Benigno, Eggertsson, It's Baaack: The Surge in Inflation in the 2020s and the Return of the Non-Linear Phillips Curve (NBER w31197, 2023-04)

Summary

NBER Working Paper 31197, dated April 2023, by Pierpaolo Benigno and Gauti B. Eggertsson. The abstract proposes a non-linear New Keynesian Phillips curve (Inv-L NK Phillips Curve) to explain the surge of inflation in the 2020s, with economic slack measured as firms' job vacancies over the number of unemployed workers. It presents a New Keynesian model with search and matching frictions and a form of wage rigidity. The authors state that appropriate monetary policy can bring inflation down without a significant recession and that the recent inflationary surge was mostly generated by an exceptionally tight labor market.

Summary drafted by a model from the document's text below and checked by script against that text before publication. It is a navigation aid, not a reading of what the document proves. Where AI is used

Cite this document

“Benigno, Eggertsson, It's Baaack: The Surge in Inflation in the 2020s and the Return of the Non-Linear Phillips Curve (NBER w31197, 2023-04).” Original at www.nber.org. Archived at Pandemic Darlings, https://pandemicdarlings.com/source-documents/benigno-eggertsson-it-s-baaack-the-surge-in-inflation-in-the-2020s-and-5c96f99458/, SHA-256 3e2614f996ae6eaad1719ccc4388007095a89fa5632f20288fcea5a772dc5e57.

Full text

NBER WORKING PAPER SERIES

IT’S BAAACK: THE SURGE IN INFLATION IN THE 2020S AND THE RETURN
OF THE NON-LINEAR PHILLIPS CURVE

Pierpaolo Benigno
Gauti B. Eggertsson

Working Paper 31197
http://www.nber.org/papers/w31197

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 2023

We are grateful to Luca Benati, Don Kohn, Pascal Michaillat, Emi Nakamura and Chris Sims for
helpful discussions and suggestions, to seminar participants at UPF-CREI, EFIP and Brown, and
to GaOn Kim for providing excellent research assistance. We have no financial relationships to
declare. The views expressed herein are those of the authors and do not necessarily reflect the
views of the National Bureau of Economic Research.

NBER working papers are circulated for discussion and comment purposes. They have not been
peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies
official NBER publications.

© 2023 by Pierpaolo Benigno and Gauti B. Eggertsson. All rights reserved. Short sections of text,
not to exceed two paragraphs, may be quoted without explicit permission provided that full
credit, including © notice, is given to the source.
It’s Baaack: The Surge in Inflation in the 2020s and the Return of the Non-Linear Phillips
Curve
Pierpaolo Benigno and Gauti B. Eggertsson
NBER Working Paper No. 31197
April 2023
JEL No. E12,E3,E30,E40,E50,E60

ABSTRACT

This paper proposes a non-linear New Keynesian Phillips curve (Inv-L NK Phillips Curve) to
explain the surge of inflation in the 2020s. Economic slack is measured as firms' job vacancies
over the number of unemployed workers. After showing empirical evidence of statistically
significant nonlinearities, we propose a New Keynesian model with search and matching
frictions, complemented by a form of wage rigidity, in the spirit of Phillips (1958), that generates
strong nonlinearities. Policy implications include the thesis that appropriate monetary policy can
bring inflation down without a significant recession and that the recent inflationary surge was
mostly generated by a labor shortage -- i.e. an exceptionally tight labor market.

Pierpaolo Benigno
Dipartiment of Economics
University of Bern
3003 Bern
Switzerland
pierpaolo.benigno@vwi.unibe.ch

Gauti B. Eggertsson
Department of Economics
Brown University
64 Waterman Street
Providence, RI 02912
and NBER
gauti_eggertsson@brown.edu
10 10

9 9

8 8

7 7

6 6

5 5

4 4

3 3

2 2

1 1

0 0

-1 -1
0.9 0.95 1 0.9 0.95 1

Figure 1: The crude Keynesian Phillips curve versus the original Phillips curve proposed and esti-
mated by Phillips in 1958

1 Introduction

This paper proposes to replace the canonical New Keynesian Phillips curve with an Inverted-L New
Keynesian Phillips curve (Inv-L NK Phillips curve), which is nonlinear. This Inv-L NK Phillips curve
can explain the sharp, unexpected rise in inflation in the U.S. in the early 2020s following the COVID-
19 pandemic. Demand shocks are seen to have played a greater role than supply shocks in explaining
the surge, which carries policy implications. Another implication is that appropriate monetary policy
can engineer a “soft landing”. That is, for each percentage point decrease in inflation, the Federal
Reserve can achieve a smaller increase in unemployment compared to the Volcker recession, which
successfully curbed inflation following the 1970s’ Great Inflation but at the expense of a substantial
increase in unemployment. The U.S. data show statistically significant and quantitatively important
support for a non-linear Phillips curve. Below we set out the broader motivation for this approach,
the historical background, and the main contribution of this paper.

The early Keynesian literature assumes a stark relationship between prices and output, namely the
inverted L in the left-hand panel of Figure 1. Blinder (2022) labels this “crude Keynesianism”. When
output is below potential, there are idle workers and empty factories, in which case prices are fixed.
Once all workers are employed and factories are in full swing, output can go no higher, and aggre-
gate supply becomes vertical. In this case the economy enters a neoclassical world of flexible prices.

2
Keynesianism took a sharp turn with Phillips’ seminal work (1958), one of those papers that are so
well known that they are rarely cited even in the literature that builds on them. Instead, the author’s
name has simply become synonymous with the Phillips curve.1

The Phillips curve is the best known and most controversial equation in macroeconomics. Its most
common version is:2

πt = κxt + µ̂t + βEt πt+1 (1)
| {z } | {z }
Keynesian New Keynesian
Phillips Curve Phillips Curve

where πt is inflation, κ is a coefficient, and xt is some measure of economic activity such as the output
gap, in which case κ > 0, or the unemployment gap, in which case κ < 0. The coefficient β is between
0 and 1, Et is an expectation operator and µ̂t is a supply shock.

The first part of equation (1), indicated by the first curly bracket, is the Keynesian Phillips curve
popular in the 1960s. Suggested by Samuelson and Solow (1960), it implied a stable trade-off between
inflation and the output gap. The second term, which includes a supply shock and expectations on
future inflation, was emphasized by the literature that developed after the 1970s. The formulation in
equation (1) is known as the New Keynesian Phillips curve, and it is currently the backbone of most
macroeconomic models.

In the course of the 1970s the Keynesian Phillips curve collapsed as a stable statistical relationship
(see Figure 12 in the Appendix). This empirical failure had a decisive impact on macroeconomics. Ar-
guably, it gave birth to the rational expectations revolution and made microfoundations for macroe-
conomic models mainstream, partially because they offer an explicit account of how expectations are
formed. What made the demise of the Keynesian Phillips curve dramatic – a watershed moment for
macroeconomics – was that it had been prominently foreseen by Phelps (1967) and Friedman (1968).
They predicted that a breakdown of this kind should occur – as a matter of theory — if the govern-
ment tried to exploit the inflation-unemployment trade-off, because inflation expectations catch up
and endogenously shift the relationship as in equation (1). With inflation rising in the late 1960s, it
looked to many observers as if the government was indeed accepting excessive inflation in return for
higher employment. That the relationship broke down in the 1970s, just as predicted, lent consider-
able credibility to Friedman and Phelps’s prophecy.

The main explanation now put forward for the demise of the Keynesian Phillips curve is the combi-
nation of the unanchoring of inflation expectations in the 1970s and supply disruptions. The presence
of expected inflation in (1), which is central to any modern Phillips curve, explains nowadays central
banks’ focus on anchoring inflation expectations. As this relationship makes clear, higher inflation
expectations have an effect similar to that of a negative supply shock.
1 Like the original paper by John Nash, which is seldom cited when Nash equilibrium is applied in game theory.
2 See e.g. Woodford (2003) and Galí (2015) for textbook treatments.

3
Someone reading Phillips’ classic paper today will realize immediately that modern research pays
little or no attention to its central proposition. The Phillips of 1958 would hardly recognize the linear
relationship (1) as his own construction, even apart from the terms that represent expectations.3 First,
the relationship Phillips suggests is between unemployment and wage inflation. Second, and more
importantly, his point is that this relationship is strongly non-linear. The curve proposed in Phillips
(1958) is c
1
πtw = a + b
ut
where πtw is wage inflation, ut is the unemployment rate and a, b, c are estimated coefficients.

The original Phillips curve is plotted in the right-hand panel of Figure 1 using the coefficient values
Phillips estimated for the period 1861-1913, with 1 − ut plotted on the x-axis representing a higher
level of economic activity to relate it to the crude Keynesianism on the left.4 As the figure reveals,
this relationship is strongly non-linear and in fact resembles the inverted L of "crude Keynesianism"
much more than it does the linear curve that became synonymous with his name.

While Phillips’ paper is empirical, his theoretical argument for nonlinearity is straightforward. He
writes that with “very few unemployed we should expect employers to bid up wages quite rapidly,
each firm and each industry being continually tempted to offer a little above the prevailing wage.”
But why does the asymmetry arise when unemployment is high? Phillips suggests that “workers are
reluctant to offer their services at less than the prevailing rate”so “wages fall only very slowly”.5 A
very non-linear curve is Phillips’s central proposition and the concluding line of his opening para-
graph.

The main objective of this article is to resurrect Phillips’ original idea in order to explain the surge in
inflation in the 2020s. We first present some suggestive empirical evidence, then propose a theoretical
model for a non-linear Phillips curve and explore policy implications.

The recent increase in inflation took policy makers by surprise, at least if we go by the Summary of
Economic Projections (SEP) of the Federal Reserve. It was also unexpected by private forecasters, as
judged by Survey of Professional Forecasters (SPF), as shown in Figures 13 and 14 in the Appendix.
Neither SEP nor SPF anticipated the surge in prices in 2021. And both consistently projected it to de-
cline rapidly to the Fed’s 2 percent target rate. Instead, inflation continued to increase in the following
quarters.

The reason why both private forecasters and policy makers were caught flatfooted is largely explained
by three broad observations.

First, the conventional wisdom, forged by the realization of Phelps and Friedman’s prophecy in the
1970s, is that persistent increase in inflation are triggered when inflation expectations become unan-
chored. During the 2020s, however, there was no increase in inflation expectations comparable to that
3 Phillips did, however, recognize that supply shocks could be an important factor in shifting his proposed empirical

relationship.
4 This is Figure 1 in Phillips (1958). The paper estimates b=9.636 and c=1.394 via least squares and sets a=-0.9 via "trial and

error" based on data from the U.K. between 1861 and 1913.
5 See also the discussion by Tobin (1972).

4
14

% 6

-2
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004 2008 2012 2016 2020

Figure 2: Inflation: CPI inflation rate at annual rates. 12-month Livingston inflation expectations.

of the 1970s, and both policymakers and market participants accordingly thought that 2021 surge was
transitory. Figure 2 shows inflation expectations according to the Livingstone survey, which asks par-
ticipants what they expect inflation to be in a year. Whereas this measure had peaked at 10 percent
in the late 1970s, in the 2020s it rose only modestly. Several alternative measures of expectation that
proxy longer-run inflation expectations are even more striking. These measures suggest that the Great
Inflation of the 1970s unanchored expectations so drastically that at their peak people were expecting
inflation of 10 percent to persist even over a five-to-ten year horizon. Several such proxies, reported in
the Appendix, are used for robustness checks of the empirical analysis in the next section.6 In short,
one of the key suspects for driving the Great Inflation of the 1970s was simply missing in the 2020s.

A second main reason why forecasters missed the surge is that most estimates of the slope of the
Phillips curve, i.e. κ in equation (1), are very low. This view is reflected in virtually all the models
used for inflation projections at policy institutions (see e.g. the discussion in Gopinath (2023) of
the IMF model).7 According to the widely cited recent estimate of Hazzell, Herreno, Nakamura and
Steinsson (2022), for example, a 1-percentage-point reduction in unemployment generates only a 0.34-
point increase in inflation. Their analysis uses a carefully designed identification strategy based on a
6 In the Appendix we show the Cleveland Fed’s five-year inflation expectations since 1982. The expectations were at 6

percent in 1982 even though current inflation was lower. As of this writing (April 2023), they stand at 2.2 percent and have
never gone above 2.5 percent during the surge. Even more compelling is the evidence on the five-year five-year forward
inflation rate, what markets expect inflation to be five years ahead. It has held remarkably stable throughout the 2020s surge.
While a market-based measure such as this is only available from 1997, Groen and Middledorp (2013) identifies 108 dataseries
to back-cast this measure to 1970 using partial least-squares. They find that it peaked at 10 percent during the Great Inflation of
the 1970s indicating that market participants thought the high inflation was there to stay (see Figure 15 in the Appendix). The
Blue Chip Economic indicators of ten-year inflation expectation (dating back to 1980), maintained by the Philadelphia Federal
Reserve, show similar results. At the end of 1980, for example, ten-year inflation expectations were 8.4 percent.
7 Gopinath draws the conclusion that the failure of the models currently used by major institutions to predict the inflation

surge are plausibly explained by the fact that there may be "important nonlinearities in the Phillips curve slope: price and
wage pressures from falling unemployment become more acute when the economy is running hot than when it is below full
employment." This is precisely the type of mechanism our model formalizes.

5
2

1.5

=1
1

0.5

0
19
19
19
19
19
19
19
19
19
19
1960
62
64
66
68
70
72
74
76
78
19
19
19
19
19
19
19
19
19
2080
82
84
86
88
90
92
94
96
98
00
20
20
20
20
20
20
20
20
20
20
2002
04
06
08
10
12
14
16
18
20
22

Inflation (%)
8

-2
60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 00 02 04 06 08 10 12 14 16 18 20 22
19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 20 20 20 20 20 20 20 20 20 20 20 20

Figure 3: Top panel: (θ) vacancy-to-unemployed ratio. Bottom panel: CPI inflation rate at annual
rates.

cross-section of Metropolitan Statistical Areas in the United States for 1978-2018. To summarize, while
a number of commentators suggested that demand was considerably above potential output, the
mainstream macroeconomic models predicted that this should have a modest inflationary impact.8

Finally, a third culprit in the 1970s was supply disruptions. However, conventional measures, such
as the difference between headline and core inflation or the difference between the rates of change
in the import-price and GDP deflators did not increase nearly as much in the 2020s as in the 1970s.
Accordingly, supply disruptions are a less plausible candidate for the inflation surge in the 2020s
relative to the 1970s, especially as inflation expectations have remained anchored, together with the
persistence of the surge.9 So what is the missing piece?

This brings us to the labor market. A key statistic both in our empirical analysis and in the modeling
Job Vacancies
framework is labor market tightness, captured by θt = Unemployed workers , which is a standard measure
of labor market tightness in the search and matching literature (for a recent contribution in this lit-
erature see e.g. Michaillat and Saez, 2023). The numerator, i.e. the number of vacancies at firms,
summarizes how many jobs firms are seeking to fill, while the denominator is the number of job
seekers. This variable is plotted in the top panel of Figure 3 for the period 1960-2022, which gives the
reader a hint to our main hypothesis.10 The average value of θ, conditional on its being lower than 1,
8 Most prominently, in a Washington Post op-ed of 5 February 2021, Lawrence Summers (2021) issued an early warning

that the fiscal stimulus planned was big enough to push demand substantially above potential output and “set off inflationary
pressures of a kind we have not seen in a generation”.
9 During the early part of the surge, for example, there were widespread reports of temporary bottlenecks in the supply

chain, which led many see the inflation spike as transitory.
10 See the empirical section and the appendix for a description of how this variable is constructed.

6
6 15

4 10

Inflation (%) Inflation (%)
2 5

0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2

6 10

8
5
6

Inflation (%) Inflation (%)
4
4
3
2
2 0

0 0.5 1 1.5 2 0 0.5 1 1.5 2

Figure 4: Inflation: CPI inflation rate at annual rates. θ: vacancy-to-unemployed ratio.

is 0.56. This is relatively close to our empirical assessment of the state of the labor market as neither
inflationary nor deflationary, even if the exact estimate is subject to uncertainty.

On two occasions, however, this metric is greater than 1. We define θ > 1 as labor shortage, meaning
that there are more firms vacancies to be filled than workers looking for jobs. In the last 63 years,
situations of labor shortage have emerged on two occasions: during the Vietnam War in the late
1960s a period also marked by a substantial demand stimulus, as documented by Blinder (2022), and
the period around the COVID-19 epidemic. Both periods lie outside the typical econometric estimates
of the slope of the Phillips curve, i.e. the influential work of Hazell et al. (2022) cited above. But they
also correspond to inflation surges, as shown by the shaded region in the lower half of Figure 3. In
other words, labor shortages go hand in hand with inflationary surges.

Over the past century U.S. economic history documents three other major episodes of acute labor
shortages (i.e. θ > 1): World War I, World War II, and the Korean War. All three episodes were also
associated with inflation surges as shown in Figure 16 in the Appendix. However, as documented
by Rockoff (1981), they were also accompanied by comprehensive price controls, which complicates
empirical inference. For this reason, we focus on the period from 1960 to 2022.11
11 Rockoff (1981) also discusses the price controls from August 1971 to April 1974 implemented by President Nixon, as

shown in Figure 3. This period of price controls is less problematic for the empirical analysis since there was no labor shortage
at the time, and our emphasis is on documenting nonlinearity when θ > 1.

7
How can these hints lead to the assertion that there is statistically significant empirical evidence for
a nonlinear Phillips curve? Sometimes a figure is worth more than a thousand words. Figure 4
shows a scatter plot between the two major data-series at the heart of the empirical analysis: raw
annualized inflation rates and labor market tightness at quarterly frequency. The periods of labor
shortage, namely 1960-1969 and 2008-2022, strongly suggest that an inverted L shaped Phillips curve
lurks behind the scene. The other two periods, those typically used in studies of Phillips curves, show
θ < 1 and – at least at first glance – a relatively flat Phillips curve.

Scatter plots, of course, are only suggestive at best. In the empirical analysis we add more structure
to the data, in line with the earlier literature, by considering a series of regressions with both fixed
and time-varying coefficients, adding controls, creating proxies for expectations, and suggesting two
distinct ways of testing for nonlinearities.

The second main contribution of the paper is to resurrect Phillips’ original idea of an inverted-L curve
as in Figure 1, and within the canonical New Keynesian framework. We think it is important to retain
the core elements of the New Keynesian framework, which highlights the role of the anchoring of
inflation expectations. This is because our objective is a theory that can account for both the Great
Inflation of the 1970s (when the labor market was slack but inflation expectations unanchored) and
the inflation surge of the 2020s (when the labor market was tight but inflation expectations firmly
anchored).

While the elements of our model that generate inflation expectations in the Phillips curve are stan-
dard, drawn from the New Keynesian literature, the nonlinearities arise due to an imperfect labor
market. In the model, households decide how many people enter the labor force. The main friction is
that some workers cannot find jobs. This is modeled via standard search and matching frictions. In
the spirit of Phillips, we focus on the role of nonlinearities in wage setting. The key asymmetry is that
while firms are tempted to outbid one another other when the labor market is tight and workers are
more than happy to accept higher wages, the margin for cutting wages given high unemployment is
limited, because workers will not accept jobs that pay much below the prevailing wage, even in the
face of high unemployment. An extensive empirical literature documents that wages are “downward
rigid,”which provides support to Phillips’ hypothesis.12

We derive a Phillips curve of the same form as (1) with four major differences. First, instead of the
output or unemployment gap, the explanatory variable, i.e. xt in equation 1, is now labor market
tightness, measured by the log deviation of θ from its steady state. Second, when there is labor
shortage, i.e. θ > 1, then the slope κ changes and becomes steeper than normal. This captures the key
nonlinearities that Phillips emphasized and arises due to the same force that he proposed. Third, in
normal circumstances, the Phillips curve has a state variable (lagged wages), and thus moves more
gradually than when there is a surge in inflation. Finally, when the labor market is tight, supply
12 The idea of downward rigid nominal wages dates back at least to Malthus who noted that “it very rarely happens that the

nominal price of labour universally falls”, Malthus (1798). Bewley (1999) interviewed corporate executives documenting their
reluctance to cut nominal wages. More recently, substantial nominal wage rigidity has been studied in U.S, administrative data
by Fallick, Lettau and Wascher (2011), in worker surveys by Barattieri, Basu and Gottschalk (2014), and in cross-country data
by Schmitt-Grohe and Uribe (2016).

8
shocks are transmitted to inflation with greater force. Meanwhile, inflation expectations play a major
role in either regime, their relative strength depending on the details of the model specification.

Our work is related to several strands of the literature. We contribute to labor market models in the
spirit of the search and matching framework of Mortensen and Pissarides (1994), among others. We
introduce employment agencies that optimally chose how many vacancies to post, which pins down
wages, resolving the classic problem of wage indeterminacy (see for one, Hall, 2005). In addition, we
impose a constraint on the employment agencies optimization problem in the form of a wage norm
whereby under normal circumstances wages adjust only gradually.

We cast our model in an NK framework with price rigidities, thus presenting a theory of the rela-
tionship between unemployment and monetary policy alternative to Blanchard and Galì (2010), Galì
(2009) and Michaillat (2014) among others. We also contribute to the extensive literature following
from Phillips (1958), which has formalized the relationship between inflation and various measures
of economic activity. Our contribution is to derive the Inv-L NK Phillips curve when the measure
of economic slack is given by the vacancy-to-unemployed ratio, which also engenders nonlinearities
when there is labor shortage. Gagliardone and Gertler (2023) also seek to account for the recent infla-
tion surge, but where we explain the inflation surge via nonlinearities in the Phillips curve, they see
the cause as oil price spikes coupled with easy monetary policy. Another work highlighting nonlin-
earities in the Phillips curve is Harding, Linde and Trabandt (2023). A major difference from our work
is that instead of generating nonlinearities through labor shortage, they trace them to quasi-kinked
demand for goods, as in Kimball (1995).

There is substantial empirical literature estimating the Phillips curve using time-series data. Recent
closely related works include Ball et al. (2022), Barnichon and Shapiro (2022), Blanchard et al. (2015),
Domash and Summers (2022), Furman and Powell (2021), Gordon (1977, 2013) and McLeay and Ten-
reyro (2019). Within this literature, some have also emphasized nonlinearities, including Ball et al.
(2022) and Gagnon and Collins (2019). Other have stressed time variation in the slope, such as Benati
(2010), Blanchard et al. (2015), Blanchard (2016) and Matheson and Stavrev (2013).

Given the limited sample size of time-series data, cross sectional data analysis is becoming increas-
ingly used. Cerrato and Gitti (2023) produce a new analysis in the spirit of Hazzell et al. (2022)
with cross-sectional data of Metropolitan Statistical Areas, finding that the slope of the Phillips curve
tripled in the wake of the pandemic. In a similar vein, Smith et al. (2023) use both cross sectional U.S.
and E.U. data and find evidence of a kink point when the labor market is running hot. Generally, we
interpret these results as broadly consistent with our evidence.

This work is structured as it follows. Section 2 presents our empirical motivation. Section 3 sets out
the theoretical model. Section 4 discusses the model’s implications for the Phillips curve, Section 5 its
policy implications. Section 6 concludes.

9
2 Empirical Motivation

Although Figure 4 hints at the possibility of a nonlinear Phillips curve, each point on the scatter plot
represents an equilibrium outcome when viewed through the prism of a general equilibrium model.
In the model introduced in the second part of the paper, for instance, each data point is determined
by the intersection of the Phillips Curve (aggregate supply) and the aggregate demand. Aggregate
demand is influenced by the spending decisions of households and the government, as well as by
monetary policy. Consequently, from the perspective of our theory, we do not have strong reasons to
believe that the scatter plot, when viewed in isolation, provides compelling evidence one way or the
other.

The identification problem inherent in the scatter plot has been well understood for over a century,
dating back at least to Lenoir (1913).13 If only the supply curve shifts, the data traces out demand.
Conversely, if only demand shifts, the data traces out aggregate supply. Viewed in this light, the data
for the 1960s, displayed in the upper-left panel of Figure 4, is particularly intriguing. According to the
measures discussed below, the factors influencing the Phillips curve’s shift (such as various proxies
for inflation expectations and supply disturbances) remained relatively stable during this period.
Meanwhile, shifts in aggregate demand during the 1960s are well documented (see, for example,
Blinder, 2022). The figure speaks for itself.

The literature on Phillips curve estimation and the conditions under which it is identified is extensive.
We do not attempt to survey it here. McLeay and Tenreyro (2019) provide a recent, lucid account that
highlights various challenges to identification and possible solutions. For our purposes, an important
takeaway (see Section 5 of their paper) is that if supply shocks are adequately controlled for, the
Phillips curve can be empirically recovered.14

Our preferred benchmarks consist of two empirical specifications. The first is an ordinary least-
squares regression with constant coefficients, allowing for piecewise nonlinearity through a dummy
variable. The second benchmark specification permits time variation in all coefficients and is esti-
mated using a Kalman filter. In summary, our main conclusion is that we find statistically significant
nonlinearity in the Phillips curve when there is a labor shortage under both specifications (i.e. θ > 1).
Furthermore, this result is robust across various alternative specifications, which are summarized in
Section 2.1 and detailed in the Appendix.

In the first benchmark we consider the following ordinary least squares regression:

πt = β c + β π πt−1 + ( β θ + β θd Dt ) ln θt + β µ µt + β π e πte + ε t , (2)

where β c , β π , β θ , β θd , β µ , β π e are parameters, and ε t is a zero-mean normally-distributed error. Dt is a
dummy variable that takes value one if θt ≥ 1. πt ≡ ln Pt / ln Pt−1 is inflation, πt−1 is its one-quarter
13 For the historical context of the identification problem in macroeconomic models, see Christ (1994).
14 See also two recent survey articles: Mavroeidis, Plagborg-Moller, and Stock (2014) and Coibion and Gorodnichenco (2019).

10
lag, ln θt is the logarithm of the vacancy-to-unemployed ratio, µt is a supply shock, and πte is inflation
expectations.

In formulating (2), we closely follow the recent literature.15 Our empirical contribution is straightfor-
ward: accounting for nonlinearity. The nonlinearity takes a special form, as can be seen from equation
(2). When θ > 1, the slope of the regression can differ from that under normal circumstances. The
contribution most closely related to ours is Ball, Leigh, and Mishra (2022). They instead allow for
nonlinearity that applies at all times by including squared and cubed terms for θt . They find the
non-linear terms to be statistically significant. We view their findings as complementary to ours. The
motivation for our alternative approach is the suggestive evidence in Figure 4, where the data ap-
pears as if it can be closely approximated by a piecewise linear regression. Moreover, the model we
present in the next section naturally lends itself to this formulation.

Table 1 presents the estimates of an OLS regression of U.S. quarterly data from 1960 Q1 to 2022 Q3.
The dependent variable is the core Consumer Price Index (CPI), which excludes food and energy
prices. For ease of interpretation, all inflation variables are annualized quarterly rates and are ex-
pressed relative to a constant 2 percent annual inflation rate.16 We have already discussed the key
explanatory variable θ, which is expressed in logs. The next subsection discusses how we proxy
the other explanatory variables -â i.e., supply shocks and inflation expectations â- in the context of
various alternatives to our benchmark proxies to check for robustness.

The first major takeaway from Table 1 is that the nonlinearity of the Phillips curve is statistically
significant (at the 1% level) and large. This is shown by the third row of Table 1, in column (3) for
the full sample and column (4) for the sub-sample 2008-2022. The slope of the curve when θ >
1 is the sum of the second and third rows.17 The estimate for the slope when θ < 1 is given by
the second row of columns (3) and (4). While the point estimate has the expected sign, one cannot
reject the hypothesis that the Phillips curve is completely flat when θ < 1, as conjectured by the
crude Keynesians (see panel (a) of Figure 1 in the Introduction). The slope coefficient is larger and
statistically significant in columns (1) and (2) when one does not allow the slope to change when
θ > 1. This suggests that the slope in columns (1) and (2) is statistically significant thanks exclusively
to the periods in which θ > 1.
15 Furman and Powell (2021) argue that the best measure of economic slack for forecasting nominal wage and price growth

is the vacancy-to-unemployed ratio, θ. Recent literature has corroborated this finding; see, for example, Ball et al. (2022),
Barnichon and Shapiro (2022), and Domash and Summers (2022).
16 Estimates are invariant to the adjustment, except for the constant. However, as clarified below, considering the data as a

deviation from 2 percent (the current inflation target of the Federal Reserve) is meaningful for interpreting the estimates of the
constant.
17 For example the slope is 0.222+3.8957=4.1177 in column (4).

11
Table 1: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.3690∗∗∗ 0.2758 0.2623∗∗∗ 0.0322
(0.0965) (0.2560) (0.0928) (0.2348)

ln θ 0.6493∗∗∗ 0.6909∗ 0.2220 0.4864
(0.1887) (0.3791) (0.1930) (0.3670)

θ≥1 3.8957∗∗∗ 4.2684∗∗∗
(0.8231) (1.3704)

Supply shock 0.0390∗∗ 0.0126 0.0469∗∗ 0.0170
(0.0192) (0.0381) (0.0198) (0.0390)

Inflation expectations 0.6614∗∗∗ 1.0470 0.7991∗∗∗ 0.5274
(0.1085) (0.6228) (0.1020) (0.6776)

Constant 0.5423∗∗∗ 1.0146∗∗ 0.1922 0.4680
(0.1630) (0.4662) (0.1652) (0.4146)

R2 adjusted 0.816 0.463 0.827 0.511
Observations 251 57 251 57

· ***,**,* denote statistical significance at the 1, 5, and 10 percent level, respectively.
· Newey-West standard errors.
· (1) and (3): sample 1960 Q1 – 2022 Q3
· (2) and (4): sample 2008 Q3 – 2022 Q3

12
To illustrate the quantitative meaning of these results, consider the impact of increasing θ without
taking into account the estimated nonlinearity (columns 1 and 2). For example, core inflation in 2020
Q4 was 1.52%, while θ = 0.46. Suppose θ increases to 1.9, its value in 2022 Q2. This corresponds
to a 1.41 log point increase. Abstracting from the effect of lagged inflation, the full sample predicts
that inflation rises by 0.92 percentage points, while the smaller sample predicts 0.98. Now consider
taking the nonlinearities into account. The increase of θ from 0.4 to 1 (0.7 log-points) has only a trivial
effect. It increases inflation by just only 0.17 p.p. for the full and 0.38 p.p. for the smaller sample. The
picture changes completely for the increase in θ from 1 to 1.9 (0.64 log-points). Now, taken together,
core inflation is projected to reach 4.81% in 2022 Q2 using the full sample estimate and 5.4% using the
restricted sample. This is not far from the 5.5% core inflation registered in 2022 Q2.

A second major takeaway from Table 1 is that the coefficient for lagged inflation is statistically signif-
icant for the entire sample, but not for the period 2008-2022, when θ > 1 for a significant part of the
period. As we will see, this is a key prediction of the theoretical model.

A third major takeaway is the evidence of a smaller role for supply shocks in the inflationary surge of
2020s than in the 1970s, with a short-run pass-through coefficient of between 0.039 and 0.047 for the
full sample. During the first oil crisis of the 1970s, the magnitude of the shock was 0.3 according to our
metric (discussed in detail below). Accordingly the short-run pass-through on inflation is estimated
between 1.17 p.p. and 1.41 p.p.. During the current inflation surge, however, this shock peaked at
0.05, implying trivial effects on inflation between 0.2 p.p. and 0.23 p.p.. Allowing for time variation
in the coefficients, however, gives supply shocks a greater role, as we will see shortly.

Finally, inflation expectations remained relatively stable throughout the inflation surge of the 2020s.
The measure in the benchmark model peaked at 0.9 points above the 2 percent inflation target of
the Federal Reserve (in 2022 Q2). With the full sample pass-through estimate in the range 0.66-0.799
points, the regression suggests that a rise in inflation expectations could at most account for a 0.72-
point increase in inflation. This is in sharp contrast to the 1970s, when by various metrics, including
our benchmark one, inflation expectations were close to 10 percent, implying a substantial role.18

The constant of the regression suggests an interesting economic interpretation. It can be related to
the value for θ at which inflation is equal to 2% in the absence of shocks, which we denote by θ̄. We
obtain θ̄ = 0.42 for the full sample and 0.38 for the 2008-2022 sub-sample.19

Figure 5 provides a visual interpretation of the regression that parallels Figure 4. The left panel
combines all the data points from Figure 4 into a single scatter plot. The right panel, on the other
hand, subtracts from inflation all the right-hand side explanatory variables, with the exception of θ
and the constant.
18 While inflation expectations have a positive coefficient across all specifications, they lose statistical significance in the

2008-2022 sub-sample. This may be because they held stable, and so did not generate sufficient variation to obtain a statistically
significant estimate.
19 To see this, first observe that the mean of the supply shock is close to zero, and all inflation measures are considered as

deviations from the 2% target rate. The variable ln θ, however, is not measured as a deviation from its average value. This
implies that we can back it up through the equation ln θ̄ = − β c /β θ , obtaining θ̄ = 0.42 and θ̄ = 0.38, for the third and fourth
specification, respectively.

13
Sample 1960-2022 Sample 1960-2022
14 14

12 12

10 10

Inflation deviations (%)
8 8

Inflation (%)
6 6

4 4

2 2

0 0

-2 -2

-4 -4

-6 -6
0 0.5 1 1.5 2 0 0.5 1 1.5 2

Figure 5: Left panel: scatter plot of the raw data of inflation and θ used in the regression (3) of Table
1, sample 1960 Q1 - 2022 Q3. Right panel: scatter plot of ’inflation deviations’ and θ, sample 1960 Q1
- 2022 Q3. ’Inflation deviations’: inflation in deviations from its lagged value, the supply shock and
inflation expectations using regression (3) of Table 1.

The second benchmark empirical specification allows for time-varying coefficients, focusing on the
period from 2008 Q3 to 2022 Q3. We follow closely the existing literature, see Blanchard, Cerutti
and Summers (2015). We consider the regression reported in Table 1, but the parameters are now
allowed to vary over time by a random walk. The model is estimated by a Kalman filter using as
initial conditions the OLS estimates generated by a regression up to 2008 Q2. Figure 6 shows how
the estimated coefficients vary over time from 2008 Q3 to 2022 Q3 with red lines. The blue lines
correspond to one-standard-deviation confidence bands.

The main conclusion here is that the estimated coefficients shifts sharply towards the end of the sam-
ple, once θ > 1. The slope of the curve steepens significantly in the post-COVID period, ending with
a value of 1.78. This is consistent with the results in Table 2, although different in magnitude. A key
difference relative to Table 2 is that the supply-shock coefficient increases from near zero to over 0.18.
This is more than four times higher than the OLS estimates in Table 2. This is of considerable inter-
est, because an increase in the supply-shock pass through is another key prediction of the theoretical
model in situations of θ > 1.

The inflation-persistence coefficient declines over time and hovers near zero at end of the sample.
This, too, is one of out model’s main predictions when θ > 1.

Both benchmark specifications highlight how poorly forecasters perform if they fail to recognize the
estimated nonlinearities. The left panel of Figure 7 applies the OLS regression (2) without the dummy

14
1 3

2
0.5

1
0
0

-0.5
-1

-1 -2
2008 2012 2016 2020 2008 2012 2016 2020

0.3 2

0.2 1.5

0.1 1

0 0.5

-0.1 0

-0.2 -0.5
2008 2012 2016 2020 2008 2012 2016 2020

Figure 6: Estimates of the Kalman Filter with time-varying parameters on sample 2008 Q3 – 2022 Q3
with one-standard-deviation confidence bands.

to the sample 2008 Q3 – 2021 Q1, i.e. stopping before the inflation surge, to forecast inflation for the
period 2021 Q2 to 2022 Q3. This forecast, indicated by the red line, is stable around 2%, despite the
additional supply shocks and the tight labor market. A flat Phillips curve cannot predict the actual
inflation surge, traced by the black line. This result broadly corresponds to the failure of professional
forecasters and policymakers to predict the increase in inflation, despite the evidence of a tight labor
market and supply disruptions. The blue line in the Figure shows the forecast, now in-sample, when
the regression is estimated for 2008 Q3 – 2022 Q3 taking into account the dummy, as in column 2 of
Table 1. This adjustment improves the forecast substantially.

The right-hand panel conducts the same experiment using the second benchmark specification with
time-varying coefficients. This forecast comes slightly closer to the actual inflation rate. We decom-
pose the contribution of the three regressors – labor tightness, supply shock, and change in inflation
expectations. All three components, especially the supply shocks, contribute to the initial surge of in-
flation in the second and third quarters of 2021, but the main reason for the ongoing inflation during
2022 is the tight labor market.

15
7 7

6 6

5 5

4 4

2021 Q1 2021 Q1
% %

3 3

2 2

1 1

0 0
2019 2020 2021 2022 2019 2020 2021 2022

Figure 7: Left panel: CPI inflation rate at annual rates (black line); out-of-sample inflation forecast
(red line) using OLS regression (2) without the dummy variable on the sample 2008 Q3 – 2021 Q1;
in-sample inflation forecast (blue line) using OLS regression (2) on the sample 2008 Q3 – 2022 Q3.
Right panel: CPI inflation rate at annual rates (black line); in-sample inflation forecast (purple line)
using Kalman-Filter estimation with time-varying coefficients on the sample 2008 Q3 – 2022 Q3. The
three dashed lines represent the inflation forecasts using the Kalman-Filter estimates by restricting
only to the variable θ, or the supply shock or the inflation expectations, respectively.

2.1 Robustness to alternative measures of supply shocks and inflation expecta-
tions

It is well known that Phillips curve estimates are highly sensitive to the exact empirical specifica-
tion and choice of variables; for a recent survey, see Mavroeidis, Plagborg-Moller, and Stock (2014).
Although an exhaustive exploration is beyond the scope of this paper, we do summarize the exact
specification for each variable in Table 1 and show how alternative assumptions change the results.
The bottom line is that while the magnitudes of the coefficients are sensitive to the exact specification,
consistent with the existing literature, the evidence in favor of nonlinearity when θ > 1 is robust.

To determine the slope of the Phillips curve, it is critical to control for supply shocks. The literature
has considered several proxies. One approach is to use so-called "headline shocks", i.e. the difference
between headline and core CPI inflation. The Personal Consumption Expenditures (PCE) price index
is an alternative to CPI to generate these shocks. Another common approach is to compute the dif-
ference between the change in the import prices and the GDP deflators. Figure 18, in the Appendix,
presents the raw data using these three approaches. We are agnostic about the appropriate measure

16
of supply shocks. In the benchmark specification we do a principal component analysis and use the
first principal component of these three series as a proxy. As shown in Appendix A, the main result
of interest (evidence of non-linearity when θ > 1) is robust to considering each measure separately
(see Tables 2 and 3).

Another explanatory variable is inflation expectations, which we derive through a direct measure of
expectations as in the recent literature.20

The benchmark analysis uses the two-year quarterly inflation expectations measure provided by the
Federal Reserve Bank of Cleveland, i.e. the rate that inflation is expected to average over the next
two years.21 This series is available only since 1982 Q1. For the earlier period we use the Livingston
inflation expectations survey; since this survey is conducted only twice yearly, we interpolate it to
obtain quarterly observations.22 Figure 17 in the Appendix shows this data.

Appendix A shows that our main results are robust to various alternative measures of inflation expec-
tations: 1-year CPI inflation expectations in the Survey of Professional Forecasters, five-year inflation
expectations of the Federal Reserve Bank of Cleveland and a five-year five-year forward measure
developed by Groen and Middledorp (2013); see Tables 4-6.

Finally, Tables 7 and 8 consider two alternatives to our main variables of interest: core PCE instead of
core CPI in the baseline regression and the level of θ rather than its log.

3 The model

Just like the canonical New Keynesian model, ours is designed to capture the forward-lookingness of
firms, which gives rise to expectations about future inflation as in equation (1). The main difference
in our model with respect to the standard New-Keynesian account involves the labor market, which
is the factor that generates nonlinearities.

The labor market is modeled via search and matching. As we know, in such models there is no definite
way of determining real wage, since each match generates a “surplus” shared between firms and
20 In the canonical Keynesian model, the expectation on the right-hand side of the Phillips curve, as in equation (1), is

expected inflation in the next quarter. Since we consider various types of expectation, a simple way of interpreting regressions
using longer-term expectations is to posit the one-quarter-ahead inflation expectation as

Et πt+1 = h · πt + (1 − h ) · πte
|{z} |{z}
Current Inflation Longer term inflation

where h is the weight of current inflation in predicting next-quarter inflation, while πte represents a measure of some longer-
term expectation, which serves as an anchor. What is the best measure then becomes an empirical question. As shown below,
the key result is robust to various alternative measures of inflation expectations as proxies of πte . Bernanke (2007), in discussing
the inflation forecasting model of the Federal Reserve, argues that long-term expectations seem more important for the price-
setting behavior.
21 In their estimates, they use Treasury yields, inflation data, inflation swaps, and survey-based measures of inflation expec-

tations.
22 The 12-month CPI forecast of the Livingston Survey better represents a 14-month forecast (see Carlson, 1977).

17
7

ln max

ln min
1

0
-1.5 -1 -0.5 0 0.5

Figure 8: A Model of an Inv-L NK Phillips Curve as a function of labor market tightness.

workers. Here we propose a wage-setting mechanism that is directly motivated by the observation of
Phillips (1958), that wages appear to be highly flexible when the market is tight but fall only slowly
under regular circumstances.

A road map, i.e. statement of the bottom line up-front, enables the reader to anticipate what the
model delivers. Using the same notation as in the introduction, the Inv-L NK Phillips curve is given
by
 tight

 κ tight θ̂t + κu ût + βEt πt+1 labor shortage (θ > 1)
πt = (3)

κw ŵt−1 + κ θ̂t + κu ût + κ β Et πt+1 normal ( ≤ 1 ) .

θ

We plot this relationship in Figure 8 using the regression results (column (4) of Table 1) to capture
the slope of the Phillips curve and the point at which θ hits the inflation target. The key result is
that κ tight > κ, as shown in Figure 8. At a given level of tightness, denoted by the vertical line, a
tighter labor market increases inflation. However, the curve is clearly quite flat. Once θ reaches 1 (or
ln θ = 0), inflation is barely above target at 2.5 %. It is only when θ crosses θ = 1 and enters a regime
of labor shortage that it exerts significant inflationary pressure. Increasing θ from 0.38, which is the
estimate for labor tightness being neither inflationary nor deflationary, to 1 only increases inflation
from 2% to 2.5%. But, in labor shortage territory, raising θ from 1 to 2 increases inflation from 2.5% to
6%.

18
When there is labor shortage, the response of inflation to supply shocks, ût , too is more pronounced,
tight
i.e. κu > κu . The effect of inflation expectations, however, is not clear-cut. It depends on how
wages react to expectations of future inflation. As we will see, there are some parameter values that
capture the idea of a wage-price spiral, in which case, in a slack labor market, inflation is even more
responsive to expectations of future inflation than during labor shortage, i.e. κ β > β. Finally, the û is
a composite exogenous disturbance that includes productivity, mark-up and labor-force participation
shocks.

Section 4 summarizes the microfoundations of equation (3). Section 5 discusses policy implications
where the model is cast in a form familiar to most readers.

3.1 Households

There is a continuum of representative households of measure one. The members have different disu-
tilities of working. No decision is made about each member’s hours of work (intensive margin), but
the household does decide how many members work (extensive margin). In other words, the house-
hold chooses the labor market participation rate. The utility flow at time t is given by preferences as
in Greenwood, Hercowitz and Huffman (1988):23
Z Ft 1− σ
1
U (Ct , Ft , χt , Ψt , ξ t ) = Ct − χt f d f + Ψt
ω
ξt (4)
1−σ 0

where Ct is consumption and Ft is the number of members who decide to participate in the labor
market. Each household member is indexed by f and has fixed disutility f ω from taking part in
the labor force, as in Galì (2009), with ω > 0. The variable χt is an exogenous shock to labor force
participation, and σ > 0 is a parameter. The variables Ψt and ξ t are treated as exogenous by the
household. Ψt is introduced to simplify the Euler Equation for consumption as is clarified in Section
5, and ξ t is an intertemporal disturbance that moves the natural rate of interest.

Household members are ordered by their disutility from working. For example, it may be more costly
to have an aging grandmother in the labor force than a prime age woman. Integrating the cost of labor
force participation yields
F 1+ ω
Z Ft
f ωd f = t . (5)
0 1+ω
The household decides labor force participation. Not all of the labor force is employed, however,
owing to frictions in the labor market, which are modeled by a search and matching function.

The labor force consists of the employed and the unemployed

Ft = Nt + Ut , (6)
23 The use of GHH preferences allow us to abstract from wealth effects in labor force participation, which simplifies the

algebra.

19
where Nt is workers employed by firms and Ut is unemployed workers at the end of period t – after
job search in period t.

At the beginning of each period a fraction (1 − s) of the labor force is attached to firms. Implicitly, we
think of this as employment based on existing relationships. For the purpose of this paper, however,
we do not model how these relationships are formed or keep track of them over time.24

The remaining fraction of the labor force, s, is jobless at the start of the period and is denoted by
Utb = sFt . These people search for work in period t. Their ability to enter employment is determined
by the matching function
1− η
Mt = m(Utb )η Vt , (7)

where the parameter m > 0 is matching efficiency and Vt is vacancies posted by employment agencies
(described shortly); η is a parameter with 0 < η < 1. That s can range from zero to one is attractive,
because it allows us to nest two special cases. The standard NK model assumes that s = 0 so that
the labor market is perfectly flexible. On the opposite extreme s = 1 represents the case in which all
people have to search for a job in every period.

Define the tightness of the market by θt ≡ Vt /Utb . A tight labor market means higher θt , as there are
relatively more vacancies than job seekers. Mt represents the unemployed at the start of period t who
are considered suitable for work after the search and matching process, which is carried out by an
employment agency. The number of people actually hired is Ht . As we will see, in equilibrium Mt =
Ht . Yet as it will be made clear shortly, when a hiring agency maximization problem is introduced it
will be useful to bear in mind that employment agencies can in principle post more vacancies than
the number of people firms want to hire. The hiring agency, for example, can post a number Mt of –
say – electricians at wage wt – even if the firms actually only want to hire Ht < Mt electricians.

Since in equilibrium Ht = Mt , the probability of an unemployed worker being hired at the beginning
of time t is
Ht 1− η
b
= mθt = f (θt ).
Ut
Given that sFt is the number of people searching for a job in each period, sFt f (θt ) says how many job
seekers find jobs. Total employment is therefore

Nt = (1 − s) Ft + sFt f (θt ) = Ft (1 − s + s f (θt )). (8)

Members of the household looking for a job pay a fee γb proportional to their income to the employ-
ment agency. The household’s flow budget constraint is

Bt + Pt Ct + Tt = (1 + it−1 ) Bt−1 + [1 − s + s(1 − γb ) f (θt )]Wt Ft + ZtF + ZtE , (9)

where Bt is a risk-free nominal bond denominated in units of currency at time t at the nominal interest
24 It is as if each labor force member entered the period unattached.

20
rate it , Pt is the price index associated with consumption basket Ct , Tt are lump-sum taxes, and ZtF
and ZtE are the firms’ and the employment agencies’ profits.

Considering equation (5), the household maximizes (4) subject to (9) by its choice of Ct , Bt , Ft . The
household’s optimal labor-force participation implies:

!1
(1 − s + s(1 − γb ) f (θt )) ω
Ft = wt , (10)
χt

which says that participation is increasing both in tightness and in the real wage, defined as wt ≡
Wt /Pt , but can be negatively affected by the shock χt . The optimal consumption decision implies:

1 ξ t +1
Xt−σ = β(1 + it ) Et Xt−+σ1 Π−
t +1 ξ , (11)
t

in which β is the rate of time preference, Πt ≡ Pt /Pt−1 and

Ft1+ω
Xt ≡ Ct − χt + Ψt . (12)
1+ω

Finally, a necessary condition for optimality is that the household’s intertemporal budget constraint
holds with equality.25

We assume that Ct is a consumption basket given by a Dixit-Stiglitz aggregator of the form
Z 1 ϵt −1

Ct ≡ ct (i ) ϵt di ,
0

where i indexes consumption of a good of variety i, and ϵt > 1 is the elasticity of substitution among
the differentiated goods. The household’s optimal choice of good variety i at time t implies
− ϵt
p t (i )
c t (i ) = Ct ,
Pt

where pt (i ) is the price of variety i and

Z 1 1−1ϵ
t
Pt ≡ pt (i )1−ϵt di .
0

3.2 Firms

There is a continuum of firms of measure one, each firm i producing a good of variety i. The source
of demand is household and government consumption, Ct and Gt plus the cost to the employment
agency of posting vacancies, γc Vt , measured in terms of the consumption good, where γc > 0 is a
25 Or equivalently we can state a transversality conditition; see e.g. Woodford (2003) for a discussion.

21
parameter. We assume that the government spending bundle and the vacancy cost, γc Vt , take the
same form as the Dixit-Stiglitz household consumption basket.

A generic firm i thus faces the following demand, yt (i ), for its output
− ϵt
p t (i )
y t (i ) = Yt , (13)
Pt

where pt (i ) is the price of goods of variety i and Yt = Ct + Gt + γc Vt . Firms produce according to a
decreasing-return technology

yt (i ) = At Nt (i )α (14)

where 0 < α ≤ 1 and At is labor productivity.

The problem of the firm is to maximize the expected discounted value of profits:

∞
( 2 )
ς p T (i )
Et ∑ Qt,T p T (i )y T (i ) − WT NT (i ) − −1 PT YT
T =t
2 p T −1 ( i )

subject to demand (13) and (14) where Qt ,T ≡ β T −t ( XT−σ /PT )/( Xt−σ /Pt ) is the stochastic discount
factor the household uses at time t to value future nominal income at time T. As in the price-
adjustment model of Rotemberg (1983), ς is a parameter measuring the cost of adjusting prices.26

As all firms face the same problem, there is a symmetric equilibrium in which pt (i ) = Pt and yt (i ) =
Yt , yielding the aggregate Phillips curve:

! ( −σ )
ϵt − 1 1 ϵt Wt Nt1−α X t +1 Yt+1
( Π t − 1) Π t = −1 + βEt ( Π t +1 − 1 ) Π t +1 . (15)
ς α ϵt − 1 Pt At Xt Yt

3.3 Employment agencies

There is a continuum of employment agencies of measure one, whose function is to find workers suit-
able for employment Consider the problem of a representative agency. It charges a fee proportional
to the salary of the worker whom it helps to find a job, i.e. γb Wt Mt , but has to pay a cost to post the
vacancy, γc Vt . Its problem is to maximize real profits, ZtE , by choosing the number of vacancies to
post
ZtE = γb wt Mt − γc Vt ,

taking as given its ability to find suitable workers given by (7).
26 Calvo’s price-setting model leads to the same AS equation in a first-order approximation. However, we use Rotemberg’s

assumption in order to simplify the presentation.

22
The employment agency takes both Utb and the real wage rate wt ≡ W t
Pt as given. The firm’s demand
for workers from the employment agency is Ht .27

There is in principle nothing in this problem that says Mt and Ht cannot differ, that is that the em-
ployment agency cannot produce more qualified workers than firms are prepared to hire. We assume,
however, that the agency never posts a vacancy unless it knows there is sufficient labour demand
given the matching technology. Doing otherwise would imply paying vacancy cost without any ben-
efit. Therefore we make the restriction that

Mt ≤ Htd

where Htd is an upper bound on the number of people the firms hire through the agency at time t –
the source of which we will clarify shortly. Substituting the matching function into the profit function
and using the definition of θt the problem can be written as:28

max γb wt Utb f (θt ) − γc Utb θt
θt

subject to
Utb f (θt ) ≤ Htd . (16)

When wages are perfectly flexible, they adjust so that (16) is not binding. This yields the interior
condition for optimality:

f lex
γb wt f ′ ( θt ) = γc (17)
| {z } |{z}
Marginal benefit Marginal cost

f lex
where wt denotes the flexible wage. This condition indicates that in equilibrium the marginal
benefit of posting a vacancy is equal to the marginal cost. Using the functional form for the matching
function we can express this as
f lex 1 γc η
wt = θ . (18)
m (1 − η ) γ b t
That is, the higher the wage rate, the tighter the market conditions θt . In other words, a higher wage
rate incentivizes the employment agencies to post more vacancies for a given number of unemployed
workers.

The upper bound Htd is relevant when we allow for the possibility of rigid wages. We denote the rigid
wage by wtnorm . In this case, for a given wage, the number of hires is demand-determined, i.e. given
by the number of workers firms are willing to hire at the prevailing wage wtnorm , which we denote by
27 Note, however, that the firm’s total demand for workers, i.e. N , is going to be split between the attached labor force
t
(1 − s) Ft and the labor force hired through the employment agency Ht .
28 Since the agency takes U b as given, it can control θ by its choice of V .
t t t

23
Htd . In this case the inequality constraint (16) is binding and Mt = Htd . The marginal value of each
additional hire is given by
γb wtnorm f ′ (θt ) − γc
ψt = ,
f ′ (θt )
where ψt corresponds to the Lagrange multiplier on (16) in the employment agency’s problem. This
means that when employment is demand-constrained due to rigid wages, the agency passively posts
vacancies to satisfy whatever labor demand is expressed by the firms. In the process, however, the
marginal profits are positive. Thus, at that wage rate it would be optimal for the agency to post a
new vacancy if the firms were willing to hire more workers. In the absence of a rigid wage the hiring
agencies would do this, which would drive wages downward until (17) is satisfied.

3.4 The wage norm

We now come to the heart of the nonlinearities in the model. It is common in the search literature
to simply assume that the real wage rate is fixed; see e.g. Hall (2005). Alternatively, authors such
as Blanchard and Galì (2010) posit the real wage as a function of aggregate output. Here, instead,
following Phillips (1958), we assume that how wages are set depends on how tight the labor market
is. A tight labor market is defined as more vacancies than unemployed workers, i.e.

Number of vacancies > number of job seekers.

In this case we assume that wages are fully flexible and given by (18).

An index of tightness equal to 1 corresponds to what Michaillat and Saez (2022) define as “efficient”.
This is a natural benchmark, as it is open to the intuitive interpretation of there being more “unem-
ployment at firms” than workers searching for jobs: what can be called “labor shortage” – a term
commonly used during the surge of inflation in 2021, and the one we have adopted here. As we have
seen in the empirical motivation, it also provides a reasonable point at which nonlinearities in the
Phillips curve become most important in the U.S. data.

Consider now a situation that is more typical in the U.S. labor market, i.e. one in which there are
more job seekers than vacancies, i.e. θ ≤ 1. In this case, we assume that wages are set according to a
wage “norm” given by
f lex
Wtnorm = (Wt−1 (Πet+1 )δ )λ ( Pt wt )1−λ . (19)

To understand the nature of this assumption, consider first the classic norm proposed by Keynes,
namely that wages could never fall below their previous-period value. This idea is a special case of
(19) with λ = 1 and δ = 0, so that Wtnorm = Wt−1 . This norm only prevents nominal wages from
falling, not from rising. The firm may pay wages above the wage norm in equilibrium: if the labor
market is sufficiently tight, i.e. θt > 1, then wages are given by (18).

Our specification of the norm is more general than Keynes’s and more in line with Phillips, thanks to
f lex
the introduction of λ and the fact that the norm can be driven down by market forces via wt . To see

24
this, consider the norm allowing for λ < 1 but maintain the assumption δ = 0. Nominal wages are
no longer completely rigid downward: they may now fall “very slowly” as suggested by Phillips. In
f lex
a weak labor market, i.e. θt < 1, then wt < W t
Pt so how quickly nominal wages fall depends on
how far λ is from 1. If λ = 0 then wages are completely flexible and wages will fall immediately in
f lex
response to shocks and W t
Pt = wt .

We have introduced one additional feature to the norm by including the variable Πet+1 which is rele-
vant if δ > 0. This variable captures the notion that inflation expectations can affect wage-setting be-
havior. These expectations could, for example, be anchored by the inflation target of the central bank.
Alternatively, this features allows us to model the sort of “price-wage” spiral commonly thought to
have played a role in the 1970s.

Writing the wage norm in real terms implies:
 f lex

 wt for θt > 1
wt = (20)
f lex 1−λ
( w t −1 Π −

1 λ
t ) ( Π t +1 ) ( w t
 e δλ ) for θt ≤ 1.

When wages are flexible, the optimizing behavior of the employment agency plays a central role
in determining them by equating the marginal cost with the marginal benefit of posting vacancies.
This solves the classic wage indeterminacy of standard labor-market search-matching models. This
flexible wage rate is also important in the wage norm, as it serves as an anchor towards which the
norm is pulled by a factor 1 − λ.

An alternative to (20) that we have considered, and which leads to a similar result, is to assume that
nominal wages are given by
f lex
Wt = max(Wtnorm , Pt wt ),

as in Eggertsson, Mehrotra and Robbins (2019). This means workers will never accept a job at less
than what they see as the prevailing wage (Wtnorm ), while they are naturally happy to earn higher
wages. In this case, what defines a tight market (and thus the cut-off for θ̂t ) is endogenous.29

In (20), instead, we separate the two regimes via θt ≥ 1. The main reason for choosing this modeling
strategy is its relative simplicity. Further, the data seems consistent with a tight market being well
approximated by θ > 1, as we saw in the empirical section.

To summarize: If the wage norm is binding, households are not willing to work for less. At the same
time, they are making an active labor market decision by choosing their participation rate. For any
given wage rate and labor market tightness θt , the household will choose Ft optimally. Meanwhile, the
wage norm constrains the number of workers that firms employ. Wages fall only gradually towards
the flexible wage rate when the market is weak, so that Nt is determined by firms’ demand for labor
29 Benigno and Ricci (2011) consider instead a framework in which wage setters choose their wage optimally by constraining

it not to fall.

25
at the prevailing wage. This constraint, in turn, becomes binding for the employment agency, as
discussed in Section 3.3. The result is that the market tightness θt (via vacancy postings, which are
determined by firms’ demand for additional workers) adjusts endogenously to satisfy that demand.

3.5 Equilibrium definition

To close the model, aggregate output is given by

Yt = At Ntα , (21)

since all firms behave in the same way. Finally, goods market equilibrium implies:30

Yt = Ct + Gt + sγc θt Ft , (22)

abstracting from the resources taken by the quadratic cost of price setting; see Eggertsson and Singh
(2019) for various ways in which this has been justified. The specification of the monetary policy rule
closes the model, for example with

it = ϕ(Πt , ξ t , At , χt , Gt ) (23)

for some functional form ϕ(·). This completes the description of the model. An equilibrium is defined
f lex
by a collection of stochastic processes for {it , Xt , Πt , Yt , Ft , θt , Nt , Ut , wt , wtnorm , wt }∞ t=0 that satisfy
(6), (8), (10), (11), (12), (15), (18), (20), (21), (22), and (23) given an exogenous process for { At , Gt ,
ξ t , χt , Ψt }∞
t =0 .

4 The Inv-L NK Phillips Curve

The model we have just sketched out lends itself to a natural representation that is more in the spirit
of Phillips’ original suggestion (depicted in Figure 1) than later incarnations. The central result is
that this modified curve now takes an inverse-L shape, as in Figure 1, in a first order approximation
around the steady state. Two main economic propositions underlie this inverse-L representation. The
first is simply that while, given sufficient time and price incentives, most factors of production can
typically be increased in one way or another, one factor will always be in limited supply over any
reasonable time horizon: the number of people who can work. Second, it has long been recognized
that more than other prices the price of labor (wages) falls less quickly in conditions of excess supply
(high unemployment). Together, these two observations imply that over some range, higher inflation
brings increased production, as more people are drawn into employment by firms benefiting from
lower real wages. However, given the first proposition, this process is bound to hit a wall once the
labor force is fully employed.
30 We have used Y = C + G + γc V , noticing that U b = sF . Therefore V = sF V /U b = sF θ .
t t t t t t t t t t t t

26
Define µt ≡ ϵtϵ−t 1 . A first-order approximation of the optimal pricing condition of firms, equation (15),
implies that

( ϵ − 1)
πt = (ŵt − Ât + µ̂t ) + βEt πt+1 , (24)
ς
where a hat denotes the log-deviation of a variable with respect to the steady state and πt = ln Πt .

The nonlinearities of the Phillips curve depend on how the wage rate, ŵt , is determined, i.e. whether
it is flexible as in the standard NK model or instead a wage norm applies.

The flexible wage (17) is given in log deviation by

f lex
ŵt = η θ̂t , (25)

while the wage norm (19) is

f lex
ŵtnorm = λ(ŵt−1 − πt + δEt πt+1 ) + (1 − λ)ŵt , (26)

assuming πte+1 = Et πt+1 .

Both equations for wages are exact and involve no approximation error. Using this, we can write the
Phillips curve as:

tight
labor shortage θ̂t > θ ∗


 κ tight θ̂t + κu ût + βEt πt+1
πt =
normal θ̂t ≤ θ ∗

κw ŵt−1 + κ θ̂t + κu ût + κ β Et πt+1


where all the parameters are detailed in Appendix D together with the definition of the variable ut .
Here θ ∗ ≡ − log θ̄ > 0 where θ̄ < 1 is the steady state we expand the model around so that θ̂t > θ ∗
means θt > 1.

Note that when λ > 0 the following inequalities hold:

• κw > 0,

• κ tight > κ,
tight
• κu > κu ,

• κ β > β if δ > β.31

Clearly, when λ = 0, κw = 0 and the two curves are identical.
31 See the Appendix for details.

27
There are several important differences between the Phillips curve in a tight labor market and under
normal circumstances. First, under normal conditions it is marked by persistence via its dependence
on the past wage rate, κw > 0. Second, it is flatter under normal circumstances than in labor shortage,
i.e. (κ < κ tight ). Third, supply shocks have a greater impact when there is labor shortage, i.e. (κu <
tight
κu ). Finally, the response to inflation expectations is ambiguous, depending critically on the degree
to which the wage norm incorporates any dependence of expected inflation.

Let us now turn to the policy implications.

5 The policy framework for the Inv-L NK Phillips Curve

We have relied on the fact that the Phillips curve can be written in terms of θt in the vein of recent
empirical work suggesting that this forecasts inflation better than some other common proxies of
aggregate demand. There is, however, a direct link between θ and output. Given that most policy
discussion turns on output and inflation, there is some advantage in casting the model in these terms.

To further streamline the model and focus more closely on the key points, we make the simplify-
ing assumption that in equilibrium Xt = Ct = Yt − Gt .32 This implies that the Euler Equation for
consumption takes the traditional NK form:

Ŷt − Ĝt = Et Ŷt+1 − Et Ĝt+1 − σ−1 (ı̂t − Et πt+1 − r̂te ),

in which r̂te ≡ ξ̂ t − Et ξ̂ t+1 is the natural interest rate component that is driven by demand shocks. As
a second simplification, we assume a simple wage norm of the form
 f lex

 wt for θt > 1
wt =
f lex 1−λ
wλ (Πt−1 )λ (Πet+1 )δλ (wt

) for θt ≤ 1.


in which w is the steady-state real wage. Thanks to this assumption, we can omit the lagged vari-
able in the Phillips curve when the labor market is slack, which, while empirically relevant, is not
essential to the policy analysis. Making the model perfectly forward-looking allows a tight analytic
characterization, which will be helpful, as we will see.

The Phillips curve is now given by

tight (Ŷ + ω −1 χ̂ − Â ) + k tight ( µ̂ − Â ) + βE π if θ̂t > θ ∗

 k
 t t t µ t t t t +1
πt =
if θ̂t ≤ θ ∗ ,

k(Ŷt + ω −1 χ̂t − Ât ) + k µ (µ̂t − Ât ) + k β Et πt+1


32 To obtain this result, we assume that in equilibrium Ψ 1+ ω
t = Ft / (1 + ω ).

28
tight
where the coefficients, which are detailed in Appendix E, satisfy ktight > k > 0, k µ > k µ if λ > 0.
The relationship between β and k β is again ambiguous. To simplify the analysis we set β = k β = 1 so
that there is no long-run trade-off between inflation and output, but this is not essential to our main
results.33

We close the model with a simple policy rule:

ı̂t = r̂te + ϕπ (πt − π ∗ ) + ϵt

where ϕπ > 1 is a reaction coefficient of inflation deviating from the target, denoted by π ∗ . Recall
that r̂te contains movements in aggregate demand explained by the demand disturbance ξ t . In other
words, we assume that the central bank fully offsets any exogenous demand shock, but we leave open
the question of whether it will do so in response to supply shocks, or demand shocks like government
spending. The central bank response to these variables can be incorporated into the monetary policy
shock ϵt , which for now we leave unspecified.

One prominent hypothesis on the causes of inflation in the 1970s, set forth by Clarida, Gali and Gertler
(2000), is that the central bank did not react strongly enough to inflation by raising the interest rate, so
ϕπ < 1. This assumption leads to equilibrium indeterminacy, meaning that there are infinite possible
paths of inflation that qualify as a solution. Our purpose here, however, is to highlight the effect of
inflation expectations becoming unanchored. That is, we are interested in comparative statics with
respect to the central bank’s long-run inflation target. If the model has an infinite number of equilibria,
comparative statics are meaningless. So it is useful to assume that ϕπ > 1. Moreover, since we allow
for a policy shock ϵt , the policy rule is in any case still rich enough to encompass the possibility that
monetary policy did not respond strongly enough to the rise in inflation, thus capturing the spirit of
their hypothesis.

5.1 Understanding the inflation and disinflation of the 1970s

We first consider how the model explains the increase in inflation in the 1970s, often referred to as
the “Great Inflation”. This will be a familiar story to most readers, and our version does not differ
substantially from the conventional account (see e.g. Erceg and Levin (2003) and Goodfriend and
King (2006)). Nevertheless, it is helpful to spell it out clearly within our model, so that we can contrast
with our account of the inflationary surge of the 2020s. A convenient analytical device is to split the
model into the short run, denoted by S, and the long run, denoted by L. A major simplification, which
allows us to illustrate the main points analytically, is to assume that all shocks occur in the short run
and in the long run revert to zero with a fixed probability (1 − τ ). The short run is described by the
supply shock µS .
33 By assuming that β = k = 1 then in steady state, where all variables take a constant value, the Phillips curve implies
β
that inflation cancels out and output is equal to zero.

29
We consider the possibility that the public can have a different belief about the central bank’s long-
run inflation target, which we assume is π ∗ . This means that even if long-run inflation will eventually
stabilize, i.e. π L = π ∗ , as was eventually the case of the U.S., we want to allow for the possibility that
people’s beliefs in the 1970s, denoted π Le , were different from what turned out to be the case ex post.
The motivation for this is straightforward: several long-term expectation measures in the 1970s and
early 1980s cited in the introduction suggested that five-to-ten year inflation expectations reached 10
% and only declined gradually during the 1980s (see especially footnote 6).

In the long run, we assume the labor market is back to "normal", i.e. θ̂t < θ ∗ . In the absence of shocks,
the unique bounded solution is simply given by π L = π ∗ and ŶL = 0.

Let us now consider the short run. Suppose, for simplicity, that expected long-term inflation expecta-
tions are above π ∗ , as observed in the data, and constant at π Le > π ∗ . Then inflation expectations in
the short run are

ES πS+1 = τπS + (1 − τ )π Le .

Since our model is characterized by long-run monetary policy neutrality, then even though people
expect π Le > π ∗ , it remains the case that a unique bounded solution is given by ŶL = 0.

Setting all shocks to zero (except the cost push shock) the Euler Equation for consumption is:

ϕπ − τ
ŶS = −σ−1 (πS − π ∗ ) + σ−1 (π Le − π ∗ ) (27)
1−τ

while the Inv-L NK Phillips Curve is:

tight

kµ
ktight
 1−τ ŶS + 1−τ µ̂S + π Le − π ∗

 θ̂t ≥ θ ∗
∗
πS − π = (28)
< θ∗.

 k kµ e ∗
1−τ ŶS + 1−τ µ̂S + π L − π
 θ̂t

See Appendix E for further details of the derivations.

These two equations are plotted in Figure 9, the aggregate demand equation (27) in red and the Inv-L
NK Phillips Curve in blue. Consider the initial equilibrium when inflation is at target and output at
potential at point A. While the figure is meant to be purely illustrative, it captures the fact that the
Inv-L NK Phillips curve is very flat. Moreover, entering the 1970s, θ was less than one and held at
that level during that period. Thus the equilibrium is determined on the flat part of the Phillips curve.
Consider first the effect of a supply shock. This will directly shift the blue curve upward, with higher
inflation and lower output. The oil shocks of the 1970s clearly played a role in the inflation surge.
Of greater interest, however, is the effect of a rise in the central bank’s inflation target perceived

30
Figure 9: Inflation and output determination in response to an increase in long-run inflation expecta-
tions.

by the private sector, i.e. π Le > π ∗ . If the public believes that the central bank will set a higher
inflation target in the long run, this results in a one-to-one upward shift of the blue curve. Hence,
it immediately generates inflation just like a supply shock, for a given level of demand. This is not
the whole story, however. As shown by equation (27), the rise in long-term inflation expectations
also increases demand in the short run by reducing the real interest rate and so making borrowing
cheaper (real rates were negative through much of the 1970s.) This shifts the red curve up. It is easy
to show, however, that the effect on output is always negative if the model has a unique equilibrium,
i.e. under the condition that ϕπ > 1.34

We can use the same framework to see why disinflation can be very costly if the private sector believes
that the long-term inflation target is high, despite the central bank’s claims to the contrary. How
costly this is depends fundamentally on how the perceptions are formed. Consider the possibility
that people will only reconsider their perception of the long-run inflation target if they see some
significant reduction in current inflation. In this case, because of the flatness of the Phillips curve when
θ̂ < θ ∗ , it will be very costly to bring inflation down.

5.2 Understanding the inflationary surge and a potential soft landing in the 2020s
34 It would be wrong, however, to draw the conclusion that ϕ < 1 implies that higher long-term inflation expectations
π
increase output. If ϕπ < 1 there is an infinite number of equilibria. The model cannot predict which one will beequilibrium is
to be chosen in response to a change in any of the exogenous variables if ϕπ < 1. Comparative statics are thus meaningless.

31
Figure 10: Inflation and output determination in response to an increase in demand.

Let us now contrast this fairly conventional account of the Great Inflation with the surge in inflation
in the 2020s, in the framework of our model. The fundamental difference is that in this case θˆt > θ ∗
so that the demand intersects the steeper segment of the Phillips curve.

The model can be simplified as before, dividing it into short and long run. Now, however, we assume
that long-run inflation expectations are anchored, so that π Le = π L = π ∗ . Moreover, we now focus on
the short-run demand (fiscal) shock ĜS , the supply shock µ̂S and the monetary policy shock ϵS . We
will also consider a shock to labor force participation, χ̂S 35

As shown in Appendix E, in the model the short run can then be summarized by the following two
equations:

σ −1 ϕπ − τ
ŶS = ĜS − ϵ S − σ −1 ( π S − π ∗ ), (29)
1−τ 1−τ
| {z }
Demand Stimulus

tight

tight κµ
 κ1−τ (ŶS + ω1 χ̂S ) + 1−τ µ̂S

 θ̂t ≥ θ ∗
πS − π ∗ = (30)
< θ∗.

 κ 1 κµ
1−τ (ŶS + ω χ̂S ) + 1−τ µ̂S
 θ̂t
35 This can in principle be permanent, which involves some minor complications.

32
The equilibrium is shown in Figure 10. Consider first the equilibrium at point A. We think of this as
representing the pre-pandemic situation, say 2018. As we have seen, inflation took off in 2021. Our
main hypothesis is that this was due to the labor market tightness, which in the current framework
shows up as an increase in aggregate demand through government spending, GS , and an expansion-
ary monetary policy, represented by negative ϵS . Note also that supply shocks, such as a reduction in
labor force participation, will also produce labor market tightness – a point we return to shortly. That
there had been a substantial demand stimulus was relatively well known by 2021. Indeed, it was the
ground for widespread criticism of the administration, most notably a series of articles by Lawrence
Summers (2021). Our hypothesis on why the surge in inflation nevertheless caught policymakers and
private forecasters by surprise is that they assumed the Phillips curve was given by the flat part of
Figure 10, so that even if the stimulus was indeed excessive, the impact on inflation would be minor,
as illustrated by point B, the intersection of demand and the flat part of the Phillips curve. With labor
market tightness measured by θ rising to a level not seen since measurements have been available,
our key hypothesis is that the economy was instead on the upward sloping segment of the Phillips
curve, a region that was a key prediction of Phillips himself in the original article (1958), but one that
may have been overlooked, since the Great Inflation of the 1970s was driven by different forces. This
degree of tightness or labor shortage had not been seen since the Korean and Vietnam wars.

If one accepts our basic premise that the inflationary surge was driven by labor shortage, there is a
silver lining. If the surge is driven not by expectations – which may be hard to rein in – but instead by
a steep Phillips curve, it should be much less costly to bring inflation down to target. A steep Phillips
curve implies "easy up" – i.e. a relatively small output gain is associated with the inflation – but
also the converse, "easy down" – small output losses from bringing inflation under control. Leaving
aside the trade-off implied by the supply shock, all that is needed is for the central bank to raise the
interest rate to offset the increase in demand, which is summarized by ĜS plus the shock to labor
force participation χ̂S . It is easy to confirm that if we select ϵS = σ(1 − τ )( ĜS + ω1 χ̂S ) then inflation
is on target and the output gap is zero. Translated to interest rates, this means that if a central bank
finds itself on the upward-sloping segment of the Phillips curve it can hit its inflation target and attain
potential output by raising interest rates according to the formula:

1
ît = r̂te + σ(1 − τ )( ĜS + χ̂ ). (31)
ω S

The nonlinearity of the Phillips curve implies that reaching point D in Figure 10 can reduce inflation
significantly with a relatively small sacrifice of output – a soft landing. It is an open question, how-
ever, how far the non-inflationary level of output is from the kink point, defining as non-inflationary
output as that consistent with target inflation. What that output actually is, and the degree of labor
tightness associated with it, remains an open question to which we return in the conclusion.

33
63.5

62.5

% 62

61.5

60.5
20 20 20 20 20 20 20 20 20 20
13 14 15 16 17 18 19 20 21 22

Figure 11: Labor force participation in the U.S. during the last decade.

6 Conclusion

In this paper we have proposed a reformulation of what has become known as the canonical New
Keynesian Phillips curve and replaced it with one that admits significant nonlinearities. Our hypoth-
esis is that the nonlinearity is responsible for the increase in inflation in the 2020s. We conjecture that
a key reason why policymakers and market participants alike failed to foresee the surge in inflation,
or its persistence is that they implicitly or explicitly assumed a “flat” Phillips curve. Even after sub-
stantial inflation had already occurred, the reassurance that expectations were holding stable further
induced the belief that the surge was merely transitory. One question is why the Federal Reserve
did not raise interest rates more quickly. Possibly the new policy framework announced in 2020 put
greater emphasis on the employment side of the Fed’s dual objective. Yet, at the same time, it ac-
knowledged that there was no agreement on any precise measure of how close the US economy was
to full employment at any given point in time. This, of course, contrasts very sharply with the other
side of the mandate, i.e. inflation, for which there is broad consensus on how the Fed can attain its
objective.

Figure 11 sheds some light on why policymakers may have believed in 2021 that even though the
traditional gauge of labor slack, i.e. unemployment, was very low, this did not capture the full picture.
The unemployment rate only tells us how many active job seekers there are. As the figure reveals,
however, participation collapsed with the COVID-19 epidemic, which might have suggested to many
that there was still considerable room for employment to grow further. Moreover, given the flat
Phillips curve – the professional consensus at the time – and stable inflation expectation, it might have
been tempting for policymakers to explore the possibility that the US economy could attract greater
labor force participation, e.g. similar to pre-pandemic level with relatively low risk of inflation. In
terms of the dual mandate, conditional on a flat Phillips curve, this could easily have been seen at the

34
time as a situation with possible high reward and relatively limited downside risk. The bottom line of
this paper, however, is that the inflationary risk of allowing the labor market to tighten too much, to
a degree we have defined as labor shortage, generates much greater upside risk for inflation than has
been commonly thought. An important reason for this underestimation of inflation risk is no doubt
the unprecedented labor shortage, historically unprecedented except in wartime, and the countless
estimates of the slope of the Phillips curve that did not incorporate wartime. We have sought first to
show this empirically and then to build a model to explain it. The good news, in any case, is that if
our theory is correct the cost of taming inflation triggered by a labor shortage, but with stable inflation
expectations, can be expected to be much lower than it was in the 1970s.

35
References
[1] Ball, Laurence, Daniel Leigh and Prachi Mishra. 2022. Understanding U.S. Inflation During the
COVID Era. Brookings Papers on Economic Activity.

[2] Barattieri, Alessandro, Susanto Basu, and Peter Gottschalk. 2014. Some Evidence on the Impor-
tance of Sticky Wages. American Economic Journal: Macroeconomics, 6(1): 70-101.

[3] Barnichon, Regis. 2010. Building a Composite Help-Wanted Index. Economics Letters 109(3):
175-178.

[4] Barnichon, Regis and Adam Hale Shapiro. 2022. What’s the Best Measure of Economic Slack?
FRBSF Economic Letter 2022-04.

[5] Benati, Luca. 2010. Evolving Phillips Trade-Off. ECB Working Paper No. 1176.

[6] Benigno, Pierpaolo, and Luca Antonio Ricci. 2011. The inflation-output trade-off with downward
wage rigidities. American Economic Review 101(4): 1436-1466.

[7] Bernanke, Ben S. 2007. Inflation Expectations and Inflation Forecasting. Federal Reserve Bank of
Governors, https://www.federalreserve.gov/newsevents/speech/bernanke20070710a.htm.

[8] Bewley, Truman F. 1999. Why Wages Don’t Fall During a Recession. Harvard University Press.

[9] Blanchard, Olivier. 2016. The Phillips Curve: Back to the 60s? American Economic Review,
106(5): 31–34.

[10] Blanchard, Olivier, and Jordi Galí. 2010. Labor Markets and Monetary Policy: A New Keynesian
Model with Unemployment. American Economic Journal: Macroeconomics 2 (2): 1-30.

[11] Blanchard, Olivier, Eugenio Cerutti, and Lawrence Summers. 2015. Inflation and Activity – Two
Explorations and their Monetary Policy Implications. NBER Working Paper No. 21726.

[12] Blinder, Alan. 2022. A Monetary and Fiscal History of the United States, 1961-2021. Princeton
University Press: Princeton, NJ.

[13] Carlson, John A. 1977. A study of price forecasts. In: Annals of Economic and Social Measure-
ment, Volume 6, number 1, 27-56.

[14] Cerrato, Andrea and Giulia Gitti. 2023. Inflation Since COVID: Demand or Supply. Unpublished
manuscript, University of California at Berkeley.

[15] Christ, Carl F. 1994. The Cowles Commission’s Contributions to Econometrics at Chicago, 1939-
1955. Journal of Economic Literature 32.1: 30-59.

[16] Clarida, Richard, Jordi Galí and Mark Gertler. 2000. Monetary Policy Rules and Macroeconomic
Stability: Evidence and Some Theory. Quarterly Journal of Economics 115(1): 147-180.

36
[17] Coibion, Olivier and Yuriy Gorodnichenko. 2015. Is the Phillips Curve Alive and Well after All?
Inflation Expectations and the Missing Disinflation. American Economic Journal: Macroeco-
nomics 7(1): 197-232.
Coibion, Olivier, Yuriy Gorodnichenko and Rupal Kamdar. 2018. The Formation of Expectations,
Inflation, and the Phillips Curve. Journal of Economic Literature, 56(4): 1447-1491.

[18] Domash, Alex and Lawrence Summers 2022. How Tight are U.S. Labor Market? NBER Working
Paper No. 29739.

[19] Erceg, Christopher, and Andrew Levin. 2003. Imperfect Credibility and Inflation Persistence.
Journal of Monetary Economics, 50: 915-944.

[20] Eggertsson Gauti B., Neil R. Mehrotra and Jacob Robbins. 2019. A Model of Secular Stagnation:
Theory and Quantitative Evaluation. American Economic Journal: Macroeconomics 11 (1).

[21] Eggertsson Gauti B. and Sanjay Singh. 2019. Log-Linear Approximation versus an Exact Solution
at the ZLB in the New Keynesian Model. Journal of Economic Dynamics and Controls 105.

[22] Fallick, Bruce, Michael Lettau, and William Wascher. 2011. Downward Nominal Wage Rigidity
in the United States During the Great Recession. Mimeo, Federal Reserve Board.

[23] Friedman, Milton. 1968. The Role of Monetary Policy. The American Economic Review 58 (1):
1-17.

[24] Furman, Jason and Wilson Powell III. 2021. What’s the Best Measure of Labor Market Tightness?
Peterson Institute for International Economics, November 22, 2021.

[25] Gagliardone, Luca and Mark Gertler. 2023. Oil Prices, Monetary Policy and Inflation Surges.
Unpublished manuscript, New York University.

[26] Gagnon, Joseph and Christopher G. Collins. 2019. Low Inflation Bends the Phillips Curve. Peter-
son Institute of International Economics, W.P. 19/6.

[27] Galì Jordi. 2009. Unemployment Fluctuations and Stabilization Policies: A New Keynesian Per-
spective. MIT Press, Boston MA.

[28] Galì Jordi. 2015. Monetary Policy, Inflation, and the Business Cycle. Princeton University Press,
Princeton NJ.

[29] Goodfriend, Marvin, and Robert G. King. 2005. The Incredible Volcker Disinflation. Journal of
Monetary Economics, 52: 981-1015

[30] Gopinath, Gita. 2023. Crisis and Monetary Policy. Finance Development, International Monetary
Fund.

[31] Gordon, Robert J. 1977. The Theory of Domestic Inflation. The American Economic Review, 67(1):
128–134.

37
[32] Gordon, Robert J. 2013. The Phillips Curve is Alive and Well: Inflation and the NAIRU During
the Slow Recovery. NBER Working Paper No. 19390.

[33] Greenwood, Jeremy, Zvi Hercowitz, and Gregory W. Huffman. 1988. Investment, Capacity Uti-
lization, and the Real Business Cycle. The American Economic Review: 402-417.

[34] Groen, Jan and Menno Middledorp. 2013. Creating a History of U.S. Inflation Expectations. Lib-
erty Street Economics.

[35] Hall, Robert E. 2005. Employment Fluctuations with Equilibrium Wage Stickiness. American
Economic Review 95 (1): 50-65.

[36] Harding, Martin, Jesper Linde and Mathias Trabandt. 2023. Understanding Post-COVID Infla-
tion Dynamics. Unpublished manuscript, Bank of Canada.

[37] Hazell, Jonathon, Juan Herreno, Emi Nakamura and Jon Steinsson. 2022. The Slope of the Phillips
Curve: Evidence from U.S. States. The Quarterly Journal of Economics 137(3): 1299-1344.

[38] Keynes, John Maynard. 1936. The General Theory of Employment, Interest, and Money. New
York.

[39] Kimball, Miles S. 1997. The Quantitative Analytics of the Basic Neomonetarist Model. Journal of
Money, Credit, and Banking 27(4): 1241-1277.

[40] Lenoir, Marcel. 1913. Etudes sur la formation et le mouvement des prix. Paris: Giard et Briere.

[41] Malthus, Thomas R. 1798. An Essay on the Principle of Population. J. Johnson, London.

[42] Mavroeidis, Sophocles, Mikkel Plagborg-Moller, and James H. Stock. 2014. Empirical Evidence
on Inflation Expectations in the New Keynesian Phillips Curve. Journal of Economic Literature
52(1): 124-188.

[43] Matheson Troy and Emil Stavrev. 2013. The Great Recession and the inflation puzzle. Economics
Letters 120(3): 468-472.

[44] McLeay, Michael and Silvana Tenreyro. 2019. Optimal Inflation and the Identification of the
Phillips Curve. NBER Macroeconomics Annual 34, 199-255.

[45] Michaillat, Pascal. 2014. A Theory of Countercyclical Government Multiplier. American Eco-
nomic Journal: Macroeconomics, 6 (1): 190-217.
√
[46] Michaillat, Pascal and Emmanuel Saez. 2022. u∗ = uv. Unpublished Manuscript. Brown Uni-
versity.

[47] Mortensen, Dale T. and Christopher A. Pissarides. 1994. Job Creation and Job Destruction in the
Theory of Unemployment. Review of Economic Studies 61: 397–415.

[48] Petrosky-Nadeau, Nicolas and Lu Zhang. 2021. Unemployment crises. Journal of Monetary Eco-
nomics 117: 335-353.

38
[49] Phelps, Edmund S. 1967. Phillips Curves, Expectations of Inflation and Optimal Unemployment
over Time. Economica 34: 254-281.

[50] Phillips, Alban W. 1958. The Relation between Unemployment and the Rate of Change of Money
Wage Rates in the United Kingdom, 1861-1957. Economica 25(100): 283–299.

[51] Rockoff, Hugh. 1981. Price and Wage Controls in Four Wartime Periods. The Journal of Economic
History 41(2): 381-401.

[52] Rotemberg Julio. 1982. Monopolistic Price Adjustment and Aggregate Output. The Review of
Economic Studies 49(4): 517–531.

[53] Samuelson, Paul A., and Robert M. Solow. 1960. Analytical Aspects of Anti-Inflation Policy. The
American Economic Review 50 (2): 177–94.

[54] Schmitt-Grohe, Stephanie, and Martin Uribe. 2016. Downward Nominal Wage Rigidity, Cur-
rency Pegs, and Involuntary Unemployment. Journal of Political Economy, 124(5): 1466-1514.

[55] Smith, Simon C., Allan Timmermann, and Jonathan Wright. 2023. Breaks in the Phillips Curve:
Evidence from Panel Data. NBER Working Paper No. 31153.

[56] Summers, Lawrence. 2021. The Biden Stimulus is Admirably Ambitious. But it Brings
Some Big Risks, Too. Washington Post, February 5th. Retrieved on April 1, 2023 at:
https://www.washingtonpost.com/opinions/2021/02/04/larry-summers-biden-covid-
stimulus/

[57] Tobin, James. 1972. Inflation and Unemployment. The American Economic Review 62 (1/2);

[58] Woodford, Michael. 2003. Interest Rate and Prices. Princeton University Press: Princeton NJ.

39
A Appendix: Additional Tables

Table 2 presents the OLS estimates of regression (2) with the same variables as Table 1, except that we
proxy the supply shock with the four-quarter average of the CPI headline shock.

Table 2: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.4149∗∗∗ 0.2474 0.3341∗∗∗ 0.0268
(0.1013) (0.2515) (0.1100) (0.2306)

ln θ 0.5863∗∗∗ 0.7160∗ 0.2601 0.5130
(0.1751) (0.3759) (0.1899) (0.3747)

θ≥1 3.1219∗∗∗ 3.9785∗∗∗
(0.9679) (1.4907)

µ shock 0.1317 0.2020 0.1175 0.1583
(0.0816) (0.1990) (0,0816) (0.1974)

Inflation expectations 0.6207∗∗∗ 0.8187 0.7313∗∗∗ 0.4308
(0.1086) (0.6578) (0,1153) (0.6839)

Constant 0.5091∗∗∗ 0.9409 0.2323∗∗∗ 0.4507
(0.1554) (0.4353) (0.1609) (0.4136)

R2 adjusted 0.808 0.478 0.816 0.518
Observations 259 57 259 57

· ***,**,* denote statistical significance at the 1, 5, and 10 percent level, respectively.
· Newey-West standard errors.
· (1) and (3): sample 1958 Q1 – 2022 Q3
· (2) and (4): sample 2008 Q3 – 2022 Q3

40
Table 3 presents the OLS estimates of regression (2) with the same variables as Table 1, except that we
proxy the supply shock with the four-quarter average import-price shock.

Table 3: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.4035∗∗∗ 0.2762 0.3100∗∗∗ 0.0324
(0.0952) (0.2561) (0.0949) (0.2349)

ln θ 0.5623∗∗∗ 0.6899∗ 0.1611 0.4856
(0.1798) (0.3791) (0.1876) (0.3666)

θ≥1 3.6080∗∗∗ 4.2738∗∗∗
(0.8122) (1.3691)

µ shock 0.0385∗∗∗ 0.0118 0.0462∗∗ 0.0169
(0.0194) (0.0389) (0,020) (0.0399)

Inflation expectations 0.6009∗∗∗ 1.0544 0.7195∗∗∗ 0.5310
(0.1072) (0.6214) (0,1068) (0.6771)

Constant 0.5689∗∗∗ 1.0202∗∗ 0.2525 0.4749
(0.1587) (0.4726) (0.1592) (0.4200)

R2 adjusted 0.811 0.462 0.821 0.511
Observations 261 57 261 57

· ***,**,* denote statistical significance at the 1, 5, and 10 percent level, respectively.
· Newey-West standard errors.
· (1) and (3): sample 1957 Q3 – 2022 Q3
· (2) and (4): sample 2008 Q3 – 2022 Q3

41
Table 4 presents the OLS estimates of regression (2) with the same variables as Table 1, except that
2-year Cleveland-Fed inflation expectation is replaced by the 1-year CPI inflation expectations of the
U.S. Survey of Professional Forecasters.

Table 4: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.3086∗∗∗ 0.1240 0.2202∗∗ −0.0325
(0.0984) (0.2798) (0.0987) (0.2523)

ln θ 0.6865∗∗∗ 0.4441 0.3236∗ 0.3668
(0.1851) (0.3532) (0.1800) (0.3681)

θ≥1 3.3129∗∗∗ 3.9512∗∗
(0.8372) (1.8451)

µ shock 0.0361∗ 0.0113 0.0426∗∗ 0.0159
(0.0196) (0.0372) (0.0200) (0.0363)

Inflation expectations 0.7636∗∗∗ 1.8287∗∗∗ 0.8789∗∗∗ 0.9948
(0.1107) (0.6043) (0.1096) (0.7480)

Constant 0.4732∗∗∗ 0.3084 0.1656 0.1253
(0.1507) (0.3936) (0.1457) (0.4207)

R2 adjusted 0.821 0.479 0.830 0.516
Observations 251 57 251 57

42
Table 5 presents the OLS estimates of regression (2) with the same variables as Table 1, except that
inflation expectations are proxied by the 5-year inflation expectations of the Cleveland Fed until 1982
Q2, which are patched with PFS 1-year inflation expectations for the GDP deflator until 1970 Q2 and
the interpolated 12-month Livingston inflation expectations until 1960 Q1.

Table 5: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.3947∗∗∗ 0.3336 0.2895∗∗∗ 0.0189
(0.1031) (0.2586) (0.1031) (0.2281)

ln θ 0.7136∗∗∗ 0.8250∗∗ 0.3182 0.5253
(0.1920) (0.4080) (0.1983) (0.3639)

θ≥1 3.7656∗∗∗ 5.0882∗∗∗
(0.9003) (1.358)

µ shock 0.0496∗∗ 0.0316 0.0595∗∗∗ 0.0301
(0.0191) (0.0399) (0.0198) (0.0394)

Inflation expectations 0.6451∗∗∗ 0.4761 0.7864∗∗∗ −0, 1318
(0.1239) (0.7638) (0.1227) (0.8294)

Constant 0.5734∗∗∗ 0.9050∗ 0.2415 0.2262
(0.1641) (0.4689) (0.1705) (0.3756)

R2 adjusted 0.808 0.424 0.819 0.502
Observations 251 57 251 57

43
Table 6 presents the OLS estimates of regression (2) with the same variables as Table 1, except that
the 2-year Cleveland-Fed inflation expectations are replaced by the five-year five-year forward infla-
tion expectations back-casted by Groen and Middledorp (2013) until 1971 Q4. The expectations are
patched with the interpolated 12-month Livingston inflation expectations until 1960 Q1.

Table 6: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.5877∗∗∗ 0.1489 0.5744∗∗∗ −0.0241
(0.0851) (0.2468) (0.0896) (0.2722)

ln θ 0.8516∗∗∗ 1.8833∗∗∗ 0.7433∗∗∗ 1.4149∗∗
(0.2014) (0.5825) (0.2133) (0.6458)

θ≥1 1.0394 3.3607∗
(0.7077) (1.8404)

µ shock 0.0677∗∗∗ −0.0013 0.0711∗∗∗ 0.0015
(0.0216) (0.0262) (0.0219) (0.0263)

Inflation expectations 0.3291∗∗∗ 1.7207∗∗∗ 0.3429∗∗∗ 1.3148∗∗
(0.0833) (0.5205) (0.0857) (0.6084)

Constant 0.5384∗∗∗ 0.9237∗∗ 0.4451∗∗∗ 0.5526
(0.1283) (0.4314) (0.1421) (0.4538)

R2 adjusted 0.784 0.531 0.784 0.511
Observations 251 57 251 57

44
Table 7 presents the OLS estimates of regression (2) with the same variables as Table 1, except that
PCE core inflation rate replaces CPI core inflation as the dependent variable.

Table 7: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.5373∗∗∗ 0.3617 0.4373∗∗∗ 0.2269
(0.075) (0.2446) (0.0809) (0.2813)

ln θ 0.2638∗∗ 0.3946 −0.0313 0.2914
(0.1202) (0.2813) (0.1483) (0.3026)

θ≥1 2.5690∗∗∗ 1.9776
(0.7126) (1.3985)

µ shock 0.0367∗∗∗ 0.0256 0.0438∗∗∗ 0.0357
(0.0134) (0.034) (0,0138) (0.0321)

Inflation expectations 0.3971∗∗∗ 0.7925 0.5015∗∗∗ 0.5391
(0.0726) (0.545) (0.0812) (0.5439)

Constant 0.1832∗ 0.5216∗ −0.0699 0.2310
(0.106) (0.2738) (0.1362) (0.3306)

R2 adjusted 0.865 0.488 0.871 0.498
Observations 251 57 251 57

45
Table 8 presents the OLS estimates of regression (2) with the same variables as Table 1, except that the
level of θ is used rather than its log.

Table 8: Phillips Curve Estimates
(1) (2) (3) (4)
1960-2022 2008-2022 1960-2022 2008-2022

Lagged inflation 0.3355∗∗∗ 0.1677 0.2754∗∗∗ 0.1452
(0.0936) (0.2561) (0.0913) (0.2612)

θ 1.1832∗∗∗ 1.5252∗∗ 0.2054 0.9361
(0.2725) (0.7491) (0.3687) (0.6318)

θ≥1 0.9185∗∗∗ 0.5612
(0.3123) (0.7061)

µ shock 0.0397∗∗ 0.0169 0.0477∗∗ 0.0129
(0.0195) (0.0379) (0,020) (0.0342)

Inflation expectations 0.7080∗∗∗ 0.7379 0.7982∗∗∗ 0.7152
(0.1037) (0.6844) (0,0995) (0.7246)

Constant −0.5876∗∗∗ −0.5394 −0.1187 −0.3397
(0.1611) (0,4470) (0.2150) (0.3779)

R2 adjusted 0.822 0.494 0.828 0.495
Observations 251 57 251 57

46
B Appendix: Additional Figures

12
79 80
74

81
78
8 73

6 77
90
69
70
76
68 89 82
4 88 87
84
66
72 71 85
83
67
2
65
64 60
63
62 86
61
0
3 4 5 6 7 8 9 10

Figure 12: Empirical breakdown of the Phillips Curve in the 1970s as discussed in the Introduction,
sample 1960-1990.
7

4
%
3

2
2% Inflation Target
1

0
2019 2020 2021 2022 2023 2024 2025 2026

Figure 13: PCE-index inflation at annual rate (red line) and the inflation forecast of the Summary of
Economic Projections (SEP) (dashed lines) of the Federal Reserve up to and during the inflation surge.

47
7

4
%
3

2
2% Inflation Target
1

0
2019 2020 2021 2022 2023 2024 2025 2026

Figure 14: PCE-index inflation at annual rate (red line) and the inflation forecast of the Survey of
Professional Forecasters (SPF) (dashed lines) up to, and during, the inflation surge.
14

(%) 6

-2
70 74 78 82 86 90 94 98 02 06 10 14 18 22
19 19 19 19 19 19 19 19 20 20 20 20 20 20

Figure 15: This figure contrasts CPI inflation at annual rates with the five-year expected inflation
rate compiled by the Cleveland Fed and five-year five-year forward inflation expectations, which are
market-based from 1997 and back-casted by Groen and Middledorp (2013) to 1970.

48
7

2
=1
1

0
19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19 19
13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59

Inflation (%)
10

-5

-10

-15
19 19 19 19 19 19 19 19 19 19
1 19 19 19 19 19 19 45 19 19 19 19
3 55 57 19
13 15 17 19 21 23 25 27 29 3 33 35 37 39 41 43 19 47 49 51 5 19 19 59

Figure 16: Top panel: (θ) vacancy-to-unemployed ratio. Bottom panel: CPI inflation rate at annual
rates. Sample 1913-1959.

(%)
6

-2
60 64 68 72 76 80 84 88 92 96 00 04 08 12 16 20
19 19 19 19 19 19 19 19 19 19 20 20 20 20 20 20

Figure 17: CPI core inflation (annualized quarterly rates). 2-year inflation expectations of the Cleve-
land Fed patched, before 1982 Q1, with 12-month Livingston inflation expectations. 1-year CPI infla-
tion expectations of SPF patched, before 1981 Q3, with 12-month Livingston inflation expectations.

49
(%) (%)

19 -60 -40 -20 0 20 40 60 19 -10 -5 0 5
60 60
19 19
64 64
19 19
68 68
19 19
72 72
19 19
76 76
19 19
80 80
19 19
84 84
19 19
88 88
19 19
92 92
19 19
96 96
20 20
00 00
20 20
04 04
20 20
08 08
20 20
12 12
20 20
16 16
20 20
20 20

50
(%)
(%)
19 -20 -10 0 10 20 30 19 -8 -6 -4 -2 0 2 4 6
60 60
19 19
64 64
19 19
68 68
19 19
72 72
19 19
76 76
19 19
80 80
19 19
84 84
19 19
88 88
19 19
92 92
19 19
96 96
20 20
00 00
20 20
04 04
20 20
08 08
20 20
12 12
20 20
16 16
20 20
20 20
Figure 18: Measures of supply shock and their principal component (four-quarter average)
C Appendix: Data Description

Table 1

Table 1 presents the estimates of equation

πt = β c + β π πt−1 + ( β θ + β θd Dt ) ln θt + β µ · µt + β π e · πte + ε t , (C.1)

in which πt is the annualized quarterly inflation rate, computed as log changes, in deviation from
a 2% inflation target. The rate is computed using the CPI core component (net of energy and food);
πt−1 is its lagged value. Data on CPI are from FRED, collected quarterly, using the average of monthly
observations for each quarter.

ln θt is the log of the ratio of vacancies to unemployed workers provided by Barnichon (2011) and
updated by the author. Data are monthly. Accordingly, the quarterly series is the average of the
relevant monthly observations. Dt is a dummy variable taking the value one when θt ≥ 1.

µt is the four-quarter average of the principal component of the following three series: headline
shocks, both to CPI and PCE, and import shock. The CPI or PCE headline shock is the difference
between the annualized quarterly inflation rate computed using the CPI or PCE price index and that
computed using the CPI or PCE price index excluding energy and food. The import shock is the
difference between the annualized quarterly inflation rate computed using the import-price deflator
and that computed using the GDP deflator. Data are from FRED and collected quarterly, using the
average of the relevant monthly observations. We proxy the supply shock with the four-quarter
average of the principal component of the three series. Let zt be the principal component of the three
series described above; then µt is given by:

µt = (zt + zt−1 + zt−2 + zt−3 )/4.

We proxy inflation expectations (π e ) with the 2-year inflation expectations of the Federal Reserve
Bank of Cleveland, which are collected from FRED quarterly and available since 1982 Q2. The series
is patched backward to 1960 Q1 with the 12-month inflation expectations from the Livingston survey.
Since the latter is twice yearly, missing observations are interpolated though a spline curve-preserving
function. In all regressions, πt and πte are deviations with respect to a 2% annual inflation target.

Table 2

Table 2 uses as a measure of supply shock the four-quarter average of the CPI headline shock, de-
scribed under Table 1.

51
Table 3

Table 3 uses as a measure of supply shock the four-quarter average of the import-price shock, de-
scribed under Table 1.

Table 4

Table 4 uses as a proxy of inflation expectations the 1-year CPI inflation expectations of the U.S.
Professional Forecasters Surveys, retrieved from Thompson Reuters Datastream, which starts in 1981
Q3. We patch this series backward to 1960 Q1 again using interpolated 12-month Livingston inflation
expectations.

Table 5

Table 5 uses as a proxy of inflation expectations the 5-year inflation expectations of the Federal Re-
serve of Cleveland, which starts in 1982 Q1, collected from the FRED database. The series is patched
with the 1-year GDP-deflator inflation expectations of the U.S. Professional Forecasters Surveys, re-
trieved from Thompson Reuters Datastream, which starts in 1970 Q2, and finally patched backward
to 1960 Q1 again using interpolated 12-month Livingston inflation expectations.

Table 6

Table 6 uses as a proxy of inflation expectations the 5-year 5-year forward inflation expectations back-
casted by Groen and Middleddorp (2013) until 1971 Q4. The series is patched backward to 1960 Q1,
again using interpolated 12-month Livingston inflation expectations.

Table 7

Table 7 uses inflation measures from the core PCE at annualized quarterly rate. The core PCE price
index is collected from the FRED database quarterly, as the average of the relevant monthly observa-
tions.

Table 8

Table 8 uses the level of θ rather than the ln θ.

52
Figure 2

Figure 2 plots the annual inflation rate computed using the quarterly CPI. CPI quarterly observations
are the average of the relevant monthly observations. Inflation expectations are from the Livingston
Survey of the Federal Reserve Bank of Philadelphia for the 12-month horizon on CPI. The frequency
of the graph is twice yearly, like the Livingston Survey data.

Figure 3

Figure 3 plots the annual inflation rate computed using the quarterly CPI (top panel). CPI quarterly
observations are the average of the relevant monthly observations. θ is the ratio of vacancies to
unemployed workers (bottom panel) provided by Barnichon (2011) and updated by the author. Data
are monthly. Accordingly, the quarterly series is the average of the relevant monthly observations.

Figure 4

Figure 4 uses the same data as Figure 3.

Figure 5

The left panel of Figure 5 uses the same data for inflation and θ as in Table 1, described above. The
variable ’inflation deviations’, πtd , on the right panel is built as

πtd = πt − β π πt−1 − β µ µt − β π e πte

using the estimates of Table 1, column (3).

Figure 6

Figure 6 presents the estimates through Kalman Filter of the measurement equation

πt = β c,t + β π,t πt−1 + β θ,t ln θt + β µ,t µt + β π e ,t πte + ε t ,

in which ε t is distributed as N (0, σε2 ) with the state equations given by

β c,t = β c,t−1 + ϵt
β π,t = β π,t−1 + ϵπ,t
β θ,t = β θ,t−1 + ϵθ,t
β µ,t = β µ,t−1 + ϵµ,t
β π e ,t = β π e ,t−1 + ϵπ e ,t

53
in which ϵt ∼ N (0, σϵ2 ), ϵπ,t ∼ N (0, σϵ2π ), ϵθ,t ∼ N (0, σϵ2θ ), ϵµ,t ∼ N (0, σϵ2µ ), ϵπ e ,t ∼ N (0, σϵ2πe ). The
Kalman Filter is initialized by running an OLS regression of the measurement equation with constant
coefficients on the sample period 1960 Q1 – 2008 Q2. Then, the Kalman Filter estimation runs from
2008 Q3 to 2022 Q3. σε2 is initialized as the variance of the residuals of the OLS regression on the
pre-sample; β c , β π , β θ , β µ and β π e are initialized with OLS estimates of the respective coefficients on
the pre-sample; σϵ2 , σϵ2π , σϵ2θ , σϵ2µ , σϵ2πe are initialized with the variance of the respective coefficients of
the OLS regression on the pre-sample. Figure 6 plots the estimated time-varying coefficients β π,t , β θ,t ,
β µ,t and β π e ,t using the Kalman Filter and their one-standard-deviation confidence bands.

Figure 7

The black line of the left panel of Figure 7 is the annual CPI inflation rate excluding food and energy
sectors. The red line represents the out-of-sample prediction of equation

πt = β c + β π πt−1 + β θ ln θt + β µ µt + β π e πte + ε t ,

estimated for the sample 2008 Q3 – 2021 Q1 for the forecast period 2021 Q2 – 2022 Q3. The model
produces forecasts for the quarterly inflation rate. Accordingly, we build the corresponding predicted
inflation at annual rates.

The blue line represents the in-sample prediction of equation

πt = β c + β π πt−1 + ( β θ + β θd Dt ) · ln θt + β µ µt + β π e πte + ε t ,

estimated for the sample 2008 Q3 – 2022 Q3 for the forecast period 2021 Q2 – 2022 Q3. The model
produces forecasts for the quarterly inflation rate. Accordingly, we build the corresponding predicted
inflation at annual rates.

The black line in the right panel of Figure 7 is again the annual CPI inflation rate excluding food
and energy. The purple line is the in-sample prediction using the non-linear Kalman Filter estimates,
while ‘θ component’, ‘Supply Shock component’, ‘Inflation Expectations component’ correspond, re-
spectively, to the in-sample prediction derived from the following three equations using the Kalman
Filter estimates.

πtθ = β π,t πtθ−1 + β θ,t ln θt ,
µ µ
πt = β π,t πt−1 + β µ,t µt ,
πt∗ = β π,t πt∗−1 + β π e ,t πte .

The model produces the forecast of the inflation rate at quarterly frequency, so we build the corre-
sponding annual inflation forecasts plotted in the Figure.

Figure 11

54
Data are taken from the FRED Database.

Figure 12

Data are taken from the FRED Database. Inflation is computed using the CPI annual inflation rate
(Q4 on Q4) for the reference year. The unemployment rate is the annual average.

Figure 13

Data for the PCE-index inflation and its forecasts of the Summary of Economic Projections are from
the FRED database.

Figure 14

Data on PCE-index inflation and its forecasts of the Survey of Professional Forecasters are from the
FRED database.

Figure 15

Figure 15 plots the annual inflation rate computed using the quarterly CPI. CPI quarterly observations
are the average of the relevant monthly observations. Inflation expectations are the 5-year inflation
expectations of the Federal Reserve of Cleveland. Data are from the FRED database. The Figure also
plots the 5-year 5-year forward inflation expectations of Groen and Middledorp (2013).

Figure 16

Figure 3 plots the annual inflation rate computed using the quarterly CPI (top panel). CPI quarterly
observations are the average of the relevant monthly observations. θ is the ratio of vacancies to un-
employed workers (bottom panel) derived by Petrosky-Nadeau and Zhang (2021) back to 1919. Data
are monthly. Accordingly, the quarterly series is the average of the relevant monthly observations.

Figure 17

Figure 17 plots the inflation rate and inflation expectations used in Table 1 and Table 4. Inflation rate
is the annualized quarterly inflation rate computed using core CPI. Inflation expectations, used in

55
Table 1, are the 2-year inflation expectations of the Federal Reserve Bank of Cleveland, collected from
FRED quarterly and available since 1982 Q2. The series is patched backward to 1960 Q1 with the
12-month inflation expectations from the Livingston survey. Since the latter is twice yearly, missing
observations are interpolated though a spline curve-preserving function. Inflation expectations in
Table 4 are the 1-year CPI inflation expectations of the Survey of Professional Forecasters, retrieved
from Thompson Reuters Datastream, which starts in 1981 Q3. We patch this series backward to 1960
Q1 again using interpolated 12-month Livingston inflation expectations.

Figure 18

Figure 18 presents the three different measures of the supply shock that we use to build the proxy for
µt , namely the four-quarter averages of the principal component of the two headline shocks (using
CPI and PCE price index) and the import-price shock, as described under Table 1.

56
D Appendix: The Model

D.1 Steady state

Let us first consider a steady state assuming flexible wages. The steady-state versions of equations
(8), (10), (15) and (17) give us four equation in four unknowns ( N, F, θ, w)

N = F (1 − s + s f (θ )),

χF ω
w= ,
1 − s + (1 − γb )smθ 1−η
ϵ−1 A
w=α ,
ϵ N 1− α

γc
w(1 − η )mθ −η = .
γb

We substitute out for w and F to obtain two equations in two unknowns, i.e. ( N, θ ). Combining the
first three yields:

ϵ−1 A
N ω +1− α = α (1 − γ b ) (1 − s + smθ 1−η )ω (1 − s + (1 − γb )smθ 1−η ),
ϵ χ

which shows a positive correlation between employment and market tightness. Now combine the
last two to obtain
γb ϵ − 1
N 1− α = α c A(1 − η )mθ −η ,
γ ϵ
which is a negative correlation between employment and market tightness. The above equations can
now be solved for steady-state N̄ and θ̄, which in turn gives us F̄ and w̄ from the other conditions,
thus solving for a steady state in which wages are flexible.

Let us now assume that, instead, the wage norm is binding. In steady state we have

wnorm = (wΠ−1 (Πe )δ )λ (w f lex )1−λ

and wnorm = w. Let us furthermore impose zero inflation so that Π = Πe = 1. Then it follows that
wnorm = w f lex which suggests that we obtain exactly the same steady state as we just derived. We
make assumptions such that θ̄ < 1.

Once we have determined N̄ and F̄, then we have Ū = F̄ − N̄. To simplify notation it is useful to define
ū ≡ Ū/ F̄ which can be interpreted as representing unemployment rate in steady state. Observe that
this definition implies that N̄/ F̄ = 1 − ū so that we have

57
1 − ū = 1 − s + smθ̄ 1−η .

which will simplify some of the notation in the log-linearized system.

D.2 Phillips curve characterization

We characterize first the Phillips curve when the wage norm is not binding. We take a first-order
approximation of equations (8), (10) and (17) to obtain

N̂t = F̂t + ϱ(1 − η )θ̂t , (D.2)

ŵt = χ̂t + ω F̂t − ζ (1 − η )θ̂t , (D.3)

ŵt = η θ̂t , (D.4)

in which we have defined
s − ū
ϱ≡ ,
1 − ū
and
(1 − γ b ) s f ( θ )
ζ≡ .
1 − s + (1 − γ b ) s f ( θ )
We can combine equations (D.3) and (D.4) to obtain

1 1
F̂t = (η + ζ (1 − η )) θ̂t − χ̂t .
ω ω

We can substitute this into (D.2) to write

1 1
N̂t = (η + ζ (1 − η )) θ̂t − χ̂t + ϱ(1 − η )θ̂t
ω ω
1
= nθ θ̂t − χ̂t
ω

in which we have defined
1
nθ ≡ [η + ζ (1 − η )] + ϱ(1 − η ),
ω
as the elasticity of employment with respect to labor-market tightness.

Use now the Phillips curve

( ϵ − 1)
πt = (ŵt + (1 − α) N̂t − Ât + µ̂t ) + βEt πt+1 , (D.5)
ς

which can be written, having used the above equations to substitute for ŵt and N̂t , as

( ϵ − 1) (1 − α )

πt = (η + (1 − α)nθ )θ̂t − χ̂t − Ât + µ̂t + βEt πt+1 .
ς ω

58
We can simplify this to:
tight
πt = κ tight θ̂t + κu ût + βEt πt+1 ,

given the following parameters

(ϵ − 1)(η + (1 − α)nθ )
κ tight =
ς

tight ( ϵ − 1)
κµ = ,
ς
having defined
(1 − α )
ût ≡ µ̂t − Ât − χ̂t .
ω
Now consider the state in which the wage norm binds:

f lex
ŵtnorm = λŵt−1 − λπt + λδEt πt+1 + (1 − λ)ŵt
= λŵt−1 − λπt + λδEt πt+1 + (1 − λ)η θ̂t .

Note that equations (D.2) and (D.3) hold

N̂t = F̂t + ϱ(1 − η )θ̂t ,

ŵt = χ̂t + ω F̂t − ζ (1 − η )θ̂t ,

implying that
1 1
F̂t = (ŵt − χ̂t ) + ζ (1 − η )θ̂t
ω ω
and therefore
1 ζ
N̂t = (ŵt − χ̂t ) + ϱ + (1 − η )θ̂t .
ω ω
We can then substitute the above equation into (D.5) to write it as

( ϵ − 1) (1 − α )

ζ
πt = ŵt + (ŵt − χ̂t ) + (1 − α) ϱ + (1 − η )θ̂t − Ât + µ̂t + βEt πt+1
ς ω ω

and hence using the wage norm, as

( ϵ − 1) (1 − α ) (1 − α )

πt = 1+ (λŵt−1 − λπt + λδEt πt+1 + (1 − λ)η θ̂t ) − χ̂t +
ς ω ω

ζ
(1 − α ) ϱ + (1 − η )θ̂t − Ât + µ̂t ) + βEt πt+1 ,
ω

which can be written more compactly as

πt = κw ŵt−1 + κ θ̂t + κu ût + κ β Et πt+1 ,

59
given the following parameters
κw = 1 − ϑ,
( ϵ − 1) (1 − α )

ζ
κ=ϑ 1+ (1 − λ ) η + (1 − α ) ϱ + (1 − η ) ,
ς ω ω
tight
κu = ϑκu ,

κ β = (1 − ϑ )δ + ϑβ,

with ϑ being a positive parameter with 0 < ϑ ≤ 1 defined as

1
ϑ≡ .
( ϵ −1) (1− α )
1+ ς 1+ ω λ

Note that we can write κ as

( ϵ − 1) (1 − α ) (1 − α )

ζ
κ = ϑ 1+ η + (1 − α ) ϱ + (1 − η ) − λ 1 + η
ς ω ω ω
( ϵ − 1) (1 − α )

= ϑ η + (1 − α ) n θ − λ 1 + η
ς ω
( ϵ − 1) (1 − α )

= ϑκ tight − ϑ λ 1+ η.
ς ω

Since from the first line it follows that κ > 0, we obtain from the last line that κ < ϑκ tight and therefore
κ < κ tight .

E Derivations of Section 5

In Section 5, we make the assumption that the wage norm is
 f lex

 wt for θt > 1
wt =
f lex 1−λ
wλ (Πt−1 )λ (Πet+1 )δλ (wt

) for θt ≤ 1,


in which w is the steady-state real wage. As discussed in the text, for the sake of simplicity we abstract
from persistence in real wages and center the norm around the steady-state real wage. In this section
We assume that α = 1.

We now express the Phillips curve in terms of output. Consider first the case θt > 1. We can combine
the following four equations
N̂t = F̂t + ϱ(1 − η )θ̂t ,

ŵt = χ̂t + ω F̂t − ζ (1 − η )θ̂t ,

ŵt = η θ̂t ,

60
Ŷt = Ât + N̂t

to obtain
χ̂t + ω (Ŷt − Ât )
θ̂t = .
η + (1 − η )(ζ + ωϱ)
Recall:
tight
πt = κ tight θ̂t + κu ût + βEt πt+1 .

Under the assumption α = 1 and noting that ût = µ̂t − Ât , we can write it as

tight
πt = ktight (Ŷt + ω −1 χ̂t − Ât ) + k µ (µ̂t − Ât ) + βEt πt+1 ,

given the following parameters

( ϵ − 1) ωη
ktight = ,
ς η + (1 − η )(ζ + ωϱ)

tight ( ϵ − 1)
kµ =
ς
When instead θt ≤ 1, we use the following three equations:

N̂t = F̂t + ϱ(1 − η )θ̂t ,

ŵt = χ̂t + ω F̂t − ζ (1 − η )θ̂t ,

Ŷt = Ât + N̂t ,

to obtain
χ̂t + ω (Ŷt − Ât ) − ŵt
θ̂t = .
(1 − η )(ζ + ϱω )
We can plug the latter expression into

ŵt = η (1 − λ)θ̂t − λ(πt − δEt πt+1 ),

to substitute for θ̂t to obtain

ŵt = (1 − ϖ )(χ̂t + ω (Ŷt − Ât )) − λϖ (πt − δEt πt+1 )],

in which we have defined
(1 − η )(ζ + ϱω )
ϖ≡ .
(1 − η )(ζ + ϱω ) + η (1 − λ)
We can then substitute this into

( ϵ − 1)
πt = (ŵt − Ât + µ̂t ) + βEt πt+1 , (E.6)
ς

61
to obtain
πt = k(Ŷt + ω −1 χ̂t − Ât ) + k µ (µ̂t − Ât ) + k β Et πt+1

( ϵ − 1)
k= (1 − ϖ )ωϑk
ς
( ϵ − 1)
kµ = ϑk
ς
k β = ( 1 − ϑk ) δ + ϑk β

in which
1
ϑk = ( ϵ −1)
.
1 + ς ϖλ
tight
It follows that ktight > k and k µ > k µ when λ > 0.36

E.1 The 1970s

To characterize the 1970s, we consider a short run in which the only shock is a supply shock, µ̂S > 0,
and we allow for a shock to the policy rate ϵS . Shocks revert to normal in the long run, which is an
absorbing state that occurs with probability 1 − τ. In the long run ŶL = 0 π L = π ∗ in which π ∗ is the
central bank inflation target. In the short run, however, we also assume that private agents fear that
the central bank may have changed its long-run inflation target, so their belief is π Le > π ∗ .

In this case, the short-run Euler equation, substituting for the policy rule, is given by

ŶS = τ ŶS − σ−1 (π ∗ + ϕπ (πS − π ∗ ) + ϵS − τπS − (1 − τ )π Le )

which implies that

ϕπ − τ σ −1
ŶS = −σ−1 (πS − π ∗ ) − ϵ + σ−1 (π Le − π ∗ ).
1−τ 1−τ S

while the Inv-L NK Phillips curve is:

tight Ŷ + k tight µ̂ + τπ + (1 − τ ) π e θ̂t ≥ θ ∗

 k
 S µ S S L
πS =
θ̂t < θ ∗

kŶS + k µ µ̂S + τπS + (1 − τ )π Le


implying
tight

kµ
ktight
 1−τ ŶS + 1−τ µ̂S + (π Le − π ∗ )

 θ̂t ≥ θ ∗
∗
πS − π = .
< θ∗

 k kµ e ∗
1−τ ŶS + 1−τ µ̂S + ( π L − π )
 θ̂t

36 To see that k tight > k note that (1 − ϖ ) is decreasing in λ and that (1 − ϖ ) = η/ ((1 − η )( ζ + ϱω ) + η ) when λ = 0.

62
During the 1970s, θ was below the unitary value, so that θ̂t < θ ∗ . Therefore, the inflation rate, looking
at the flat segment of the Inv-L NK curve, is given by

kµ σ −1 −1 e ∗ 1− τ e ∗
∗ k µ̂S − 1−τ ϵS + σ ( π L − π ) + k ( π L − π )
πS − π = .
1− τ −1 ϕπ −τ
k +σ 1− τ

Short-run inflation is pushed above the target by the supply shock, the disanchoring of inflation
expectations, and an accommodative monetary policy.

E.2 The 2020s

To characterize the 2020s, we consider a short run in which ĜS > 0, χ̂S > 0, µ̂S > 0 and allow for
variations in the policy shock ε S . Shocks revert to zero in the long run. This is an absorbing state that
occurs with probability 1 − τ. In the long run ŶL = 0 and π L = π ∗ .

The short-run Euler equation, substituting for the policy rule, can accordingly be written as

ŶS = ĜS + τ (ŶS − ĜS ) − σ−1 (π ∗ + ϕπ (πS − π ∗ ) + ϵS − τπS − (1 − τ )π ∗ )

which implies:
ϕπ − τ σ −1
ŶS = ĜS − σ−1 (πS − π ∗ ) − ϵ
1−τ 1−τ S
while the Inv-L NK Phillips curve is:

tight Ŷ + χ̂S + k tight µ̂ + τπ + (1 − τ ) π ∗



 k S ω µ S S θ̂t ≥ θ ∗
πS =

θ̂t < θ ∗

k ŶS + χ̂ωS + k µ µ̂S + τπS + (1 − τ )π ∗



implying  ktight
ktight ∗ θ̂t ≥ θ ∗
χ̂S µ
 1−τ ŶS + ω + 1−τ µ̂S + π



πS = .
< θ∗

k
k
+ 1−µτ µ̂S + π ∗

ŶS + χ̂ωS θ̂t


1− τ

We can now combine the two equations to obtain
  tight 
−1 χ̂ kµ
ĜS − σ1−τ ϵS + ωS + tight µ̂S


 k
θ̂t ≥ θ ∗

1−τ + σ −1 ϕπ −τ

  
−
 tight 1 τ
k

πS − π ∗ =
θ̂t < θ ∗


 −1 χ̂ kµ
ĜS − σ1−τ ϵS + ωS + k µ̂S




 1−τ + σ −1 ϕπ −τ
k 1− τ

63
showing that inflation will be higher when the curve is steeper and that a monetary policy in which

χ̂
ϵS = σ(1 − τ ) ĜS + S
ω

will be able to stabilize inflation and output, conditional on µ̂S = 0.

File and source

File: it-s-baaack-the-surge-in-inflation-in-the-2020s-and-the-return-of-the-non-linear.pdf
Size: 897,399 bytes
SHA-256: 3e2614f996ae6eaad1719ccc4388007095a89fa5632f20288fcea5a772dc5e57
Our copy: it-s-baaack-the-surge-in-inflation-in-the-2020s-and-the-return-of-the-non-linear.pdf
Original: www.nber.org