Calvo (1983). They argued that real marginal costs and
inflation expectations are significant determinants of inflation, as opposed to the backward-
looking behavior that had been considered quantitatively important in the existing literature.
Several articles then used this microfounded Phillips curve, also called the New Keynesian
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Figure 1
The Phillips Curve for the United States and United Kingdom, 1900-40
United States
United Kingdom
20
30
25
15
20
15
Inflation
Inflation
10
5
0
10
5
0
–5
–5
–10
–10
–15
–15
0
5
10
15
20
–20
0
25
2
4
Unemployment rate
6
8
10
12
14
16
18
Unemployment rate
SOURCE: FRED®, Federal Reserve Bank of St. Louis; Annual Estimates of Unemployment in the United States, National Bureau of Economic
Research; and Monthly Labor Review, Census Bureau.
Figure 2
The Phillips Curve in the 1960s for Select Countries
7
6
5
1965 1966
1962
4 1961
3
1964
2
1
1960
0
2.0 2.2 2.4 2.6 2.8
United States
1969
1970
Inflation
Inflation
United Kingdom
1968
1967
1963
3.0
3.2
3.4
3.6
3.8
4.0
Unemployment rate
7
6
5
4
3
2
1
0
3.0
1970
1969
1968
1966
1967
1960
1965
1964 1963
3.5
4.0
4.5
5.0
5.5
1962
6.0
1961
6.5
7.0
Unemployment rate
Inflation
France
7
6
1962
5
1963
1960
4
1964
1965
3
1966
2
1961
1
0
1.0
1.2
1.4
1.6
1969 1970
1968
1967
1.8
2.0
2.2
2.4
2.6
Unemployment rate
SOURCE: FRED®, Federal Reserve Bank of St. Louis.
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Figure 3
Evolution of Inflation and Unemployment Rate in the United States, 1955-2019
Percent
16
14
12
10
8
6
4
2
0
–2
1960:Q1
1961:Q4
1963:Q3
1965:Q2
1967:Q1
1968:Q4
1970:Q3
1972:Q2
1974:Q1
1975:Q4
1977:Q3
1979:Q2
1981:Q1
1982:Q4
1984:Q3
1986:Q2
1988:Q1
1989:Q4
1991:Q3
1993:Q2
1995:Q1
1996:Q4
1998:Q3
2000:Q2
2002:Q1
2003:Q4
2005:Q3
2007:Q2
2009:Q1
2010:Q4
2012:Q3
2014:Q2
2016:Q1
2017:Q4
2019:Q3
Unemployment rate
CPI inflation
Quarter
SOURCE: FRED®, Federal Reserve Bank of St. Louis and authors’ calculations.
Phillips curve, to argue that successful monetary policy is responsible for flattening the slope
of the Phillips curve by anchoring inflation expectations (Williams, 2006; Bernanke, 2007;
and Mishkin, 2007). Several articles also support the argument that even in a purely static
setting without expectations, the structural relationship between unemployment and inflation can be masked by the conduct of monetary policy (Bullard, 2018; Roberts, 2006; and
Krogh, 2015).
3 EMPIRICAL PHILLIPS CURVE FOR THE UNITED STATES
We begin by looking at the evolution of U.S. inflation and the unemployment rate over
the past 60 years. As we can see from Figure 3, the times when inflation falls below trend coincide with the times with higher unemployment. This is suggestive of a negative correlation
between the series. In the 1960s, for instance, there seems to be a clear negative correlation,
as well as in the 1980s. We next use some scatterplots to understand the correlation between
these variables better.
Figure 4 shows the behavior of the Phillips curve in the United States. Each dot in the
graph represents a quarter. From the graph, it is clear that the slope of the Phillips curve has
not been stable over the past 60 years. In the 1960s, there was a sizeable negative correlation
between inflation and the unemployment gap, which has since flattened. We observe that
over the past two decades, the negative relationship has returned somewhat, but the slope is
not as steep as it was in the 1960s.
We formally estimate the empirical Phillips curve using aggregate data. The equation
that we estimate is
(
)
π t = α + β1 U t −U t* + β 2 Et π t+1 + Σ 3k=1β 2+kπ t−k + ε t ,
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Figure 4
The U.S. Phillips Curve
CPI inflation
15
1960−70
1981−90
2001−19:Q1
1971−80
1991−2000
10
5
0
−2
0
2
4
6
Unemployment gap
NOTE: The unemployment gap is defined as the difference between the unemployment rate and the CBO estimate of
natural rate of unemployment.
SOURCE: BLS, CBO, and authors’ calculations.
Table 1
Phillips Curve Using Aggregate U.S. Data (1990:Q1-2018:Q1)
Inflation
Unemployment gap
Bivariate
–0.222***
(–3.96)
With lags
–0.0171
(–0.41)
New Keynesian
–0.159***
(–3.95)
1.006***
(12.37)
Inflation expectations
Hybrid
–0.112*
(–2.46)
0.634***
(3.91)
Inflation lags
First lag
0.424***
(4.32)
0.186
(1.65)
Second lag
0.461***
(4.28)
0.137
(1.36)
Third lag
0.0884
(0.86)
Constant
Observations
Adjusted R 2
2.587***
(26.32)
–0.000435
(–0.00)
–0.339
(–1.51)
–0.111
(–0.47)
118
118
107
107
0.085
0.957
0.517
0.534
NOTE: The t-statistics are in parentheses. *p < 0.05, **p < 0.01, ***p < 0.001.
SOURCE: BLS, CBO, the Survey of Professional Forecasters, and authors’ calculations.
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where πt is the inflation rate and (Ut – Ut* ) is the unemployment gap. We compute inflation
as the annualized change in quarterly core consumer price index (CPI) inflation (obtained
from the Bureau of Labor Statistics [BLS]); the unemployment gap is defined as the difference
between the unemployment rate (from the BLS) and the natural rate of unemployment (from
the Congressional Budget Office [CBO]). Inflation expectations are obtained from the Survey
of Professional Forecasters.1 We estimated four specifications of this Phillips curve: a bivariate
model, in which we simply regress inflation on the unemployment gap (β2 = β3 = β4 = β5 = 0);
a model with lags, in which we regress inflation on the unemployment gap and three lags of
inflation (β2 = 0); a model with inflation expectations, in which we regress inflation on the
unemployment gap and expectations (β3 = β4 = β5 = 0); and a hybrid model where both past
inflation and inflation expectations can influence the inflation rate.
Across all specifications, the regression coefficient for the unemployment gap is negative,
which agrees with our previous discussion on the Phillips curve. The estimates in Table 1 show
that the steepest slope is with the bivariate specification (–0.2), while the one with lags yields
the flattest slope (–0.02). Also worth noting is the fact that inflation expectations are significant and positive in the third regression, indicating that if inflation is expected to increase in
the future, current inflation responds by increasing as well.
4 THE BASIC NEW KEYNESIAN FRAMEWORK
To better understand the problems and limitations of a simple empirical estimation of
the slope of the Phillips curve, we now study the issue using a macroeconomic model: the New
Keynesian model. We first describe the canonical New Keynesian model and then discuss its
equilibrium conditions. We follow closely Clarida, Gali, and Gertler (1999), as well as Chapter 3
of Gali (2008).
Within the New Keynesian macro-model framework, monetary policy affects the real
economy in the short run. The basic model consists of identical households that make decisions about labor supply, consumption, and savings. Households are risk averse and dislike
fluctuations in consumption. Households can borrow or save in one-period bonds that pay
an interest rate. These bonds are in zero net supply. Firms produce a range of differentiated
goods using only labor with a constant returns-to-scale production function. Firms set prices
for their goods, taking into account that they face competition from close substitutes of their
goods and that they are unable to change prices every period and able to do so only after some
random period. In the basic model, we abstract from investment and capital and from government consumption and interactions with the rest of the world. The total production cannot
be stored, and total consumption has to equal total production. We also use a version of the
model that abstracts from money holdings, sometimes referred to as the cashless limit economy.
Since all households are identical, in equilibrium the interest rate must be such that households are indifferent between saving and consuming their income.2
The dynamic conditions that characterize equilibrium in this model can be approximated
by two equations: a dynamic IS curve that relates the output—in deviations from trend—to
the real interest rate, and a Phillips curve that relates inflation positively to the output. It is
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important to note this dynamic aggregate behavior evolves from optimization by firms and
households. Moreover, these equations approximate the dynamics around a stationary
equilibrium.
For any variable z, let ẑ denote the deviation of the variable from its steady state or longrun value. The New Keynesian Phillips curve is given by
(1)
π̂ t = β Et [π̂ t+1 ] + κ ŷt + ε t .
Here, πt refers to the aggregate inflation at time t, ŷ t refers to the output in deviations
from trend or potential at time t, and εt is a cost-push shock that follows an AR(1) process
(εt = ρεt–1 + et ).
This equation reflects the aggregate relationship between the inflation rate—in deviations
from a long-run value—and output, and it tells us that an increase in output is associated
positively with a positive change in inflation. This equation evolves from the individual firm’s
problem: Firms are monopolistically competitive, and each firm chooses its price to maximize
profits subject to constraints on the frequency of future price adjustments. The aggregate
Phillips curve is simply capturing the aggregation of individual pricing decisions by a log-linear
approximation. Another way of interpreting this equation is by thinking in terms of marginal
costs. Excess demand (output above the potential) is associated with marginal costs above
average, and the reverse is true in the case of excess supply. Thus, firms set nominal prices
based on their expectations of future marginal costs, as well as current marginal costs. The
cost-push shock is meant to capture other forces that may affect inflation not already captured
in the model.
The next ingredient of the model is the dynamic IS curve, which inversely relates the aggregate output (or consumption since they are the same in this model) to the real interest rate,
and tells us that aggregate output evolves as a function of the expected future output, the
interest rate, the expected inflation level, and demand shocks:
(
)
ŷt = Et [ ŷt+1 ] − it − Et [π̂ t+1 ] − zt .
Here, zt is a demand shock that follows an exogenous AR(1) process with persistence ρz (i.e.,
zt = ρz zt–1 + γt ). This equation is a result of log linearizing the consumption Euler equation
that arises from optimizing the household’s utility. The term it – Et [π̂t+1] is the (expected) real
interest rate since it is the nominal rate. The mechanism by which the real interest rate affects
output is somewhat intuitive: Higher interest rates imply higher borrowing costs or a higher
return to savings, which lead to lower consumption today and thus lower current output.
This is also known as the intertemporal substitution of consumption. The reason that expected
future output affects current output is that individuals prefer a smooth path of consumption.
If individuals expect higher income and consumption in the next period, they prefer to consume more today, thus resulting in an increase in output today. Overall, output demand
depends negatively on the real interest rate and positively on expected future output. One
possible interpretation for the demand shock zt is a change in households’ preferences for
consuming today versus tomorrow, like a sudden increase in impatience.
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This simple model does not have a direct link to unemployment. We add an exogenous
relationship between the output and the unemployment rate, or, more precisely, the unemployment rate gap, which is best known as Okun’s law:
ût = α ŷt .
Finally, as is usual in most New Keynesian models, we add a monetary authority with the
objective to control inflation and promote high levels of employment. The monetary authority
sets the nominal interest rate to achieve these objectives. In particular, we assume that the
short-term nominal interest rate is set according to a Taylor rule of the form
it = r * + π * + φπ π̂ t + φ y ŷt ,
where r* is the long-run, or natural, real interest rate, and π* is a long-run value of inflation,
both of which we assume are exogenous. We assume that the coefficients ϕπ and ϕy are positive.
This policy function implies that the monetary authority will increase the nominal interest
rate if inflation is above its long-run value—that is, when π̂t is positive—or when output is
above its long-run value—that is, when ŷ t is positive. A solution to this model includes decision rules for inflation, output, interest rate, and unemployment, as a function of the two exogenous shocks, demand shocks, and cost-push shocks. Assuming that the Taylor principle holds
and that the parameters of the model are within normal ranges, we can solve for the equilibrium inflation and output by plugging this optimal policy rule into the Phillips curve. We then
find that both the equilibrium inflation and output gap are functions of the demand shock and
cost-push shock:
ŷt = c yε ε t + c yz zt ,
ît = ciε ε t + ciz zt ,
π̂ t = cπε ε t + cπ z zt , and
ût = cuε ε t + cuz zt ,
where coefficients in the equations are constants that depend on model parameters. In the
analysis that follows, we focus on the last two equations that say that the equilibrium values
of inflation and unemployment at time t depend on the realization of shocks. In particular, it
can be shown that both of the coefficients in the inflation decision rule are positive; thus,
inflation increases with positive demand shocks or positive cost-push shocks. On the other
hand, unemployment decreases with positive demand shocks but increases with cost-push
shocks, as these last shocks will induce a contraction in economic activity.
We now use this model as a data-generating process to simulate shocks to the economy.
For this, we need to specify values for the parameters of the model. We use the same calibration as Gali (2008).3 We can use the New Keynesian Phillips curve in equation (1), together
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with the equation for Okun’s law, to derive a relationship between inflation and unemployment,
which we can estimate. In particular, we have
(2)
π̂ t = β Et [π̂ t+1 ] + δ ût + ε t ,
where the coefficient δ is a combination of parameters κ and α. A direct estimation of equation (2) is problematic. As the decision rules clearly show, a simple regression of inflation on
the unemployment gap would give us biased estimates because the unemployment gap is
correlated with the cost-push shock εt. In a similar way, the expected value of inflation in the
next period would also be correlated with today’s cost-push shock if the shocks were persistent.
McLeay and Tenreyro (2019) discuss several assumptions and estimation methods
under which estimates would be correct. First, in an economy without cost-push shocks and
in which expected inflation and unemployment are observed, a regression would recover the
slope δ. Alternatively, if there are good instruments for both the unemployment rate and the
expected inflation that are uncorrelated to the cost-push shock, then an instrumental variables regression would be valid. Finally, they propose a different way to tackle the problem,
which we discuss later.
We now use the model to understand possible shortcomings in the empirical estimation
of Phillips curves. The exercises that we describe next use the decision rules for inflation and
unemployment together with simulated values for the demand and cost-push shocks. These
give a simulated time series for the two variables in the Phillips curve. Figure 5 shows the results
of simulating various combinations of shocks. To simplify the analysis and make our point,
we assume that the shocks are not persistent. In other words, we assume that ρ = ρz = 0, which
would then imply that Et [πt+1] is not affected by the current realization of the shocks.4
4.1 Case I: Only Demand Shocks
We now start our analysis by focusing on the effects of different forces on the slope of the
Phillips curve, one by one. In this case, we allow for only demand shocks. As Figure 5 shows,
all the different equilibrium values of inflation and unemployment reflect a typical negative
relationship between inflation and unemployment.
This is what we expect to see given the decision rules obtained before and that only demand
shocks are allowed. When a demand shock hits, output increases, lowering unemployment
but also increasing inflation. The monetary authority counteracts positive demand shocks by
increasing the interest rate, which pushes both output and inflation to an otherwise lower level.
In this case, the monetary authority does not face a trade-off between output and inflation.
Note, however, that the slope of the empirical Phillips curve in Figure 5, in general, will not
be equal to the slope of the structural Phillips curve—that is, the coefficient δ in equation (2).
The reason is that the actions of the monetary authority affect the movements in inflation and
the unemployment rate, which in turn affect the empirical slope, as we highlight in Case III.
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Figure 5
Simulated Inflation and Unemployment with Only Demand Shocks
Aggregate inflation gap
0.1
0
–0.1
–0.4
–0.2
0
0.2
0.4
Aggregate unemployment gap
SOURCE: Authors’ calculations.
Figure 6
Simulated Inflation and Unemployment with Only Cost-Push Shocks
Aggregate inflation gap
0.5
0
–0.5
–0.4
–0.2
0
0.2
0.4
Aggregate unemployment gap
SOURCE: Authors’ calculations.
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Figure 7
Simulated Inflation and Unemployment with Different Policy Parameters
A. Less aggressive toward inflation
B. More aggressive toward inflation
Aggregate inflation gap
Aggregate inflation gap
0.10
0.10
0.05
0.05
0
0
–0.05
–0.05
–0.10
–0.10
–0.15
–0.4
–0.2
0
0.2
0.4
–0.15
–0.4
Aggregate unemployment gap
–0.2
0
0.2
0.4
Aggregate unemployment gap
SOURCE: Authors’ calculations.
4.2 Case II: Only Cost-Push Shocks
We next set all demand shocks equal to zero in our simulations and allow for only costpush shocks. Figure 6 shows the results of our simulations for this case. In the decision rules,
the coefficient that multiplies the cost-push shock is positive for both variables—unemployment and inflation. Thus, with a positive cost-push shock, both inflation and unemployment
increase and we observe a positive relationship between inflation and the unemployment gap
in the graph. In the background, it is the optimal decisions of agents and the monetary authority that drive this behavior. In particular, the monetary authority is aggressive at fighting
inflation resulting from the cost-push shock, increasing the interest rate, thus lowering total
consumption and production and increasing unemployment.
4.3 Case III: Changes in the Policy Rule Over Time
In a recent discussion, Bullard (2018) studies how the empirical Phillips curve would look
in a simple New Keynesian model if the monetary authority reacts more aggressively toward
increases in inflation. Figure 7 shows this comparison. Panel A presents simulations using a
value of ϕπ = 1.5, and Panel B uses a value of ϕπ = 5.5, which implies a more aggressive stance
toward increases in inflation. As the figure shows, with a more aggressive policy, the empirical
slope of the Phillips curve is flatter. The main takeaway of this exercise is that if the monetary
authority becomes more aggressive toward inflation over time, then, all else equal, the empirical
Phillips curve will be flatter. However, this flattening of the empirical Phillips curve occurs even if
the structural Phillips curve of equation (1) did not change, as we are keeping δ constant.
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Figure 8
Simulated Inflation and Unemployment with Different Variance of Cost-Push Shocks
A. Higher variance of demand shocks
B. Lower variance of demand shocks
Aggregate inflation gap
Aggregate inflation gap
0.10
0.10
0.05
0.05
0
0
–0.05
–0.05
–0.10
–0.10
–0.15
–0.4
–0.2
0
0.2
0.4
–0.15
–0.4
Aggregate unemployment gap
–0.2
0
0.2
0.4
Aggregate unemployment gap
SOURCE: Authors’ calculations.
This exercise illustrates an important pitfall in empirical analysis using reduced-form
estimations of the Phillips curve: The actions of the monetary authority affect estimates of
the slope of the Phillips curve. Even if the slope of the structural Phillips curve remains constant (parameter δ), changes in monetary policy over time affect the empirical slope of the curve,
and a more aggressive stance will make the empirical curve flatter.
4.4 Case IV: Changes in the Variance of Shocks
As Cases I and II show, the slope of the empirical Phillips curve will be affected by the type
of shock that impacts the economy. We next simulate both shocks and compute the corresponding time series behavior for inflation and unemployment according to the decision rules.
Figure 8 presents the results for two different sets of parameters. In particular, Panel A shows
the simulations for an economy with a relatively higher variance of the cost-push shock, while
Panel B shows the same simulations but for a lower variance of these shocks. As is clear from
the figure, an economy with larger cost-push shocks will display a flatter (or even an upward-
sloping) empirical Phillips curve. Once again, it is worth stressing that the flattening of the
empirical Phillips curve would occur even if the structural Phillips curve of equation (2) does
not change and δ remains constant, which also shows a potential pitfall in empirical analysis
using reduced-form estimations of the Phillips curve. To the extent that empirical estimates
cannot control properly or instrument correctly for cost-push shocks, differences in the variances of the shocks will be reflected in the estimates of the slope.
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Case III highlights that the empirical Phillips curve may flatten over time if the monetary
authority adopts a more aggressive stance toward inflation. Case IV, on the other hand, highlights that the flattening could be due to a decrease in the importance (size) of demand shocks
relative to cost-push shocks.
5 REGIONAL PHILLIPS CURVES
As we mentioned before, McLeay and Tenreyro (2019) discuss several avenues to properly
recover the slope of the Phillips curve. The first method they suggest is to control for costpush shocks, to the extent that these are observable. In essence, this would allow us to replicate Case II from the previous section by removing the effect of cost-push shocks from both
inflation and the unemployment rate, such that any remaining variation in the unemployment
gap and inflation gap must be due to movements in aggregate demand. There are articles,
such as Roberts (1995), which use oil prices to control for cost-push shocks. This approach,
while reasonable, is difficult to implement in practice since there is a large number of potential cost-push shocks that would need to be controlled for, which in many cases may not be
observable.
Another method is to use an instrumental variable estimation for the unemployment gap.
The instrument should be correlated with unemployment but uncorrelated with cost-push
shocks. This is also hard to do in practice because it is difficult to find good instruments—
that is, a macroeconomic variable that is truly exogenous to cost-push shocks but correlated
with the unemployment rate and inflation expectations. Moreover, strong temporal dependence or persistence in the shocks may also affect the validity of the instruments.5
Finally, the third method they mention, and that we analyze in this article, is to exploit
the cross-sectional heterogeneity in regional data. Fitzerald and Nicolini (2014) also implement this approach in their article. As highlighted in Case III, the actions of the monetary
authority affect the movements in unemployment and inflation and thus affect the estimates
of the slope of the Phillips curve, even if properly instrumenting for cost-push shocks. How
ever, in a large country like the United States, comprising several states and cities, the monetary
authority only reacts to fluctuations in aggregate inflation and unemployment, and monetary
policy is independent of local conditions. Thus, the main idea behind a regional approach to
estimating the Phillips curve is to use fluctuations in regional economic conditions, in deviations from the aggregate, which the monetary authority will not influence by its actions. In
this way, by properly instrumenting or controlling for cost-push shocks, we should be able to
recover the structural relationship between inflation and the unemployment rate as we are
also controlling for monetary policy. We first discuss empirical evidence and then use a model
to analyze this approach.
5.1 Empirical Analysis
To explore the empirical implications of this regional approach to the Phillips curve estimation, we use semi-annual CPI inflation and unemployment data from the BLS for 20 metropolitan statistical areas (MSAs). We divide our sample into three periods: 1990-2000, 2000-10,
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Figure 9
Empirical Relation between the Inflation Gap and Unemployment Gap for Different MSAs
Deviation from decadal means
(1990-2000)
Deviation from decadal means
(2000-10)
Deviation from decadal mean: CPI inflation
Deviation from decadal mean: CPI inflation
5
5
0
0
−5
−5
−5
0
5
10
Deviation from decadal mean: Unemployment rate
Correlation coefficient = 0.05
−5
0
5
10
Deviation from decadal mean: Unemployment rate
Correlation coefficient = −0.38
Deviation from decadal means
(2010-18)
Deviation from decadal mean: CPI inflation
4
2
0
−2
−4
−4
−2
0
2
4
6
Deviation from decadal mean: Unemployment rate
Correlation coefficient = −0.42
SOURCE: BLS and authors’ calculations.
and 2010-19. For each decade, we compute the mean inflation and unemployment for each
MSA and use this as a proxy for the steady-state level of unemployment and inflation for each
MSA. In Figure 9 we use simple scatterplots of the deviation of inflation from the decadal
mean versus the deviation of unemployment from the decadal mean for each MSA. We interpret these as the unemployment and inflation gaps for each MSA.
As can be seen from Figure 9, the slope of the scatterplots varies over time. The curve was
positively sloped in the 1990s, which contradicts our understanding of the theoretical relationship between these variables.
Formally, we estimate
π it = α i + β1Et π it+1 + β 2U it + γ t + ε it ,
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Table 2
Estimates of the Empirical Phillips Curve Using MSA-Level Data, 1990-2018
Metro area
FE only
Year FE only
Metro area and
year FE
Inflation
Pooled
Unemployment rate
–0.141***
(–7.20)
–0.168***
(–6.57)
–0.215***
(–5.81)
–0.406***
(–5.91)
0.390***
(8.00)
0.367***
(6.97)
0.127*
(2.58)
0.0570
(1.18)
Within R2
0.252
0.254
0.435
0.446
Observations
1,191
1,191
1,191
1,191
MSA FE
No
Yes
No
Yes
Year FE
No
No
Yes
Yes
Seasonal dummies
Yes
Yes
Yes
Yes
Inflation lag
First lag
NOTE: FE, fixed effects. The t-statistics are in parentheses. *p < 0.05, **p < 0.01, ***p < 0.001.
SOURCE: BLS.
where subindex i denotes an MSA and subindex t denotes the time period. As before, π is the
inflation rate and U is the unemployment rate. The regression includes both time fixed effects
and MSA fixed effects. Including MSA fixed effects allows us to remove the effect of the state-
specific mean unemployment rate, which can be seen as a proxy for the equilibrium rate of
unemployment for each MSA. The time fixed effects help control for time-varying changes in
the aggregate equilibrium unemployment rate and help us overcome the bias caused by the
correlation between the regional unemployment rate and aggregate unemployment rate.6 In
Table 2, we present results from estimating the above regression, with a combination of fixed
effects.
Column (2) shows the results of regressing inflation on unemployment without controlling
for time fixed effects or seasonality. However, once we begin to control for these factors, we
observe that the coefficient increases in magnitude; from Column (5), we observe that the
coefficient is highly negative and significant.
5.2 A Simple New Keynesian Model of a Monetary Union
To understand the factors that influence the empirical estimates of the slope of the Phillips
curve using regional data, we turn to a simple model of a monetary union with a common
monetary authority, specified by Gali and Monacelli (2008). With this model, we proceed in
a similar fashion as before, finding decision rules and then simulating shocks and time series
for inflation and unemployment, in this case at both the aggregate and regional levels.
Gali and Monacelli (2008) model the currency union as a continuum of small (atomistic)
open economies that are subject to imperfectly correlated shocks. Each of these small economies, which we call regions, share identical preferences, technology, and market structures.
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Let Ptj represent region j’s price index for goods produced locally, and let Ptf be the price index
for goods purchased (imported) from other regions. In this way, the CPI for a region is the
geometric weighted average of domestic and imported price indexes, with weights given by
consumption shares Pc,tj = (Ptj )1–α(Pt* )α. For the monetary union as a whole, the price index is
the aggregator across all regions.
Pf
The bilateral terms of trade between regions j and k can be defined as S fj ,t = t j , which is
Pt
the price of region k’s goods in terms of region j’s goods. Then, the CPI inflation rate of the
region (πc,tj ) can be approximated as follows:
π cj,t = π tj + α Δstj .
The model assumes complete markets and perfect risk-sharing across all regions, together
with a zero net savings for all regions. This means that regional consumption and consumption for the whole country can be approximated as
ctj = ct* + (1− α ) stj ,
where ct* is the (log) aggregate consumption for the whole country.
A log-linear approximation of the model around a symmetric steady state for all regions
leads to similar dynamic equilibrium conditions as before. For region j, the Phillips curve is
j
⎤⎦ + κ ŷtj + ε tj .
π tj = β Et ⎡⎣π t+1
In addition, in each region an IS equation holds:
(
)
j
⎤⎦ − it* − π cj,t − ztj ,
ctj = E ⎡⎣ct+1
where it* is the interest rate for the whole country. Aggregating these expressions across regions
can form a Phillips curve and an IS curve for the whole country, which leads to the same
expressions as in Section 4. We further assume that an expression like Okun’s law holds in
every region and that the monetary authority follows the same Taylor rule as in Section 4,
reacting only to fluctuations in aggregate inflation and the aggregate output gap. As before,
we use this model to simulate shocks to the economy.7
5.3 Case V: Only Demand Shocks—Regional Analysis
As in Case I, here we allow for only demand shocks. We assume a few possible scenarios.
First, we allow for only demand shocks, both at the aggregate level and for individual regions.
Second, we allow for only demand shocks at the regional level, but no aggregate shocks of any
kind. In this last case, aggregate inflation and unemployment are constant, and the monetary
authority will not change the interest rate. Figure 10 presents the results. Panel A shows the
effect of aggregate demand shocks at the aggregate level—that is, for aggregate inflation and
the aggregate unemployment rate. Panel B shows the effect of aggregate and regional demand
shocks at the regional level. That is, each dot in the figure shows the inflation rate and the
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Figure 10
Effects of Regional and Aggregate Demand Shocks
A.
B.
Aggregate inflation
0.10
Regional inflation
0.10
0.05
0.05
0
0
–0.05
–0.15
–0.01
–0.05
0
0.05
0.10
–0.05
–0.15
–0.01
–0.05
0
0.05
0.10
Regional unemployment gap
Aggregate unemployment gap
C.
Regional inflation
0.10
0.05
0
–0.05
–0.15
–0.01
–0.05
0
0.05
0.10
Regional unemployment gap
SOURCE: Authors’ calculations.
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Figure 11
Effects of Regional and Aggregate Cost-Push Shocks
A.
B.
Aggregate inflation
0.10
Regional inflation
0.10
0.05
0.05
0
0
–0.05
–0.04
–0.02
0
0.02
Aggregate unemployment gap
0.04
–0.05
–0.04
–0.02
0
0.02
0.04
Regional unemployment gap
C.
Regional inflation
0.10
0.05
0
–0.05
–0.04
–0.02
0
0.02
0.04
Regional unemployment gap
SOURCE: Authors’ calculations.
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unemployment rate in one individual region. Finally, Panel C shows the effect of only regional
demand shocks at the regional level. In this case, the aggregate inflation and unemployment
rate are constant at their steady-state value.
Clearly, for aggregate inflation and unemployment displayed in Panel A, allowing for
only demand shocks shows a similar picture as Case I (shown in Figure 5). Adding regional
shocks on top of this (Panel B) introduces some noise around the empirical relationship, but
the main pattern still holds. More interesting, Panel C shows the relationship in the absence
of aggregate shocks. In this case, we can see that regional demand shocks can help identify a
negative empirical slope between inflation and unemployment at the regional level.
5.4 Case VI: Only Cost-Push Shocks—Regional Analysis
Similar to the previous case, we now analyze the effects of allowing for only cost-push
shocks. Figure 11 shows the results. In Panel A, and similar to Figure 6, cost-push shocks
generate a positive-sloping empirical relationship between inflation and unemployment for
the aggregate economy. While the monetary authority intervenes to stabilize the economy, it
faces the trade-off between lower inflation at the expense of a larger contraction and higher
unemployment. At the regional level, the effects of both aggregate and regional cost-push
shocks, shown in Panel B, generate a more dispersed pattern. Yet, the empirical slope between
unemployment and inflation is still positive when using only regional data. Finally, Panel C
shows the results for an economy with only regional cost-push shocks. In this case, aggregate
variables do not move and there is no intervention by the monetary authority; the positive
slope is more pronounced.
This exercise leads to an important conclusion. The empirical analysis using regional data
must also control or instrument for regional cost-push shocks. Otherwise, estimates of the
Phillips curve will be affected.
5.5 Case VII: Changes in the Policy Rule Over Time—Regional Analysis
Similar to our methods in the simple model, here we ask if changes in the monetary
authority’s preferences for how aggressively to fight inflation affect the empirical slope of the
Phillips curve estimated using regional data. For this we use an economy where all shocks are
active (aggregate and regional; demand and cost-push), and we compare whether and how
the empirical slope of the Phillips curve estimated using regional data changes. We show this
comparison in Figure 12, where we use simulated data from the model.
Panels A and C show the relation between unemployment and inflation at the aggregate
level for two economies with different degrees of monetary policy aggressiveness, with Panel
C showing the results for an economy with a more aggressive stance. These graphs are similar
to those in Case III analyzed above, where the empirical slope of the Phillips curve was affected
by changes in the monetary authority’s preferences. Panels B and D show the relationship
between unemployment and inflation using regional data, where Panel D shows the case of a
monetary authority more aggressive toward inflation. In this case, the changes in the policy
rule do not translate into noticeable changes in the slope of the empirical Phillips curve at the
regional level. In other words, while the slope estimated with regional data does change with
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Figure 12
Effects of Changes in the Policy Rule at Aggregate and Regional Levels
A.
B.
Aggregate inflation
Regional inflation
0.06
0.05
0.04
0.02
0
0
–0.02
–0.05
–0.04
–0.10
–0.05
0
0.05
0.10
–0.2
–0.1
0
0.1
0.2
Regional unemployment gap
Aggregate unemployment gap
C.
D.
Aggregate inflation
0.06
Regional inflation
0.05
0.04
0.02
0
0
–0.02
–0.05
–0.04
–0.10
–0.05
0
0.05
Aggregate unemployment gap
0.10
–0.2
–0.1
0
0.1
0.2
Regional unemployment gap
SOURCE: Authors’ calculations.
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Figure 13
Effects of Changes in the Variance of Shocks at Aggregate and Regional Levels
A.
B.
Aggregate inflation
Regional inflation
0.06
0.05
0.04
0.02
0
0
–0.02
–0.05
–0.04
–0.10
–0.05
0
0.05
0.10
–0.2
–0.1
0
0.1
0.2
Regional unemployment gap
Aggregate unemployment gap
C.
D.
Aggregate inflation
0.06
Regional inflation
0.06
0.04
0.04
0.02
0.02
0
0
–0.02
–0.04
–0.10
–0.02
–0.05
0
0.05
Aggregate unemployment gap
0.10
–0.04
–0.2
–0.1
0
0.1
0.2
Regional unemployment gap
SOURCE: Authors’ calculations.
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changes in monetary policy, the changes are very small. This result lends some support to the
use of regional data for estimating the slope of the curve, as it is almost invariant to changes
in policy.
5.6 Case VIII: Changes in the Variance of Shocks—Regional Analysis
Finally, we analyze the effect of changes in the relative variance of demand and supply
shocks to answer the following question: Can regional data help estimate the slope of the
empirical Phillips curve if the relative variance of shocks changes? We proceed in a similar
way as in Case IV presented earlier and increase the variance of cost-push shocks relative to
the variance of demand shocks, at both the aggregate and regional levels. Figure 13 shows the
results of this exercise. Panels A and B display the baseline case for aggregate and regional
variables, respectively; and Panels C and D have the same information but for an economy
with a larger variance of cost-push shocks. Comparing the top and bottom panels makes it
clear that an increase in the relative variance of cost-push shocks will affect the estimated slope
of the empirical Phillips curve, whether estimated using aggregate data or regional data.
This highlights an important limitation with the use of regional data to estimate Phillips
curves. The empirical slope of the curve does not need to capture the structural slope in equation (2). Moreover, the empirical slope may flatten over time even if the structural slope did
not change.
6 CONCLUSION
We use a simple New Keynesian framework to illustrate the main problems in estimating
the slope of the structural Phillips curve. Some of these problems arise due to the actions of
monetary policy, as it affects economic activity to fight inflation, which leads to biases in simple
estimations. Recent proposals have favored estimations using regional or city data in an effort
to overcome these issues, as monetary policy will react to only aggregate economic conditions
and not regional. We use a simple model of a monetary union with a continuum of economies
and a common monetary policy authority that reacts to aggregate conditions. When we use
this model as a data-generating process, we find that the main drawbacks of the aggregate
model are still present in a cross-section of many regions in a monetary union. The relative
importance of the demand and supply shocks will largely determine the empirical relation
between unemployment and prices in both the aggregate and the cross-section of regions.
Our analysis shows potential pitfalls in estimating the slope of the Phillips curve, even if using
regional data. n
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NOTES
1
We follow McLeay and Tenreyro (2019) and use 10-year-ahead mean forecasts for CPI inflation as our measure of
inflation expectations. In their words, “we use five to ten year ahead inflation expectations, as suggested by
Bernanke (2007) and Yellen (2015) as having a stronger empirical fit with the data” (p. 32).
2
For further details of the model, see Gali (2008).
3
The specific values we use in our quantitative analysis are not important. Our goal is to make a qualitative point,
not a quantitative one.
4
Note, however, that this assumption is not without loss of generality, since, as we mentioned before, with persistent shocks future expected inflation is correlated with the current realization of the shocks.
5
And, as said before, one must also instrument for expected inflation.
6
Note that in the regional Phillips curve case, we abstract from inflation expectations. The reason is that we do not
have data on inflation expectations at the regional level, since the Survey of Professional Forecasters contains
information on inflation for the whole U.S. economy.
7
Note that we assume regions within the country are small and do not affect aggregate variables. This assumption
is not without loss of generality. In fact, this assumption gives the best chance of success to the hypothesis that
using regional data would recover the slope of the Phillips curve. Our point is that, even in this case, the assumption may fail in some circumstances.
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