Slow Learning
Lawrence J. Christiano

Martin Eichenbaum

Benjamin K. Johannsen∗
May 24, 2024

Abstract
This paper investigates what features of an economy determine whether convergence under learning is fast or slow. In all of the models that we consider,
people’s beliefs about model outcomes are central determinants of those outcomes. We argue that under certain circumstances, convergence of a learningequilibrium to the rational expectations equilibrium can be so slow that policy
analysis based on rational expectations is very misleading. We also develop new
analytic results regarding rates of convergence in learning models.

∗The analysis and conclusions set forth are those of the authors and do not indicate concurrenceby the Board of Governors or anyone else. The authors thank Klaus Adam, George-Marios Angeletos,
Ed Herbst, Albert Marcet, Michael Massman, Alexander Mayer, Damjan Pfajfar, Bruce Preston, TomSargent, Robert Tetlow, Ivan Werning, Mike Woodford and participants in a number of seminars forhelpful comments and suggestions.

11Introduction

Rational expectations may be a useful modeling strategy in tranquil times like the Great
Moderation. This strategy is less appealing, however, when people are confronted withnovel events, such as the Great Recession or the COVID-19 pandemic. This fact was
self evident to the founders of rational expectations. For example, referring to themodel in his seminal paper on asset prices, Robert E. Lucas, Jr. writes:....The model described above “assumes” that agents know a great deal
about the structure of the economy, and perform some non-routine computations. It is in order to ask, then: will an economy with agents armed
with “sensible” rules-of-thumb, revising these rules from time to time soas to claim observed rents, tend as time passes to behave as described...”Lucas (1978, p. 1437)
This paper analyzes the evolution of economic aggregates after a novel event. We
assume that people must learn about their environment by forming beliefs about future
economic outcomes and update those beliefs as the data come in.
There is a voluminous literature that addresses the question of whether economies inwhich people are learning about their environment converge to a rational expectations
equilibrium (REE).1 In contrast, much less attention has been paid to the question of
how long it takes to converge to an REE.2
The answer to this question is critical to assessing the usefulness of rational expectations for understanding the effects of shockson the economy and the efficacy ofpolicies to deal with the repercussions of those shocks. As Vives (1993, p. 329) writes,in a changing world, for all practical purposes, “‘slow’ convergence may mean no convergence.” This paper investigates what features of an economy determine whether
learning is fast or slow. Critically, in all of the models we consider, people’s beliefs
about model outcomes are central determinants of equilibrium outcomes.3
Our central finding is that when beliefs are partially self-fulfilling, learning equilibria
1See, for example, Bray and Savin (1986), Marcet and Sargent (1989b), and Evans and Honkapohja(2001).
2
Some exceptions include Vives (1993), Marcet and Sargent (1995), Ferrero (2007), and Chien etal. (2021).
3
There is a large literature that explores the speed with which people learn the parameters ofexogenous stochastic processes. For example, Erceg and Levin (2003), Gust et al. (2018) and Farmer
et al. (2021) describe an empirically relevant set of time series representations with hard-to-learn low
frequency components.

2

converge slowly to rational expectations. Indeed, learning can be extraordinarily slow,
with progress being measured in millennia. Under these circumstances, policy analyses
based on rational expectations can be very misleading.
We begin by considering a model developed by Bray and Savin (1986) that is aworkhorse in the learning literature. An important virtue of the model is that it is avery simple setup in which people’s beliefs determine market outcomes. The reduced
form of the model encompasses cases in which beliefs are partially self-fulfilling andcases in which beliefs are self-defeating. A particular parameter, which we denote byb, controls how beliefs about market outcomes affect actual market outcomes. When0 < b < 1, beliefs are partially self fulfilling. The Lucas (1973) supply model, in whicha higher expected price level leads to a higher actual price level, falls into this case.
When b < 0, beliefs are self defeating. Muth (1961)’s version of the classic Cobweb
model, in which a higher expected price level leads to a lower actual price level, falls
into this case.
In the REE of the Bray and Savin (1986) model, people’s beliefs about economic
aggregates are constants, independent of past data. Bray and Savin (1986) show thatwhen people behave like Bayesians, their beliefs converge almost surely to rational
expectations. In contrast, we focus on the rate at which beliefs converge under both
Bayesian and classical (least squares) learning. In both cases, beliefs are a stochastic
process that depends on past data. Building on Ljung (1977), Marcet and Sargent(1989c) and Evans and Honkapohja (2001) show that in a class of learning models,the eigenvalues of a particular ordinary differential equation (ODE) determine whether
or not the system converges asymptotically. Those eigenvalues can also be used tocharacterize the amount of time it takes for learning to converge. In the Bray and
Savin (1986) model, there is only one eigenvalue and it is equal to b. Convergence canbe very slow, depending on the value of b.
Consistent with results in Christopeit and Massmann (2018), for moderately highvalues of b, we show that it takes an extraordinarily large number of periods to close
two-thirds of the expected gap between the initial priors and the rational expectations
belief. The intuition behind the possibility of slow convergence is as follows. Whenpeople’s expectations of a variable are largely self-fulfilling, they are slow to adjust
their priors, and it takes a long time for them to converge to rational expectations. In
contrast, when people’s expectations lead to outcomes different from their beliefs, they
are quick to change those beliefs, and convergence is fast. For convenience, we refer to3this intuition as the learning principle.4
The possibility of slow convergence is not just a theoretical curiosum. The binding
zero lower bound (ZLB) during the Great Recession was a novel event that few people
understood when it first occurred. We argue that when the ZLB is binding, learning
is particularly slow. We also argue that the implications of slow learning for policy
are substantial: The predicted effects of monetary and fiscal policies are very different
under rational expectations than under slow learning. We make these arguments using
a simple version of the New Keynesian (NK) model that was widely used to understandthe Great Recession and the effect of the binding ZLB.5
We begin by considering learning equilibria in the ZLB in the absence of government
interventions. Our key result is that convergence is very slow. Indeed, in the benchmark
parameterization of the model the ZLB would almost certainly be over long before
learning has come close to converging. The reason for slow convergence is that in the
ZLB, the expectations of households and firms tend to be self-fulfilling. To understandwhy, suppose that firms and households expect lower inflation in the future. Becauseof price-setting adjustment costs, firms are incentivized to cut prices today. In the
ZLB, low inflation expectations mean households believe the real interest rate is high.
Consequently, households reduce their demand for consumption, which leads to a fallin the marginal cost of production. Hence, the actions of both households and firms
lead to lower current inflation. With learning, low current inflation shifts expected
inflation down in the next period. The previous mechanism repeats itself in the next
period so that actual inflation in the next period is also low. We conclude that, in the
ZLB, deflation expectations are partially self fulfilling, and the NK model behaves likea high b economy.
In the NK model, when people have rational expectations, a shock that triggers
a binding ZLB leads to a sharp decline in inflation and output (see Eggertsson andWoodford (2004)). The large effects arise because the shock triggers high expected
deflation and real interest rates. In contrast, under learning the same shock leads only
to a moderate and gradual decline in inflation because, with learning, expectations
are partially backward-looking. So, if people begin the episode not expecting a large
deflation, then the actual decline in inflation will be relatively moderate.
4

A version of the learning principle has been studied by Heemeijer et al. (2009) and Hommes(2011).
5
Much of the work in the initial aftermath of that event combined rational expectations with the
NK model. See, for example, Eggertsson and Woodford (2004), Christiano et al. (2011) and Del Negro
et al. (2023).

4

Next, we consider the effects of fiscal policy in the ZLB. We find that the efficacy
of fiscal policy is much smaller under learning than under rational expectations. Under
rational expectations, the multiplier is very large in the ZLB because an increase ingovernment purchases causes a rise in expected inflation (see Christiano et al. (2011)).
Because the nominal interest rate is fixed, this rise generates a fall in the real interest
rate, a rise in consumption, and a multiplier substantially larger than unity. Under
learning, expected inflation is partially backward-looking and does not move much after
an increase in government purchases. So, the real interest does not fall by very much,and consumption rises by only a small amount. As a result, the key driver of the large
REE multiplier is effectively eliminated, and the multiplier is close to unity.
Next, we turn to the efficacy of monetary policy in the wake of a shock to thediscount rate, under learning. We begin by considering the effects of a simple form of
forward guidance: The monetary authority commits to keeping the nominal interest
rate at zero for one period after the shock that makes the ZLB binding returns to
its steady-state level. Interestingly, the number of REEs proliferates under forward
guidance. However, we show that only one equilibrium is stable under learning. Consistent with the existing literature (for example, Del Negro et al. (2023) and Woodford(2012)), we find that forward guidance is powerful under rational expectations. As
is well-known, the power of forward guidance under rational expectations reflects its
strong effect on expected inflation. Under learning, the effects of forward guidance havevery little influence on expected inflation because expectations are partially backwardlooking. It follows that under learning, forward guidance is not very powerful. So,
as with fiscal policy, a rational expectations-based analysis of monetary policy can bevery misleading.
In our analysis, people fully integrate the fact that they are learning when they solve
their problems. For convenience, we refer to this approach as internalized learning.
We formulate households’ and firms’ problems in recursive form. Given the recursive
structure of learning, this approach seems natural: People start a period with an
initial set of beliefs, then see data and update their beliefs using Bayes’ rule. In ourenvironment, aggregate prices adjust in a given period to clear markets. In our learningequilibrium, however, we do not require that planned future individual decisions aremarket clearing or that the sum of expected future individual decisions coincides withthe corresponding expected aggregate outcomes. People’s value functions incorporate
their understanding that they will continue learning and adapting their behavior as

5new data arrive.
As it turns out, implementing internalized learning in the nonlinear solution of themodel is computationally very challenging. In a learning equilibrium, the parametersthat characterize beliefs are also state variables and this greatly exacerbates the curseof dimensionality.6 See Appendix C for details.
Next, we study the asymptotic properties of the model by linearizing its solution.
We establish three sets of results. First, we identify the analog of b in the linearized
solution, which determines the asymptotic rate of convergence of the NK model. It
is the largest real part of the eigenvalues of the matrix that maps beliefs about thestate of the economy into their realized values. This property is not specific to thesimple NK model. It applies to a broader class of multivariate models and extends
some results in Christopeit and Massmann (2018) to those multivariate models. So our
result about the analog of b allows an analyst working with a model falling within thatclass to quickly determine whether learning is slow or fast. Second, we show that thelinearized solution to the NK is a good approximation to the nonlinear solution in a
nontrivial neighborhood about the point at which the approximation is taken. Third,
we show that the asymptotic rate of convergence in mean beliefs is a good guide to thesmall t (finite sample) rate of convergence.
In contrast to internalized learning, much of the learning literature works with aversion of Kreps (1998)’s Anticipated Utility approach. In this approach, people update
their beliefs every period as new data come in. But, when they make their decisions,people proceed as though their beliefs will never be revised again. This approach has
been criticized for its internal inconsistency (see Cogley and Sargent (2008) and Adamand Marcet (2011)).
We simulate our model under both internalized learning and anticipated utility. We
find that the results under both approaches are qualitatively similar. However, for someexperiments there are important quantitative differences between the two approaches
due to the more prominent role played by uncertainty under internalized learning.
These results are consistent with those obtained by Cogley and Sargent (2008), who
studied a stochastic endowment economy with a storage technology.
The remainder of this paper is organized as follows. Section 3 analyzes learning in
the Bray and Savin (1986) environment. Section 4 discusses our approach to learningin the NK model. Section 5 characterizes the set of minimal state variable REE while6

We solve our model using a compiled programming language (c++) and we make use of morethan 300 processors.

6the representative household’s discount rate is low. Section 6 analyzes the local andglobal learnability of those equilibria. In Section 7, we analyze the speed of convergenceof the learning equilibrium in the NK model after a drop in the discount rate. In that
section, we also compare the internalized learning and anticipated utility approaches
to learning. Section 8 assesses the sensitivity of the efficacy of fiscal policy and forward
guidance to learning. Section 9 extends our analytic results about rates of convergencereported in section 9 to the vector case that encompasses the NK model. Section 10
contains concluding remarks.

2

Related Literature

Our paper relates to a number of literatures. The first is the literature that studies theproperties of recursive stochastic estimators in learning models. As noted above, Ljung(1977) establishes that a recursive estimator, θbt , converges almost surely to a limiting
value, θ, if a particular ordinary differential equation, ODE, which is determined by
the underlying system, has eigenvalues with real parts that are less than unity. (1989c;
1989a), Woodford (1990), (2000; 2001) and others build on Ljung (1977) to study theconditions under which learning equilibria converge to rational expectations.
Marcet and Sargent (1995) study the rate at which learning equilibria converge
to rational expectations using results from Benveniste et al. (1990), who show that
if the real parts of the eigenvaluesof the ODE identified by Ljung (1977) are less

1/2 b
1
θt − θ has an asymptotic Normal distribution with finite, nonthan /2, then t
zero variance. However, eigenvalues with real parts greater than 1/2 can easily arise inpractice. We show that the NK model analyzed below has this property in the ZLB.
In addition, Marcet and Sargent (1995) use numerical simulations to study versions of
the Cagan (1956) model of hyperinflation. In some of those simulations, eigenvalues
are substantially larger than 1/2. Christopeit and Massmann (2018) extend the resultsin Benveniste et al. (1990) for the Bray and Savin (1986) model to the case of 1/2 ≤b < 1. We extend some of the results in Christopeit and Massmann (2018) to a classof multivariate models that nests our NK model.
Ferrero (2007) discusses learning in the context of a linear NK model in which the
ZLB on interest rates is not binding. He uses the simulation methods proposed by
Marcet and Sargent (1995) to study convergence rates of learning equilibria. Ferrero(2007) adopts the so-called Euler-equation approach to learning as opposed to our7approach; see Evans (2021) for a definition of the Euler-equation approach to learning.7
Another difference with Ferrero (2007) is that we compare rates of convergence in anonlinear NK model when the ZLB on interest rates is and is not binding. Ferrero(2007) only considers the latter case.
Cogley and Sargent (2008), Adam and Marcet (2011), and Adam et al. (2017)
develop the internalized learning approach in the context of endowment economies.
Adam and Merkel (2019) use this approach to analyze a real business cycle model inwhich peoples’ beliefs do not nest an REE. In contrast, we study an NK model inwhich peoples’ beliefs do nest an REE and we characterize rates of convergence to thatequilibrium.
Preston (2005) and Eusepi et al. (2022) study the effects of monetary policies inlinearized NK models under learning, both in and out of the ZLB. They use the anticipated utility approach to modeling how people make decisions. In contrast, wework with a nonlinear model and adopt the internalized learning approach to decision
making. Additionally, we characterize how quickly, under learning, monetary and fiscal
policies have effects similar to those obtained under rational expectations.
A different literature investigates prices’ information content for fundamentals observed with noise. In this context, Vives (1993) asks a question similar to ours: How
quickly do people’s beliefs converge? Specifically, he studies a model in which people
use price signals and other noisy observations to learn about an object (a cost parameter) whose value is independent of beliefs. In our model, the values of the objects
that people are learning about–for example, aggregate output and inflation–depend on
their beliefs. Heemeijer et al. (2009) and Hommes (2011) study positive and negative
feedback loops from expectations to outcomes using laboratory experiments and univariate models with constant gain. By contrast, we consider learning in the context ofthe multivariate NK model under Bayesian learning and analytically characterize therate of convergence of beliefs.
Our paper is also related to a recent game-theoretic grounded literature that analyzes the implications of bounded rationality for the effectiveness of fiscal and monetary
policy. Farhi and Werning (2019) use k -level thinking models to study how deviations
from rational expectations affect the effectiveness of forward guidance. Garcı́a-Schmidt
and Woodford (2019) study forward guidance and interest rate pegs using reflective
expectations. Iovino and Sergeyev (2023) apply k-level thinking and reflective expec7

Our approach is an example of what Evans (2021) calls the agent-based approach to learning.

8tations to analyze the effects of quantitative easing. Angeletos and Lian (2017) develop
the idea that a lack of common knowledge can attenuate general-equilibrium effects anddamp the effects of government spending. Angeletos and Lian (2017; 2018) analyze theconsequences of bounded rationality for the size of fiscal multipliers.
Farhi and Werning (2019), Farhi et al. (2020) and Woodford and Xie (2019; 2022)
use different models of bounded rationality to study the size of the government-spending
multiplier. Vimercati et al. (2021) assess the implications of bounded rationality for theeffectiveness of tax and government spending policy at the ZLB. They do so throughthe lens of a standard NK model in which people are dynamic k-level thinkers.
In all of the papers just cited, individuals have a limited ability to understand thegeneral equilibrium consequences of monetary and fiscal policies. Like learning, thistype of deviation from rational expectations can limit the power of forward guidance.
Our paper studies a form of deviation from rational expectations different from thosecited in the previous two paragraphs. Moreover, in contrast to our analysis, these
papers do not analyze rates of convergence to rational expectations.

3

Simple Example

We consider a workhorse model used in the learning literature; see, for example, Brayand Savin (1986) and Evans and Honkapohja (2001). We use this model to exposit ourbasic intuition about the factors determining how fast learning models converge. That
intuition is summarized by what we refer to as the learning principle. First, when people’s expectations are partially self-fulfilling then convergence to REE is slow. Second,
when people’s expectations lead to outcomes that are different from their expectations, then convergence is quick. We exposit this principle using different measures of
convergence.
Suppose a variable, xt , for t = 1, 2, . . . , is determined as follows:
xt = a + bEt−1 xt + εt .

(1)Here, εt has mean zero, variance σ 2 < ∞, and is not correlated over time. The
operator, Et−1 , denotes the cross-sectional average of expectations based on the historyof observations on xt up to period t − 1. Evans and Honkapohja (2001) show that whenb > 0, equation (1) is the reduced form of the Lucas (1973) supply model. When b < 0,equation (1) is the reduced form of the Cobweb model analyzed in Muth (1961).
9

We consider two specifications of Et−1 corresponding to whether people have rational expectations or use past data to learn about the data-generating process for xt .
When b ̸= 0, the way people form their beliefs affects the law of motion for xt .
In the REE, Et−1 corresponds to the mathematical expectation, Et−1 , and xt is
given bya.
(2)
xt = µ + εt , µ =1−bThat is, in the REE xt is distributed iid with mean a/ (1 − b) and variance σ 2 . For
example, if εt ∼ N (0, σ 2 ), then xt ∼ N (a/ (1 − b) , σ 2 ).

3.1

Beliefs about the Mean of xt

As in Bray and Savin (1986) and Evans and Honkapohja (2001), people assume that xt
is Normally distributed with mean µ and variance σ 2 , but they do not know the valueof µ.8 For ease of exposition, we assume people know the value of σ 2 . The analysis
below is unchanged if we assume that people must also learn the value of σ 2 so longas they have Normal-inverse-gamma priors about µ and σ 2 , which would result in thesame equations for µt .
Another approach to learning used in the related literature is constant gain learning.
In the model considered here, constant gain learning is not an optimal approach tolearning from the perspective of households and firms. As a result, it does not fit into aframework of internalized learning. See Appendix B for further discussion of constantgain learning and for a characterization of the rate of convergence of beliefs to REE.
We assume that before observing xt , people’s prior belief about µ is given by theNormal distribution
N µt−1 , σ 2 /λt−1 .
(3)Here, λt−1 characterizes the precision of the prior about µ.9 After seeing xt people’sposterior belief about µt is N (µt , σ 2 /λt ), whereµt = µt−1 +1
λt−1 + 1(xt − µt−1 ) ,λt = λt−1 + 1 = λ0 + t,
8

(4)
(5)

Recall, we have not assumed that εt actually has a Normal distribution. People in the model
assume Normality of the likelihood when they derive Bayes’ rule.
9
We can interpret µ0 as the average of person-specific priors, but, we require that all people havethe same value for λ0 .

10for t = 1, 2, ... , where 1/ (λ0 + t) is the optimal weight on new information.10 The
parameter, λ0 , is finite and non-negative. If λ0 = 0 then µt in equation (4) corresponds
to the time t least-squares estimator of µ.
Substituting from equation (1), and rearranging, we obtainµt =a + εt + (b + λt−1 ) µt−1.
λt−1 + 1(6)

After repeated substitution, we obtain a decomposition of µt in terms of the shocks
and µ0 . LettYzt =(1 − bj ) ,
(7)
j=1

where
bj ≡

1−b
.
λ0 + j(8)

The following Lemma summarizes the time-series representation of µt that we workwith:Lemma 1. The variable, µt , in equation (6), has the following representation:



t 
X
a
zt εj
a
µt =
+
+ zt µ0 −,1 − b j=1 zj λ0 + j
1−b

(9)

where zt is defined in equation (7).
For the proof, see Appendix A.

3.2

Characterizing Rates of Convergence

Bray and Savin (1986) prove that if b < 1, then µt → a/ (1 − b) almost surely. In
contrast, we are interested in the rate at which µt converges. To this end, we focus on
the mean of the posterior distributions of µt . The parameter b is the critical determinantof the rate of convergence.
10

This result about posteriors is well known; see, for example, Hamilton (2020).

11

3.2.1

Rate of Convergence of Mean of µt

We now consider the rate of convergence of Eµt , where E is the unconditional mathematical expectation. According to Lemma 1,
zt = Eaµt − 1−baµ0 − 1−b

!
,

(10)

for each t ≥ 1. We can interpret 1 − zt as the fraction of the initial gap, µ0 − a/ (1 − b) ,closed by period t.
From equations (7) and (8), we see that zt depends only on λ0 and b. The smaller
the precision, λ0 , the larger the gain at all dates (see equations (4) and (5)). This gain
effect implies that the less precise people’s initial priors are, the more weight they giveto the data and, in our simple model, the more quickly their views converge.
We now analyze numerically how the speed of convergence of Eµt depends on b.
Our metric is the amount of time it takes to close two-thirds of the initial gap. Thatis, we calculate T, the value of t such that zT ≈ 1/3. In the following calculations, we
set λ0 = 1.11 When b = 0, 0.5, 0.75, 0.85, and .95, then T = 3, 11, 113, 2201, and 5.2
billion, respectively. Note how the speed of convergence decreases nonlinearly with b.
When b = 0, people’s beliefs converge very quickly. In contrast, when b is large andpositive, for example, 0.95, beliefs essentially take forever to converge.
Suppose 0 < b < 1. Then, learning injects a positive feedback loop into the data.
For higher b, a large value of µt−1 implies a large value of xt (see equation (1)). That, inturn implies a higher value of µt (see equation (4)). The higher is b the more powerful isthe feedback loop. This positive feedback loop explains why the higher value of b leads
to a slower speed of convergence.12 As we discuss below, this intuition also applies forb < 0. That is, for all b satisfying b < 1, convergence is faster for smaller values of b.
To discuss the rate of convergence of Eµt , we need to define what it means fortwo sequences– xt and at , both of which converge to zero–to have the same rate ofconvergence. Loosely, two sequences have the same rate of convergence when theirratio does not diverge or converge to zero. This condition is satisfied when (i) the ratio
converges to a finite, non-zero constant or (ii) the ratio oscillates in a bounded set.
Case (i) is relevant to our analysis of the Bray and Savin (1986) model. As it turns
11

Note that the value of a is irrelevant for zt .
If b is too large, then the feedback loop is too strong, so that the process would not converge.
That is the reason why we focus on b < 1 in these simulations.
1212

out, case (ii) is relevant to our analysis of the NK model. Our definition accommodates
both cases.
Definition 1. Consider two series, xt and at > 0 that converge to zero–that is, limt→∞ xt =limt→∞ at = 0. We say that xt and at converge at the same rate if (a) there exists an
A < ∞ such that |xt | /at ≤ A for all t, and (b) there exists an ϵ > 0 such that for
any T ≥ 1 we have supt≥T |xt | /at > ϵ. If conditions (a) and (b) are satisfied we write
x t ≃ at .
Conditions (a) and (b) correspond to the requirements that |xt | /at does not diverge
and does not converge to zero, respectively.
The following proposition follows from results in Christopeit and Massmann (2018)
and establishes the rate at which zt converges to zero:13
̸= 1 for all t, then zt ≃ tb−1 .
Proposition 1. For any b < 1 and any 0 ≤ λ0 < ∞, if λ1−b0 +t∗
̸= 1 for all t is necessary because if λ1−b
The requirement that λ1−b
∗ = 1 for some t ,
0 +t
0 +t
then zt = 0 for all t ≥ t∗ (see equation (7)). The assumption that (1 − b) / (λ0 + t) ̸= 1
only rules out isolated values of b and λ0 . While the proof of Proposition 1 is somewhat
involved, the intuition can be seen from the following approximations:log (zt ) ≈ −tXbj ≈ c1 + (b − 1)

j=1

t
X
1

j
j=1

≈ c2 + (b − 1) log (t) ,where c1 and c2 are finite constants. The first approximation is the first-order Taylor
series approximation, log (1 − x) ≈ −x. The second approximation puts the effectof λ0 in a constant (c1 ). The third approximation captures well-known properties of
the harmonic sum. If these approximations held with equality then the result of the
proposition would follow trivially by exponentiating log zt . Thus, the power convergence result, that zt eventually evolves as tb−1 , occurs because of the decreasing gain inequation (4). In particular, if the gain were instead constant then zt would display geometric convergence, i.e., it would eventually evolve according to Λt where |Λ| < 1. It is
well known that power convergence is qualitatively slower than geometric convergence.
The analog to the ODE considered in Ljung (1977) that is associated with equation(4) is given by µ̇ (τ ) = x (τ ) − µ (τ ) , where x (τ ) = bµ (τ ) and τ denotes notional time.
13

See equation B.7 in the appendix of Christopeit and Massmann (2018). See also Christopeit andMassmann (2019).

13

The eigenvalue of the mapping from µ (τ ) to x (τ ) is b. The solution to the ODE isµ (τ ) = e(b−1)τ µ (0) . Consistent with Ljung (1977), whether µ (τ ) converges to zerois determined by the value of b. Proposition 1 shows that b also determines the rateof convergence, in actual time, of Eµt . Note that the rate of convergence in actual
time is a power function of t, where the power is determined by b. This is notablebecause convergence in notional time is geometric, which is always faster than power
convergence.
Proposition 1 says that for large enough t, |µt − a/ (1 − b)|, is approximately κtb−1
for some finite constant, κ ̸= 0. So, we can compute how many periods, Tt , it takes toclose two-thirds of an initial gap, κtb−1 , in period t. It is easily verified that:
i
h 1
1−b
− 1 t.
Tt = 3(11)

Note that Tt only depends on b, and not on other objects like λ0 . We also computethe time required, T, to close the initial gap in period 0, obtained by simulating theactual zt ’s. When b = 0, 0.5, 0.75, 0.85, and 0.95 then T1 (T ) = 2 (3), 8 (11), 80 (113),
1516 (2201), and 3.5 billion (5.2 billion). It is striking how well T1 tracks T . We
infer that the asymptotic result in Proposition 1 is informative about the behavior ofµt − a/ (1 − b), even for small values of t.
We redo these calculations for the case of λ0 = 0–that is, the case of least-squares
learning. For b = 0, 0.5, 0.75, 0.85, and 0.95, we obtain T1 (T ) = 2 (1), 8 (3), 80 (36),1516 (745), and 3.5 billion (1.9 billion). When λ0 = 10, we obtain T1 (T ) = 2 (21),8 (83), 80 (831), 1516 (15, 804), and 3.5 billion (36.5 billion). The results for all threevalues of λ0 are qualitatively similar. For small values of b, convergence is relatively
fast, and for large values of b, the time required to converge explodes.
The following corollary follows immediately from Lemma 1 and Proposition 1.
Corollary 1. For any b < 1 and any 0 ≤ λ0 < ∞: (i) if µ0 ̸= a/ (1 − b) and1−b
̸= 1 for all t, then E (µt − a/ (1 − b)) ≃ tb−1 ; and (ii) if µ0 = a/ (1 − b) then
λ0 +t
E (µt − a/ (1 − b)) = 0.
Corollary 1 establishes the rate of convergence of Eµt .
3.2.2

Rate of Convergence of Distribution of µt

In this paper, we generally focus on convergence in terms of the rate of convergence inmean. Marcet and Sargent (1995) propose an alternative measure of convergence, the14rate of convergence in the distribution of µt . They characterize that rate by the valueof δ for which tδ (µt − a/ (1 − b)) converges to a non-degenerate distribution as t → ∞.
If we knew what that distribution is, then the distribution approach to convergencewould provide us with an approximation to the entire distribution of tδ (µt − a/ (1 − b))for small t.
At the time that Marcet and Sargent (1995) was written, there were no analytic results that could be used to develop an approximation for δ, for b ≥ 1/2. So, Marcet andSargent (1995) developed a numerical approach. Christopeit and Massmann (2018)
develop the required analytic results for the univariate case when b ≥ 1/2. Their results reveal some limitations of the distribution approach to the rate of convergence.
First, when b < 1/2, δ is a constant, equal to 1/2. So, Marcet and Sargent (1995)’s
characterization of the rate of convergence in distribution does not capture the factthat the rate of convergence of Eµt accelerates as b falls below 1/2. In addition, theircharacterization misses the fact that the variance of the asymptotic distribution isincreasing in b when b < 1/2.14 Second, when b = 1/2, there is no value δ for whichtδ (µt − a/ (1 − b)) converges to a non-degenerate distribution as t → ∞. Third, when
1/2 < b < 1, Eµ converges at the same rate as its standard deviation, implying thatta
, is not zero. So, thethe mean of the asymptotic distribution, limt→∞ Et1−b µt − 1−basymptotic distribution may be misleading for small values of t because it is not centered on the point, a/ (1 − b), to which µt converges almost surely. Finally, although
Christopeit and Massmann (2018) show that an asymptotic limiting distribution exists
when 1/2 < b < 1, they do not provide a characterization of that distribution. Unless
we use numerical simulations, we cannot at this time use the distribution approach toapproximate the distribution of tδ (µt − a/ (1 − b)) for finite t, when 1/2 < b < 1.

4

Learning in the New Keynesian ModelIn this section, we describe a simple NK model. As in Eggertsson and Woodford (2003),
we allow for a shock to the household’s discount rate that can cause the ZLB on theinterest rate to be binding. To study the properties of the model under learning, it isconvenient to express people’s problems in recursive form.
In the current period, households discount next period’s utility by 1/ (1 + r). In
steady state, r = rss > 0. We assume that initially, the economy is in the unique14

See Christopeit and Massmann (2018) for the asymptotic distribution of µt when b < 1/2.

15

non-stochastic rational expectations steady state in which the nominal interest rate is
positive. Then, unexpectedly, r = rℓ < rss . People correctly understand that next
period’s discount rate, r′ , is drawn from a two-state Markov chain, r′ ∈ [rℓ , rss ] , with
an absorbing state:
Pr [r′ = rℓ |r = rℓ ] = p, Pr [r′ = rss |r′ = rℓ ] = 1 − p,(12)

Pr [r′ = rℓ |r = rss ] = 0.
Once r = rss , the economy returns to the initial rational expectations steady state.
There is another rational expectations steady state in which there is deflation and thenominal interest rate is unity (see Benhabib et al. (2001)). We abstract from thatsteady state equilibrium because it is not stable under the learning that we consider.
Moreover, focusing on one steady state greatly simplifies our analysis. See Arifovic etal. (2018) for a discussion of stability for other learning models in which the deflationary
steady state is learnable.

4.1

Fiscal and Monetary Policy

Monetary policy is given by
R = max {1, 1 + rss + α (π − 1)} ,(13)

where α/ (1 + rss ) > 1 and the max operator reflects the ZLB constraint. Later, we
discuss other variations on monetary policy including forward guidance.
We consider two specifications for G. In the baseline specification, G = Gss , its
nonstochastic steady-state value. We also consider a policy where G = Gℓ > Gss whiler = rℓ . The government finances its expenditures with lump-sum taxes, G + νwN,
where νwN represents a subsidy paid to intermediate goods firms.

4.2

Private Agents’ Problems

Below, we define the household and firm problems.

16

4.2.1The Household’s Problem When r = rℓ

The household enters a period with a stock of bonds, bh = Bh,t−1 /Pt−1 . Here, Bh,t−1
denotes the beginning-of-period t payoff on nominal bonds acquired in the previous
period, when the price of consumption goods was Pt−1 . At the beginning of a period, before markets open, the household also knows the value the vector, Θ, whichsummarizes its beliefs about the distribution of a vector, x:
"

C
x=
π

#
.

Here, C and π denote the current period’s aggregate consumption and aggregate inflation. The variable, π, corresponds to Pt /Pt−1 , where Pt and Pt−1 denote the current
and previous period’s aggregate price level, respectively.
In a standard recursive equilibrium, people know current-period market prices andprofits when they make their current decisions. Typically, when markets open in thesemodels, people can deduce the prices and profits from a small set of variables. In ourcontext, these variables are the two components of x. In this spirit, we assume thatpeople observe x when markets open, and they make their current consumption, saving,and labor decisions. In making those decisions, households internalize the effect of x
on their beliefs about the distribution x′ –that is, the value of x in the next period.
Those beliefs, Θ′ , are given byΘ′ = L (Θ, x) .
(14)
The form of L depends on the model of learning being analyzed. The household isinternally rational in the sense of Adam and Marcet (2011). Specifically, when making
decisions, it takes into account uncertainty about the distribution of x and the factthat beliefs about that distribution will evolve as new data arrive (see Section 4.3.2).
Let Ch , Nh , b′h denote the representative household’s consumption, hours worked
and end-of-period bond holdings. The household solvesnχ
maxCh ,Nh ,b′h log (Ch ) − (Nh )2
2

1
′
′
′
′
+
[(1 − p) Vh,ss (bh ) + pEΘ′ Vh (bh , Θ , x )]1 + rℓ17(15)

subject to
Ch +

b′h
bh
≤
+ w (x) Nh + T (x) .
R (x)
π (x)

(16)

Here, T (x) denotes profits net of lump-sum taxes, w (x) denotes the real wage, R (x)denotes the nominal rate of interest, and π (x) denotes the inflation rate.15 In equation(15), Vh,ss (b′h ) denotes the value function of the household conditional on r′ = rss andVh (b′h , Θ′ , x′ ) denotes the value conditional on r′ = rℓ . The expectation operator, EΘ′ , isevaluated using the marginal data density for x′ implied by Θ′ = L (Θ, x) and r′ = rℓ .
Using the first order optimality condition for Nh and equation (16), we reduce thehousehold problem to finding an optimal decision rule, b′h (bh , Θ, x), for bond holdings.
The function, Vh,ss (bh ), satisfies the following fixed point:Vh,ss (bh ) = max ′Ch ,Nh ,bhsubject to
Ch +


χ
1
2
′
log (Ch ) − (Nh ) +Vh,ss (bh ) ,2
1 + rss

(17)

b′h
bh
≤
+ wss Nh + Tss ,Rss
πss

where Tss denotes steady-state profits net of taxes in steady state, wss denotes thesteady-state real wage, Rss denotes the steady-state nominal interest rate, and πss
denotes the steady-state inflation rate.
The function, Vh , in equation (15) has the fixed point property:nχlog (Ch ) − (Nh )2
Ch ,Nh ,bh
2

1
′
′
′
′
+
[(1 − p) Vh,ss (bh ) + pEΘ′ Vh (bh , Θ , x )] ,1 + rℓVh (bh , Θ, x) = max ′(18)

where the maximization is subject to equation (16) and the law of motion for Θ inequation (14).
15 
We constrain the choice of b′h to a compact set b, b , which we discuss in Appendix C.

18

4.2.2The Firm’s Problem When r = rℓA final homogeneous good, Y, is produced by competitive and identical firms using thetechnology
ε
Z 1
 ε−1
ε−1
Y =
Yf ε df,(19)
0

where ε > 1. The representative firm chooses inputs, Yf , to maximize profits Y P −R1
Y P df , subject to (19). The firm’s first order condition for the f th input is
0 f f

Yf =

Pf
P

−ε
Y.

(20)

The f th intermediate good is produced by a monopolist with production technology
Yf = Nf , where Nf is labor hired by firm f . Let pf denote the f th firm’s price in theprevious period, scaled by that period’s aggregate price index–that is, Pf,t−1 /Pt−1 .
Also, let p′f denote the firm’s current choice of price scaled by the current aggregate
price index. In our scaled notation,
p′f
Pf,t
π=
.
pf
Pf,t−1(21)

Firms value a unit of real profits by the marginal utility of consumption, 1/C.
Prices are sticky as in Rotemberg (1982). When r = rℓ the current-period problem offirm f is to set its price p′f so thatp′f (pf , Θ, x) = argmaxp′f


−ε
1 n ′
pf − (1 − ν) w (x) p′f
Y (x)
C (x)
)

2
ϕ p′f
π (x) − 1 (C (x) + G (rℓ ))
−
2 pf


1 
+
(1 − p) Vf,ss p′f + pEΘ′ Vf p′f , Θ′ , x′ .
1 + rℓ(22)

Here, Vf,ss p′f denotes the value of the firm’s problem conditional on r′ = rss andVf,ss p′f , Θ′ , x′ denotes its value conditional on r′ = rℓ .16 Firms and households havethe same information sets and update priors in the same way. Thus, the expectations
operator is the same as the one in the household’s problem. In equation (22), we
16

 
 
We constrain the choice of log p′f to a compact set p, p . See Appendix C for a discussion.

19

follow the literature by scaling price adjustment costs by real GDP.17 Also, ν is a tax
subsidy on employment designed to eliminate the effect of monopoly distortions in
steady state.18
The function, Vf,ss (pf ) , has the fixed-point property ′ −ε1  ′Vf,ss (pf ) = max
pf − (1 − ν) wss pf
Y (x)
p′f
Css
)

2

1
1 ϕ p′fπss − 1 (Css + Gss ) +Vf.ss p′f.
−
Css 2 pf
1 + rss(23)

The function, Vf , in equation (22) has the fixed point property



−ε
1
p′f − s p′f
Y (x)
pf
C (x)
2

1 ϕ p′f−π (x) − 1 (C (x) + G (rℓ ))C (x) 2 pf



1 
′
′
′
′
(1 − p) Vf,ss pf + pEΘ′ Vf pf , Θ , x+.
1 + rℓVf (pf , Θ, x) = max′(24)

The maximization takes into account the law of motion of Θ, L, in equation (14).
4.2.3

The Mapping from x to Aggregate Variables

For individual households’ and firms’ problems
to be well defined,hi they must know
the values of seven aggregate variables, C π R Y N w T . We assume thateach agent knows theh model’si static equilibrium conditions so they can deduce thosevariables from x = C π . We denote this mapping by F (x). Households derive Rfrom π using equation (13). The mappings from x and r to Y , N , and w are given byϕ2
Y = (C + G (r)) 1 + (π − 1) , N = Y, w = χN C.
2
The first two equalities correspond to goods market clearing and the aggregate production function. The third equality corresponds to the belief that the labor supply curveof the individual household holds as an aggregate condition. These equalities hold in
every period of our learning equilibria (described in the next sub-section).
17
18

See, for example, Kaplan and Violante (2018, page 711).
That is, (1 − ν) ε/ (ε − 1) = 1.

20Aggregate firm profits net of taxes implied by x and r are
T = (1 − w) Y −4.3ϕ(π − 1)2 (C + G (r)) − G (r) .
2Equilibrium and Beliefs

The equilibrium for our model is a learning equilibrium for the duration of time thatr = rℓ , followed by a jump to the positive interest rate, steady state REE. The learningequilibrium is a sequence of period equilibria.
4.3.1

Equilibrium Definitions

We now define a temporary equilibrium.
Definition 2. Given Θ and rℓ , a period equilibrium is a set of values of x and Θ′ =L (Θ, x) such that(i) households and firms solve their optimization problems, defined in equations(15) and (22), respectively(ii) labor, goods and bond markets clear(iii) p′f = 1, Ch = C, Nh = N
Because firms are identical, in a learning equilibrium, no firm will ever inherit apf ̸= 1. Then, equation (21) and the first part of condition (iii) imply that people’sviews about inflation, π, are correct. The second and third parts of condition (iii)
imply that people’s views about C and N are correct.
The only new conditions in Definition 2 relative to those imposed by F (x) are thatbond markets clear (b′h = 0) and firms choose p′f = 1. These two conditions determine
the two elements of x.
Two comments about the period equilibrium are worth emphasizing. First, people have perfect foresight regarding current aggregate variables. Second, in general,they do not have perfect foresight about future aggregates. It follows that the period
equilibrium under learning is, in general, different from what it would be if people hadrational expectations.
We now define a learning equilibrium.
Definition 3. A learning equilibrium is :(i) a sequence of period equilibria in which beliefs are updated according to equation(14) when r = rℓ ,21

(ii) a steady state REE with R > 1, when r = rss .
In a learning equilibrium, the value of Θ in the first period when r = rℓ is exogenous. We assume that in the case of an unprecedented event, people’s priors about theeconomic variables, x, are very diffuse. Below, we describe how our parameterizationof the initial Θ captures this property.
4.3.2

Beliefs and Equilibrium

We now describe in detail how households’ and firms’ common beliefs evolve, startingin the first period that r = rℓ . People assume that each of the two elements of log (x)is drawn from a Normal distribution:
"
# "
# "
#
log (C)µC
εClog (x) =
=
+
,
(25)
log (π)µπ
επ
EεC = Eεπ = 0, Eε2C = σC2 and Eε2π = σπ2 . These distributions are independent acrosstime and the elements of log (x). People are uncertain about the values of µi , σi2 fori ∈ {C, π}. Their prior about µi conditional on σi2 is Normal, parameterized with amean, mi , and variance, σi2 /λi , where λi characterizes the precision of the prior aboutµi . The marginal density of their prior for σi2 is proportional to an inverse-gamma
distribution, with shape and scale parameters, αi and (ψi2 (αi + 1/2)), respectively. The
prior for σi2 is not exactly an inverse-gamma distribution because we truncate thesupport of σi2 so that E [C] and E [π] have finite values. We find it convenient toexpress the scale parameter in this way because ψi is a consistent estimator for σi . The
joint density of µi , σi2 is proportional to the Normal inverse-gamma distribution. We
collect the parameters of the priors in the vector Θ:Θ=

mC mπ 1/λC 1/λπ ψC ψπ 1/αC 1/απ



.

(26)

The posterior distribution is also proportional to the Normal inverse-gamma distribution, and the function, L, in equation (14) can be constructed using standard updating
formulas, which are detailed in Appendix C.
4.3.3

Anticipated Utility

Virtually all of the related literature works with a version of Kreps’ Anticipated Utility
approach to how people integrate learning into their decisions. While this approach22

has computational advantages, it has been criticized for being internally inconsistent(see Cogley and Sargent (2008) and Adam and Marcet (2011)). We assess the robustness of our results to using the anticipated utility approach. In our context, thatapproach assumes that when households and firms make their state-x contingent decisions, they assume that in the current and all future periods, log (x) will be drawn froma Normal distribution with mean and variance fixed at the values of mi and ψi2 fromthe beginning-of-period Θ. We make two changes to the household and firm decision
problems to implement this assumption. First, we set Θ′ = Θ in their next-period
value functions. Second, in evaluating the expectation operator, EΘ′ , that appears in
the household and firm problems, we use the log Normal density for x with mean andvariance fixed at the values of mi and ψi2 from Θ. Importantly, at the beginning of thenext period, firms and households set Θ′ = L (Θ, x).
In sum, anticipated utility differs from internalized learning in two ways. First, inmaking their state-x contingent decisions, people ignore the fact that after they see
current x, they will update their views, using Θ′ = L (Θ, x). Second, they ignore theiruncertainty about the mean and variance of the distribution of log (x).

5Multiple Rational Expectations Equilibria

In this section, we describe the equilibria in our model when agents have rational
expectations.
An equilibrium is a set of values for output, employment, inflation, and consumption, Yℓ , Nℓ , πℓ , Cℓ , respectively, when r = rℓ . We assume that the economy reverts
to the unique rational equilibrium steady state, Yss , Nss , πss , Css , with Rss > 1 whenr = rss .19
The four equilibrium conditions associated with the four unknowns, πℓ , Cℓ , Rℓ , Nℓ ,19

Throughout the paper, we only consider equilibria in which quantities and prices are constant fora given value of r. For example, we do not consider sunspot equilibria.

23

are


1
Cℓ
1
p + (1 − p),
1=
1 + rℓ πℓCssε−1(πℓ − 1) πℓ (Cℓ + Gℓ ) =(χNℓ Cℓ − 1) Nℓ
ϕ1p (πℓ − 1) πℓ (Cℓ + Gℓ ) ,+1 + rℓϕ2Nℓ = (Cℓ + Gℓ ) 1 + (πℓ − 1) , and2
Rℓ = max {1, 1 + rss + α (πℓ − 1)} .

(27)
(28)

(29)
(30)

In equations (27) and (28) we have taken into account that πss = 1. In addition, we
verify and use the fact that Rℓ = 1. Equation (27) can be expressed as one equationin the unknown, πℓ , after using equations (27) and (29), to express Cℓ and Nℓ as
functions of πℓ . We compute Css using the steady state of the model. Then, we canfind a candidate equilibrium by finding a value of πℓ that sets a function, f (πℓ ) = 0.
To verify that a candidate value of πℓ is an equilibrium, we must verify that the implied
aggregate quantities and firm values are non-negative.
Our baseline parameters are:p = 0.80, rℓ = −0.0015, Gss = G (rss ) = 0.20, β = 0.995,
ε = 4, ϕ = 110, χ = 1.25, α = 1.5
In the R > 1 steady-state REE, Css = 0.8,πss = 1, Nss = 1. In the alternative
specification of government purchases,
Gℓ = G (rℓ ) = 1.05 × G (rss ) ,(31)

while r = rℓ .
Figure (1) displays the function f (πℓ ) for a range of values of πℓ in the baseline(solid blue line) and alternative (dashed blue line) cases. In each case, there are twovalues of πℓ for which f (πℓ ) = 0. Table 1 reports the values of Cℓ , wℓ , Nℓ , Rℓ and πℓat these zeros of f . Each crossing corresponds to an interior equilibrium in which the
ZLB binds. The values of the variables corresponding to the equilibria in Figure (1)are reported in Table 1.

24

Figure 1: f (π) Corresponding to the Target-Inflation Steady-State Equilibrium
0.002

f( )

0.000
0.002

A

B

0.004

G = Gss
G = 1.05 × Gss

0.006
0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Note: The function, f, is defined in the text. The dashed line is discussed in Section 8.1 below.
The range of πℓ in the figure includes the two values of πℓ that correspond to an equilibrium.
Source: Authors’ calculations.

The economy’s response to a drop in r is the result of two countervailing forces.
First, the drop in r leads to an increase in desired savings. In the first best equilibrium, the real interest rate would drop enough to undo the increased desire to savecompletely, allowing market clearing in the bond and goods market without any change
in consumption and employment. When monetary policy is operated by a Taylor rule,and prices are sticky, then we know that policy goes only part-way towards achieving
the first best equilibrium. The real interest rate falls, but not by enough so that market
clearing must be accomplished in part by a drop in output and income, which reduces
the desire to save, as long as the low-r spell is expected to be short enough (that is, p
is small enough).20 If the required fall in the nominal interest rate is sufficiently large,
then the ZLB on the nominal interest rate binds. When the ZLB binds, a form of
deflation spiral is triggered. The fall in output leads to a drop in marginal cost that
reduces actual and expected deflation. The latter raises the real interest rate, amplifying the desire to save, leading to an additional drop in actual and expected inflation.
An important countervailing force limits the extent of this spiral. As output drops,
consumption smoothing leads people to save less. The lower is p, the shorter is the
expected duration of the ZLB and the stronger is the consumption smoothing motive.
Three observations about the ZLB follow. First, the logic of the deflation spiralprovides intuition into why the fall in output can be very large when the ZLB is binding.
20

Further discussion of this point appears below.

25

Table 1: Equilibrium Values While rt = rℓ , Returning to Target-Inflation Steady State
Bad ZLB Good ZLB
A
B
Label
ℓ
400(π − 1)
-35.78-6.60
00400(Rℓ − 1)
ℓ
C
0.48
0.74
0.98
0.95
Nℓ
ℓ
w
0.59
0.88
(a) Gℓ = Gss

Bad ZLB Good ZLB
Label
A
B
ℓ
400(π − 1)
-36.99-3.00
400(Rℓ − 1)
0
0
ℓ
0.47
0.77
C
Nℓ
1.00
0.98
ℓ
0.58
0.95
w
∆C+∆G
-0.17
3.95
∆G
(b) G = 1.05 × Gss
Note: This table reports {πℓ , Rℓ , Cℓ , Nℓ , wℓ } for two equilibria indicated by A and B when G = Gss(2a) and when G = 1.05Gss (2b). Each equilibrium returns to the target-inflation steady state as soon
as r = rss . The government purchases multiplier reported in the last line of panel is the change in
GDP per unit increase in G within each of the type A and B equilibria. Source: Authors’ calculations.

The larger the expected deflation in an REE, the larger is the drop in output. Second,
the interplay between the deflationary spiral and consumption smoothing provides
intuition for why there can be multiple REEs in the ZLB. Third, if p is sufficiently
large, the consumption smoothing motive is very weak. When the deflationary spiralis too dominant, an REE does not exist.21
Turning to the fiscal multiplier, we calculate the effect of an increase in G comparing
A to A′ and B to B ′ –that is, comparing two Bad-ZLB equilibria and two Good-ZLB
equilibria (see Figure 1). Table 1 shows that the multiplier is very large in the lattercase and very small in the former. Consistent with this observation, expected deflation
is much larger at A′ than at B ′ .
In sum, this section highlights the central role that expected deflation plays indetermining the properties of an REE in the ZLB. We expect that because expectations
are backward looking, the properties of the learning equilibrium will be very different
21

See Werning (2012), who also discusses the possibility of nonexistence of equilibrium in the ZLB.

26

from those of the REE.

6

Equilibrium Selection

In this section, we consider whether the multiplicity of REEs can be resolved by learnability. We analyze the learnability of an REE by considering a small perturbation in
the REE beliefs. We consider these perturbations by analyzing learning equilibria withinitial values of Θ that are not REE beliefs, but are close to the REE values. We saythat an REE is learnable if learning equilibria that begin with beliefs in a neighborhoodof the REE beliefs converge to the REE. In this section, we conduct the analysis numerically and consider initial values for Θ that have mC and mπ equal to log (Cℓ ) andlog (πℓ ), respectively, where Cℓ and πℓ are the REE values of C and π. Importantly,
the variance of the priors is greater than zero. If learning equilibria starting with thesevalues of Θ converge to the associated REE, then we say that the REE is learnable.
We have also considered learning equilibria that begin with a vector Θ in which mC
and mπ are near, but not equal to, the associated REE values. In these cases, we findsimilar results and our conclusions about learnability are unchanged.
Other initial values of Θ are of particular interest. For example, beliefs with mC andmπ equal to log (Css ) and log (πss ), respectively, are natural candidates in the initialvalues of Θ. If an REE is learnable and learning equilibria beginning with these initialvalues of Θ also converge to that REE, then we say that the REE is quasi-globallylearnable. In a model with multiple REEs (like the NK model), any particular REE
cannot be globally learnable. This result obtains because if beliefs are consistent withanother REE, then beliefs will not diverge from that equilibrium.
We initially consider the learnability of the Bad-ZLB equilibrium by examining alearning equilibrium with mi set to the Bad-ZLB equilibrium values. Figure 2a suggeststhat the learning equilibrium deviates from the Bad-ZLB equilibrium. The red dot
shows where that equilibrium is after 10,000 periods and indicates that it is headed
toward the Good-ZLB equilibrium. In Section 9, we use linearization methods to provethat at the assumed parameter values, the learning equilibrium cannot converge to the
Bad-ZLB equilibrium.22 We conclude that the Bad-ZLB equilibrium is not learnable.
22

Our proof is by contradiction. We linearize our learning model around the Bad-ZLB equilibrium.
Suppose the Bad-ZLB equilibrium is stable. Then, the learning equilibrium would eventually (aslong as r = rℓ ) arrive in an arbitrary small interval, U , about the Bad-ZLB equilibrium, where ourlinearized system is arbitrarily accurate. We show that that model satisfies the conditions of Theorem

27

Figure 2: Equilibrium Selection in the ZLB, by Learning(a) Non-learnability of Bad-ZLB Equilibrium

Aggregate Inflation Expectations
REE
Learning

1020

t=100,000 (blue square)
t=10,000 (red dot)

30
40

0

0

4

8

12

16

Aggregate Consumption Expectations

10
Elog(C/Css) × 100

Elog( ) × 400

0

20
30
40
50
60

20

0

4

8

12

16

20

(b) Learnability of Good-ZLB Equilibrium

Aggregate Inflation Expectations
REE
Learning

24
t=100,000 (blue square)
t=10,000 (red dot)

6
0

4

8

12

16

Aggregate Consumption Expectations

2
Elog(C/Css) × 100

Elog( ) × 400

0

8

0
4
6
8
10
12

20

0

4

8

12

16

20

(c) Learnability of Good ZLB Equilibrium

Aggregate Inflation Expectations

0

REE
Learning

0

Elog(C /Css) × 100

Elog( ) × 400

22
4

t=10,000 (red dot)

6

t=100,000 (blue square)
0

4

8

12

16

Aggregate Consumption Expectations

2
4
6
8

20

0

4

8

12

16

20

Note: In the panels (a) and (b), mi is initially set to the associated REE value. In panel (c) mi is
initially set to the steady state REE value. In all sub-figures, ψi = 0.02, λi = 1, αi = 2. Source:
Authors’ calculations.

Figure 2b shows that the learning equilibrium is converging to the Good-ZLB equilibrium. In Section 9, we use linearization methods to prove that at the assumed
7.2 in Evans and Honkapohja (2001) for beliefs to leave U . This outcome contradicts the hypothesisthat the Bad-ZLB equilibrium is stable.

28parameter values, the learning equilibrium will converge to the Good-ZLB equilibrium
if beliefs start in a neighborhood of that REE. Figure 2c shows that the learning equilibrium converges to the Good-ZLB equilibrium when the beliefs are initially centeredon the steady-state REE. Taken together, these results indicate that the Good-ZLB is
quasi-globally learnable.

7

Speed of Convergence

In this section, we consider how quickly the learning equilibrium converges to theunique learnable REE. In the first subsection, we consider our results for the baseline
parameterization of the model. In the second subsection, we consider the effect of the
ZLB on the interest rate on the speed of learning.

7.1

Baseline Results

We now consider the effects of a drop in r in our learning model. Our basic assumption
is that when people are confronted with an unprecedented observation, here modeled
as a drop in r, they become very uncertain about how market-determined variables will
evolve. We set the initial value of Θ to the following vector:=mC mπ 1/λC 1/λπ ψC ψπ 1/αC 1/απlog (Css ) , log (πss ) , 1, 1, 0.02, 0.02, 1/2, 1/2

′
′

.

Figure 3 displays the marginal density of log C and log π associated with anticipatedutility (that is, the Normal distribution evaluated at the prior estimates of the meansand variances) and with internalized learning (that is, the marginal data density associated with the Normal-inverse-gamma prior on the parameters of the Normal distribution). Note the fatter tails on the density function associated with internalized
learning. The tails are fatter for consumption than inflation because we set a higher
upper bound on σC (0.05) than on σπ (0.025). The bounds on the standard deviations
correspond to typical period-by-period shock sizes equal to about 6 percent for aggregate consumption and about 10 percentage points for annualized aggregate inflation.

29

Figure 3: Data Density Under Two Models of the Interaction of Beliefs and Decisions
Aggregate Inflation

25

Aggregate Consumption

25

Anticipated Utility
Internalized Learning

20

20

15

15

10

10

5

5

0

0
0.10

0.05

0.00
log( )

0.05

0.10

0.10

0.05

0.00
0.05
log(C) C

0.10

Note: The dashed line corresponds to Normal density functions with means mi and standard deviations
ψi . The solid line corresponds to the marginal data density of log (x) at time one, using Θ before it
is updated by the time one value of x is realized. Source: Authors’ calculations.

The thin and thick solid lines in Figure 4 display the evolution of inflation, consumption, and the real interest rate after the drop in r under REE and learning, respectively.
First, consider Figure 4a, which reports results for the REE and internalized learning.
Two key features are worth noting. First, in the REE, there is a very large drop in
inflation and consumption, and the real interest rate rises sharply. The fall in inflationand consumption and the rise in the real rate are much smaller under learning. Second,
the learning economy converges very slowly to the REE. As shown in Figure 2c, after
people initially change their views somewhat quickly, the rate at which they change
their views slows dramatically. For example, the dot labeled T = 10, 000 displays people’s views about the variables after 10,000 quarters. Given our value, p = 0.8, r is only
expected to be low for about five quarters. Whether convergence to the REE happens
after 20 quarters or 10,000 quarters is irrelevant. The crucial point is that in a typical
ZLB episode, people’s beliefs are very far from rational expectations.
Now consider Figure 4b. This figure compares the evolution of the learning equilibrium under anticipated utility (dashed line) and internalized learning (solid line). The
key takeaway is that we obtain the same slow-learning result qualitatively regardless ofwhich approach we take to learning. However, consumption and inflation fall somewhat
more under internalized learning.
30

Figure 4: Simulations of Benchmark Model(a) Speed of Convergence in the Benchmark Model

Inflationlog( ) × 400

24
t=10k (red dot)
6 t=100k (blue square)04

8

log(C /Css) × 100

REE
Learning

0Consumption

0

Real Rate
62

Annualized Percent

2

4
6
8

12 16 20

0

4

8

4
2
0
2

12 16 200

4

8

12 16 20(b) Comparison, Speed of Convergence Under Anticipated Utility and Benchmark

Inflation2
4

2

Annualized Percentlog( ) × 4000log(C /Css) × 100

Anticipated Utility
Internalized Learning
REE

2

Consumption

0

4
6
8

6
0

4

8

12 16 200

4

8

12 16 20

7
6
5
4
3
2
1
0
1

Real Rate

04

8

12 16 20

Source: Authors’ calculations.

7.2

The Role of the ZLB in the Baseline Results

Figure 5 reports a simulation of our benchmark model in which the ZLB on the interest
rate is ignored. For convenience, we reproduce the results from Figure 5 in which the
ZLB is binding. The key result is that the learning economy converges very quickly
when the ZLB is not binding. The reason is that the Taylor rule weakens the connection
between expected and realized inflation. To understand why, suppose people’s prioris that inflation will be high in the next period, causing firms to want to raise prices
in the current period. When the Taylor principle is operative, the central bank takes
actions in the current period that make actual inflation lower. Because expectations
are less self fulfilling, the learning principle implies that people will quickly adjust theirbeliefs. The speed with which they do so depends very much on the value of α, a pointthat we return to in Section 9.

31

Figure 5: Benchmark Simulations with and without Binding ZLBInflationlog( ) × 400

02

t=10,000

4

6

2
4
6
8

6
0

4

8

12 16 20

Real Rate
Annualized Percent

REE
Learning
REE, no ZLB
Learning, no ZLBConsumption0log(C /Css) × 100

2

0

4

8

12 16 20

4
2
0
2

0

4

8

12 16 20

Source: Authors’ calculations.

8

Learning and Government Policy

In this section, we analyze the sensitivity of monetary and fiscal policy analysis in the
ZLB to deviations from rational expectations. We juxtapose that sensitivity to thelack of sensitivity when the ZLB is not binding.

8.1

Government Purchases Multiplier

We begin by analyzing the effect of learning on the government purchases multiplierwhen the ZLB binds. We compute the multiplier by considering the effect on GDP,
C +G, of a 5 percent rise in government purchases relative to its steady-state level–that
is, G (rℓ ) = 1.05 × G (rss ). We denote the difference in consumption and government
purchases across the two equilibria by ∆C and ∆G = 0.05×G (rss ). Since the Bad-ZLB
equilibrium is not stable under learning, we focus on ∆C across Good-ZLB equilibria.
We define the multiplier as∆C + ∆G.
(32)∆G
In the REE, the multiplier in the ZLB is equal to 3.95 (see Figure 6b). The multiplieris large when the ZLB is binding because the rise in G generates an increase in expected
inflation (see the left panel in Figure 6b). Because R is fixed, this rise generates a fallin the real interest rate and a rise in C (see the middle panel). So, in this case, themultiplier is bigger than one.

32

Figure 6: Equilibria with and without Jump in G
Inflation
REE
Learning
REE, high G
Learning, high G

Consumption

0
2

Annualized Percent

1
0
1
2
3
4
5
6
7

log(C /Css) × 100log( ) × 400(a) Increase in Government Purchases During ZLB

4
6
8

0

4

8

12 16 200

4

8

12 16 20

7
6
5
4
3
2
1
0
1

Real Rate

04

8

12 16 20(b) Government Purchases Multiplier in ZLB

G Multiplier

4
3
2
1

REE
Learning
REE, no ZLB
Learning, no ZLB

0
1
2

0

4

8

12

16

20

Notes: The solid lines in Figure 6a reproduce the results based on G = Gss in Figure 4a. The dashedlines report the simulation of the model when G = 1.05 × Gss . Figure 6b displays the government
purchases multiplier under internalized learning and in the REE. That figure reports results for thecase in which the ZLB is imposed and not imposed (‘no ZLB’).

Under learning, expected inflation is backward-looking and does not move muchwith a rise in G (compare the thick dashed and thick solid lines in the left panel of
Figure 6a). Hence, the real interest rate does not fall very much and the response in
consumption is small (middle panel).
Figure 6b displays the value of the multiplier over time in the REE and underlearning in the ZLB. Consistent with the results above, the multiplier in the learningcase is small compared with what it is in the REE. Significantly, there is very little
convergence of the learning multiplier to its REE value over the 20 quarters displayed.
We now turn to the case when the ZLB is not binding. The REE multiplier, in thiscase, is much smaller, 0.80, than when the ZLB is binding (see Figure 6b). When the33

ZLB is not binding, the rise in inflation causes the monetary authority to raise the realinterest rate, which leads to a fall in C. That rise is why the REE multiplier is less thanunity outside the ZLB. Figure 6b displays the government purchases multiplier in thelearning equilibrium when the ZLB is ignored. Note that the value of the multiplieris very similar to its value in the REE. This result is not surprising in light of ourdemonstration that when the ZLB is not binding, the learning equilibrium converges
quite quickly to the REE.

8.2

Forward Guidance

In this subsection, we consider the sensitivity of the effects of forward guidance tolearning. Under such a policy, the monetary authority commits to keeping the nominalinterest rate at the ZLB for J periods after the discount rate has returned to its steadystate level. To make our point as simply as possible, we consider the case J = 1. In thefirst subsection we show that the number of REE proliferates under forward guidance.
Only one of those equilibria is stable under learning. Second, we analyze the effect of
forward guidance.
8.2.1

Rational Expectations Equilibria

We construct the REEs with forward guidance by working backward in three steps.
First, we compute the unique non-stochastic steady state with R > 1. Second, we
compute the continuation equilibrium in the period, I, in which r switches from rℓ to
rss , where I ∈ [2, 3, ...]. Third, we compute the equilibrium allocations in the periods
before I, denote by I−1 .
In period I, R = 1 even though r = rss . People know that the economy willtransition to steady state in period I + 1. The equilibrium conditions in period I are
equations (27) through (29) adjusted for forward guidance:
1=CI
1
1 + rss πss Css(πI − 1) πI (CI + Gss ) −ε−1(χNI CI − 1) NI = 0
ϕ



ϕ
2
NI = (CI + Gss ) 1 + (πI − 1) .
2
34

(33)

(34)

(35)

Equations (33) and (35) define functions mapping πI to CI and NI . These functions
allow us to express the left-hand side of equation (34) as a function of πI . We denote
this function by fI (πI ) . A candidate continuation equilibrium in period I is a value ofπI such that fI (πI ) = 0 along with the associated values of CI , NI , wI and the present
value of the intermediate good firm in period I. For a candidate equilibrium to be an
equilibrium, the four variables must be non-negative. Figure 7 displays the fI function
for a range of values of πI . We find two continuation equilibria corresponding to thetwo zeros of fI displayed in the figure (see points A and B).23
Figure 7: Equilibria in Period of Switch from r = rℓ to r = rss Under One-Period
Forward Guidance
0.02
0.00

fI ( I )

0.02
0.04
0.06

0

0.08

1

0.10
0.12

0.90

0.95

1.00

1.05

I

1.10

1.15

1.20

1.25

Notes: Graph of the function, fI (πI ) , discussed after equation (35). The two crossings with the zeroline correspond to equilibria in period I, the date when r switches from r = rℓ to r = rss . Monetary
policy in period I corresponds to one-period forward guidance–that is, the interest rate is held atzero in period I and then reverts to Rss . The red star indicates the level of inflation in period I in
the absence of forward guidance.

We now compute the equilibrium allocations in the periods before I−1 conditional on
the continuation equilibrium starting in period I. The period I−1 equilibrium conditions
are the appropriate analog of equations (27) through (29):


1
Cℓ
Cℓ
p
+ (1 − p)
1=
1 + rℓ πℓ Cℓ
πI C I
23(36)

From equation (33) we see that CI does not vary with πI . It follows that fI is quadratic functionof πI , so that the two solutions displayed in Figure 7 are the only zeros of fI .

35

ε−1
(πℓ − 1) πℓ (Cℓ + Gℓ ) −(χNℓ Cℓ − 1) Nℓ
ϕ



1
Cℓ
ℓ
−
p (πℓ − 1) πℓ Cℓ + G + (1 − p) (πI − 1) πI(CI + Gss ) = 0
1 + rℓCIϕ
2
1 + (πℓ − 1)Nℓ = Cℓ + G
2
ℓ



(37)

(38)

Here, we impose the condition that Rℓ = 1. In effect, we assume that the ZLB is
binding in periods I−1 , and the Taylor rule holds. In all of the examples that we havestudied, this assumption is satisfied.
We now compute the equilibrium allocations in the periods before I, which wedenote by I−1 . Given CI and πI , equations (36) through (38) define a mapping fromπℓ to Cℓ and Nℓ . Now, we can express the left-hand side of equation (37) as a functionof πℓ . We denote this function by fI−1 (πℓ ; πI , CI ). There are two functions, fI−1 ,conditional on the πI , CI associated with the period I continuation equilibria, A and
B.
Figure 8 displays both fI−1 functions for a range of values of πℓ ; see the dotted anddot-dashed lines. We chose the range of πℓ so that the graph only displays zeros offI−1 that correspond to equilibria. We find two equilibria corresponding to the fI−1
associated with A (see D and E in Figure 8) and one associated with B (see C in Figure8). So there are three REEs with forward guidance. The two REEs without forward
guidance can be seen in the solid line in Figure 8 (this curve is taken from Figure 1).

36

Figure 8: REE Equilibria at the ZLB with and without Forward Guidance

Going to in period I
Going to in period I
No forward guidance

0.004

C

fI 1 ( )

0.002
0.000

A

0.002

B

0.004
0.006
0.800

0.825

0.850

0.875

0.900

0.925

0.950

0.975

1.000

Notes: The solid line reproduces the solid line in Figure 1 and corresponds to the case of no forward guidance. The
dashed and dot-dashed lines correspond to the case of forward guidance. The dashed line corresponds to the case inwhich the economy goes to point B in the period of the switch in r to rss (that is, period I). It crosses the zero line
more than once, but the other crossing involves very high inflation and is not an equilibrium because the present valueof intermediate goods monopolists is negative. The dot-dashed line corresponds to the case in which the economy goesto point A in period I (see Figure 7).

8.2.2

Learning Equilibria

In the period of forward guidance, r = rss , R = 1. In all periods when r = rℓ (thatis, I−1 ), people understand that the economy reverts to an REE when r = rss . As
discussed, there are two REEs starting in period I, the first date when r = rss (seepoints A and B in Figure 21).
We are interested in three questions. First, do any of the learning equilibria converge
to a particular REE in I−1 ? Second, if any do converge, how quickly do they do
so? Third, are the effects of forward guidance different under learning and rational
expectations?
Consider the first question. Two of the three REEs in I−1 are not learnable. These
are the equilibria associated with points A and C in Figure 8. In contrast, the equilibrium represented by B is learnable. Thus, learnability selects a unique REE.
We now consider a learning equilibrium using the same initial values for Θ as in
Section 7.1. Interestingly, forward guidance has no measurable effect on the learningequilibrium. The two learning equilibria are indistinguishable in Figure 9. It follows
that the learning equilibrium converges slowly and that there is no forward guidance
puzzle under learning.24 Promises about interest rates in the future do not have implau24

For a discussion of the forward guidance puzzle, see Del Negro et al. (2023).

37

sibly large effects on current economic outcomes. Indeed, under internalized learning,these effects are virtually zero.25

Inflation
REE
Learning
Learning, FG

Consumption

0
2

Annualized Percent

1
0
1
2
3
4
5
6
7

log(C /Css) × 100log( ) × 400

Figure 9: Forward Guidance Under Learning and REE

4
6
8

0

4

8

12 16 200

4

8

12 16 20

7
6
5
4
3
2
1
0
1

Real Rate

04

8

12 16 20

The power of forward guidance under REE reflects its strong effect on expected
inflation. Under learning, the effects of forward guidance have very little influence on
expected inflation expectations, because expectations are backward-looking.

9

The Analog of b in the NK Model

In analyzing our nonlinear model. we used the ‘learning principal’ that emerged fromthe Bray and Savin model: The larger the parameter, b, that controls how self-fulfilling
expectations are, the longer it takes to converge. In this section, we accomplish twotasks. First, we demonstrate that the analog of b in the linearized solution of our NK
modelh is the largest
real part of the eigenvalues of the matrix that maps beliefs aboutix =C π into the realized values of x. Specifically, we develop the analog of
Proposition 1 for the NK model, which characterizes the asymptotic rate of convergenceof the learning equilibrium as a function of b. Second, we argue that the asymptotic
rate of convergence is a good guide to the small t rate of convergence.
We base our analysis below on linearized versions of the policy functions defined inequations (15) and (22). Here, we find it convenient to use time notation rather thanrecursive notation. The details of our linearization appear in Appendix D. Recall thatthe household problem can be reduced to finding an optimal decision rule for bond
25

Under anticipated utility (not displayed) forward guidance has a slight effect, but not large enoughto be economically meaningful.

38

holdings, b′ (bh , Θ, x), denoted here by bh,t . Log-linearizing this decision rule, we obtainb̂h,t = γb,b b̂h,t−1 + γb,π π̂t + γb,C Ĉt + γb,µπ m̂π,t + γb,µC m̂C,t .

(39)

With one exception, the hat notation, q̂t , denotes log (qt /q) where q denotes the REE
value of qt about which the linearization is done. The exception is household bondholdings, bh,t , in which case b̂h,t denotes bh,t − bh . Also, µ̂t = [m̂π,t , m̂C,t ] represents
the log deviation of people’s time t posterior of Et xt+1 and the REE value of Ext+1
conditional on rt+1 = rℓ .26 We use the posterior means (Et xt+1 ) rather than theprior means (Et−1 xt+1 ) because, with Θ and x, households and firms can compute Θ′ .
Variance of beliefs do not appear because of the certainty equivalence implied by thelinearization. The parameters in equation (39) are functions of model parameters andthe point about which the linearization is done. These points correspond to different
REEs when r = rℓ . Similarly, the linearized price decision rule, p′f (pf , Θ, x), of thefirm (denoted by p̂f,t ) isp̂f,t = γp,p p̂f,t−1 + γp,π π̂t + γp,C Ĉt + γp,µπ m̂π,t + γp,µC m̂C,t .

(40)

The time t realized value of x̂t enters the decision rules, equations (39) and (40),by two channels. The first channel reflects that people use x̂t to determine the periodt values of the exogenous variables in their period t budget constraint. The second
channel reflects that µ̂t depends on x̂t , µ̂t−1 , and the gain in the Bayesian updating
equation.
In each period we compute a linearized period equilibrium (see Definition 2), sothat (i) bbh,t−1 = p̂f,t−1 = 0 and (ii) x̂t is determined by the requirements, bbh,t = 0 andp̂f,t = 0:0 = γb,π π̂t + γb,C Ĉt + γb,µπ m̂π,t + γb,µC m̂C,t0 = γp,π π̂t + γp,C Ĉt + γp,µπ m̂π,t + γp,µC m̂C,t .
26

Note that the only exogenous random variable in our model is the natural rate of interest. Given
the model structure, in the REE, E xt+1 |rt+1 = rℓ is equal to the value of xt while rt = rℓ . We
discuss posterior and prior means because when the natural rate of interest is low households andfirms believe that there are additional sources of variation (see equation 25). See Mayer (2021) for a
discussion of rates of convergence in univariate learning models with additional stochastic regressors.

39

Assuming the relevant matrix inverse exists, x̂t is given byx̂t = B µ̂t ,where27"B=−

γb,π γb,C
γp,π γp,C

#−1 "

(41)

γb,µπ γb,µC
γp,µπ γp,µC#.

The law of motion of µ̂t is a stacked version of the updating expressions in equation(4). For simplicity, we impose the same gain, γt = 1/ (λ0 + t) , on the two equations.
Here, λ0 denotes the initial precision of beliefs about the mean of inflation and consumption. Combining the vector Bayesian updating expression with equation (41) we
obtain:µ̂t = (I − Bt ) µ̂t−1 ,

(42)



where Bt = γt I − B (1 − γt ) (I − γt B)−1 is the analog of bt in equation (8).28 A difference is that (1 − γt ) (I − γt B)−1 does not appear in bt , reflecting the timing differences
between the two models. These timing differences are negligible for our purpose; if
we replaced (1 − γt ) (I − γt B)−1 by I, then the asymptotic convergence result statedbelow would be unchanged.
The mapping from beliefs about x̂t –that is, µ̂t−1 –to realized values of x̂t is obtainedby multiplying equation (42) by B and using equation (41):x̂t = B (I − Bt ) µ̂t−1 .

(43)

The period equilibrium of the linearized model corresponds to the values of x̂t , µ̂t , γtwhich solve equations (42) and (43).
For t large enough, equation (43) is approximately x̂t = B µ̂t−1 . In a slight abuseof notation, we let b denote the largest real part of the eigenvalues of B. The central
result of this subsection is that b characterizes the asymptotic speed at which thelearning equilibrium converges to the stable REE. Therefore, it plays the same role as27

In the examples that we have considered, we have not encountered an exception to the invertibility
assumption.
28
According to the Bayesian updating equations, µ̂t = µ̂t−1 + γt (x̂t − µ̂t−1 ) . Substituting out for x̂tusing equation (41), we obtain µ̂t = µ̂t−1 + γt (B µ̂t − µ̂t−1 ). Equation (42) follows after rearranging.

40

the parameter, b, in the Bray and Savin (1986) model.
Our analysis of the speed of convergence of µ̂t holds for any finite-dimensional µ̂t .
We extend Definition 1 to the vector case of x̂t as follows:29
Definition 4. The vector series, x̂t , and the scalar series, at > 0, converge to zero at
the same rate if (i) for all j either (a) x̂j,t ≃ at or (b) limt→∞ |x̂j,t | /at = 0, and (ii) forat least one j, condition (a) holds.
Our definition requires that all elements of x̂t converge to zero at least as fast as at
and that at least one element converges to zero at the same rate as at .
In what follows, it is useful to denote the eigenvalue-eigenvector decomposition of B
as B = QΛQ−1 , where Λ is a diagonal matrix with the eigenvalues of B in the diagonal
elements.30 In the NK model, one cannot rule out the possibility that the eigenvalues,Λj , of B are complex. Let Λj = Λr,j + iΛc,j , where Λr,j and Λc,j denote the real andcomplex parts of Λj , respectively. Let rj,t denote the modulus of the j th eigenvalue of
I − Bt , and let j ∗ denote a value of j for which Λr,j attains its maximal value, b. Also,
define µ̃t = Q−1 µ̂t , where µ̂0 is given. The following proposition establishes the rate ofconvergence of µ̂t .
Proposition 2. Suppose that (i) B has an eigenvalue-eigenvector decomposition with
Λr,j < 1 for all j, (ii) µ̃j ∗ ,0 ̸= 0, and (iii) rj ∗ ,t ̸= 0 for each t. Then, µ̂t ≃ tb−1 .
See Appendix A for a proof.
Some comments about Proposition 2 are in order. First, violations of (ii) or (iii)
are isolated special cases. Condition (ii) rules out the case in which the initial priors
are orthogonal to the left eigenvector associated with the eigenvalue, Λj ∗ . In thatcase, Λj ∗ plays no role in the system’s dynamics. Condition (iii) is analogous to therequirement in Proposition 1 that γt (1 − b) ̸= 1. Second, our proposition is consistent
with the result of Evans and Honkapohja (2001, Theorem 7.1) that limt→∞ µ̂t = 0. The
novelty of Proposition 2 is that it establishes the rate at which µ̂t converges to zero.
Third, as in our analysis of the Bray and Savin (1986) model, the rate of convergenceof learning in the NK model is decreasing in b. Fourth, the fact that b plays a similarrole in the Bray and Savin (1986) and NK models can be seen by noting that premultiplication of equation (42) by Q−1 diagonalizes the system into a set of first-order
See Definition 1 for ‘≃′ .
A sufficient condition for this decomposition to exist is that the eigenvalues of B are distinct. Thiscondition is satisfied in all the examples that we consider.
29

30

41

scalar difference equations in µ̃j,t that are independent across j. Each of these equations
resembles equation (6) in Bray and Savin (1986). So the representation of µ̃j,t has theform given in equations (7) and (9) (though µ̃j,t is potentially complex valued), and its
behavior is determined by the j th eigenvalue of B. Since µ̂t = Qµ̃t and the columnsof Q are linearly independent, it follows that the largest real part of the eigenvalues of
B determines the rate of convergence of at least one element of µ̂t . Fifth, when the
eigenvalues of B are complex, µ̂t can exhibit sinusoidal fluctuations. That is why ourdefinition of the rate of convergence (Definition 1) needs to accommodate the possibility
that the µ̂t /tb−1 oscillates in a bounded set.31
Table 2 displays the eigenvalues of B corresponding to the Good- and Bad - ZLB
equilibria for the benchmark parameter values. The maximal eigenvalue (‘Eigenvalue1’), b, associated with the Good-ZLB and Bad-ZLB equilibria are 0.92 and 1.26, respectively. Consistent with the claim in Section 6, the Bad-ZLB equilibrium is not locallylearnable because b > 1. The Good-ZLB equilibrium is locally learnable because, inthat case, b < 1.
Asymptotic convergence to the Good-ZLB REE is slow because b is large. According
to Proposition 2 |µ̂j ∗ ,t | is approximately κtb−1 for some κ ̸= 0 and t sufficiently large.
The amount of time it takes to close two-thirds of a gap, µ̂t , for t sufficiently large, isgiven by Tt in equation (11). Table 2 reports values of T1 for different variants of themodel. In the benchmark model, when b = 0.92, then T1 = 920, 482. This large valueof T1 is qualitatively consistent with the basic prediction of the nonlinear solution to
the model–namely, that the rate of convergence is quite slow (see Figure 4a). Similarly,
the small value of T1 reported in the table for the case in which the ZLB is not bindingand α = 1.5 is qualitatively consistent with the finding for the nonlinear solution to themodel (see Figure 5). In this sense, the asymptotic result in Proposition 2 is a useful
guide about the rate of convergence, even for small t.
A different way to assess the usefulness of the asymptotics is to calculate the actual
amount of time, T, required to close two-thirds of the initial gap between priors andsteady state according to the linearized solution to the model.32 To this end, we
simulate the linearized solution to the model when the ZLB is binding and when we
ignore the ZLB. In the latter case, we consider α = 1.5 and 3. The results are reported
31

A discussion about the possibility of oscillations follows the proof of Proposition 2 in Appendix

A.
32

The initial gap in log xi , i = 1, 2, corresponds to the log-deviation of xi in the initial steady state
and the REE while r = rℓ .

42

in Table 4a. We find that, for the benchmark model, when the ZLB is binding, T =944, 710. In sharp contrast, when the ZLB is not binding and α = 1.5 and 3, we findthat T = 3 and 1 periods, respectively. These results about the importance of the
ZLB and the value of α in determining the speed of convergence are qualitatively thesame as our results using T1 . Thus, the rule of thumb, equation (11), suggested by
Proposition 2 is informative about actual rates of convergence in the linearized solution
to the model.
Table 2: Eigenvalues of B
Eigenvalue 1 Eigenvalue 2
T1
TGood ZLB
0.92-0.48
920,482 944,710
Bad ZLB
1.26
-1.21
NA
NA
No ZLB, α = 1.5 0.054+0.44i
0.054-0.44i
23
No ZLB, α = 3 -0.135+0.84i -0.135-0.84i
21
Note: The matrix, B, is defined in equation (41). The scalar, b, discussed in the text is the largest real part of theeigenvalues of B. The reported values of T are based on simulations of the linearized solution to the model. For thedefinitions of T and T1 see the text.

10

Conclusion

In this paper, we consider the speed with which people learn about their environment
after an unusual event. We do so in a non-linear NK model with internally rational
households and firms that are learning about how the economy will evolve after the
event. To characterize the speed of convergence of people’s beliefs, we analyticallyextend results in the literature to encompass circumstances when learning is very slow.
We argue that the slow convergence result arises naturally in the NK model when the
ZLB is binding. Under these circumstances, analyses of fiscal and monetary policies
under rational expectations can be very misleading. Since inflation declined by a modest amount during the Great Recession, learning moves the model toward the data
relative to rational expectations. In this sense, learning provides a possible resolution
to the ‘missing deflation puzzle’ (see Del Negro et al. (2023)). It would be interesting
to pursue this possibility in an empirically plausible version of the NK model of thesort considered by Christiano et al. (2016) or Del Negro et al. (2023).
Finally, we note that there are other circumstances in which slow learning could
arise. For example, plausible parameterizations of Cagan (1956)’s model of money demand under hyperinflation map into high b economies. Results in Marcet and Sargent43

(1995, Table 6.3) imply that estimates of the elasticity of money demand in hyperinflations (see, for example, Christiano (1987) and Taylor (1991)) map into high valuesof b. More generally, the learning principle suggests that any model with strong strategic complementarities could exhibit slow convergence to rational expectations underlearning.

44

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48

A Proofs of Lemmas and Propositions
A.1

Proof of Lemma 1

Lemma 1 follows from the following Proposition.
Proposition 3. Suppose bt ̸= 1 for all t, then µt can be written as



t 
X
a
zt
εj
a
µt =
+
bj
+ zt µ0 −1 − b j=1 zj 1 − b
1−band has meanaa
+ zt µ0 −1−b1−band variance

( 
)
t
2
2
X
zt
σ
ε
.
b2j
2
z
(1
−
b)
j
j=1

Proof. Note that


a
µt −
1−b





a
εt
+ (1 − bt ) µt−1 −
=bt
1−b
1−b


εt
εt−1
a
=bt
+ (1 − bt ) bt−1
+ (1 − bt ) (1 − bt−1 ) µt−2 −
1−b
1−b
1−b
εt
εt−1
εt−2
=bt
+ (1 − bt ) bt−1
+ (1 − bt ) (1 − bt−1 ) bt−2
1−b
1−b
1−b


a
+ (1 − bt ) (1 − bt−1 ) (1 − bt−2 ) µt−3 −
1−b
(" j
#
)


t−1
X
Y
εt−j
εt
a
=bt
+
(1 − bt−k+1 ) bt−j
+ zt µ0 −1 − b j=1
1
−
b
1−b
k=1



t 
X
zt
εj
a
=
bj
+ zt µ0 −.
zj 1 − b1−bj=1

The results of the proposition follow immediately from the properties of εt .

A.2

Proof of Proposition 1

We first state a number of lemmas that will be useful in the proof.

49

Lemma 2. For any b < 1 and any 0 ≤ λ0 , there exists a t∗ so that 0 < bt < 1 for allt ≥ t∗ .
Proof. Let t∗ = max {1, ⌈2 − b − λ0 ⌉} where ⌈x⌉ is the smallest integer larger than x.
The result follows immediately.
P
Lemma 3. Define Hλ0 ,T = Tt=1 λ01+t . Suppose 0 ≤ λ0 < ∞, thenlim {Hλ0 ,T − log (λ0 + T )} = cλ0 ,T →∞where cλ0 is a finite constant.
Proof. Note that Hλ0 ,T > 0 and
1+
Hλ0 ,T ≤λ0 + 1Z λ0 +T
λ0 +1

1
1
dt =
+ log (λ0 + T ) − log (λ0 + 1) .
t
λ0 + 1

Define the sequence yλ0 ,T = log (λ0 + 1) + Hλ0 ,T − log (λ0 + T ) . The above inequality
and the convexity of t−1 imply 0 < yλ0 ,T ≤ λ01+1 for all T ≥ 1. Also,yλ0 ,T +1 − yλ0 ,T =1
+ log (λ0 + T ) − log (λ0 + T + 1) < 0
λ0 + T + 1

Thus, yλ0 ,T is a monotone, decreasing series. The result of the lemman follows by themonotone convergence theorem.
P
Lemma 4. Define Hλ0 ,T = Tt=1 λ01+t . Suppose 0 ≤ λ0 < ∞, then there exist positive,
finite constants cλ0 and cλ0 , and a finite constant cλ0 so thatcλ 0
cλ< log (T + λ0 ) + cλ0 − Hλ0 ,T < 0 .
T
TProof. Using cλ0 from Lemma 3, noting that t−1 is convex, and following the geometriclogic in Young (1991) “Euler’s Constant” The Mathematical Gazette, Vol. 75, No. 472(Jun., 1991) we have1
1
< log (λ0 + T ) + cλ0 − Hλ0 ,T <.
2 (λ0 + T )λ0 + TThe result of the lemma follows immediately.
We are now in a position to prove Proposition 1.
50



Proof. Consider 0 < b < 1. Note that 0 < bj < 1 for all j. Define yt = log (λ0 + t)
Note that,
yt+h − yt = (1 − b) log 1 +h
λ0 + t= (1 − b) log 1 ++h
λ0 + t



h
X

1−b

zt .

log (1 − bt+k )

k=1
h
X

1
−
λ0 + t + k
k=1!
+ Rt,t+h

where Rt,t+h is the remainder term from Taylor’s Theorem in the representation ofP
log (·). By Lemma 3, log (λ0 + t) − tk=1 λ01+t is a Cauchy sequence. So, for any ϵ > 0
there exists a t1 so that if t ≥ t1 then for any h ≥ 0hlog 1 +
λ0 + t


−

h
X

ϵ
1
<
.
λ
4
−
4b
0+t+k
k=1


Ph

1
λ0 +t+k

2

for some 0 < K < ∞. Because
From Taylor’s Theorem, |Rh,t+h | ≤ k=1 K
P∞ −2
converges, for any ϵ > 0 there exists a t2 so that if t ≥ t2 then for any h ≥0
t=1 t
|Rh,t+h | < ϵ/4. It follows that if t ≥ max {t1 , t2 }, then |yt+h − yt | < ϵ/2. As a result,if t ≥ max {t1 , t2 } for any j, k > 0, |yt+j − yt+k | < ϵ. That is yt is a Cauchy sequence.
The conclusion of the proposition follows for 0 < b < 1 is then immediate.
Now consider b ≤ 0. By Lemma 2 there exists a t∗ so that for all t ≥ t∗ we haveP
0 < bt < 1. Define the sequence yt∗ = (1 − b) log (λ0 + t) + tj=t∗ log (1 − bj ) . Using
the same argument as in the case when 0 < b < 1, we have that yt∗ is Cauchy, so it
converges to a finite constant. For t ≥ t∗ , we havet1−b zt =



t
λ+t

1−b

exp (yt∗ )"t∗ −1Y#(1 − bj ) .

j=1

The conclusion of the proposition for b ≤ 0 follows by noting that we have ruled out
Q ∗ −1the possibility that tj=1
(1 − bj ) = 0 through our assumption that bt ̸= 1 for all t.

A.3

Proof of Proposition (2) and related results

Throughout, we assume that B has an eigenvalue-eigenvector decomposition meaningthat B = QΛQ−1 where the columns of Q are linearly independent eigenvectors of Band Λ is a diagonal matrix with eigenvalue Λj,r + Λj,c i in the jth diagonal element. To
51

prove the proposition, we will need some Lemmas and notation.
−1 


1 − λ01+t has an eigenvalue-eigenvector decomposiLemma 5. Ωt = B I − λ01+t B
tion so that Ωt = QΛt Q−1 where Q is the same matrix as in the eigenvector-eigenvalue
decomposition of B.
Proof. From the definition of Ωt
−1 

1
1
B
1−
Q
λ0 + tλ0 + t

−1 

1
1
−1
−1
−1
QQ −
= QΛQ
1−
QΛQ
Q
ν0 + tν0 + t

−1


1
1
−1
−1
= QΛQ Q I −ΛQ1−Qν0 + tν0 + tΩt Q = B I −

= QΛt
−1 


1 − ν01+t is a diagonal matrix.
where Λt = Λ I − ν01+t Λ
Lemma 5 says that the columns of Q are eigenvectors of Ωt , though Ωt has different
eigenvalues than B. Denote the jth diagonal element of Λt by Λj,r,t + Λj,c,t i. The
following Lemma follows from tedious algebra.
Lemma 6. Suppose Λj,r,t , Λj,r , Λj,c,t , and Λj,c are as defined above and that for all j1−Λeither λ0 +tr,j ̸= 1 or Λc,j,t ̸= 0 for all t. Then limt→∞ Λj,r,t = Λj,r , limt→∞ Λj,c,t =Λj,c and there exists a finite, positive constant K so that |Λj,r,t − Λj,r | < K (λ0 + t)−1
and |Λj,c,t − Λj,c | < K (λ0 + t)−1 .
It will be useful to define µ̃t ≡ Q−1 µt , which is given byµ̃t = I −

1(I − Λt ) µ̃t−1 =λ0 + t

t 
Y
I−
k=1

1
(I − Λk )λ0 + k

!
µ̃0 .

(44)



The modulus of the jth diagonal element of I − λ01+t (I − Λt ) is
rj,t =

2 
2 !1/2
1
1
1−(1 − Λj,r,t ) +Λj,c,tλ0 + tλ0 + tWe will use the following Lemma.
52

(45)

Lemma 7. Suppose rj,t ̸= 0 for all t and Λj,r < 1 , then1−Λj,rttY

rj,k → κj > 0

k=1

where rj,k is given by equation (45).
Proof. For a given j, define1−Λj,ryj,t ≡ (λ0 + t)

t
Y

rj,k

k=1

ỹj,t ≡ log (yj,t ) .
Note thatỹj,t+h − ỹj,t = (1 − Λj,r ) log 1 +h
λ0 + th1X1(Λj,r,t+k − 1)+log 1 + 2
2 k=1
λ0 + t + k

2 
2 !
1
1
(Λj,r,t+k − 1) +Λj,c,t+k+
λ0 + t + k
λ0 + t + k
!

 X
h
h
1
+ Rt,t+h= (1 − Λj,r ) log 1 +−λ0 + t
λ
0+t+k
k=1
h 
X
1
+
(Λj,r,t+k − Λj,r )
λ
0+t+k
k=1

2

2 !
1
1
1
1
+
(Λj,r,t+k − 1) +Λj,c,t+k
2 λ0 + t + k
2 λ0 + t + k
where Rt,t+h is the remainder term from Taylor’s theorem. For any ϵ, there exists a t1
(which does not depend on h) so that if t ≥ t1 then for any k ≥ 0 by Lemmas 3 and 653

we havelog 1 +k
ν0 + t


−

k
X

1
ϵ
<
ν +t+n6 |1 − Λj,r |n=1 0|Λj,r,t+k − Λj,r | <K
λ0 + t + k(Λj,r,t+k − 1)2 < (1 − Λj,r )2 + ϵ
Λ2j,c,t+k < Λ2j,c + ϵ.
for some finite, positive K. Then
2 

h 
 1 2

1
1
ϵ X
2
K+
(Λj,r − 1) + ϵ +Λ + ϵ + |Rt,t+h ||ỹt+h − ỹt | ≤ +6 k=1 ν0 + t + k
2
2 j,c
P
−2
converges, for any ϵ there exists a t2 (which does not depend on h)
Because ∞t=1 t
so that if t ≥ t2 ≥ t1 then
∞ 
X
k=1

1
ν0 + t + k

2 

 1 2

1
ϵ
2
K+
(Λj,r − 1) + ϵ +Λj,c + ϵ < .
226Now, consider
h
X


1
1
1
2
(Λj,r,t+k − 1)
|Rt,t+h | =
2
4
λ
(1
+
x
)
0+t+k
k
k=0

2 
 2 #2
1
1+(Λj,r,t+k − 1) +Λj,c,t+k
λ0 + t + k
λ0 + t + kfor"1(Λj,r,t+k − 1) +xk ∈ 0, 2
λ0 + t + k1(Λj,r,t+k − 1)
λ0 + t + k

54

2


+

1
Λj,c,t+k
λ0 + t + k2 #.

For any h, if t ≥ t2
|Rt,t+h | <

h 
X

1
λ0 + t + k

2

1
2



1
4

1
1 − 2 λ0 +t(Λj,r − 1 − ϵ)
2



2
× 2K + 2 (1 − Λj,r + ϵ) + (Λj,r − 1)2 + ϵ + Λ2j,c + ϵ .
k=0PBecause t t−2 converges to a finite constant, for any ϵ, there exists a t3 ≥ t2 (which
does not depend on h) so that if t ≥ t3 thenϵ|Rt,t+h | < .
6Combining results, for any ϵ > 0, there exists a t3 so that if t ≥ t3 , then for any hϵ|ỹt+h − ỹt | < .
2Then forr any n, m ≥ 0|ỹt3 +m − ỹt3 +n | < ϵmeaning that ỹt is a Cauchy sequence, and thus converges to a finite constant. It
follows immediately that yt converges to a finite, non-zero constant.
We are now in a position to prove the proposition.
Proof. Note thatt1−b µt = t1−b Qµ̃t = Q t1−b

t 
Y
I−
k=1

The jth element of t1−b

Q
t

k=1

t1−b



1
(I − Λk )λ0 + k

!

!
µ̃0 .


I − λ01+k (I − Λk ) µ̃0 is given by

t
Y

!
rj,k

exp i

k=1

t
X

!
φj,k

µ̃j,0

k=1


Qt
If Λj,r = b, then by Lemma 7 limt→∞ t1−br
= κj for some finite, positive κj .
j,k
k=1

Qt
1−b
If Λj,r < b then limt→∞ t
k=1 rj,k = 0. The conclusion of the proposition follow
by noting that µ̃j ∗ ,0 ̸= 0 by assumption, that exp (ϕi) is bounded for all ϕ, and thatthe columns of Q are linearly independent.
55

A comment related to the possibility of sinusoidal fluctuations in t1−b µ̃j ∗ ,t is inorder. From the definition of φj ∗ ,k and the power series representation of sin−1
sin (φj ∗ ,k ) =sin−1 (x) =Λj ∗ ,k,c
λ0 +k

rj,k
∞
X

(2n)! x2n+1.
(2n n!)2 2n + 1
n=0

By Lemma 6, for large enough k all of the terms in the power series representationof sin (φj ∗ ,k ) are of the same sign. It follows that for large enough t it must be thatPt
k=1 φj ∗ ,k is either strictly increasing or strictly decreasing in t and that|Λj∗ ,t,c ||φj ∗ ,t | >λ0 +krj ∗ ,t.

P
By Lemma 6 and because limt→∞ rj ∗ ,t = 1, if |Λj ∗ ,c | ≠ 0 we have limt→∞ tk=1 |φj ∗ ,k | =∞. This result, along with the observation that limt→∞ φj ∗ ,t = 0, implies sinusoidal
fluctuations in t1−b µ̃j ∗ ,t .

B

Constant gain learning

Another, widely-used way to model learning based on past data is to have people
update beliefs using a constant gain. When people update their beliefs using constantgain learning, equation (4) becomesµt = µt−1 + γ (xt − µt−1 ) ,for 0 < γb < 1. Rearranging, we obtain the analog of equation (9)




t−1
X
a
a
εt−j
t
j
=
γb + (1 − γb ) µ0 −(1 − γb )
µt −
1−b
1
−
b
1−b
j=0

(46)

where γb = (1 − b) γ. As long as |1 − γb | < 1 the impact of µ0 on µt goes to zero
eventually. As in the case of Bayesian learning, the rate of convergence is decreasing in
b. So, the positive feedback loop discussed in the previous section continues to operate.

56

Rewriting equation (46)aµt − 1−btEa = (1 − γb ) .
µ0 − 1−bHere, convergence occurs at a geometric rate, λt , 0 ≤ λ < 1. In contrast, convergence
under Bayesian learning proceeds at a power rate, t−δ , δ > 0 ( Proposition 1). Power
convergence is well known to be slower than geometric convergence for any δ > 0 and
λ < 1.
Nevertheless, convergence can be very slow under constant gain learning. For example, when γ = 0.5 and b = 0, 0.5, 0.75, 0.85, .95. Again, let T satisfy (1 − γb )T ≃ 1/3.
Then T = 2 (3), 4 (11), 9 (113), 15 (2201), and 44 (5.2 billion), respectively. Numbers inparentheses reproduce the results under Bayesian learning when λ0 = 1. The variable,T, is increasing at an increasing rate as b gets larger. While learning under constantgain learning can be very slow for large b, it is much faster than when the gain isdecreasing.

C

Solution algorithm for non-linear NK model

In this Appendix we detail our solution strategy for the non-linear NK model we consider in our paper. We exploit the model’s structure to simplify its solution. In particular, because the steady state is an absorbing state for the REE and learning equilibria
that we consider, we can solve the steady state decision rules without reference to theperiod when r = rℓ . With this solution in had, we then turn to the period when r = rℓ ,which is where we consider learning.
Our main model code is implemented in c++, with reliance on the Eigen, boost,and nlopt libraries. Our computations were conducted using nearly 400 processors
with heavy reliance on MPI. Our computations took roughly two weeks to complete.
Details related to our model code are available in the README file associated withthe replication materials. This Appendix outlines the strategy used to solve the modelthat is implemented in that code.

C.1

Steady state

In the steady state, there is no uncertainty. However, households still face a bondholding choice and firms still face a relative-price choice. In an REE, households will

57

choose to hold zero bonds and firms will choose to set their price to the aggregate pricelevel.
C.1.1

Household problem

In the steady state, the household value function is given byonχ
log (Ch ) − (Nh )2 + βVh,ss (b′h )Ch ,Nh ,bh2

Vh,ss (bh ) = max ′subject to
Ch +

b′h
bh
≤
+ wss Nh + Tss .
Rss
πss

Here, bh and b′h are household h’s real bond holdings chosen in the previous and current
period, respectively. The variables Ch and Nh are household h’s consumption and labor
supply. The aggregate variable Rss , πss , wss , and Tss are the gross nominal interest
rate, the gross inflation rate, the real wage, and taxes net of transfers and profits.
The values of these aggregate variables are known to the household. We constrain
 
households so that b′h ∈ b, b . Implicitly, we have functions Ch,ss (bh ), Nh,ss (bh ), andb′h,ss (bh ). Assuming the constraint on b′h is not binding, household maximization implies
1
Ch,ss (bh )= βRss1
Ch,ss (b′h (bh )) πss

χCh,ss (bh ) Nh,ss (bh ) = wss
Ch,ss (bh ) +b′h,ss (bh )
Rss

=

bh
+ wss Nh,ss (bh ) + Tss
πss

(47)
(48)
(49)

 
We define a grid over b, b and approximate the functions Ch,ss (bh ), Nh,ss (bh ), andb′h,ss (bh ) on that grid in the following way.33(i) We conjecture a value for b′h,ss (bh ) at each grid point. Call the conjectured valueb′ih,ss (bh ).
33

In our implementation, we set −b = b = 1, which is equal to steady state output. We use asymmetric grid with 25 points that includes zero and places more points near zero than at more
extreme values because bh = b′h = 0 in both REE and in learning equilibria.

58

(ii) Note that equations (48) and (49) can be written asb′ih,ss (bh )
bh
−
− Tss
χCh,ss (bh ) Ch,ss (bh ) +
Rss
πss

!
2
= wss
.

The left-hand-side is increasing in Ch,ss (bh ) ≥ 0. For every bh , we solve for thevalue of Ch,ss (bh ) that makes this hold with equality. We call this the conjecturedii(bh ), we can(bh ). Note that with Ch,ssvalue for Ch,ss (bh ) and denote it by Ch,ssiback out Nh,ss (bh ) from equation (48).
(iii) For each grid point, bh , find b′h that solves the following version of equation (47)Ch βRss1
iCh,ss(b′h )−1=0

where Ch ≥ 0 solves


b′h
bh
2
χCh Ch +−− Tss = wss.
Rss πssi
We use linear interpolation to compute Ch,ss(b′h ) for values of b′h that fall between
grid points. If the procedure would set b′h > b or b′h < b, we set b′h to the respective
endpoint of the grid. We record the value of b′h in by updating the conjectured′
rule for b′h,ss (bh ) using b′i+1h,ss (bh ) = bh .

(iv) Having computed b′i+1h,ss (bh ) for every grid point, we check to see if′ib′i+1h,ss (bh ) − bh,ss (bh ) < ϵat every grid point for some small ϵ. If yes, we say that we have solved thehousehold problem in steady state. If no, we set b′ih,ss (bh ) = b′i+1h,ss (bh ) and repeat
steps (ii), (iii), and (iv).
ss= 1, it is not surprising that we find that b′h,ss (bh ) = bh .
Because β Rπss

C.1.2

Firm problem

In the steady state, the firm value function is given by59Vf,ss (pf ) = max
′
pf


−ε
1
p′f − (1 − ν) wss p′f
Yss
Css

2
1 ϕ p′fπss − 1 (Css + Gss )−Css 2 pf+βVf,ss p′f .

Here, pf and p′f are the ratio of firm f ’s price to the aggregate price level in the previous
and current period, respectively. The aggregate values πss , wss , Css , Gss , and Yss are  
known to the firm. We constrain firms so that log p′f ∈ p, p . Implicitly, we havea function p′f,ss (pf ). Assuming the constraint on p′f is not binding, firm maximization
implies ′pf (pf )1
ϕπss − 1πss (Css + Gss ) =
pf
pf
!
−ε
wss
(ε − 1)− 1 p′f (pf )Yss′
pf (pf )!
p′f p′f (pf )p′f p′f (pf )πss − 1+ βϕ
2 πss (Css + Gss )p′f (pf )p′f (pf )

(50)

 
We define a grid over p, p and approximate the function p′f,ss (pf ) on that grid in thefollowing way.34(i) We conjecture a value for p′f,ss (pf ) at each grid point. Call the conjectured value
p′if,ss (pf ).
34

In our implementation, we set −p = p = 1. We use a symmetric grid with 25 points that includes
 
zero that places more points near zero than at more extreme values because log (pf ) = log p′f = 0
in both REE and in learning equilibria.

60

(ii) For each grid point, pf , find p′f that solves the following version of equation (50)
 ′

pf
1
ϕ
πss − 1πss (Css + Gss ) =
pf
pf
!

wss
′ −ε
−
1
p
Yss
(ε − 1)fp′f!
p′if,ss p′fp′if,ss p′f+ βϕ
πss − 12 πss (Css + Gss )
p′f
p′f



We use linear interpolation over log p′f to compute p′if,ss p′f for values of log p′fthat fall between grid points. If the procedure would set log p′f > p or log p′f <p, we set p′f to the respective endpoint of the grid. We record the value of p′f in′by updating the conjectured rule for p′f,ss (pf ) using p′i+1f,ss (pf ) = pf .
(iii) Having computed p′i+1f,ss (pf ) for every grid point, we check to see if′ip′i+1f,ss (pf ) − pf,ss (pf ) < ϵat every grid point for some small ϵ. If yes, we say that we have solved the firm
problem in steady state. If no, we set p′if,ss (pf ) = p′i+1f,ss (pf ) and repeat steps (ii)and (iii).

C.2

Solution when r = rℓTo address the case when r = rℓ , we assume that we have the steady state decisionrules in hand and that households and firms know these decision rules with certainty.
C.2.1

Beliefs

Before presenting the household and firm problems, some comments about beliefs are
in order when r = rℓ . To simplify the model, we assume households and firms have thesame beliefs (though they do not know that they have the same beliefs). Households
and firms believe that so long as r = rℓ the log of aggregate consumption, log (C),and the log of aggregate inflation, log (π), have uncorrelated Normal distributions with61

unknown means and variances. That islog (π) ∼ N µπ , σπ2log (C) ∼ N µC , σC2 .
We assume that households and firms have beliefs about the means and variances ofthe distributions for log (C) and log (π) that are characterized by density functions
that are proportional to Normal-inverse-gamma distributions. These beliefs are notexactly Normal-inverse-gamma distributions because the households and firms embedin their beliefs an upper bound on the variances. This upper bound is importantbecause if variances were unbounded, E [π] = E [C] = ∞, which would challenge theapplicability of an expected utility framework. The distributions characterizing beliefs
are independent across C and π. That is, for i ∈ {π, C}, µi ∈ (−∞, ∞) and σi2 ∈ [0, σ 2i ]we haveαPr σi2 |αi , βi =Pr

µi |σi2 , mi , λiβi iΓ(αi )

 αi +11
σi2

exp − σβ2i

Γ(αi ,βi /σ 2i )Γ(αi )

i

,

√


λi
λi
2
=p
exp − 2 (µi − mi ) .
2σi
2πσi2

Here, Γ (·) is the gamma function and Γ (·, ·) is the incomplete gamma function. Notethat Γ (·) = Γ (·, 0). Again, the advantage of truncating the support of σi2 is that
E [π] < ∞ if σ 2π < ∞ and E [C] < ∞ if σ 2C < ∞.
Even though we truncate the distributions for σi2 , we maintain conjugacy between
prior and posterior beliefs as well as the usual recursive updating equations because thelikelihoods associated with observations of π and C are not truncated. Beliefs aboutlog (i) are parameterized by four values, αi , βi , mi , and λi . So, we have 8 total values
for both π and C. The standard recursive updating formulas for these variables are
λ′i = λi + 1
λmi + log (i)λ+1′αi = αi + 1/2

m′i =λi (log (i) − mi )2′
βi = βi +.
2 (λi + 1)
62

Here, a prime indicates the value taken after having observed log (i).
We need to include variables in Θ that will fully capture the values αi , βi , mi , andλi for i ∈ {π, C}. First, we keep t1ℓ in Θ, which is the inverse of the number of periods
that r has been equal to rℓ . We keep the inverse because it is bounded between zeroand one, which will be useful. From this value, we can trivially back out λi and αi ,given their values in the first period when r = rℓ . We set the initial value of λi = 1
and the initial value of αi = 2. We keep mC and mπ in Θ. And we also keep
s
ψi′ =

ψi2

2αi′
λi
1
+
(log (i) − mi )2 .
′′2αi + 1 λi + 1 2αi + 1

Note that by setting βi = (ψi )2 αi′ it is clear that we recover the exactly recursive
structure of βi (given above). An advantage of using ψi in Θ rather than βi is that
ψi is a consistent estimator for the standard deviation, whereas βi generally grows
without bound (except when the standard deviation is zero). Keeping the values of Θ
withini be important for the purposes of approximation. In total,h bounded grids will
Θ = τ1ℓ , mπ , mC , ψπ , ψC has five elements and we have a mapping from Θ to αi , βi , mi ,and λi for i ∈ {π, C}. We also have a law of motion for Θ so that Θ′ = L (Θ, [π, C]) .
An advantage of the Normal-inverse-gamma setup detailed above is that we can haveanalytic expressions for the distribution for the variables log (π) and log (C) conditionalon Θ. In particular
Pr (log (i) |µi , σi2 , Θ) Pr (µi , σi2 |Θ)Pr (log (i) |Θ) =Pr (µi , σi2 | log (i) , Θ)
√ 1 2 exp − 2σ1 2 (log (i) − mi )2
2πσi
i
= √

 ′ α′i  α′i +1
 ′
′
λi
β
λ′1
2 (βi )
1′
√ 2 exp − 2σ2 (µi − mi )
exp
− σ2i
2
′
σ
Γ
α
2πσi
( i) i
i
i
√


λi
λi
×p
exp − 2 (µi − mi )2
2
2σi
2πσi




αi +1
βiαi
1
βi κ′i×exp − 2Γ (αi ) σi2
σi κi
where
κi =

Γ (αi , βi /σ 2 ).
Γ (αi )

63

Then

Pr (log (i) |Θ) =λi α iβi (λi + 1)1/2Γ (αi + 1/2)√Γ (αi ) 2παi
−α0 −1/2

1 (log (i) − mi )2 × 1 +
−1 2αi
λi α i

κ′i
.
κi

(51)

βi (λi +1)

Notice that κ′i depends on the point of evaluation for log (i). Evidently, if we ignored
the ratio κ′i /κi , which would be correct in the case when σ̄i2 = ∞, the pdf for log (i)−1/2λi α iis a t distribution with location parameter mi , scale parameter βi (λ, and 2αii +1)
2′degrees of freedom. If σ i is large, κi /κi ̸= 1 but is close to unity. For finite σ̄i2 , theratio κ′i /κi serves to thin the tails of the distribution of log (i) by down-weighting theprobability of extreme values for log (i).35 Because the density function of the t distribution is readily available and reliably computed in statistical software and because
κi and κ′i are easily computed using readily available implementations of the gamma
and incomplete gamma functions, we can use equation (51) for quadrature weighting.
We use Gauss-Hermite quadrature with seven nodes when computing approximations
to integrals based on equation (51).
C.2.2Household problem

When r = rℓ , the household value function is given bynχVh (bh , Θ, x) = max ′ log (Ch ) − (Nh )2
Ch ,Nh ,bh
2

1
′
′
′
′
+
[pEΘ′ Vh (bh , Θ , x ) + (1 − p) Vh,ss (bh )]1 + rℓsubject to
Ch +

b′h
bh
≤
+ wNh + T.
RπHere, x = [π, C]′ , Vh,ss (·) is the steady state value function for the household, which
is defined above, and EΘ′ denotes expectations of the household computed conditionalon Θ′ . Given x and Θ, we have Θ′ = L (Θ, x). So, the expectation of the household is
35

We set σ̄i2 equal to the squared maximum value on the grid for ψi (described below).

64

taken with respect to x′ , which is believed to be iid. We assume that households know
the monetary and fiscal policy rules. We also assume that they correctly think that Y =(C + G) 1 + ϕ2 (π − 1)2 , N = Y , and w = χCY . Given x, with these assumptions
R, π, w, and T can be computed. The steady state values of aggregate variables are 
known to the household. We constrain households so that b′h ∈ b, b . The household
optimization problem gives us implicit functions for Ch (bh , Θ, x), Nh (bh , Θ, x), andb′h (bh , Θ, x). Considering interior solutions for b′h , we have



1
1
1
=
R pEΘ′Ch (bh , Θ, x)
1 + rℓπ ′ Ch′ (b′h , Θ′ , x′ )
1
+ (1 − p)
πss Ch,ss (b′h )w = χCh (bh , Θ, x) Nh (bh , Θ, x)
Ch (bh , Θ, x) =bhb′ (bh , Θ, x)+ wNh (bh , Θ, x) + T − h
.
π
R

(52)
(53)
(54)

Instead of approximating Ch (bh , Θ, x), Nh (bh , Θ, x), and b′h (bh , Θ, x) directly, we approximate1vh (bh , Θ) = EΘ.
πCh (bh , Θ, x)
We take this approach because we can eliminate x as a state variable in the approximation. We define grids on the elements of Θ and use the grid defined for bh in thesteady state. We then approximate vh (bh , Θ) in the following way.36(i) We conjecture a value for vh (bh , Θ) at each grid point in the cross product of thegrids over the elements of bh and Θ.37 Call the conjectured value vhi (bh , Θ).
(ii) For a given grid point we use quadrature to get a value for EΘ (πCh )−1 .To
solve for the expectation of interest, we need to solve for Ch given many different
values for x. Conditional on a value for x, equations (53) and (54) can be writtenas
b′h bh−− T = w2
χCh Ch +R
π36

The grids for mi contain 12 points that are are not evenly spaced. They include each REE point
as well as the target-inflation steady state. The remaning points are bunched relatively close to the
REE points. The grid for ψC contains 11 points that are evenly spaced from 0 to 0.1. The grid for ψπcontains 11 points that are evenly spaced from 0 to 0.05. Note that inflation is expressed in quarterly
terms, so a change of 0.05 would be 20 percent if annualized.
37
There are 435, 600 = 12 × 12 × 11 × 11 × 25 points in the cross product of the grids for mi , ψi ,and bh . The grid for t−1
is handled in a way discussed below.
ℓ65

The left-hand-side is increasing in Ch ≥ 0. For a given b′h , we solve for the valueof Ch that makes this hold with equality. We then search for the value of b′h thatmakes the following version of equation (52) hold with equality:


1
1
1
′
′
i
=
R pvh (bh , Θ ) + (1 − p).
Ch
1 + rℓπss Ch,ss (b′h )We use linear interpolation to compute vhi (b′h , Θ′ ) for values of b′h and Θ′ that fall
between grid points. If the procedure would set b′h > b or b′h < b, we set b′h to therespective endpoint of the grid for bh . We record the associated value of Ch and
use it in the quadrature to compute vhi+1 (bh , Θ) = EΘ (πCh )−1 .
(iii) Having computed vhi+1 (bh , Θ) for every grid point, we check to see if
vhi (bh , Θ) − vhi+1 (bh , Θ) < ϵat every grid point for some small ϵ. If yes, we say that we have solved thehousehold problem when r = rℓ . If no, we set vhi (bh , Θ) = vhi+1 (bh , Θ), repeat
steps (ii) and (iii).
 1 1

The grid that we use on t1ℓ is special. In particular, we let that grid be 0, 99, 98 , . . . , 1 .
The first element of the grid corresponds to the case when infinite time has past.
In this case households think that they would update their beliefs so that Θ′ = Θbecause mi and ψi are consistent estimators for the means and variances. In our
numerical computations, we utilize this fact to first approximate vh in this case. We
then approximate vh in the case where tℓ = 99. When tℓ = 99, we need to interpolate
between the solution to the case when tℓ = ∞ and the conjectured value of vhi (bh , Θ)when tℓ = 99 to evaluate vhi (b′h , Θ′ ). That is, when tℓ = 99 we have to find a fixed
point of this interpolation, which is computationally intense. To do the interpolation,= 1/99 and t−1= 0. When tℓ = 98, having
we linearly interpolate between t−1ℓℓapproximated vh (bh , Θ) for tℓ = 99 means that can evaluate vh (b′h , Θ′ ) exactly att′ℓ = 99 without reference to vhi (bh , Θ). We approximate for vh (bh , Θ) when tℓ = 98
and work work back in this way to tℓ = 1. This strategy fits this into the structure ofsteps 1-3 because we know that the value of vh (bh , Θ) will not depend on its value at
any any tℓ that is smaller than implied by Θ. So, we have a block dependent structure
to vh (bh , Θ) . Additionally, we know that tℓ will only take integer values.

66C.2.3Firm problem

When r = rℓ , the firm value function is given by

!
 ′
2
p
f
p′f − (1 − ν) w pf
π − 1 (C + G)
Vf (pf , Θ, x) = max
p′f
pf


1 
+
pEΘ′ Vf p′f , Θ′ , x′ + (1 − p) Vf,ss p′f
1 + rℓ
(

1
C




′ −ε

ϕ
Y −
2

Here, x = [π, C]′ , Vf,ss (·) is the steady state value function for the firm, which is defined
above, and E denotes expectations of the firm. Given x and Θ, we have Θ′ = L (Θ, x).
So, the expectation of the firm is taken with respect to x′ , which is believed to be iid.
We assume that firms know the monetary and fiscal policy rules. We also assume that
they correctly think that Y = (C + G) 1 + ϕ2 (π − 1)2 , N = Y , and w = χCY . Givenx, with these assumptions π, w, G, and Y can be computed. The steady state values of  aggregate variables are known to the firm. We constrain firms so that log p′f ∈ p, p .
Implicitly, from firm optimization we have a function p′f (pf , Θ, x). Considering interior
solutions for p′f , firm maximization implies ′pf (pf , Θ, x)1π−1π (C + G) =
ϕ
pf
pf
!
−ε
w
′
(ε − 1)
−
1
p
(p
,
Θ,
x)
Y
f
f
p′f (pf , Θ, x)!

p′f p′f (pf , Θ, x) , Θ′ , x′ ′p′f p′f (pf , Θ, x) , Θ′ , x′ ′ ′
1
C
+
π −1
pEΘ′ ′ ϕπ (C + G′ )
2′′
1 + rℓCpf (pf , Θ, x)
pf (pf , Θ, x)!

p′f,ss p′f (pf , Θ, x)p′f,ss p′f (pf , Θ, x)
1C+(1 − p) ϕ
πss − 12 πss (Css + Gss ) .
1 + rℓ Css
p′f (pf , Θ, x)p′f (pf , Θ, x)(55)
Instead of approximating p′f (pf , Θ, x) directly, we approximatevf (pf , Θ) = EΘ

1
ϕ
C

 ′
 ′

pf
pf
π−1
π (C + G) .
pf
pf

We take this approach because we can eliminate x as a state variable in the approximation. We use the same grids on the elements of Θ that we use for the household problem
67

and the grid defined for log (pf ) in the steady state and we approximate vf (pf , Θ) in
the following way.
(i) We conjecture a value for vf (pf , Θ) at each grid point in the cross product of thegrids over the elements of pf and Θ. Call the conjectured value vfi (pf , Θ).
(ii) For a given grid point we use quadrature to get a value for

E

1
ϕ
C

 ′
 ′

pf
pf
π−1
π (C + G) .
pf
pf

To solve for the expectation of interest, we need to solve for p′f given manydifferent values for x. Conditional on a value for x, we find a value of p′f thatsolves the following version of equation (55)
 ′

pf
1
ϕ
π−1
π (C + G) =
pf
pf
!

w
′ −ε
−
1
p
Y
(ε − 1)
f
p′f
C
1
+
pvfi p′f , Θ′ ′1 + rℓpf
1C+(1 − p) ϕ
1 + rℓ Css!

p′f,ss p′f
p′f,ss p′f
πss − 12 πss (Css + Gss ) .
p′f
p′f



We use linear interpolation over log p′f to compute vfi p′f , Θ′ for values of
log p′f and Θ′ that fall between grid points. If the procedure would set log p′f >p or log p′f < p, we set p′f to the respective endpoint of the grid for pf . We record
the value of p′f in and the associated aggregate variables so that the quadratureprocedure can approximate
vfi+1 (pf , Θ) = EΘ



1
ϕ
C

 ′
 ′

pf
pf
π−1
π (C + G) .
pf
pf(iii) Having computed vfi+1 (pf , Θ) for every grid point, we check to see if
vfi (pf , Θ) − vfi+1 (pf , Θ) < ϵat every grid point for some small ϵ. If yes, we say that we have solved the68household problem when r = rℓ . If no, we set vfi (pf , Θ) = vfi+1 (pf , Θ), repeat
steps (ii) and (iii).
Our use of the same grids as in the household problem allows us to exploit the same
block dependent structure in t−1
ℓ .

C.3

Learning equilibria

Here we detail how we construct learning equilibria, given the solutions to the household
an firm problems—vh and vf .
(i) Set r = rℓ and assume a value for Θt for t = 1.
(ii) Conjecture a value for πt .
(a) Find the value of Ct that would make the following equation hold


1
1
1
=
Rt pvh (0, f (Θt , [πt , Ct ])) + (1 − p).
Ct
1 + rℓπss Ch,ss (0)Note that with π1 and C1 the values of all other aggregate variables can becomputed.
(b) Check to see if the following equation holds
ϕ (πt − 1) πt (Ct + Gt ) =1(ε − 1) (wt − 1) +pvf (1, f (Θt , [πt , Ct ])) Ct
1 + rℓ1 Ct+(1 − p) ϕ (πss − 1) πss (Css + Gss ) .
1 + rℓ Css
If yes, we have a period equilibrium for period t and we record πt and Ct . If
no, conjecture a different value for πt .
(iii) Set Θt+1 = L (Θt , [πt , Ct ]) and repeat step (ii).
When we consider “anticipated utility,” we define Θ̃t to be Θt , but with t1ℓ = 0. We
then perform step 2 with Θ̃t instead of Θt . However, in step 3 we continue to use Θt .
The switch between Θ̃t and Θt highlights the way in which “anticipated utility” is notinternally rational.

69

D Linearized NK Model
Here we describe our strategy for linearizing the NK model around an REE. We findit convenient to use t notation, rather than recursive notation.

D.1

Household problem

The household have a flow budget constraint
Ch,t +bh,tbh,t−1=+ wt Nh,t + τtRtπt

and optimality conditions given by
1 1
1= βt Eh,t
Ch,t Rt
Ch,t+1 πt+1
χNh,t Ch,t = wt .
1Here, Eh,t is EΘ′ in our recursive notation. We assume that βt takes two values: β̃ = 1+rℓ
and β, with β̃ > β. The high value happens at period 1 and goes back to the low value
with probability 1 − p. The low value is the absorbing state.
Let’s first consider the absorbing state. Log-linearize (except for bh,t , which islinearized) the equilibrium conditions around the zero inflation steady state (note thatthe aggregate variables take their steady state value and the households know this, so
their log-deviation is zero).

bh,t + βbbh,t = bbh,t−1 + N
bh,t
CC
h
i
bh,t = Eh,t C
bh,t+1C
bh,t + C
bh,t
0=N
Evidently,
1 b
β bbh,t−1 −bh,t
C +h1
i C +1bh,t = Eh,t C
bh,t+1C
bh,t =
C

1 + βb1 b
β bbh,t+1 =bh,t −
bh,t−1
C +1
C +1
C +1
70

meaninghi
1 b
β + 1bβbbh,t−1 −bh,t = −Eh,t bh,t+1
C +1
C +1
C +1
We consider solutions of the form
bbh,t = ωb,bbbh,t−1where ωb,b satisfies
1
β+1
β
−
ωb,b +
ω 2 = 0.
C +1 C +1
C + 1 b,b
The solutions to this equation areωb,b =

β+1
±
C+1

q

β+1 2
β
1
− 4 C+1
C+1
C+1
β
2 C+1

We focus on ωb,b = 1 because that is the value that corresponds to the solution of thenonlinear model. So,
1bh,t(C + 1) C
1−βbbh,t = bbh,t−1 .

bbh,t−1 =Let’s next consider the case where βt = β̃. Let x̃ be the RE aggregate quantity while
βt = β̃ and x̃ˆt be the (log-)linearized quantity around x̃. We haveb̃ + bb = bbh,t−1 + w̃Ñ w
b̃ + τ̃ b̃
b̃ t + w̃Ñ N
C̃ C
τt
h,t
h,t
h,t
π̃ 


 (1 − p) 1 − β
p
1 b̃
b̃
bbh,t
b̃t+1 +−
C h,t = −β̃Eh,t C h,t+1 + π
Cπ C + 1
C̃ R̃
C̃ π̃
b̃ + Cb̃b̃ = N
w
t

h,t

h,t

Note that we have imposed Rt = 1 while βt = β̃, which is true in the REE. In this71

sense, the system is local. We assume that households know thatΦ2
Nt = (Ct + G) 1 + (πt − 1)2
wt = χNt Ctτt = (1 − wt ) Yt −Φ
(πt − 1)2 (Ct + G) − G.
2

These relations are true in the period equilibrium, and are log-linearized to be


C̃ b̃
Φ
2
b̃
Ct + Φ
N t = 1 + (π̃ − 1)2ÑC̃ + G̃Ñ!
ˆt(π̃ − 1) π̃ π̃b̃ + Cb̃b̃ t =N
w
t
t
Φ
τ̃ b̃
τ t = (1 − w̃) Ỹ Ỹˆt − w̃Ỹ w̃ˆt − (π̃ − 1)2 C̃ C̃ˆt

2
ˆt .
− Φ (π̃ − 1) π̃ C̃ + G̃ π̃
The household optimality conditions and aggregate relations that are known to thehousehold can be written as a single equation of the form =b̃ − κ π̃ˆ−κb,t−1bbh,t−1 + κb,tbbh,t − κC,t C
t
π,t t


=κb,t+1 Eh,t bbh,t+1 − κµC ,t mC,t − κµπ ,t mπ,t72

whereκb,t−1 =1π̃1−p1−βpκb,t =1 + β̃ R̃
+ C̃ C̃ + w̃Ñπ̃ 2
Cπ C + 1
!


Φ
C̃
2
κC,t =2w̃Ñ
1 + (π̃ − 1)
+1
2
Ñ


Φ
2
− (2w̃ − 1) 1 + (π̃ − 1) C̃
2


Φ
2
− w̃Ỹ + (π̃ − 1) C̃2κπ,t =0pπ̃pκµC ,t =β̃ R̃ κC,tπ̃p
κµπ ,t =β̃ R̃ C̃ + w̃Ñπ̃κb,t+1 =β̃ R̃

Here, mπ,t is m′π in our recursive notation and mC,t is m′C in our recursive notation. Notethat the time subscripts on the κ’s is to denote if the coefficient multiplies, for example,
bt or bt−1 . The time subscript does not indicate time-variation in the coefficient.
We consider solutions to this equation of the form
ˆt + γb,C C̃ˆt + γb,µπ mπ,t + γb,µC mC,t .
b̂h,t = γb,b b̂h,t−1 + γb,π π̃Note that this is a linear approximation to b′h (bh , Θ). Our approximation does notinclude ψπ or ψC because of the certainty equivalence of the linearized model.
Using the linear decision rule for b̂h,t , γb,b is determined by2−κb,t−1 + κb,t γb,b = κb,t+1 γb,bwhich is given byκb,t ±
γb,b =qκ2b,t − 4κb,t−1 κb,t+1
2κb,t+1

Both of the solutions for γb,b are larger than unity. However, the smaller value is closer
to the solution of the nonlinear model at the REE, so we focus on that value. We73determine the other four values of γb,i using the following equations.
κb,t γb,π − κπ,t = κb,t+1 γb,b γb,πκb,t γb,C − κC,t = κb,t+1 γb,b γb,C
κb,t γb,µπ = κb,t+1 γb,b γb,µπ + κb,t+1 (γb,µπ + γb,π ) − κµπ ,t
κb,t γb,µC = κb,t+1 γb,b γb,µC + κb,t+1 (γb,µC + γb,C ) − κµC ,t .
The first two equations imply
κπ,t
κb,t − κb,t+1 γb,bκC,t.
γb,C =
κb,t − κb,t+1 γb,b
γb,π =

Then the third and fourth equations imply
κµπ ,t − κb,t+1 γb,πκb,t+1 (γb,b + 1) − κb,t
κµC ,t − κb,t+1 γb,C
γb,µC =
κb,t+1 (γb,b + 1) − κb,t
γb,µπ =

This gives a solution to the household problem.

D.2

Household problem ignoring the ZLB

We wanted to know what would happen if we ignored the ZLB. In that case, thenominal interest rate is set so thatR̃t =1b̃ = α π̃ πb̃t
+ α (π̃t − 1) ⇒ R
tβR̃Then κπ,t becomesκπ,t = C̃ + w̃Ñ R̃−1 απ̃and the rest of the analysis in the previous sub-section goes through.

74

D.3

Firm problem

The firm’s optimality condition is−ε(pf,t − wt ) (pf,t )


pf,t
pf,t
πt − 1πt (Ct + Gt )pf,t−1pf,t−1Ct Φpf,t+1
pf,t+1= βt Ef,t
πt+1 − 1
πt+1 (Ct+1 + Gt+1 )Ct+1 ε − 1
pf,t
pf,t

Φ
Yt +
ε−1



We log-linearize this condition for the case when βt = β and firms know the steadystate values of the variables to get
pbf,t +

ΦΦ(bpf,t − pbf,t−1 ) = β(bpf,t+1 − pbf,t )ε−1
ε−1

We assume a solution of the form
pbf,t = ωp,p pbt−1
so that



Φ 2
Φ
Φ
0=β
ωp,p − 1 + (1 + β)
ωp,p +ε−1
ε−1
ε−1

which has solutions

ωp,p =

Φ
1 + (1 + β) ε−1



±

q

Φ
1 + (1 + β) ε−1
Φ
2β ε−1

2

− 4β


Φ 2
ε−1

.

Only one of these solutions is less than 1 in absolute value and we use that solution
because it resembles our non-linear solution.
Now we will consider the case when βt = β̃. In this case,




Φ
p̃f,t
p̃f,t
(p̃f,t − w̃t ) (p̃f,t ) Ỹt +π̃t − 1π̃t C̃t + G̃t =ε − 1 p̃f,t−1p̃f,t−1p̃f,t+1p̃f,t+1C̃t Φpβ̃Ef,tπ̃t+1 − 1π̃t+1 C̃t+1 + G̃t+1
p̃f,t
p̃f,t
C̃t+1 ε − 1


C̃t Φ
pf,t+1
pf,t+1+ (1 − p) β̃Ef,t
πt+1 − 1
πt+1 (Ct+1 + Gt+1 )Ct+1 ε − 1
p̃f,t
p̃f,t
−ε

We log-linearize this to be75

Ỹ (1 + ε (w̃ − 1)) p̃ˆf,t + (1 − w̃) Ỹ Ỹˆt − w̃Ỹ w̃ˆt






Φ
Φ
ˆ
ˆ
ˆ
ˆ
ˆ
π̃ C̃ + G̃ (2π̃ − 1) p̃f,t − p̃f,t−1 + π̃t +(π̃ − 1) π̃ C̃ C̃t + G̃G̃t =
+
ε−1
ε−1



Φ
(π̃ − 1) π̃ C̃ + G̃ C̃ˆt − Ef,t C̃ˆt+1 +
pβ̃
ε−1




Φ
ˆt+1
π̃ C̃ + G̃ (2π̃ − 1) Ef,t p̃ˆf,t+1 − p̃ˆf,t + π̃
pβ̃
ε−1


Φ
ˆ
ˆ
(π̃ − 1) π̃ C̃Ef,t C̃t+1 + G̃Ef,t G̃t+1 +
+pβ̃
ε−1


C̃ Φ
(1 − p) β̃
(C + G) Ef,t p̂f,t+1 − p̃ˆf,t + π̂t+1
Cε−1
Using

b̃ =
Yb̃ t = NtΦ
1 + (π̃ − 1)2
2



C̃ b̃
Ct + Φ
Ñ

C̃ + G̃Ñ!
ˆt(π̃ − 1) π̃ π̃b̃b̃ + Cb̃ t = Nwtt
we can write the firm’s optimality condition as
ˆt − ζp ,t−1 p̃ˆf,t−1 =ζpf ,t p̃ˆf,t + ζπ,t π̃fζC,t C̃ˆt − ζµC ,t mC,t + ζpf ,t+1 Ef,t p̃ˆf,t+1 + ζµπ ,t mπ,t76

wherexΦπ̃ C̃ + G̃ (2π̃ − 1)
ζpf ,t =Ỹ (1 + ε (w̃ − 1)) +
ε−1


Φ
+ pβ̃
π̃ C̃ + G̃ (2π̃ − 1)
ε−1C̃ Φπ (C + G) (1 − ωpp )+ (1 − p) β̃
Cε−1


Φ
ζpf ,t−1 =π̃ C̃ + G̃ (2π̃ − 1)
ε−1


Φ
ζpf ,t+1 =pβ̃π̃ C̃ + G̃ (2π̃ − 1)
ε−1



Φ 
C̃ + G̃ (2π̃ − 1) π̃ + (1 − 2w̃) Φ C̃ + G̃ (π̃ − 1) π̃ζπ,t =ε−1Φ2
ζC,t = − (1 − 2w̃) 1 + (π̃ − 1) C̃ + w̃Ỹ
2


Φ
Φ
−
(π̃ − 1) π̃ C̃ + pβ̃(π̃ − 1) π̃ C̃ + G̃
ε−1
ε−1
Φ
(π̃ − 1) π̃ G̃ζµC ,t =pβ̃
ε−1


Φ
ζµπ ,t =pβ̃π̃ C̃ + G̃ (2π̃ − 1) .
ε−1
As in the household problem, the time subscripts on the ζ’s is to denote if the coefficient
multiplies, for example, pf,t or pf,t−1 . The time subscript does not indicate timevariation in the coefficient. For similar reasons to the solution to the household problem,
we consider solutions to this equation of the form
ˆt + γp,C C̃ˆt + γp,µπ mπ,t + γp,µC mC,t .
p̃ˆf,t = γp,p p̃ˆf,t−1 + γp,π π̃Note that γp,p is determined by2
ζpf ,t+1 γp,p− ζpf ,t γp,p + ζpf ,t−1 = 0

So,
γp,p =qζpf ,t ± ζp2f ,t − 4ζpf ,t+1 ζpf ,t−12ζpf ,t+1.

The smaller root (which is stable) is a better approximation like the nonlinear model.

77

We then have that
ζpf ,t γp,π + ζπ,t = ζpf ,t+1 γp,p γp,πζpf ,t γp,C − ζC,t = ζpf ,t+1 γp,p γp,C
ζpf ,t γp,µπ − ζµπ ,t = ζpf ,t+1 γp,p γp,µπ + ζpf ,t+1 (γp,π + γp,µπ )ζpf ,t γp,µC + ζµC ,t = ζpf ,t+1 γp,p γp,µC + ζpf ,t+1 (γp,C + γp,µC )So,ζπ,tζpf ,t − ζpf ,t+1 γp,p
ζC,tγp,C =
ζpf ,t − ζpf ,t+1 γp,p
ζµπ ,t + ζpf ,t+1 γp,π
γp,µπ =
ζpf ,t − ζpf ,t+1 γp,p − ζpf ,t+1
−ζµC ,t + ζpf ,t+1 γp,C
γp,µC =.
ζpf ,t − ζpf ,t+1 γp,p − ζpf ,t+1
γp,π = −Because Rt does not enter the firm optimality condition, ignoring the ZLB has no effecton the linearization of the firm problem.

78

D.4

Slow convergence in the linearized solution

Figure 10: Slow convergence of beliefs is similar in linearized and non-linear solutions(a) Linearized solution

Aggregate Inflation Expectations

02

t=10,000 (red dot)

4

Aggregate Consumption Expectations

0

REE
Learning
Elog(C /Css) × 100

Elog( ) × 400

2

t=100,000 (blue square)

2
4
6
8

6
0

4

8

12

16

20

0

4

8

12

16

20

(b) Non-linear solution

Aggregate Inflation Expectations

0

REE
Learning

0

Elog(C /Css) × 100

Elog( ) × 400

22
4

t=10,000 (red dot)

6

t=100,000 (blue square)
0

4

8

12

16

Aggregate Consumption Expectations

2
4
6
8

20

0

4

8

12

16

20

Note: In the sub-figures (a) and (b)mi is initially set to the steady state REE value. In all
sub-figures, ψi = 0.02, λi = 1, αi = 2. Source: Authors’ calculations.

79