DISCUSSION PAPER SERIESIZA DP No. 14853

Zero Lower Bound on Inflation
Expectations

Yuriy Gorodnichenko
Dmitriy Sergeyev

NOVEMBER 2021

DISCUSSION PAPER SERIESIZA DP No. 14853

Zero Lower Bound on Inflation
Expectations

Yuriy Gorodnichenko
UC Berkeley and IZA

Dmitriy Sergeyev
Bocconi University

NOVEMBER 2021

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IZA DP No. 14853

NOVEMBER 2021

ABSTRACT
Zero Lower Bound on Inflation
Expectations*
We document a new fact: in U.S., European and Japanese surveys, households do not
expect deflation, even in environments where persistent deflation is a strong possibility. This fact stands in contrast to the standard macroeconomic models with rational
expectations. We extend a standard New Keynesian model with a zero-lower bound on
inflation expectations. Unconventional monetary policies, such as forward guid- ance, areweaker. In liquidity traps, the government spending output multiplier is finite, and adverse
aggregate supply shocks are not expansionary. The possibility of confidence-driven liquidity
traps is attenuated.
JEL Classification:

E5, E7, G4

Keywords:

inflation expectations, non-rational beliefs, survey data

Corresponding author:
Yuriy Gorodnichenko
Department of Economics
University of California, Berkeley
530 Evans Hall #3880
Berkeley, CA 94720-3880
USA
E-mail: ygorodni@econ.berkeley.edu* We thank Davide Businelli and Bernardo Candia for outstanding research assistance. We thank Philippe Andrade,
Carola Binder, Seung Joo Lee, Michael Weber, Francesco Zanetti, and seminar participants at Berkeley, Bocconi,
Oxford, and PSE for comments.

1

Introduction

The Great Depression has left many enduring marks not only on millions of people but
also on macroeconomic thinking and policy. Among many subsequent changes such as
launching the Keynesian framework and active countercyclical policy, the depression instilled a fear of deflation. Indeed, generations of central bankers have been concerned
about avoiding deflationary spirals and developing deflationary expectations, even if attimes deflation appeared to be a remote possibility.1 The experience of Japan in the 1990s
reinforced the view that deflations can be highly damaging and thus should be avoided at
any cost. Consistent with this view, central banks utilized everything at their disposal—
including unconventional policies such as quantitative easing, forward guidance, andnegative nominal interest rates—to escape the specter of deflation during the Great Recession and the COVID-19 crisis. These policy responses have strong support in theoretical models where economic agents have full information rational expectation (FIRE). But
given mounting evidence that households and firms exhibit significant departures from
FIRE, should one be concerned about deflation as much as predicted by the mainstreamview?
We document that, unlike professional forecasters (and financial markets), households—
and likely firms, although the evidence for firms is scarce given the dearth of high-quality
surveys of firms’ inflation expectations—see little chance of a deflation even when actual
inflation significantly dips into the negative territory. Specifically, survey evidence showsthat inflation expectations are highly asymmetric. In countries with histories of stable
and low inflation, households often predict inflation to be well above targets (e.g., five
percent or more for an inflation target of two percent) but they forecast negative inflation
very rarely. It is as if there is no ceiling for households’ expected inflation but there is afloor at zero. In other words, there could be a zero lower bound (ZLB) for expected inflation! At the same time, households’ expectations exhibit no ZLB-like property for othermacroeconomic expectations (e.g., expectations for unemployment are relatively symmetric). Using a range of surveys for the U.S. and other economies, we document that thisempirical pattern holds broadly across periods and countries thus raising the prospectthat deflationary fears are unlikely to apply to households. The lack of such fears canshed new light on a variety of phenomena that could be difficult to explain within FIRE.
For example, the lack of deflation (or disinflation) during the Great Recession could berationalized by the fact that few households anticipated deflation.
While being agnostic about sources of ZLB for expected inflation (which could occur
due to, e.g., experience-based inflation expectation, dynamics of salient prices, ambiguity
1 Although Bernanke (2002) devoted a lecture to discussing the dangers of deflation, he asserted, “thechance of significant deflation in the United States in the foreseeable future is extremely small.”2

aversion, etc.), we explore the implication of this apparent constraint for macroeconomic
dynamics and policy in the standard New Keynesian model. Specifically, we consider
scenarios where ZLB can apply to nominal interest rates (to avoid confusion, we refer tothis case as the effective lower bound, or ELB), expected inflation or both. With inflation expectations being unable to fall below zero (more generally some cutoff), an ELB
on nominal interest rates continues to limit the ability of the central bank to stimulate theeconomy, but the risk of deflationary spirals could be effectively eliminated. Intuitively,an ELB on nominal interest rates can create preconditions for a deflationary spiral because the central bank cannot respond to shocks. At the same time, a ZLB on expected
inflation can obviate this possibility by countering deflationary pressures in the economy:
via the Phillips curve, actual inflation is less likely to turn deeply negative if expected inflation is stuck at zero. This result can explain why Japan has not demonstrated elevatedmacroeconomic volatility predicted by FIRE-based models despite spending nearly three
decades at the ELB on nominal interest rates.
The ZLB constraint on expected inflation also affects the effectiveness of unconventional policy tools based on the management of expectations. For instance, the presenceof a ZLB on expected inflation weakens the effectiveness of forward guidance, since expected inflation is no longer sensitive to policy announcements. Likewise, persistent negative interest rates are less powerful when inflation expectations are constrained by its
lower bound. This is because anticipation of future negative interest rates may not lift inflation expectations above its constraint. Thus, the effect of negative interest rates policy
is limited to its contemporaneous effect only. Average inflation targeting (AIT) may also
be less attractive because inflationary shocks could be harder to control when households
do not anticipate a possibility of deflation in the future (more realistically, significant disinflations).
The ZLB on inflation expectations qualitatively modifies economy’s response to shocks.
For example, standard New Keynesian models with the FIRE beliefs predict that negative
aggregate supply shocks, such as higher taxes on firms, increase output gap during the
ELB on the nominal interest rate. This prediction appears to be at odds with the data.
Relatedly, according to the standard model, government spending multipliers can be arbitrarily large with the nominal interest rate stuck at the ELB while the empirical evidence
points to larger but still relatively modest multipliers. We show that these predictions of
FIRE-based models are no longer the case when inflation expectations are at zero.
With the ZLB constraint on expected inflation, self-fulfilling liquidity traps are lessprevalent. Typical New Keynesian models with Taylor rules that strongly react to inflation unless constrained by the ELB feature two steady states. In a conventional steadystate, inflation is at its target and output gap equals zero. In a liquidity trap steady state,3the nominal interest rate is zero, inflation and output gap are negative. Importantly, inflation expectations are also negative. The ZLB on inflation expectations rules out such a
steady state by preventing agents to believe in deflation. We also show that the ZLB on
inflation expectations reduces the possibility of a temporary confidence-driven liquidity
traps.
Our analysis contributes to several strands of research. First, our work is related to
the large literature examining properties of inflation expectation for households and firms(see Coibion, Gorodnichenko and Kamdar, 2018 for a survey). Generally, this line of workfinds that inflation expectations of households in advanced economies have little resemblance to FIRE. We add to the list of departures from FIRE by documenting that households appear to downplay the possibility of future deflation to a point when they thinkdeflations effectively can’t happen. The closest paper in this literature is Baqaee (2020)
which studies asymmetry in the pass-through of actual inflation into households’ inflation expectation. Specifically, he finds that negative inflation shocks have a weaker effecton households’ inflation expectations than positive shocks. In contrast, we focus on thezero constraint for inflation expectation. In another closely related paper, Andrade, Gautier and Mengus (2020) argue that the intensive margin for inflation expectations appears
to play a limited role in determining aggregate inflation expectations of French households. Instead, the extensive margin (i.e., the share of people expecting positive inflation)
explains most of the variation. Building on Andrade, Gautier and Mengus (2020), we notethat there is little variation in the share of people predicting deflation and hence inflation
expectations effectively have a zero lower bound.
Second, we contribute to the vast literature on the importance of inflation expectations
for macroeconomic dynamics. For example, Krugman (1998), Eggertsson and Woodford(2003), Eggertsson (2008) and others argued that raising inflation expectations can materially contribute to ending economic depressions. In contrast to this literature, we suggest
that, for many advanced economies with low inflation, it is surprisingly hard to generate deflationary expectations for households. For example, even now, most households inJapan expect inflation. To the extent households’ expectations are more important for consumption decisions than those of professional forecasters or financial markets (Coibion,
Georgarakos, Gorodnichenko and van Rooij, 2019a; D’Acunto, Hoang, Paloviita and Weber, 2019), our results call for a more nuanced interpretation of recent developments for
inflation in Japan in the 1990s and afterwards. We also add to the literature studying inflation dynamics during the Great Recession when inflation did not fall as much as one
could have expected given the rise of unemployment thus generating a missing disinflation puzzle (Hall, 2011). A variety of theories has been offered to rationalize this anomaly
with some authors (e.g., Coibion and Gorodnichenko, 2015b) suggesting that a tempo4

rary increase in households’ inflation expectations due to a hike in energy prices saved
the U.S. economy from a deflation. Our analysis is related to this reasoning but insteadof emphasizing the sensitivity of households’ inflation expectations to changes in salient
prices (e.g., the price of gasoline), we stress that disinflation pressures were attenuatedbecause few households predicted deflation due to the ZLB in inflation expectations.
Our paper adds to the literature that extends business cycle models with non-FIRE
beliefs, such as dispersed information (Lucas, 1972; Angeletos and La’O, 2013), rational
inattention (Sims, 2003; Maćkowiak and Wiederholt, 2015), statistical learning (Evans andHonkapohja, 2012; Eusepi and Preston, 2011), cognitive discounting (Gabaix, 2020), extrapolation and diagnostic expectations (Bordalo, Gennaioli, Ma and Shleifer, 2020; Bordalo, Gennaioli, Shleifer and Terry, 2021) to name a few. One application of this literature
was to close a disconnect between a tremendous power of forward guidance policy instandard New Keynesian models and a lack of it in reality (Angeletos and Lian, 2018;
Garcı́a-Schmidt and Woodford, 2019; Wiederholt, 2015). The ZLB on inflation expectations in our paper also attenuates the strength of forward guidance and generates several
other predictions consistent with empirical evidence.

2

Do People Expect Negative Inflation?

This section documents a new fact that households virtually never expect deflation. We
present three pieces of evidence. First, we examine the cross-sectional distribution ofpoint forecasts by households to illustrate that few households predict deflation. Second,
we study time series variation in the share of households who predict negative, zeroor positive inflation. Third, we explore how variations in actual inflation translate into
variations of various percentiles of inflation expectations.

2.1

Expectations across Consumers

As a first step, we present evidence for the households in the Michigan Survey of Consumers (MSC) and the Bank of Japan Opinion Survey. We start with the Michigan Surveyof Consumers, which is one of the longest surveys of households’ expectations. The top
panel of Figure 1 plots the histogram of inflation expectations in gray and realized Consumer Price Index (CPI) inflation rate in red between 1985 and 2008. There are two notable
observations here. First, the distribution of inflation expectations is highly asymmetricaround its mean of 3.7 percent with the right tale stretching to more than 20 percent andthe left tale being essentially truncated at zero. Second, while there is almost no one who
expected deflation during this period, a significant share of people did expect inflation to5

be lower than the lowest realized inflation of one percent. Hence, people are capable ofexpecting inflation which is lower than actual inflation.
With the onset of the Great Recession, the United States witnessed CPI deflation for
the first time since the 1950s. The bottom panel of Figure 1 redraws the inflation expectations and realized inflation histogram for the period between 2008 and 2019. The realized
inflation histogram shifts to the left crossing the zero mark. At the same time, the distribution of inflation expectations stays to the right of the zero border. It is as if there is azero lower bound on inflation expectations. This is a central observation of this paper. Is
this pattern specific to the U.S.?
The Japanese experience during the last three decades presents an important window
into a setting where deflation is a recurrent, persistent phenomenon. Half of all quarters
between 1994 and 2021 were in deflation and yet almost no one in Bank of Japan’s Opinion
Survey, which collects numerical inflation expectations from Japanese consumers, expects
deflation.2 Figure 2 presents a histogram similar to the one plotted for the MSC in Figure
1. Strikingly, the fraction of people who expected deflation between the first quarter of2004 and the first quarter of 2020 is only five percent, while the actual inflation was in thenegative territory approximately 40 percent of the time.3 Hence, even in a country with aprominent history of deflation, households appear to avoid predicting deflation.

2.2

Expectations over Time

Cross-sectional distributions reported in the previous section are informative, but they
do not convey potentially important time-series information. To explore this dimensionof the data, we next show time series for the fraction of consumers who expect positive,
zero, and negative inflation. We focus on two prominent deflationary episodes of Japan
during the last three decades and Greece during the European debt crisis.
Figure 3 displays inflation and core inflation (the top panel) and the evolution of crosssection inflation expectations distribution (the bottom panel) in Japan. The key observation is again that only a modest share of consumers expect deflation. Most of the time series variation in the cross-sectional distribution occurs in the shares of people who expect
positive and zero inflation. The fraction of people who expect deflation barely moves.
The only exception is the episode around the global financial crisis where many differ2 Bank of Japan’s Opinion Survey asks “By what percent do you think prices will change one year fromnow? Please choose "up" or "down" and fill in the box below with a specific figure. If you think that they
will be unchanged, please put a “0.” Prices will go up/down about
percent one year from now.”
3 There is an atom at zero in the distribution of expected inflation. To make the realized inflation histogram comparable to expected inflation histogram, we treated value of realized inflation in the interval[-0.2%;0.2%] as zero. As a result, realized inflation was negative more than 40 percent of time during thisperiod.

6

ent events unfolded that could have influenced inflation expectations, such as changes insalient prices (Yen’s appreciation and a decline in energy and food prices).
Following the European debt crisis, Greece experienced a devastating recession with
unemployment rate reaching almost 30 percent in 2013. This deep recession was associated with deflation between 2012 and 2016 as shown in Figure 4. Yet the fraction of people
who expected deflation barely moved, staying below 10 percent at all times. This stands
in contrast to the behavior of the fraction of people who expected zero inflation. It started
rising rapidly after 2010 reaching its peak of more than 40 percent in 2015.

2.3

Conditional Expectations

Our results so far focus on the central tendencies in the data. While informative, one
may be interested in the comovement of inflation expectations and actual inflation at
various percentiles. Specifically, we want to examine whether the properties of bottom
percentiles, which are close to zero, are different from the properties of top percentiles,
which are unlikely to be constrained by zero. To this end, we estimate a series of quantile
regressions
Eit [pt,t+12 ] = a(t ) + b(t ) pt 12,t + g(t ) X i,t + # i,t ,(1)
where i, t, and t index respondents, time (month), and percentiles, Eit [pt,t+12 ] is the oneyear-ahead inflation forecast of household i at time t, pt 12,t is the actual inflation rateover the previous 12 month, X i,t is a vector of respondent characteristics. Because the key
variable pt 12,t does not vary across respondents, we cluster standard errors by time. We
use the Michigan Survey of Consumers data from 1978M1 to 2019M12 and report results
for t = {10, 20, . . . , 90} in Figure 5.
Consistent with evidence presented earlier, we find that as we consider lower quantiles the sensitivity of expected inflation to actual inflation declines to zero. For example,
consider the 10th percentile. The slope estimate is zero thus suggesting that expectations
exhibit no reaction to variations in actual inflation. Furthermore, the intercept is also effectively estimated at zero meaning that inflation expectations are set at zero. Note thatstandard errors for the 10th percentiles are essentially zero. This result indicates that there
is little variation in the left tail of the distribution for expected inflation because expectations are bounded by zero. As we raise the percentile, both the slope and the intercept
increase. For example, the estimated slope for the median is close to one, i.e., expected
and actual inflation co-move roughly by the same magnitude. However, the intercept isclose to 1.5 percentage points so that the forecast is systematically biased upwards. The
top percentiles are characterized by high sensitivity of expectations to actual inflation (theslope is considerably greater than one) and a large upward bias (the intercept is more than7

five percentage points).
If actual inflation declines, as it did during the Volcker disinflation, then inflation
expectations are pushed close to zero but expectations do not appear to cross the zerobound. Instead, expectations appear to pile at zero or just above zero. In contrast, theright tail of the distribution comoves strongly with actual inflation, i.e., consistent withthe simple histograms reported in section 2.1, households’ inflation expectations have afloor at zero but have no ceiling. As a result, the average expected inflation is driven bya mix of respondents reporting zero expected inflation and respondents who predict a
positive inflation rate. This pattern is also consistent with Andrade, Gautier and Mengus (2020) arguing that variation in the average expected inflation for French households
is largely due to variation in the share of people expecting positive inflation, i.e., theextensive margin. Our findings are also consistent with Baqaee (2020). As the Volcker
disinflation achieved lower levels of inflation, negative inflation shocks are unlikely togenerate reductions in expected inflation because respondents are likely to report zero
expected inflation rather than a negative magnitude thus yielding asymmetric responses
documented in Baqaee (2020).

2.4

Subjective Uncertainty

Although consumers’ point predictions for future price changes suggest that deflation as
an unlikely outcome, consumers may still assign positive probability to deflation. The
Survey of Consumer Expectations (SCE), run by the Federal Reserve Bank of New York,asks households to report their subjective probability distributions for future inflation.
Specifically, households assign probabilities to ten bins spanning the following options:“inflation will be more than 12%”, “inflation will be between 8% and 12%”, ..., “inflation
will be between 0% and 2%”, “deflation will be between -2% and 0%”,..., “deflation will
be more than -12%”. Panel A of Figure 6 shows that households put a positive probability on deflation (on average, approx. 16%) but the distribution is skewed towards
high-inflation bins. Specifically, households believe that a double-digit inflation (which
the U.S. economy has not experienced since the early 1980s) is more likely than deflation(which the U.S. economy experienced during the Great Recession). To better understandthe properties of subjective distributions, Panel B of Figure 6 shows the time series for the
average probability assigned to each bin. We observe that the variation for deflationarybins is compressed relative to the inflationary bins.4 This pattern suggests that variationin expected inflation can be largely driven by changes in the probabilities assigned to bins
associated with positive inflation. To illustrate this point more clearly, Panel C of Figure
4 For example, the standard deviation is 0.99 for the “deflation will be between -2% and 0%” bin andand 2.5 for the “inflation will be between 0% and 2%” bin.

8

6 plots the time series for the average expected inflation as well as contributions due to
inflation and due to deflation. We find that the expected inflation tracks the contribution
due to inflation. We conclude that while households put a positive probability on deflation in response to the SCE survey question, the time-series variation in this probability
is rather modest.
Why is there so little time-series variation in the probability of deflation as perceived
by households? First, we may fail to see discernible variation because actual inflation
does not move much and specifically deflation did not happen in the sample period.
This is certainly a possibility but CPI inflation did go negative briefly in 2015. Second,
the survey question treats inflation and deflation symmetrically. If respondents are notsophisticated/informed, they may assign positive probabilities to outcomes close to the
middle of the presented options because they may assume that the middle reflects reasonable outcomes. This priming of responses may result in a pattern where some households can report a positive probability of deflation even if they do not think deflation is
likely (Coibion et al. (2021)). In other words, we may face a measurement error in survey
responses. To gauge the plausibility of this scenario, we can use survey data from theDutch National Bank. Unlike other surveys of inflation expectations, the Dutch surveyof consumers ask respondents to report a minimum expected inflation.5 One can thinkof this question as asking respondents to predict the worst-case scenario for deflation.
Figure 7 shows that only a tiny share of respondents predicts a deflation. This patternof responses is consistent with potential priming of responses in the SCE-type questions
and again highlights a special role of a zero cutoff for inflation expectations.

2.5

Other Variables and Types of AgentsTo explore whether ZLB on expected inflation applies to other types of agents, we examine inflation forecasts made by firms and professional forecasters. One may thinkthat these agents have stronger incentives to pay attention to macroeconomic developments and thus generate forecasts with properties that more closely resemble FIRE. Afterall, professional forecasts make a living by producing high-quality projections and firms
should use high-quality projections to set prices. Consistent with this view, we find (thetop left panel of Figure 8) that the distribution of one-year-ahead inflation forecasts in the
U.S. Survey of Professional Forecasters not only mimics the distribution of actual inflation
but also shows few signs of asymmetry. In other words, professional forecasters do not as5 The DNB Household Survey questionnaire provides the following English translation of the actual

question: “What will be the minimum percentage prices could increase over the next twelve months, doyou think? If you think prices will decrease, you can fill in a negative percentage by using a minus in frontof the number.”9

sign any special role to zero when they prepare their inflation forecasts. Using data from
Consensus Economics, we find similar evidence for other countries. For example, the top
right panel of Figure 8 shows that in Japan, the distribution of inflation expectations byprofessional forecasters is close to the actual distribution of inflation. This result suggeststhat the apparent unwillingness of households to predict deflation does not likely reflecta fundamental property of underlying data generating processes.
Although professional forecasters seem to behave in line with FIRE, prior work (e.g.,
Candia, Coibion and Gorodnichenko, 2021) suggests that managers’ inflation expectations could be close to households’ inflation expectations. For example, like households’
inflation expectations, managers’ inflation expectation have large dispersion, upward biases, large and frequent revisions, strong correlation of short- and long-run forecasts, etc.
Consistent with the earlier evidence, we find (the bottom left panel of Figure 8) for the
U.S. that virtually no manager in the Survey of Firm Inflation Expectations (SoFIE) predicts deflation in the third quarter of 2020 when the realized inflation was the lowest
during the COVID recession. Furthermore, the bottom right panel of Figure 8 plots thehistogram of inflation expectations by Japanese enterprises in the TANKAN survey between 2014Q1 and 2021Q3. The survey asks the participants to pick inflation expectations
from the ten values { 3, 2, . . . , 5, 6}, rather than allowing to choose continuous values.
Nevertheless the distribution of the answers is right-skewed with a sharp drop below
zero. Given the short time series dimension of the SoFIE, round number answer optionsin tha TANKAN survery, and more generally the dearth of high-quality surveys of managers’ inflation expectations, this evidence is obviously tentative but it suggests that the
ZLB on expected inflation can apply to firms as well.
When we study properties of forecasts for other macroeconomic variables reported byhouseholds in the Michigan Survey of Consumers, we observe that households have little trouble predicting negative values. For example, the Michigan Survey of Consumers
asks households to report whether they anticipate increase/no change/decrease in unemployment and nominal interest rates. Unfortunately, the questions are qualitative but
we can compare shares of people predicting decreases and increases to see if the distribution of responses is symmetric. We find (Figure 9) that these distributions are much
more symmetric (especially for unemployment) than the distribution for inflation forecasts. This pattern suggests that households generally can report negative values and the
ZLB property for expectation is constrained to price changes.

10

2.6

Potential Sources of ZLB on Expected Inflation

While expectations are often thought to be flexible variables that can take any value commanded by the economic outlook, our findings suggest that households are highly reluctant to report expected inflation below zero. We observe this pattern for a variety ofcountries and periods thus suggesting that this phenomenon may be more than a statistical fluke. But why would households hesitate to predict negative inflation? In the absenceof exogenous variation in expectations targeted to push households into thinking about
possible deflation, one may be unable to provide a beyond-reasonable-doubt account forthis empirical regularity. However, we can suggest some tentative hypotheses that canrationalize the observed behavior of inflation expectations.
First, even in countries with low and stable inflation, many households lived througha period of high inflation in the past. For example, although U.S. households have notseen double-digit inflation for nearly 40 years, some households may still remember theGreat Inflation of the 1970s and thus may find it hard to believe that inflation can infact be negative. Malmendier and Nagel (2016) document evidence consistent with thishypothesis. In a similar spirit, German households may be scarred by memories and consequences of the hyperinflation after World War I so that they can be unlikely to report anegative inflation rate. In this case, one does not have to actually live through a hyperinflation but the society should have enough memory collectively to make households
believe that the distribution of inflation is asymmetric with high inflation outcomes being much more likely than deflation outcomes. Relatedly, consumers who have not lived
through a deflation may simply not understand what deflation (negative inflation) is.
Second, inflation is a complex phenomenon and understanding sources and implications of inflation shocks may be challenging for households who often have limited economics knowledge and financial literacy (D’Acunto et al., 2019). As a result, households
may interpret the inflation dynamics through lenses of worst-case scenarios where households perceive inflation as a bad state (Kamdar, 2018). For example, Baqaee (2020) uses
ambiguity aversion to rationalize downward rigidity of inflation expectations in responseto changes in actual inflation. Relatedly, Afrouzi and Veldkamp (2019) argue that uncertainty about parameters of the data generating process that has a skewed distribution—alikely case for actual inflation—can also generate an upward bias in reported inflation
expectations via Jensen’s inequality thus reducing a chance of forecasting deflation. Thisline of reasoning is consistent with the fact that the right tail (high inflation) in the densities reported by households tends to be heavier than the left tail (deflation) and themean expectation is greater than the median (see e.g., De Bruin, Manski, Topa and Van
Der Klaauw, 2011, Coibion, Gorodnichenko and Weber, 2019b).
Third, in low inflation environments, households appear to rely on salient prices for11frequently-purchased goods such as gasoline or food to form their inflation expectations (e.g., Coibion and Gorodnichenko, 2015b, Cavallo, Cruces and Perez-Truglia, 2017,
D’Acunto, Malmendier, Ospina and Weber, 2021) as these prices provide an easy way togauge inflation when incentives for track inflation are weak. To the extent inflation forthese goods is less likely to experience deflation than the whole CPI, households may conclude that the aggregate level of prices is unlikely to fall. Consistent with this view, theinflation rate for food and beverages in the U.S. during the Great Recession barely moved
into a negative territory while the full-basket CPI declined by two percentage points. In asimilar spirit, while inflation for the full-basket CPI approached zero during the COVID
crisis, the food inflation rose to four percentage points. Obviously, prices for other goods
may be salient for other countries. For example, the exchange rate may be more relevant
for small open economies or countries with recent histories of high inflation (e.g., Coibion
and Gorodnichenko, 2015a). Because food and energy prices may be determined by the
global factors or forces that are specific to food and energy, inflation expectations may
move for reasons other than domestic business cycles.
Establishing which of these hypotheses—and there could be alternative accounts—is
an empirically relevant explanation is beyond the scope of the paper but we would liketo note that, while these hypotheses suggest departures from FIRE, they may still be consistent with rationality (e.g., households can economize on information costs to arriveat potentially worse but cheaper expectations). Moreover, the apparent unwillingnessof households to predict deflation is unlikely to be structural: U.S. households were expecting deflation in the early 1950s when a different policy regime and stable-price-levelmindset were in place (Binder and Brunet, 2021). To make further progress in understanding macroeconomic implications of the ZLB for households’ inflation expectations,
we will assume that i) households act rationally; ii) for some reason, households believethat future inflation can’t be negative; iii) households do not abandon their no-deflation
belief in response to shocks. This practical approach allows us to have minimum deviations from the standard New Keynesian model and hence permits analytical tractability
that would be very hard to achieve otherwise in a more sophisticated setting for the formation of inflation expectations.

3

ZLB on Inflation Expectations in an NK Model

This section presents a simple representative agent three-equation New Keynesian modelwith a twist. Inflation expectations is constrained to be positive even when a FIRE agent
would predict inflation to be negative. Because the model is standard otherwise, we skip
the description of the fundamentals and jump directly to equilibrium conditions, taking
12particular care of expectations.
In this economy, people may be surprised by unfolding events because their statedependent non-FIRE forecasts are different from realized outcomes in some states. As aresult, we have to separately solve for forecasts and for realized variables. Formally, there
are two sets of equilibrium equations: those that determine contemporaneous endogenous variables and those that govern the behavior of forecasts.
Contemporaneous equations. We present a log-linearization of equilibrium conditions
around steady state with log inflation p > 0 and output gap of zero. The Euler equation
takes the familiar form
xt = E t [ xet+1 ]

1
(it
s

et + 1 ]E t [prtn ) ,

(2)

where rtn is the exogenous natural interest rate with the steady state value of r ⌘ log b,
it is the nominal interest rate set by the central and its steady state value is i ⌘ r + p,
and pt is the inflation rate and p is its steady state value, xt is an output gap, and apositive parameter 1/s is the intertemporal elasticity of substitution. Tilde on top of theendogenous variables highlights that the forecasts do not necessarily coincide with FIRE
expectations. In our notation, all endogenous variables are functions of past and current
exogenous shocks (see Iovino and Sergeyev (2018) for a similar notation). Thus non-FIRE
forecasts that bear tildes, such as xet+1 , are functions that differ from the actual mappingsfrom shocks to realizations, such as xt+1 .6 The expectations operator E t [·] is mathematical
expectations conditional on information at time t over exogenous shocks that have notyet been realized. The agents are perfectly aware of the distribution of future exogenous
shocks. We assume that agents do not change their forecasting functions over time. As aresult, these functions bear only one time subscript that refer to the period for which theforecast is made.7
6 A popular alternative to our distorted beliefs notation is distorted distributions over realization ofendogenous variables. This alternative and our notation are equivalent. To see why, take a two-period
example where endogenous variable p is a function of the exogenous and random state r n in the second
e (r n ), which summarizes theperiod. With our notation, the beliefs are captured by forecasting
function pRnnne (r )] = pe (r )dF (r ), where F is a known cumulative
forecast in every future state. The mean belief is E [pRne [p ] = pd Fe(p ),distribution function (cdf) of r . Using the alternative representation, one can write Ee (r n )) = dF (r n ).
where Fe is a distorted cdf of p. The relation between these distribution functions is d Fe(pFor example, the zero lower bound on inflation expectations in our notation is a constraint on forecasting
e 0. Under the alternative notation, the same constraint can be represented by zero weightfunction, i.e., p
attached to negative values of p in the distribution function Fe(p ).
7 In principle forecasting functions x
et+1 can depend on the time the forecast is made, that is, for example,
xt,t+1 . The fact that forecasting functions do not change over time does not rule out a potential Bayesianlearning about the exogenous state of the economy. However, this rules out statistical learning, such as
constant gain learning, where the agents change the mapping from shocks to forecasts over time.

13

When firms can perfectly index their prices to the constant trend inflation p, thePhillips curve is
•

pt

k



p = kxt + Â ( bq ) E t k xet+k +
k =1

1

q
q

et + k(pp)(3)

where b > 0 is the subjective discount factor of the representative household, k > 0 is theslope of the standard New Keynesian Phillips curve, and 1 q > 0 is the probability withwhich firms re-optimize their prices. Appendix A.1 derives the Phillips curve (3). Whenthe trend inflation p is zero and the expectations are rational, equation (3) collapses to thestandard New Keynesian Phillips curve. However, when expectations are not rational,
we cannot fold equation (3) into its rational expectations form except for a few convenient
special cases.8
The central bank follows a Taylor rule of the form
it = max {0, i + f (ptp )} ,(4)

where f > 1 and the interest rate cannot become negative, which we refer to as theeffective lower bound on the nominal interest rate, or the ELB.
Forecasts. The representative agent does not have FIRE. In accordance with our empirical evidence in Section 2, we posit that the agent believes that future inflation cannot
fall below zero. Formally, beliefs satisfy the analogues of equations (2)-(4). The Euler
equations (2) at some state of future period t + s is
xet+s = E t+s [ xet+s+1 ]

1 e
(it+s
s

et + s + 1 ]E t+s [ prtn+s ),

(5)for all s 1. The key difference between the contemporaneous and future Euler equations(2) and (5) is that the contemporaneous equation has the realized output gap xt on theleft-hand side, while the future equation (5) has a non-FIRE forecast instead.

8 Adam and Padula (2011) show that log-linearized equilibrium equations of a micro-founded New

Keynesian model produce a standard expectations-augmented New Keynesian Phillips curve when agents
do not expect predictable movements in their non-FIRE beliefs. We do not obtain the same result despite
the fact that the agents in our model also do not expect predictable movements in their expectations. This isbecause the agents in our model hold beliefs that may not be consistent with equilibrium relations in somefuture periods and states. For example, equation (A2) in Adam and Padula (2011) is in general not true inour model.

14

The analogue of the Philips curve is
et+s = maxp(•p + k Â ( bq )k E t+s [ xet+s+k ] +
k =0

q •

1
q

Â ( bq )k+1Et+s [pet+s+1+k

k =0

)

p] , 0 ,(6)
for all s
1 and where the max operator introduces a zero-lower bound, or the ZLB,
constraint on inflation forecasts. The equation states that whenever the agent expects
positive inflation, it equals the first argument of the max operator. Otherwise, the agent
expects zero inflation. All endogenous variables bear tilde highlighting that forecasts may
deviate from rational expectations. Finally, note that by having the same forecasts in the
Euler equation and the Phillips curve, we effectively assume that households and firms
have identical beliefs. In particular, this means that both households and firms are subject
to the zero lower bound on inflation expectations.
Finally, the Taylor rule iseit+s = max{0, i + f (pet + sp )}.

(7)

Shocks. We consider an uncertainty structure where at the beginning, the economy is
hit with a negative shock to natural interest rate rtn = r < r that reverts back to its steadystate level with probability µ 2 [0, 1], and with remaining probability, it stays at r. The
initial negative natural interest rate shock can trigger either the ELB constraint or the
ZLB on inflation expectations constraint or both. The constant probability of reversal
to the steady state allows us to solve for Markov equilibria, which we can conveniently
visualize with diagrams.9

3.1

FIRE Benchmark

A useful starting point is a rational expectations equilibrium that does not feature zero
lower bound on inflation expectations. We use backward induction to solve the model.
First, after the natural interest shock subsides, the economy immediately returns to its
steady state because there are no state variables in this model. We denote a (stochastic)
period when this happens as T. Thus we have that xt = 0, pt = p, it = i for all t T.
For periods t < T, the Euler equation isxL =1[max {0, r + p + f (p L
µs

p )}

(1

µ)p L

µp

r] ,

(8)

9 It is straightforward to analytically solve the model in the case of a deterministic duration of shocks tonatural real interest rate.

15

where x L and p L are output gap and inflation when the exogenous natural real interest
rate rtn equals r. Output gap and inflation only depend on the natural real interest rate
because there are no endogenous state variables. Figure 10 plots a piece-wise linear Euler
equation in the ( x L , p L ) space for several values of r n . When inflation is sufficiently low,
the Taylor rule prescribes the central bank to set a zero interest rate and the output gapinflation relation turns positive.
The Phillips curve (3) gives
p=

pL

k
b (1

1

µ)

xL.

(9)

It is represented by upward-sloping line in Figure 10. When a negative shock to natural interest rate is small enough, it does not trigger the ELB on the nominal interest rate.
The Euler equation and the Phillips curve jointly determine output gap
xL =

µs[1

1
b (1

b (1 µ )µ)] + (f + µ

1)k

(r

r ) < 0,(10)which together with the Phillips curve (9) determine inflation. This solution highlightsthat output gap and the inflation rate deviations from their steady state values are both
negative, which is also represented by the intersection A1 of PC and EE0 curves in Figure10.
If a shock to the natural interest rate is sufficiently large, the ELB constraint on theinterest rate becomes binding. The curve EE00 and its intersection with the Phillips curve
PC in point A2 in Figure 10 present such a case. The Euler equation and the Phillips curveintersect at
xL =1

b (1 µ )[ i + (rJ(µ)

r )] ,(11)

where J (µ) ⌘ µs[1 b(1 µ)] (1 µ)k. The inflation rate is given by (9). For thissolution to be bounded and unique, the denominator J (µ) in equation (11) has to bepositive
J (µ) = µs[1 b(1 µ)] (1 µ)k > 0.
(12)
When the persistence of the negative interest rate shock 1 µ is lower than a certain
cutoff, this condition holds. Graphically, condition (12) requires that the upward-sloping
part of the Euler equation is steeper than the Phillips curve in Figure 10.

16

For the ELB to bind, the interest rate r must be below the following cutoffri,REE ⌘

µs[1

b (1

µ)]
kf

k (1

µ)

µs[1

r

b (1

µ)] + k (f + µ
kf

1)

p.

(13)

The first term in condition (13) is the standard one that indicates the minimal size of thenatural rate shock that is needed to generate the ELB with zero steady state inflation. The
second term shows that the shock must be larger in absolute terms in the presence of apositive steady state inflation.
When firms index their prices to trend inflation, the inflation rate need not be negativein the ELB on the nominal interest rate as illustrated by the intersection A2 of the Phillips
curve PC and the Euler equation EE00 in Figure 10. For inflation to be negative, the naturalinterest rate needs to fall below the threshold
1 b (1 µ )r p,ELB ⌘ µs + 1 p.
kThis situation occurs when the Euler equation curve is shifted to the left sufficiently as
depicted by the curve EE000 in Figure 10.
Depending on the parameters of the model, the cutoff value r p,ELB , where the inflation
turns negative, can be below or above ri,ELB , where the interest rate becomes zero. For example, when f is large, the central bank reduces the nominal interest rate strongly to fighta small decline in inflation so that the ELB becomes binding even before the inflation rate
turns negative. In this case, r p,ELB < ri,REE . When f is close to but above one, the central
bank tolerates large declines in the inflation rate so that inflation can turn negative even
before the ELB is reached. In this case, r p,ELB > limf!+1 ri,REE (f), where we explicitly
acknowledged that ri,REE depends on f.
Relatedly, even when the ELB on the nominal interest rate is slack, the inflation rate
can be negative following a negative aggregate demand shock. The natural interest rate
just needs to be below the cutoff
r p,no ELB ⌘ r + p (1

f)

p



1

b (1
k

µ)

s + 1 µ.

We will use this cutoff in the next section.

3.2

Equilibrium with ZLB on Inflation Expectations

The preceding section makes clear that inflation and inflation expectations can be below
zero in rational expectations equilibrium. This section adds the zero lower bound con17

straint on inflation forecasts and derives its consequences. We again use backward induction to solve for equilibrium. We continue assuming that the persistence of the negativeinterest rate shock is small enough so that the condition (12) holds, which will guaranteethat the equilibrium is unique. The possibility of multiple equilibria is analyzed in Section
6.
First, consider the dates when the negative demand shock subsides. Formally, take alls such that t + s
T, where t is the date when the agent solves her problem, which can
et+s = p > 0,be either before T, at T, or after T. In this case, the agent forecasts xet+s = 0, pe
and it+s = i because these values satisfy equations (5)-(7).
Second, we solve for the forecasts at dates before the shock dissipates. Formally, takes such that 0  t < t + s < T. The Euler equation is
xeL =

1
eL
[max {0, r + p + f (p
sµ

p )}

(1

eL
µ)p

µp

r] ,

(14)

eL are the forecasts of output gap and inflation in states with negative
where xeL and p
aggregate demand shock rtn = r. The structure of uncertainty and the forecasts after theshock ends allow us to write the forecast from the Phillips curve (6) in a compact form⇢

eL = max p +
p

1

k
b (1

µ)

xeL , 0 .

(15)eL ) space, the modified Phillips curve features a horizontal part as can be seenIn the ( xeL , p
in Figure 11.
Depending on the parameters of the model, the economy can be in one of the fourstates: (i) neither of the two constraints binds, (ii) only the ELB constraint binds, (iii) only
the ZLB on inflation forecasts binds, (iv) both constraints bind. We consider these casesin turn.
No constraint binds. If the constraints do not bind, then the solution to the equilibrium
equations is given by equations (9) and (10), which express output gap and inflation under rational expectations when the ELB on the nominal interest rate does not bind. Figure
11 presents the forecast Euler equation (curve EE0 ) and the forecast Phillips curve (curve
PC). Forecast inflation and output gap are in point A1 . Because the ZLB on inflation expectations does not bind, the realized output gap and inflation are identical to expectedvalues under rtn = r. For this equilibrium to occur, the natural interest rate must be larger
than max{ri,REE , r p,no ELB }. The condition that the natural interest rate is larger than ri,REE
ensures that the shock is small enough that it does not trigger the ELB on the nominalinterest rate, while the requirement that the interest rate is higher than r p,no ELB limits the18

size of the shock so that inflation forecast does not reach its zero lower bound.
Only the ELB binds. In the case with slack inflation expectations constraint, realized
output gap and inflation again coincide with their forecasts. The solution is similar to theone presented in Section 3.1. Specifically, output gap and inflation are given by equations(9) and (11). Graphically, the expected Euler equation and the Phillips curve are shownin Figure 11 by curves EE00 and PC, respectively. They intersect in point A2 , where the
ZLB on inflation expectations does not bind. Importantly, for this case to be possible
in equilibrium, the natural interest rate shock must satisfy r 2 (r p,ELB , ri,REE ] and theparameters of the model must ensure r p,ELB < ri,REE , which states that the natural interest
rate that makes inflation negative under the ELB is lower than then natural interest ratethat makes the ELB constraint binding.
Only the ZLB on inflation expectations binds. We now turn to the case when the inet + s = p
eL = 0, while the lower bound on the interest
flation forecast is zero, i.e., prate is slack. The model parameters need to satisfy ri,REE < r p,no ELB , which is equivalent to r > p (f 1), and the size of the interest rate shock need to fall in the interval(ri,REE , r p,no ELB ] for this situation to occur in equilibrium. Figure 12 depicts such a scenario, where the kink in the Euler equation is below the zero inflation level and the leftward shift in the Euler equation is large enough to generate the ZLB on expected inflation.
The future Euler equation (5) and the Phillips curve (6) intersect at
xeL =1[(f + µµs1) p + r

r] ,

(16)

The forecast output gap in equation (16) is negative. Formally, (f + µ 1) p + r r < 0when r < r p,no ELB , which ensures that the ZLB on inflation forecast binds. However, notethat the forecast output gap is higher than the one under rational expectations because theinflation rate expectations are constrained from below. This is clear from Figure 12 where
the rational expectation equilibrium would feature a completely straight Phillips curveeven when expected inflation is negative.
Conditional on zero inflation forecast and output gap forecast in equation (16), we
compute the realized output gap and inflation by taking output gap and inflation forecasts as given and solving for the intersection of the realized Euler equationxL =1[max {r + p + f (p Lsp ) , 0}19pr ] + (1✓µ) xeL +

eL
p

p
s

◆

,

(17)

and the realized Phillips curvebq (1 µ)1 qeLp L = p + kx L +k xeL +
(p
1 bq (1 µ)
q

p) .

(18)

The realized Euler equation (17) and the Phillips curve (18) are written conditional on
any level of output gap and inflation forecasts and not only those found on the previous
step of the backward induction procedure. This is to highlight how expectations influence current determination of the equilibrium variables. In particular, the second term inequation (17) and the third term in equation (18) formalize this forecasts’ effect.
There are two possibilities in equilibrium: the ELB on the nominal interest rate binds
or it is slack. First, if the natural interest rate falls below the cutoff
ri,ZLB ⌘

µs [1

bq (1

µ { s [1fk (1 µ) (1 + bqµ)rfkµ)] + kf} + (1 µ)f [k (f
fk

µ)]

bq (1

1) (1 + bqµ)µsb (1q )]p,then the ELB binds and equation (17) yields the actual output gap of
xL =

1

µ
[(f + µµs1) p + rr] +1(µp + r ) .
sand the equilibrium inflation rate is pinned down by the Phillips curve (18).10
This case is interesting. Despite the fact that agents do not expect to face the ELB on
the interest rate under negative aggregate demand shock, the ELB actually binds. The
reason is that when forecasting the future, the agents mistakenly believe that inflation
will not fall below zero, which limits its impact on the forecast interest rate. In reality,
however, the inflation rate does fall below zero pushing the interest to its lower bound.
Second, when the natural interest rate r is above ri,ZLB , the ELB does not bind in equilibrium and the realized output gap is1
xL =µ (fk + s )⇢[(f + µ1) p + r✓r] 1

bq (1 µ)fk·
1 bq (1 µ) s

◆

µf

1
1

b (1 µ )p ,bq (1 µ)and the inflation rate is again given by equation (18).
The requirement that the interest rate shock is below the cutoff r p,no ELB ensures thatthe realized inflation rate p L is negative.
10 The fact that r

when µsb < (fnon-empty.

r> ri,REE
i,REE < r p,no ELB necessarily implies that r i,ZLB < r p,no ELB . Furthermore,
T i,ZLB
1) k. Thus there are model parameters for which the set (ri,REE , r p,no ELB ] ( •, ri,ZLB ] is20

Both constraints bind. Finally, when agents forecast that both constraints bind at thesame time, the intersection of the expected Phillips curve (6) and expected Euler equationeL = 0 and xeL = (µp + r ) /(µs).
(5), depicted in Figure 11 by curves PC and EE000 , is p
Point A3 on the figure represents this solution. The following condition on the parametersof the model has to be satisfied for the ELB to bindr  (f1)p.

Importantly, this condition does not involve the size of the shock r, it just follows fromplugging zero inflation rate in the Taylor rule (7). For inflation expectations to be constrained at zero, the shock r should be below the cutoffr p,ELB ⌘

µ

s [1

b (1
k

µ)] + kp < 0,

which follows from combining the solution to the forecast output gap and the Phillips
curve (6).
To compute the realized inflation and output gap, we guess and then verify that the
ELB constraint binds contemporaneously. The Euler equation (17) and the Phillips curve(18) yield
1
x L = xeL =(µp + r ) ,sµ⇥⇤k
1 b(1 µ) + sk p + sµ
r
pL =
< 0.
1 bq (1 µ)

(19)
(20)

Note that the reason why output gap equals expected output gap under rtn = r is because
the nominal interest rate and expected inflation are zero. As a result, the realized Euler
equation is identical to expected Euler equation.
Given this solution, the interest rate prescribed by the Taylor rule isi L = i + f(p Lp )  0,which verifies our assumption that the ELB binds even contemporaneously. Furthermore,
the condition that the interest rate shock is below the cutoff r p,ELB guarantees that therealized inflation is below zero.
An important property of the solution in equations (19) and (20) is that it does not
explode when probability µ approaches its cutoff where the denominator J (µ) in equation(11) becomes zero. This property contrasts with the FIRE case. Intuitively, a typical NewKeynesian deflationary spiral under the ELB on the nominal interest rate is turned off
21

when inflation forecast remains at zero. However, note that as µ declines, output gap andinflation still become large in absolute terms but negative because r < r p,ELB < 0. This isbecause the effect of the low future natural interest rate still propagates and accumulates
through the Euler equation. It is the amplification mechanism of the Phillips curve that isturned off.
Finally, we emphasize that while deflation occurs at the ELB on nominal interest rate,
it is milder than that without the ZLB constraint on inflation expectations.

4

Unconventional Monetary Policies

The experience with unconventional monetary policies during the Great Recession andits aftermath left many economists puzzled by apparent disconnect between the powersof these policies in many macro models and the lack of such powers in reality (e.g., theso-called “forward guidance puzzle”, see Del Negro, Giannoni and Patterson, 2012). In
this section, we show that unconventional policies that have large effects during the ELB
under rational expectations have weaker effects when the ZLB on inflation expectations
binds.

4.1

Forward Guidance

We start with the policy of forward guidance, which is an attempt to affect current aggregate economic activity by promising certain policy actions in the future, most notably,
to keep a low interest rate for long. We show that forward guidance is less powerfulwhen the zero-lower bound on inflation expectation binds. Intuitively, because inflation
expectations do not respond to forward guidance, the effect is confined to the response of
output gap to the future promised interest rate changes through the Euler equation.
To formally illustrate this result, assume that after the ELB ends, the central bank keeps
the interest rate at zero for some time. The probability of continuation of zero interest rate
policy is time dependent. Specifically, with probability 1 p0 , the interest stays at zero,
while with probability p0 , it starts following the Taylor rule in period t = T, when theshock to the natural interest rate ends. In all later periods t > T, these probabilities
change to 1 p1 and p1 , respectively. It will be useful to distinguish between p0 and p1
in the analysis below. In particular, probability p1 will not only affect the duration of the“low for long” policy but also its impact on output gap and inflation after the negativeinterest rate shock dissipates. At the same time, probability p0 only affects the initialchance of low interest rate policy continuation without affecting its impact on output gap
and inflation in periods t T. As a result, we will be able to study the output effect of an22

increase in the chance of the low interest rate policy conditional on its strength in t
T.
Finally, we denote the (stochastic) period when the interest rate returns to the Taylor ruleas TTR and solve the model again by backward induction.
First, for any t TTR , we have xt = 0, pt = p, it = i. Second, for t 2 [ T, TTR ), the Euler
equation and the Phillips curve determine contemporaneous and future forecast valuesof inflation and output gapb(1 p1 )] (p + r ),b(1 p1 )] (1 p1 )kx FG
p FG = p + k> p,
1 b (1 p1 )x FG =[1
p1 s [1where x FG and p FG are output gap and inflation during the period of stimulative monetary policy. A unique bounded equilibrium exists only when (1 p1 )k < p1 s [1 b(1 p1 )],that is, the central bank does not keep the interest rate low for too long after the exit of
the ELB. Note that because inflation p FG is positive, the actual inflation coincides with
inflation forecast because the ZLB on inflation expectations does not bind.
Third, consider periods t 2 [0, T ). To jump to the core of our analysis, we focus on
the case when both the ELB on the nominal interest rate and the ZLB on inflation expectations bind provided the natural interest rate is low and absent forward guidance policy.
Furthermore, we assume that forward guidance is sufficiently modest so that the two constraints continue to bind even when the policy is implemented. As a result, the forecasts
eL = 0 and of output gap isof inflation is p


1
xeL =
(µp + r ) + (1sµp0 ) x FG +p FG p,s

The bottom panel of Figure 13 plots the forecast Euler equation and the Phillips curve thatgenerate the above values of inflation expectation and output gap in the state with rtn = r.
It is instructive to compare the effect of the forward guidance on expectations when the
ZLB on inflation expectations binds with the case of rational expectations, which we plot
in the top panel of Figure 13. It is evident that for the same size of forward guidance
policy, expected output gap increases by less under the ZLB on inflation expectations.
There are two reasons for this, both of which work through the Phillips curve. First, when
inflation expectations are stuck at zero, forward guidance does not directly affect the flatpart of the Phillips curve and inflation forecast for that state remains at zero. Second,
forward guidance does not influence expected inflation indirectly through an increase in
output gap because the expected inflation is stuck at zero.
The realized output gap is23

x L = xeL = x

no FG



+ (1

p0 ) x FG +p FG p,s(21)

where the first term of the equation xno FG ⌘ (µp + r ) /(sµ) is the output gap without
forward guidance. An example of the evolution of economic variables is presented in Figure 14 where we set random realizations of T to 5 and TTR to 8. The last term in equation(21) quantifies the effect of forward guidance. It is proportional to the probability 1 p0that the policy will be implemented as well as its strength summarized by the term in thesquare brackets.
The next step is to compare expression (21) to its rational expectations counterpart.
Assuming that the ELB constraint binds, we solve for output gap
sµ [1FG, REE
x L = xno
+
L
sµ [1


b(1 µ)] (1 p0 )x FG +b(1 µ)] (1 µ) k
1

1
b (1

1
(p FG
µ) s

·

p) ,
(22)

FG, REE
where xno
is the rational expectations output gap in the ELB expressed in equationL(11). It is clear that the strength of the forward guidance effect on output gap in rational
expectations equilibrium, the second term in equation (22), is larger than that in equilibrium where inflation expectations are constrained to be non-negative, the second term inequation (21). Formally, this is because the coefficient multiplying the square bracket inequation (22) is larger than one, and the coefficient multiplying the deviation of inflationfrom its target is also above one. Intuitively, inflation forecasts in the ELB state are notaffected by forward guidance announcements (unless they are sufficiently stimulative tolift inflation expectations above zero) and the feedback from higher expected inflation on
the ex ante real interest rate and, hence, real economic activity is switched off.
Interestingly, the strength of forward guidance increases with the persistence of theFG, REE
ELB episode in rational expectations equilibrium because ∂( x L xno)/∂(1 µ) >L
0 in equation (22), while it does not depend on the ELB duration when inflation forecasts
FG ) /∂ (1are constrained at zero because ∂( x L xnoµ) = 0 in equation (21). The strengthLdepends on the duration of the ELB in the rational expectation equilibrium because anincrease in output gap boosts inflation, which, in turn, increases output gap. This twoway feedback is stronger the longer agents expect the economy to remain at the ZLB.
With the constraint on expected inflation, forward guidance increases expected inflation
outside of the ELB, but then output gap does not affect inflation expectation during the
ELB.

24

4.2

Negative interest rates

This section investigates the possibility of allowing the short term nominal interest rate to
go below its lower bound. When the interest rate is allowed to be negative, the solution
to the output gap is similar to the one in equation (19) with the only addition being e > 0
inside the brackets, which represents the value by which the interest rate can go negative.
Taking the difference between the output gap with the interest rate set at a small negativevalue e and zero and assuming that the ZLB on inflation expectation binds, we obtainD ⌘ x L (i L =x L ( i L = 0) =

e)

1
e.
µs

The difference under rational expectations is
D REE ⌘ x L (i L =e)x L ( i L = 0) =[1

1 b (1 µ )b(1 µ)]µs (1

µ )k

e.

It is clear that D REE > D. Interestingly, in both cases the effectiveness of this policy(slightly negative interest rate) depends on the duration of the ELB. The notable difference, however, is that the ELB needs to be permanent for the negative interest rate to
become infinitely effective under the ZLB on inflation expectations, while under rational
expectations it occurs for a smaller value of the ELB persistence 1 µ that satisfies q = 0.

4.3

Optimal Inflation Target

Because the ELB on expected inflation makes unconventional policy tools potentially lesspowerful and hence the cost of deflation spirals greater, one may think that the ELB on
expected inflation justifies raising the inflation target so that the central bank is less likely
to run into this constraint. To keep our analysis close to the standard three-equation New
Keynesian model, we assumed that firms can index their prices with the trend level ofinflation. While this assumption makes the model tractable, it greatly attenuates pricedispersion, a key cost of positive trend inflation (Coibion et al., 2012), thus strengthening
the rationale for raising the target. On the other hand, the risk of deflation is greatly
reduced when economic agents do not believe that deflation is possible. Indeed, ouranalysis suggests that in this case the possibility of sunspot variation is attenuated greatly.
Because the cost of deflation is highly nonlinear in the duration of nominal interest rates
at the ELB, the benefit of raising the target is reduced as well. As a result, the calculusof tradeoffs involved in establishing the optimal inflation target is more nuanced anddepends on parameter values (frequency of price changes, frequency and size of shocks
that put the economy at the ELB for nominal interest rates, etc.) as well as other details of25

the model.

5Aggregate Supply and Demand Shocks

The effective lower bound on the nominal interest rate changes the response of the economy to shocks in standard New Keynesian models. This sections investigates how thepresence of the zero lower bound on inflation expectations alters this prediction by considering shocks to the Phillips curve and the Euler equation.

5.1

Aggregate Supply Shocks

Inflationary Phillips-curve shocks, which we will refer to as negative aggregate supply
shocks, are contractionary when the Taylor rule is active but can be expansionary under the ELB when agents have FIRE (Eggertsson, 2011, 2012). For example, shocks that
increase firms’ marginal costs for some time, such as higher taxes or positive shocks tointermediate input prices, tend to increase inflation and expectations thereof. With an active Taylor rule where the nominal interest rate responds more than one-to-one to changes
in inflation rate, the real interest rate increases thus depressing current consumption andoutput gap. By contrast, when the ELB on the nominal interest rate binds, the nominalinterest rate does not change, forcing the real interest rate to go down due to an increasein expected inflation. This, in turn, has a positive effect on consumption demand and output gap. We show that this logic does not hold when the zero lower bound on inflation
expectations binds.
To formally illustrate the effects of negative aggregate supply shocks, we add shock ut
to the contemporaneous and expected Phillips curves (3) and (6) as follows•kpt = p + kxt + Â ( bq ) E t k xet+k + ut+s +et+s = max
p

(

k =1
•

1

q
q

q •

q

k =0

p + Â ( bq )k E t+s [k xet+s+k + ut+s+k ] +
k =0

p) ,

et + k(p1

Â ( bq )k+1Et+s [pet+s+1+kp] , 0 .

The Euler equation does not directly depend on the shock to the Phillips curve. To
generate the ELB on the nominal interest rate and the ZLB on inflation expectations, we
continue using a sufficiently large negative shock to the natural interest rate rtn = r thatreverts back to its steady state value r with probability µ. In addition, we assume thatshock ut is zero in steady state and positive, i.e., ut = u > 0, as along as the natural

26)

interest rate equals its low value of r.11
The top panel in Figure 15 plots the determination of inflation and output gap under
FIRE. The shock pushes the Phillips curve up from the solid to the dashed line. The
equilibrium changes from point A1 to point A2 , which features a higher output gap andinflation. The bottom panel of Figure 15 describes the determination of expected output
gap and inflation when the ZLB on inflation expectations binds. This diagram illustrates
that when shock ut is small enough, it fails to move the intersection (point A1 on the
figure) of the Euler equation and the Phillips curve. As a result, the households continue
to expect zero inflation and the same value of output gap, which is formally expressed inequation (19). Furthermore, the actual realized output gap will not change as well because
the nominal interest rate is at its ELB, and expected output gap and inflation rates do notchange. Meanwhile, the realized inflation does go up. Formally, the contemporaneousPhillips curve together with the solved values for forecast inflation and output gap give
pL =

⇥

1

b (1
1

⇤
k
µ) + sk p + sµ
r+u
bq (1

µ)

.

The last expression shows that the actual inflation rate goes up following the shock.
The above results have some empirical support. Wieland (2019) finds that the respondents of the Michigan Survey of Consumers do not change their inflation expectations
in response to negative aggregate supply shocks (proxied by negative oil supply shocks)
at the ELB on the nominal interest rate. At the same time, professional forecasters (represented by the Consensus Economics average inflation forecast) and financial markets(represented by the difference between the nominal and real rates) do increase their inflation expectations following negative aggregate supply shocks (proxied either by theGreat East Japan earthquake or by negative oil supply shocks) in Japan. This is consistent
with our observation that the realized inflation goes up and that professional forecasters
have much more accurate inflation expectations (Section 2.5). At the same time, Wieland(2019) finds that aggregate output drops after negative supply shocks. This can be consistent with the prediction of our model that output gap is unchanged and the fact thatnegative supply shocks reduce natural output.

5.2

Aggregate Demand Shocks

In standard New Keynesian models, aggregate demand shocks, such as an increase ingovernment purchases, can have a stronger impact on the economy under the ELB on
11 It is straightforward to extend the analysis to the case when shocks u

the shocks to the natural interest rate.

27

t is more or less persistent thanthe nominal interest rate than in the normal times when the ELB is slack. Moreover, thisdifference between the ELB and normal times effects can be extremely large (Woodford,
2010; Christiano, Eichenbaum and Rebelo, 2011). The reason for the difference in theeffects is that the nominal interest rate does not react to changes in inflation following
shocks when the ELB on nominal interest rate binds. In addition, persistent shocks affectexpected inflation, which, in turn, changes current output gap and inflation, magnifying
the direct effect of the shock.
This section focuses on one example of aggregate demand shocks—changes in government purchases. We show that the ZLB on inflation expectations keeps the result that theoutput multiplier is larger under the ELB on the nominal interest rate than under active
Taylor rule. At the same time, the amplification of persistent aggregate demand shocks is
attenuated when the inflation expectations are stuck at zero.
With an additional source of spending in the economy, we need to modify the Euler equation and the Phillips curve. The contemporaneous and future Euler equations
becomext
xet+s

gt = E t [ xet+1

gt +1 ]

gt+s = E t+s [ xet+s+1

1
(it
s

gt + s +1 ]et + 1 ]E t [p

1 e
(it+s
s

rtn ) ,et + s + 1 ]E t+s [ prtn+s ),

where gt ⌘ ( Gt G )/Y, Gt is government purchases, and G is its steady-state value. The
current and expected Phillips curves are
pt = p + k ( xt
et+s = max
p

(

•

k



Ggt ) + Â ( bq ) E t k ( xet+k
k =1•

p + k Â ( bq )k E t+s [ xet+s+k
k =0

Ggt+k ) +Ggt+s+k ] +

1

q
q
q •

1
q

et + k(pp) ,Â ( bq )k+1Et+s [pet+s+1+kp] , 0 ,k =0where G < 1 is the government spending output multiplier under flexible prices.12 Observe that a change in government consumption not only directly affects the Euler equations but also the Phillips curves. This is because higher government spending makes
households poorer due to an increased tax burden and, hence, more willing to work (dueto a negative income effect on leisure). This reduces firms’ marginal costs and pushes
inflation down. We assume that gt = g > 0 as long as the natural real interest rate is r,
and gt = 0 when the interest rate reverts back to its steady-state value, and that all agents
12 G can be obtained by differentiating the intra-temporal optimality condition of the representative

household under flexible prices. When the household preferences take the form u(Ct ) v( Nt ) and the production function is linear Yt = At Nt , the intertemporal optimality condition u0 (Yt Gt ) = v0 (Yt /At )/At
yields G ⌘ dYt /dGt = u00 (Yt Gt )/[ u00 (Yt Gt ) + v00 (Yt /At ) /A2t ].

28)

are aware of this law of motion.
Figure 16 graphically presents the solution. The upper panel shows the case of FIRE.
The Euler equation and the Phillips curve both move to the right. Importantly, because
the Euler equation shifts to the right by g, while the Phillips curve shift to the right by
Gg < g, the intersection of the shifted curves is in point A2 where both inflation andoutput gap are larger than those in point A1 . Furthermore, because a change in inflationis positive, an increase in output gap is larger than g, implying that the fiscal multiplier islarger than one. Formally, the output gap in point A2 of the upper panel of Figure 16 isxL =[1 b(1 µ)] µs (1 µ) kG
no g,REE
g + xL,[1 b(1 µ)] µs (1 µ) k

no g,REE

where x L
is the rational expectations output gap under the low value of naturalinterest rate and no change in government consumption, expressed in equation (11). It isevident that the coefficient that multiplies the government spending in the first term of
the last formula is greater than one as long as the condition for the existence of the uniqueequilibrium (12) is satisfied. What is more, when µ increases so that the denominator[1 b(1 µ)] µs (1 µ) k becomes arbitrarily close to zero, the multiplier Dx L /Dg becomes unbounded.
We now contrast the FIRE case to the case of the ZLB on inflation expectations. The
bottom panel of Figure 16 plots the expected Euler equation and the Phillips curve before
the fiscal shock (solid curves) and with positive government consumption (dashed lines).
The expected output gap and inflation in the low natural interest rate state go from point
A1 to point A2 , implying that the inflation expectations does not change, while output
gap increases by g. Expected inflation does not move because the shock is small enoughso that the economy is still at the ZLB on inflation expectations. Expected output gap
moves by exactly g because the interest rate and expected inflation are constant at zero.
This is in contrast to the case of FIRE beliefs where expected (and realized) output gap
increases by more than g.
Realized inflation and output gap can be obtained by jointly solving contemporaneousPhillips curve and the Euler equations conditional on the above expectations. We findno g

x L = xeL = xeL

no gpL = pL

no g

+ g,
+

no g

k (1 G )g,
1 bq (1 µ)where xeL and p L (expressed in equations (19) and (20)) are the values of output gap
and inflation under the low natural interest rate, ELB on nominal interest rate, and ZLB
29

on expected inflation. There are two main messages from inspecting the two formulas.
First, the output multiplier is one and the inflation response is positive. Second, the output multiplier value of one does not depend on the duration of the ELB on the nominalinterest rate. At the same time, inflation response increases with more persistent ELB
episode, i.e., with larger 1 µ. However, this sensitivity does not blow up as the ELB
episode becomes more and more persistent.
The predictions of our model are consistent with empirical findings in Miyamoto,Nguyen and Sergeyev (2018) for Japan. First, they show that the fiscal multiplier is significantly larger during the ELB on the nominal interest rate than in normal time in Japan.
However, the authors cannot reject that the ELB output multiplier is equal to one. Second, they find that inflation responds significantly more during the ELB episode than innormal times following a shocks to government consumption. In addition, inflation forecast by the Japan Center of Economic Research also increases by more during the ELB
episode than during the normal period. This is again consistent with the observation thatprofessional forecasters tend to have more accurate inflation forecasts.

6

Self-fulfilling Liquidity Traps

Sections 3-5 analyzed theoretical implications of the ZLB on inflation expectations in amodel where a low inflation state, which generated a liquidity trap, resulted from a fundamental shock to a natural rate of interest. However, this is not the only possibility. Ina seminal contribution, Benhabib, Schmitt-Grohe and Uribe (2001a); Benhabib, SchmittGrohé and Uribe (2001b) showed that self-fulfilling liquidity traps with deflation can occur without any fundamental shocks when monetary policy rate reacts strongly to inflation unless it reaches the ELB.13 In this section, we show how the ZLB on inflation expectation can prevent confidence driven liquidity traps from occurring in a simple New
Keynesian model.
With FIRE, the 3-equation New Keynesian model from Section 3 has a standard steadystate where inflation equals its target value of p, output gap is zero, and the nominal interest rate is i. However, there is an additional steady state that satisfies the three equations.
First, the nominal interest rate is at the ELB which we normalize to zero. Then, because of
the Fisher identity, the inflation rate is negative and equals r. Finally, the Phillips curve
implies that output gap is (1 b)(r + p )/k.
An important assumption leading to a self-fulfilling liquidity trap steady state is thatagents anticipate that inflation will be negative in the future. If, however, inflation ex13 Mertens and Ravn (2014) analyze fiscal policies effects and Bilbiie (2018) studies positive and norma-tive implications of monetary and fiscal policies in confidence-driven liquidity traps.

30

pectations cannot fall below zero, this liquidity trap is no longer possible. To formally
illustrate this point, we assume that the economy faces only non-fundamental (“sunspot”)
shocks x t that coordinate agents’ beliefs. The stochastic nature of x t allows us not only torule out a self-fulfilling liquidity trap steady state but also its temporary version. Following Mertens and Ravn (2014), we assume that a sunspot can take on two values: “pessimistic” x L and “optimistic” x H , with transition probabilities P (x t+1 = x H |x t = x H ) = 1
and P (x t+1 = x L |x t = x L ) = 1 µx 2 [0, 1]. Hence the optimistic state is absorbing. It
corresponds to the standard steady state with (pt , xt , it ) = (p, 0, i). The economy starts
with the “pessimistic” state x L .
FIRE benchmark. We consider the FIRE assumption first. The equilibrium is describedby the Euler equation (8), where we µ replace with µx and r with r, and by the Phillips
curve (9), where µ is replaced by µx . The top panel of Figure 17 plots these two curvesin the ( x L , p L )-space. When the upward sloping part of the Euler equation is flat enough(longer ELB episodes), there are two equilibria. The intersection A H is the standard steadystate, while the intersection A L features liquidity trap. The fact that output gap and inflation are negative in point A L explains why we call this state “pessimistic.” For the two
equilibria to exist, the model parameters need to satisfy
J ( µ x ) = µ x s [1

b (1

µx )]

(1

µx )k < 0,(23)which is the opposite of the condition 12 that ensures a unique bounded equilibrium
under fundamental shock to the natural interest rate. Condition (23) holds, for example,
when the persistence 1 µx of the pessimistic state is high enough, which includes apermanent pessimistic state.
ZLB on inflation expectations. When agents expect only non-negative inflation, selffulfilling liquidity traps disappear. To demonstrate this result, we solve for beliefs bylooking for the intersection of the forecast Euler equation (14) and the Phillips curve (15),where we again replace µ with µx and r with r. The bottom panel of Figure 17 presents
the two equations.
An important feature of the Euler equation EE in Figure 17 is that it intersects zeroinflation at a positive output gap. Even when the persistence of the pessimistic stateis close to one, making the upward-sloping part of the Euler equation nearly flat, theintersection with zero inflation still occurs at a positive level of output gap. Formally, in
eL = 0, the Euler equation is xeL = (1 µx ) xeL + s 1 (µx p + r ),the case when ei L = 0 and p
implying that x L is positive regardless the value of µx . Because of this fact, there is only
one intersection of the Euler equation and the Phillips curve in the bottom panel of Figure31

eL , xeL ) = (p, 0) . Given these beliefs, the actual
17. This point is the standard steady state (prealized inflation and output gap are (p, 0). Formally, these values solve equations (17)and (18), where µ = µx and r = r.
These results suggest that the ZLB on inflation expectations can be an important factor
that prevents self-fulfilling liquidity traps.14

7

Conclusion

Is deflation on the horizon? This is a question that many central bankers are deeply concerned about in current economic conditions with depressed economy and persistently
low inflation. And they should be as the Great Depression in the 1930s and Japan since
the early 1990s are grim reminders of what deflation can inflict. Furthermore, once deflation takes hold over the minds, the escape may be difficult. For example, while discussing
the Japanese experience at the 2018 ECB conference, Haruhiko Kuroda, the governor of
the Bank of Japan, said, “The deflationary mindset that has become entrenched amongst
people is quite tenacious and it will take time to completely dispel this mindset.”
Fortunately, our findings offer some hope, at least in the short run. In contrast to professional forecasters, households do not anticipate deflation in the near future even when
there is a deflation and/or professional forecasters project deflation. Specifically, in the
cross-sectional distribution, few households predict deflation. The share of households
predicting deflation shows little variation over time. Once pushed close to zero, households’ inflation expectations are insensitive to variation in actual inflation. In short, there
seems to be a zero lower bound (ZLB) on households’ inflation expectations. Although
the lack of high-quality surveys of firms’ inflation expectations prevents us from having
definitive conclusions, the available evidence suggests that the same ZLB on inflation
expectations applies to firms as well.
Using the standard three-equation New Keynesian model as a starting point, we showthat households’ inflation expectations with a floor at zero can affect macroeconomic dynamics and the effectiveness of unconventional policy tools that rely on managing expectations. For example, if the policy rate is stuck at the effective lower bound (ELB) andinflation expectations are stuck at the ZLB, forward guidance will have weaker effects
on the economy. Intuitively, with full-information rational expectations (FIRE), forward
guidance typically raises inflation expectations even if the nominal interest rate is at the
14 We note that there can still be multiple equilibria when the economy experiences both sunspot andfundamental shocks. For example, if there is a negative real interest rate shock in addition to sunspot
shocks discussed in this section, then the Euler equation in the bottom panel of Figure 17 moves to the left.
If this shift is small, the number of equilibria does not change. With an intermediate size of the shift, there
can be three intersections. When the shift is large enough, there is only one liquidity trap equilibrium.

32

ELB. However, this can mean increasing inflation expectations from, say, -3 percent to -2
percent. If inflation expectations are stuck at the ZLB, then such an increase in FIRE expectations is infra-marginal. On the bright side, if inflation expectations can’t fall below
zero, the possibility of deflationary spirals is greatly attenuated and hence the policymaker may be in a situation that is less desperate than predicted by FIRE-based models.
While our analysis remains agnostic about the sources of the ZLB on inflation expectations, future work will likely have to rely on randomized controlled trials that create
exogenous variation pushing households’ inflation expectations toward zero. With suchvariation as well as additional survey data, one may hope to establish why households
are reluctant to predict deflation. We also hope that future work will move beyond theSurvey of Firms’ Inflation Expectations and TANKAN and make progress in understanding whether and how ZLB on inflation expectations applies to firms. Finally, while thebasic New Keynesian model offers remarkable tractability and thus important theoreticalinsights, future work can provide deeper quantitative assessments of ZLB on inflation expectations. But in any case, central bankers likely do not have to worry about dispellinghouseholds’ deflationary mindset as households do not predict deflation on the horizon.

33

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37

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Figure 2: The histogram of realized CPI inflation (in red) and Bank of Japan’s Opinion Survey one year
ahead inflation expectations (in gray) between 2004Q1 and 2020Q1. The values of realized inflation in theinterval [-0.2%;0.2%] are set to zero.

39

Figure 3: The top panel of the figure shows Japanese realized inflation (bold black line), Japanese realizedcore inflation (dashed black line), and a median inflation forecast from Bank of Japan’s Opinion Survey(solid gray line). The bottom panel presents shares of Bank of Japan’s Opinion Survey participants who
expect positive inflation (green line), zero inflation (yellow line), and negative inflation (red line).

40

Figure 4: The top panel of the figure shows realized inflation (bold black line) and realized core inflation(dashed black line) in Greece. The bottom panel presents shares of European Commission Business andConsumer Survey participants in Greece who expect positive inflation (green line), zero inflation (yellowline), and negative inflation (red line).

41

Figure 5: The figure reports estimates of intercept a(t ) (top panel) and slope b(t ) (bottom panel) estimatedfor specification 1. The intercept refers to the white, married male respondent who has no children, lives
in the West region, and whose highest educational attainment is high school diploma. The solid, thick,black lines are point estimates. The dashed, blue lines are the 95 percent confidence intervals. Confidence
intervals are based on standard errors clustered by time. The estimation sample is 1978M1-2021M4.

42

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Figure 6: Data are from the Survey of Consumer Expectations. The sample period of all panels is 2013M-2020M1. Panel A is constructed asfollows. First, we compute average (across respondents) probability for each bin in a given wave. Second, we compute average (across time) andpercentiles. The red whiskers show 90th and 10th percentiles. For Panel B, we compute the average (across respondents) probability assigned
to a inflation/deflation bin and plot time series for each inflation/deflation bin. For Panel C, we compute the contribution due to inflation ase,+pt,t+12 = 1 · P ( pt,t+12 2 (0, 2)) + 2 · P ( pt,t+12 2 (2, 4)) + 6 · P ( pt,t+12 2 (4, 8)) + 10 · P ( pt,t+12 2 (8, 12)) + 14 · P ( pt,t+12 2 (12, • )), wheree,
P (pt,t+12 2 X ) is the average (across respondents) probability for inflation being in set X. The contribution due to deflation is pt,t
+12 = 1 · P ( pt,t+12 2( 2, 0)) + 2 · P (pt,t+12 2 ( 4, 2)) + 6 · P (pt,t+12 2 ( 8, 4)) + 10 · P (pt,t+12 2 ( 12, 8)) + 14 · P (pt,t+12 2 ( •, 12)); the expected inflatione,+e,e(“total”) is computed as pt,t
pt,t
+12 = pt,t+12+12 .

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Figure 7: The histogram of one year ahead inflation expectations in Netherlands from the Dutch National
Bank Household Survey between 2003 and 2020. The values below 20 percent are set to 20 and above
20 percent are set to 20 on the histogram.

44    !  
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20

30

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forecast (1y ahead)actual

-2

-1

0

1

2

3

4

5

CPI Inflation Expectations in Japan, Tankan (2014Q1-2021Q3)

-2

Japan

6

4

Figure 8: The top left panel is the histogram of one year ahead inflation expectations by professional forecasters in the US from the Survey of Professional Forecasters between 1985 and 2020. The top right panel presents the histogram of one-year ahead inflation expectations by Consensus
Economics professional forecasters (olive bars) and realized inflation rate (black lines) in Japan using monthly data from 2005 to 2019. The consensus
economics one-year ahead expected inflation is computed as (12-month)*(forecast for the current calendar year)+month*(forecast for the next calendar year). The bottom left panel is the histogram of one year ahead inflation expectations by firm CEOs in the US in 2020Q3 from Candia et al. (2021).
The bottom right panel is the histogram of one year ahead consumer price index inflation expectations by enterprises in Japan between 2014Q1 and2021Q3 from the Short-Term Economic Survey of Enterprises in Japan (TANKAN).

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0.05

0.1

0.15

0.2

0.25

0.3

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Figure 9: One-year ahead forecast of unemployment changes (the left panel) and interest rate changes (theright panel) from the Michigan Survey of Consumers between 1985 and 2021.

46

EE'' EE'

EE

PC

EE'''

Figure 10: The diagram shows the upward-sloping Phillips curve in solid blue and four piece-wise linear
Euler equations. They correspond to r n = r (solid orange line EE), r n = r and r is small enough (in absolutevalue) such that the ELB on the nominal interest rate does not bind in equilibrium (dashed orange line
EE0 ), r n = r and r is large enough (in absolute value) such that the ELB on the nominal interest binds inequilibrium but not too large so that the inflation rate is positive in equilibrium (dash-dotted orange line
EE00 ), r n = r and r is large enough (in absolute value) such that the ELB on the nominal interest binds andthe inflation rate is negative in equilibrium (dotted orange line EE000 ), respectively.

47

EE''

EE'

EE

PC

EE'''

Figure 11: The diagram shows the upward-sloping forecast Phillips curve in solid blue and four piece-wiselinear forecast Euler equations. They correspond to r n = r (solid orange line EE), r n = r and r is smallenough (in absolute value) such that the ELB on the nominal interest rate does not bind in equilibrium(dashed orange line EE0 ), r n = r and r is large enough (in absolute value) such that the ELB on the nominalinterest binds in equilibrium but not too large so that the ZLB on inflation expectations is slack in theintersection A1 (dash-dotted orange line EE00 ), r n = r and r is large enough (in absolute value) such that
the ELB on the nominal interest and the ZLB on inflation expectations bind in the intersection A3 (dottedorange line EE000 ), respectively. The realized inflation rate and output gap values coincide with their forecast
values in the case when the Euler equation is in the positions EE, EE0 , and EE00 , but not when the Euler
equation is in the position EE000 .

48

EE

PC

EE'

Figure 12: The diagram shows the upward sloping (forecast) Phillips curve PC together with its flat part,which corresponds to the ZLB on inflation forecast and two piece-wise linear (forecast) Euler equations
corresponding to r n = r (solid line EE0 ) and r n = r (dashed line EE00 ), respectively. In the intersection, the
ELB on the nominal interest rate does not bind when r n = r, however the ZLB on expected inflation does.

49PC' PC
EE' EE''

PC' PCEE' EE''

Figure 13: The top panel plots the Euler equation and the Phillips curve when households have FIRE
beliefs, while the bottom panel plots expected Euler equation and Phillips curve when beliefs feature the
ZLB on inflation expectations. The solid Phillips curves and Euler equations in both panels are plotted
without forward guidance, while the dashed Phillips curves and Euler equations correspond to the casewith forward guidance.

50         













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   



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





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





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



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



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



Figure 14: The figure plots the evolution of equilibrium interest rate, output gap, and inflation as well as
their forecasts in the possible future states for one particular realization of the path of the natural interest
rate that reverts back to its normal level in period t = 5 and for one realization of the evolution the low forlong monetary policy. The central bank starts following the Taylor rule in period t = 8. The ELB on thenominal interest rate binds contemporaneously and in the future under the low realization of the naturalinterest rate. The expected inflation under the low realization of the natural interest rate is zero.

51

PC' PCEE'

PC' PCEE'

Figure 15: The top panel plots the Euler equation and the Phillips curve when agents have FIRE beliefs,
while the bottom panel plots expected Euler equation and Phillips curve when beliefs feature the ZLB on
inflation expectations. The solid Phillips curves in both panels show are plotted for the case of ut = 0, while
the dashed Phillips curves correspond to ut = u > 0.

52

PC'
PC''

EE''
EE'

PC'
PC''
EE''
EE'

Figure 16: The top panel plots the Euler equation and the Phillips curve when agents have FIRE beliefs,
while the bottom panel plots expected Euler equation and Phillips curve when beliefs feature the ZLB on
inflation expectations. The dashed curves in both panels are plotted for the case of gt = g > 0, while thesolid curves correspond to gt = 0.

53

EE

PC

EE

PC

Figure 17: The top panel plots the Euler equation and the Phillips curve when agents have FIRE beliefs,
while the bottom panel plots expected Euler equation and Phillips curve when beliefs feature the ZLB on
inflation expectations.

54

A

Appendix

A.1

The Phillips Curve Derivation

This appendix derives the version of the Phillips curve with the constraint on inflation expectations.

Aggregate Price Level. The contemporaneous aggregate price level is
Pt =

Z

[PPt 1 (i )]1 edi + (1h1 e= qPt1 1e P
+ (1

1
1 e

q ) ( Pt⇤ )1 eq ) ( Pt⇤ )1 e

i 1

,

✓

Pt⇤
Pt 1

1 e

where Pt⇤ is the optimal price and 1 q is the fraction of firms that change their prices in period t. Note that
we took account that those firms that do not set their optimal prices nevertheless change their prices by thesteady state level of inflation due to indexation. The contemporaneous inflation rate is
P1t e =

✓

Pt
Pt 1

◆1 e

= qP

1 e

+ (1

q)

◆1 e

.

In the steady state with gross inflation P, we have Pt = Pt⇤ and Pt /Pt 1 = Pt⇤ /Pt 1 = P. Log-linearzing
around Pt = P, we get
pt p = (1 q ) ( p⇤t pt 1 p ) ,where pt ⌘ log ( Pt /Pt 1 ) , p ⌘ log P, p⇤t = log Pt⇤ , and pt = log Pt .
Note that the same equation must hold for future forecasts of the aggregate price and inflation. Specifically,
et+1 p = (1 q ) pe⇤t+1 pt p ,pandet + spp = (1q ) ( pe⇤t+s

pet+s 1p) ,q ) pe⇤t+1

pt

p ,0 ,

when s > 1. Also note that because expected inflation cannot be negative,
et+1 = max p + (1pfor s > 1.

et+s = max {p + (1pq ) ( pe⇤t+s

pet+s 1p ) , 0} ,Reset prices. Firm i operates the following production function
1 aYi,t = At Ni,t.

This relationship holds contemporaneously and it is also expected to hold in the future.
The household labor supply optimality condition isWtj= Cts Nt ,
Pt

55

(A.1)
(A.2)

1+ j

which is derived for the households whose utility function is U = Ct1 s /(1 s ) Nt /(1 + j).
The demand that firm i expects to face in period t + s when it updated its price in period t is
s

P Pt⇤
Pet+s

ei,t+s ( Pt⇤ ) =
Y

! e

et+s
C

ei,t+s ( P⇤ ) is
As a result, the forecast of the real labor costs in period t + s of the firm that produces Yteei,t+s ( Pt⇤ )) = Wt+s N
e t+s
e t + s (YY
Pet+sei,t+s ( P⇤ )
Y
t
At

e t+s
W
=
Pet+s

! 1

1 a

ei,t+s ( P⇤ )
Y
t
et+s
A

ej
= Cets+s N
t+s

! 1

1 a

i 1
j
1+ j hes+ 1 a Ae 1 a Yei,t+s ( Pt⇤ ) 1 a .
=Y
t+s
t+s

The firm that resets its price in period t solves

h
n
•
se
s
⇤
max
q
E
Q
Pt⇤ P Ytt,t+si,t+s ( Pt )
Â
⇤
Pt

where

ei,t+s ( Pt⇤ ))e t + s (YPet+s Y

s =0

et+s
C
Ct

Qt,t+s = bs
Optimal choice of Pt⇤ yields

! s


⇢
•
ei,t+s ( Pt⇤ ) Pt⇤ Ps
0 = Â q s E t Qt,t+s Y
s =0

,

Pt
e
Pt+s

e
1

e

oi

ei,t+s ( Pt⇤ ))e 0t+s (YPet+s Y.

⇣⌘ea/(1 a)ei,t+s ( P⇤ )) = Y
ei,t+s ( Pet+s )) Ps P⇤ / Pet+se 0 (Ye t + s (YUse the fact that Y
t+s
t
t
•

s

0 = Â q Et
s =0



⇢
s
⇤
e
Qt,t+s Yi,t+s ( Pt ) ( Pt⇤ )1 ea/(1 a) Pe

8<✓ P⇤ ◆1 ea/(1 a)

s

e

1

Divide by ( Pt 1 )1 ea/(1 a)•2ei,t+s ( Pt⇤ )0 = Â q s E t 4 Qt,t+s Y
s =0t: Pt 1P⇣ ⌘ea/(1 a)
1 ea/(1 a) e 0
ei,t+s ( Pet+s )) Ps
Pet+s
Y t + s (Y

e
e

1

Pet+s
Pt 1

!1 ea/(1 a)⇣ei,t+s ( Pet+s )) P
e 0t+s (YY
s.

Log-linearize around the steady state with inflation P where Pt⇤ /Pt 1 = P, Pt⇤ = Pt+s /P , Yt = Ct =s
C = Y, Qt,t+s = bs /P , Y0t+s (Yi,t+s ( Pt+s )) = (e 1)/e (including expectations).
p⇤t

pt 1

p = (1h•c
f t+s + pet+sbq ) Â ( bq )s E t Wmc
s =0

56

pt 1

( s + 1) p

i

93

=
⌘
s ea/(1 a);5.

whereW⌘11a,a + aeandmct+s ⌘ Y0t+s (Yi,t+s ( Pt+s )),c
f t+s is the log deviation of the marginal cost from its steady state value.
while mc
Rewrite it a bit
"
#
•
s
s
⇤
c
e
f t+s +
p)p
pt 1 p = (1 bq )( bq ) E t Wmc
(p

Â

t

Â

s =0

= (1

bq ) E t= (1bq ) E t

"
"

#

•

k =0

t+k

•

c
f t + s + (1
Â ( bq )s Wmc

s =0

s

et + kbq ) Â Â ( bq )s E t (p#•cet + s
f t+s + Â ( bq )s E t [p
Â ( bq ) Wmc
•

s

s =0

p)

s =0 k =0

p] .

s =0

By repeating the same steps, we can derive the forecast versions of the last equationpe⇤t+1andp⇤t+sp = (1pt

p = (1pt+s 1bq ) E t+1"

bq ) E t+s

"

•

#

s

•

c
et + 1 + s
f t+1+s + Â ( bq )s E t+1 [pÂ ( bq ) Wmc

s =0

#
•
c
et + s + k
f t+s+k + Â ( bq )k E t+s [pÂ ( bq ) Wmc
•

p] ,

(A.3)

p] ,

(A.4)

s =0

k

k =0

k =0

for s > 1. Combining equations (A.1) and (A.2) together with (A.3) and (A.4), we can write
et+s = max
p

for s
it as

(

p + (1

q)

(1

"

#

•

!

•

)

c
et + s + k
f t+s+k + Â ( bq )k E t+s [pÂ ( bq )k Wmcp] , 0 ,

#
•
c
f
bq
W
mc
+
bq
(
)
Â
Â ( bq )k Et+s [pet+s+1+k
t+s+k

!

bq ) E t+s

k =0

k =0

et+s is present both on the left- and the right-hand sides of the last equation, we simplify
1. Because p

et+s = max
p

(

p+

1

q

(1

q

bq ) E t+s

"

•

k

k =0

)

p] , 0 .

k =0Finally, the contemporaneous Phillips curve is
pt = p +

1

q
q

(

(1

bq ) E t

"

•

#

k

•

k

c
et + 1 + k
f t+k + bq Â ( bq ) E t [p
Â ( bq ) Wmc

k =0

k =0

p]

)

.

Does the last equation reduce to the standard Phillips curve when the on ZLB on IE
does not bind? Start with the future Phillips curve
et + s
p

p=

(1

q )(1
q

h
i 1
k
c
f
(
bq
)
E
mc
t+s
Â
t+s+k +bq )W •

k =0

57

q •
q

Â ( bq )k+1 Et+s [pet+s+1+k

k =0

p] .

Make the following transformations
et + s p ( 1
p
=
1 q

=

(1

+

bq )W cet + s + 1
f t+s + bE t+s [p
mc
q

1 •
et + s + 1 + k( bq )k+1 E t+s [p
q kÂ
=1

p] +

(1

p]

p]

h
i
bq )W •
k
c
f
(
bq
)
E
mc
t
+
s
Â
t+s+k
q
k =1

bq )W cet+s+1 p ] + bq
f t+s + bE t+s [p
mc
q
(
h
i 1 •
(1 bq )W •cet + s + 2 + k
f t+s+k+1 + Â ( bq )k+1 E t+s [p( bq )k E t+s mc
Âqq k =0k =0⇢et + s + 1(1 bq )W cpet+s+1 p ] + bqE t+s
f t+s + bE t+s [p
=
mc
,
q
1 q

=

and arrive athi 1 •bq )W •
k
c
f
(
bq
)
E
mc
t+s
Â
Â ( bq )k+1 Et+s [pet+s+1+k
t+s+k +
q
q
k =0
k =0

(1

p]

)

q )(1 bq )W c
et + s + 1 p ] ,f t+s + bE t+s [pmcqwhich is the standard New Keynesian Phillips curve with non-zero inflation target.
et + spp=(1

Replacing marginal costs with output gap. We use the equationwhere

et + s
p

(

•

ct =
mc

✓

s+

j+a
1 a

k

p = max k Â ( bq ) E t+s [ xt+s+k ] +

k⌘

(1

q )(1
q

bq )W

58

xt .

q •

1

k =0

◆

q
✓

Â ( bq )

k +1

k =0

s+

j+a
1 a

◆

et + s + 1 + k
E t+s [ p.

)p] , 0 ,