Anchoring Inflation Expectations∗
Lawrence Christiano† and Yuta Takahashi‡
November 12, 2020

Abstract
We derive the restrictions on monetary policy which guarantee that inflation expectations
are anchored in a simple monetary model. The restrictions are stronger than what is required
for a globally unique rational expectations equilibrium. Loosely, the restrictions are capturedby the idea that policy should ‘lean against the wind, but not too aggressively’.

∗

We are grateful for conversations with Bernardino Adão, Marco Bassetto, Jeff Campbell, V.V. Chari, John
Cochrane, Martin Eichenbaum, Joao Guerreiro, Patrick Kehoe, Ellen McGrattan, Emi Nakamura, Armen Nurbekyan,
Matt Rognlie, Jón Steinsson, Marciano Siniscalchi and John Taylor. This draft replaces an earlier draft distributed
under the title, ‘Discouraging Deviant Behavior in Monetary Economics’.
†
Department of Economics, Northwestern University, l-christiano@northwestern.edu‡
Institute of Economic Research, Hitotsubashi University, yuta.takahashi@r.hit-u.ac.jp

1

Introduction

Economists have long debated the question of how to anchor inflation expectations. Much progresshas been made on this question since the introduction of rational expectations models in the 1970s.
This literature has much to say about the conditions under which monetary policy guarantees aunique rational expectations equilibrium. But, it was soon understood that rational expectations
leaves unanswered the key question of how, or even whether, non-cooperative agents would coordinate
their beliefs on a rational expectations equilibrium (see, for example, Lucas (1978, section 6) and
Guesnerie (1992)). We derive sufficient conditions for expectations to be anchored in a model that is
globally tractable, and yet has the flavor of conventional monetary models used to analyze monetary
policy.1 We also derive necessary conditions for expectations to be anchored using local approximation
methods that can be applied very generally. The analysis suggests a simple economic principle toguide policymakers seeking to anchor the public’s expectations to a particular target.
The focus of our analysis is on the monetary policy advocated by John Taylor in a series of papers.
We refer to this policy as the Taylor strategy. The best known part of that strategy is the Taylor
rule. According to this rule, policymakers move the market interest rate more than one-for-one in
response to an increase in inflation. Taylor (1996) argues that this policy can work well in normal
times,2 but points out that there are times when the rule might lead the economy awry. For example,
that could occur if a rise in inflation leads to a rise in the interest rate which then creates, by a
Fisher-type channel, expectations of even higher inflation. In this case, policymakers applying the
Taylor rule could inadvertently become part of an inflation spiral. Taylor (1996) notes that theconverse, a deflation spiral, could happen too. With these considerations in mind, Taylor (1996, page37) suggests what we call the Taylor strategy: follow the Taylor rule under normal circumstances
but if inflation gets out of hand, invoke an escape clause and switch to a money growth rule to get
inflation under control.3
There exist studies that lend support to the Taylor strategy, by showing that it can produce a
globally unique rational expectations equilibrium in which inflation corresponds to the policymaker’s
target.4 But, as noted above, existence and uniqueness of an equilibrium is no guarantee that agentswill coordinate on that equilibrium. That is, it does not ensure that expectations are anchored.
The literature also suggests that a monetary growth rule is not necessarily the right rule to switch
to in the event that things go wrong in the Taylor rule regime.5 For example, one could also consider
switching to an Obstfeld and Rogoff (1983; 2017)-type strategy or a variant on the fiscal theory of
1

The competitive equilibrium of the model is observationally equivalent to the models in Benhabib et al. (2001a),
Woodford (2003, chapter 2) and Cochrane (2011).
2
The model-based case that the Taylor rule works well can be made in the New Keynesian model with price-setting
frictions and shocks. For additional discussion, see Section 2.2.2 below.
3
In Taylor’s words, “...I would argue that interest rate rules need to be supplemented by money supply rules incases of either extended deflation or hyperinflation.”
4
See Benhabib et al. (2002) and Christiano and Rostagno (2001). These papers build on the work of Obstfeld andRogoff (1983). For more recent work, see Atkeson et al. (2010).
5
In many models equilibrium is not unique under a money growth rule. In these models, switching to money growth
rule would not anchor expectations.

1the price level (see, e.g., Leeper (1991)). Still, the simplicity and familiarity of a money growth rule
is convenient for illustrating principles which guarantee that inflation expectations are anchored.6
To understand how expectations come to be anchored, we obviously must drop the assumptionthat agents coordinate on the rational expectations beliefs. We instead assume that each agent isrational, understands the environment, and that these are common knowledge among agents (CK).
Agents form their beliefs independent of each other by a private, logical reasoning process. To model
the formation of beliefs by rational agents with CK, we exploit the mapping between a competitive
equilibrium and a Nash equilibrium of a non-cooperative game.7 This mapping puts us in a position
to answer our anchoring question by making use of the seminal game-theoretic papers of Bernheim(1984) and Pearce (1984). They argue that in forming their beliefs, rational agents with CK only
entertain beliefs that they deem ‘reasonable’. By reasonable, Bernheim (1984) and Pearce (1984)
mean that the belief is rationalizable. So, we ask what restrictions are required for the Taylor
strategy to ensure that policymakers’ desired rate of inflation is the unique rationalizable belief.
When those restrictions are satisfied, we say that policymakers have successfully managed to anchor
agents’ expectations.
A key principle that determines unique rationalizability is that policy gives rise to a reduced
form which features what we call ‘leaning against the wind’. In our context, the word ‘lean’ andits connotations are crucial. The word literally means to ‘to push back’, and, accordingly, we findthat having a coefficient on inflation bigger than unity in the Taylor rule is an important part ofsuccessful policy design. However, the word, ‘lean’, also carries the connotation that the push-backshould be more like a gentle nudge and not be overly aggressive. Thus, we show that to anchor
inflation expectations the coefficient on inflation in the Taylor rule should also not be too big. In
addition, it is important that when policy switches to the money growth rule in case the escape clauseis activated, the resulting recession must not be too deep. So, for the Taylor strategy to successfully
anchor expectations, both the interest rate rule and money growth rule must be consistent with theprinciple of leaning against the wind, but not too aggressively.8
The outline of the paper is as follows. Section 2 provides a simple example that summarizes
the results in our dynamic model. In addition, this section discusses the relationship between ourpaper and the literature. As part of that discussion, we derive necessary conditions for anchoring
expectations in a simple New Keynesian model. Our methods for evaluating necessary conditions canbe generalized to empirically relevant models. For space reasons, the technical details are relegated6

As is standard in the recent literature on monetary policy rules, we assume that monetary policy is active, fiscal
policy is passive and the government never reneges on its policy. By ‘active’ and ‘passive’ we have in mind the ideas
in Leeper (1991).
7
For discussions of this mapping, see Bernheim (section 7 (b), 1984), Guesnerie (1992), Evans and Guesnerie (1993),
Guesnerie (2002), Bassetto (2002; 2005), Evans and Guesnerie (2005), Atkeson et al. (2010), as well as the papers thatthey cite.
8
The desirability of ‘leaning against the wind’ is well known in the macroeconomics literature and, indeed, it isstressed by Taylor (1996). It is also known, though by a different name, in the game-theoretic research that inspires
the analysis in this paper. What we mean by ‘leaning against the wind, but not too aggressively’ corresponds to theidea behind Guesnerie (2002, page 456)’s conclusion, ‘Coordination is favored whenever agents’ actions are not too
responsive to expectations’.

2to Online Appendix E. Section 3 introduces the dynamic model and shows that the competitive
equilibrium is globally unique under the Taylor strategy. This result is already known, but weinclude it here for completeness and set the stage for the later analysis. Section 4 reformulates thecompetitive equilibrium as the Nash equilibrium of a particular game. Section 5 provides our main
results, by deriving sufficient conditions for expectations to be anchored. Section 6 introduces two
shocks into the analysis, a money demand shock and trembles. Each allows us to show, in different
ways, the superiority of the Taylor strategy to alternatives encountered in our analysis. A brief
conclusion appears at the end.

2

Static Example and Literature Review

In this section we consider a drastically simplified, static version of our model which illustrates all
the important aspects of our analysis and allows us to clarify the relationship between our work andthe literature. We use the static model to show how (i) we can map the competitive equilibriumof a market economy into the Nash equilibrium of a particular game; (ii) the game can be used toformalize how agents form beliefs; and (iii) the game allows us to identify a principle (‘leaning against
the wind, but not too aggressively’) which guarantees that the belief of each agent about inflation
is anchored to the target chosen by the policy maker. To make the discussion as simple as possible,
many of the economic relations are expressed in reduced form. In the main analysis, beginning in
the next section, all features of the model are made explicit.

2.1

ExampleEach of a continuum of intermediate good firms, i ∈ [0, 1], sets its price, pi , simultaneously andwithout communicating with the others. We assume that the optimal decision by the ith firm is toset pi = W, where W is the nominal marginal cost of production. The aggregate price index, P , isthe average over each price setter’s decision:
P =

Z 1

pi di.

(1)0Note that we can express W as P × w, where w is real marginal cost. Resources are scarce, in thesense that w is increasing in the level of aggregate economic activity. We assume that the government
can influence the level of aggregate activity by changing the nominal rate of interest, R. High R slows
down economic activity (say, by reducing demand), which in turn reduces w. Monetary policy sets
R as an increasing function of inflation, so that (for a given level of the lagged aggregate price level),
R = f (P ) . Here, f is an increasing function. In this way, we can express the nominal wage rate as
follows:
W = P w (f (P )) .
(2)

3

In a competitive equilibrium each firm faces the same wage, so equation (1) implies pi = P foreach i. A competitive equilibrium is a P ∗ that satisfies P ∗ = P ∗ w (f (P ∗ )), or,w (f (P ∗ )) = 1.

(3)

We assume that w (·) and f (·) imply that the competitive equilibrium, P ∗ , is unique.
The question of how agents form their beliefs lies at the heart of the problem of how to design
monetary policy, f, to ensure that each agents’ belief is anchored at the policy maker’s target.
This question is sidestepped in equilibrium theory by the simple and unexplained assumption called‘clairvoyance’ by Guesnerie (2002): agents believe prices take on their competitive equilibrium values.
For many purposes, this assumption about beliefs may be a powerful and useful shortcut. But, forour purposes clairvoyance assumes away the problem of interest. By transforming the environment
into a game we are able to exploit the formal tools developed in epistemic game theory for thinking
about how agents form their beliefs.
The ith firm without clairvoyance finds itself in a conundrum. Note from equation (2) that thenominal wage is itself a function of P, which is the aggregate of the prices set by the other firms(see equation (1)). As a result, the ith firm cannot actually see P (hence, W ) until after firms have
set their price. So, the ith firm must set its price based on a belief, pbi , about the prices being set
simultaneously by the other firms.9 We maintain the spirit of rational expectations in that we assume
each firm understands the functions w (·) and f (·), knows that other firms face a symmetric problem,and understands that this knowledge is common knowledge (CK).
Conditional on a candidate belief, pbi , the ith firm computes the continuation equilibrium in which
the nominal wage is pbi w f pbi .10 So, the best response for the ith firm with belief, pbi , ispi = pbi w f pbi≡ F pbi ; f ,(4)where the notation is designed to highlight the dependence of F on monetary policy, f . A Nashequilibrium is a P ∗ such that P ∗ = F (P ∗ ; f ). Since any such P ∗ satisfies equation (3), we have in
effect transformed the competitive equilibrium into the Nash equilibrium of a large game in which
each player’s best response function is given by F . For the purpose of the example, we assume that
F is linear.
In forming its belief, pbi , the ith firm is led to consider a range of possible values of pbi . If the slopeof f is sufficiently small then F is increasing and crosses the 45◦ line from above, as in Figure 1a.11
9
In this example, it is immaterial whether (i) pbi corresponds to a belief about what each other firm will do; or (ii)bpi is simply a belief about the average decision of the other firms. In our dynamic model (i) and (ii) are very different

and we adopt assumption (i).
10
The aggregate price index in the continuation equilibrium is P = pbi .
11This diagram bears a resemblance to the one associated with the seminal work of Barro and Gordon (1983). It
is indeed very similar, but recall that we assume policymakers have the ability to commit to their policy strategy,
while Barro and Gordon (1983) assume they do not have the ability to commit. Also, the object on the vertical axisof Figure 1a corresponds to the decision of an atomistic agent, while in Barro and Gordon (1983) the object on thevertical axis pertains to the aggregate price level. In addition, the graph in Figure 1a implicitly assumes that price4

The anchoring question is whether each firm, i ∈ [0, 1] , will independently and without coordination
arrive at the same belief, pbi = P ∗ . If the answer is ‘yes’, we say that expectations are anchored.
In forming its belief, pbi , will the ith agent choose the Nash equilibrium belief, or some other belief?
One justification for the Nash equilibrium belief is that it is the unique belief, pbi , such that if everyone
held that belief then no firm would ex post regret its action, pj = P ∗ . The Nash equilibrium belief
has the property that everyone has perfect foresight (‘clairvoyance’). But, ex post considerations areirrelevant for justifying a Nash equilibrium because beliefs must be formed ex ante.12 If a firm believed
pbi = 1.5 ex ante then that firm would expect to feel regret if it did not choose pi = F (1.5; f ) < P ∗ .
So, although each agent knows that P ∗ is the unique Nash equilibrium, there is no prima facie reason
to think that agents would form the belief, pbi = P ∗ . For the Nash belief (or, rational expectations
equilibrium) to occur requires that f be properly designed.
We follow the seminal work of Bernheim (1984) and Pearce (1984) in supposing that firms will
only entertain beliefs that they deem to be rationalizable. To understand this approach to belief
formation, consider the best response function, F, displayed in Figure 1a. Suppose firms only consider
beliefs in the compact interval, A. In this example the exact boundaries of A are irrelevant for theoutcome of the analysis, except that A must include the Nash equilibrium, pbi = P ∗ . Because F isonly a little flatter than the 45◦ line, monetary policy leans against the wind very slightly. The ithintermediate good firm would never choose pi ∈ A/B (see the set B on the vertical axis in Figure1a).13 Understanding that the other agents are in the same situation, the ith firm deletes the beliefs,
pbi ∈ A/B from further consideration. No rational agent would believe that its opponent would takean action that is under no circumstances a best response. Using the 45◦ line to map B onto thehorizontal axis, we see that B is strictly interior to A. The ith firm applies the same reasoning to
the set of beliefs, B, and is driven to delete additional beliefs from further consideration. It is easily
verified that, given the shape of F, this process leads to deleting all beliefs, except the Nash belief.
So, in this case beliefs are uniquely anchored at pbi = P ∗ for each i ∈ [0, 1].
The reason inflation expectations are anchored is: (a) monetary policy leans against inflation and(b) policy does so in a ‘non-aggressive’ way, in the sense that the range of best responses is smaller
than any given range of beliefs. To explore (b), consider now the case where policy, f, leans against
inflation more aggressively. Suppose the slope of f is so large that F is downward-sloping (see Figure1b), but flatter than the 135◦ line (the dashed line).14 If we ask, ‘what beliefs are rationalizable’
and we apply the same iterated deletion argument as before, we conclude once again that the only
rationalizable belief is the Nash equilibrium. However, if monetary policy is even more aggressive,
so that F is steeper than the 135◦ line, then (b) is violated. In this case all beliefs are rationalizablesetting is characterized by strategic complementarity. Here, the nature of monetary policy plays a central role indetermining whether price setting is characterized by complementarity or substitutability.
12This is a one-period model, so each firm is in a ‘first play’ situation. There is no history or ‘norm’ to coordinate
on. The first play perspective is squarely in the spirit of rational expectations because those models are used to explore
the consequences of policy strategies that differ from the ones currently in place.
13
Here the set A/B ≡ {x : x ∈ A, x 6= B} .
14
Here, we ignore any zero lower bound (ZLB) on the nominal rate of interest. In our dynamic example we do take
into account the ZLB.

5

Figure 1: Anchored Expectations(a) Mild Push Back(b) More Aggressive Push Back

pi

◦

45

pi

pi



 b ;f
pi

=F

13
5◦

B
P∗ = 1P∗ = 1B
0.5

P∗ = 1pi = F  b pi ; f

pbi

pbi

A

P∗ = 1

1.5

and inflation expectations are not anchored.
Our conclusion is that for expectations to be anchored two conditions are sufficient: the best
response function should cross the 45◦ line only once, so that there is a unique Nash equilibrium;
and the best response function lies inside the butterfly-shaped diagram formed by the area between
the 45◦ and 135◦ degree lines.15 The second condition is our formalization of the idea that policy‘leans against the wind, but not too aggressively’.16 For example, in the model described in the next
section the coefficient, φ, on inflation in the Taylor rule must be greater than unity to help guarantee
a unique equilibrium. But, if φ is too large then policy is too aggressive and inflation expectations
become unanchored despite the uniqueness of the competitive equilibrium.17 In this case, there is no
reason to expect that the coordination assumed by competitive equilibrium will occur.
An important input into the above analysis is that F pbi ; f is well-defined over a reasonable rangeof values of pbi . If the best response functions in the Nash equilibrium representation of a competitive
equilibrium are well defined, then we say that the competitive equilibrium is a strategy equilibrium.
As is evident from the example, for inflation expectations to be anchored at pbi = P ∗ for each i ∈ [0, 1]
it is necessary, but not sufficient, that the equilibrium be a strategy equilibrium.
The main finding of the paper is that for inflation expectations to be anchored, it is sufficientthat monetary policy is consistent with the ‘leaning again the wind, but not too aggressively’ prin15

For a precise statement of the ‘butterfly’, see section 5.3. That section shows that being inside the butterfly
diagram is sufficient for unique rationalizability. It also discusses necessity (see Proposition 10).
16
Condition (iii) of Proposition 9 below provides a mathematical statement of the proposition that a sufficientcondition for unique rationalizability requires that F lie inside the butterfly diagram. Lying inside the butterfly diagram
is what we mean by ‘leaning against the wind, but not too aggressively’. The discussion immediately after Proposition
9 explains that unique rationalizability depends the graph of F lying inside the butterfly, for some transformation of
the domain and range of F . Although the upper and lower bounds that we placed on beliefs in our example played
a role in the argument in the text, the precise value of those bounds does not affect the conclusion that pbi = 1 is the
only rationalizable belief. The bounds do raise technical issues, which we address in Section 5.2. This type of boundissue is also considered in Kocherlakota (2018) and we relate our analysis to his in Footnote 76.
17
A detailed discussion appears in section 5.5.

6

ciple. This principle guarantees that atomistic and non-cooperative agents arrive at the same beliefs
independently, and that the desired rational expectations equilibrium is the unique outcome.

2.2

Relation to the Literature

2.2.1

Strategy Equilibrium

Many previous authors have stressed that for policy to be successful, it is necessary that the equilibrium be what we call a strategy equilibrium. Diamond and Dybvig (1983) make this point in caseof bank runs.18 Bassetto (2005) stresses the importance of strategy equilibrium for implementationof Ramsey optimal allocations. Atkeson et al. (2010) and Cochrane (2011) consider the issue in anenvironment very similar to ours.19 Notably, Cochrane (2011) presents a model in which the Taylor
strategy produces a unique equilibrium, but it is not a strategy equilibrium. He correctly concludes
that the Taylor strategy does not anchor inflation expectations in his model. However, we show that
his conclusion is valid only because he assumes an endowment economy, in which the real interest
rate must be the same on and off equilibrium paths. In our model the real interest rate can vary on
out-of-equilibrium paths and this plays a key role to ensure that the unique equilibrium under the
Taylor strategy in our model is a strategy equilibrium.
2.2.2

Rationalizability, Implementation and Leaning Against the Wind

The papers summarized in the previous subsection do not discuss rationalizability. But, a series ofother papers has introduced rationalizability into macroeconomic analysis.20 Those papers clearlystate the insight that a relatively flat best response function helps to promote unique rationalizability.21 However, the papers do not apply the insight to the policy design problem, as we do here.
Our analysis is closely related to the literature on policy implementation. Loosely, we say that apolicy uniquely implements a desired equilibrium if the following three conditions are satisfied: (i) theunique (Nash) equilibrium is the desired equilibrium; (ii) the equilibrium is a strategy equilibrium;and (iii) the beliefs associated with the unique equilibrium are uniquely rationalizable.22 If these
conditions are satisfied then inflation expectations (as well as expectations about other variables) are18

Diamond and Dybvig (1983, section V) make a powerful case that the desired equilibrium must be a strategy
equilibrium. They argue that although it is formally true that deposit insurance rules out the bank run equilibrium,
it is important to study the off-equilibrium path in which a bank run occurs, to verify that the deposit insurance fundwould have enough resources in that case. One supposes that if the no-run equilibrium were not a strategy equilibriumthen agents would in fact not adopt the equilibrium belief that their money is safe in the bank.
19
Bassetto (2005) refers to what we call a strategy equilibrium as a Schelling timing equilibrium, or a sequential
equilibrium. Atkeson et al. (2010) call it a sophisticated equilibrium.
20
See Guesnerie (1992), Evans and Guesnerie (1993), Guesnerie (2002), and Evans and Guesnerie (2005). For areview, see Desgranges (2014).
21
Recall in particular the reference to Guesnerie (2002) in Footnote 8.
22
A formal definition of our implementation concept is provided in Definition 10. In the body of the paper, wefollow Atkeson et al. (2010) in working with one-shot sequential game representations of our dynamic equilibrium. We
extend the one-shot concept of implementation to multiple shots in Definition 11 in the Online Appendix.

7

anchored. Our approach to implementation differs from existing approaches in the macroeconomics
literature.
There exist at least two approaches to implementation in macroeconomics. One approach focuseson (i) only, by designing policy so that the rational expectations equilibrium is unique and has
desirable characteristics.23 The second approach requires (ii) in addition to (i). This is the approachadvocated in Bassetto (2005) and Atkeson et al. (2010). But, the two approaches to implementation
just described do not answer our anchoring question: ‘what is required for policy to ensure thatagents uniquely coordinate on the equilibrium beliefs?’. This question is answered by our approachto implementation. Our approach to implementation requires condition (iii), in addition to (i) and(ii).
In addition to anchoring expectations, condition (iii) has another feature that is of interest. In
particular, condition (iii) provides a formal foundation for ruling out policies that seem intuitively
peculiar, even though they satisfy (i) and (ii). We provide two examples. In the first example, highexpected inflation is ruled out by a commitment to make actual inflation even higher (see section
5.5). In terms of Figure 1a this corresponds to a scenario in which the best response function is
steeper than the 45o line. Intuitively, one might expect such an odd dynamic to manifest itself in theform of some kind of expectational instability, but (i) and (ii) offer no way to contemplate such a
possibility. In contrast, in this example the unique equilibrium fails to be uniquely rationalizable (i.e.,(iii) fails) precisely because an expectational instability prevents iterated deletion from eliminating
all but the competitive equilibrium expectations. Our second example is the opposite of the first
because that example involves a best response function with slope steeper than minus unity.24 The
example is a simple version of the New Keynesian model with Calvo-style price-setting frictions andtwo shocks. A known property of that model is that for a sufficiently large value of the coefficient
on inflation, φ, in the Taylor rule, the locally unique equilibrium allocations approach the first-best
allocations.25 Intuitively, it may seem peculiar that a central bank commitment to respond to inflation
with arbitrarily large force could move the economy arbitrarily close to first-best. One might expectthat some form of instability would occur instead. But, there is no room to consider any kind of
instability in the implementation concepts, (i) and (ii). Under condition (iii), a form of instability
indeed does occur when φ is too large.26 A policymaker contemplating high values of φ would expect
good results under (i) and (ii) alone. But, a policymaker that incorporates (iii) when designingpolicy would realize that when φ is large enough, expectations become unanchored and control overthe economy is lost.
23

For examples of this approach see what Atkeson et al. (2010, p. 49) refer to as ‘unsophisticated implementation’.
In terms of Figure 1b the best response cuts the 135o line from above.
25
The fact that, in equilibrium, a sufficiently high value of φ can close the gap between actual and first best output
reflects in part that the model incorporates a tax subsidy that neutralizes a monopoly distortion in non-stochastic
steady state (for completeness, all results for the New Keynesian model are proved in Online Appendix E). Our analysisof this model assumes that exists an escape policy which ensures that the locally unique equilibrium is also globallyunique.
26
For details, see Online Appendix E.
2482.2.3

Coordination

From a broader perspective, our analysis is part of an effort that began well before the rational
expectations revolution. Walras and Marshall in the 19th century also struggled to understand how
markets coordinate on equilibrium prices. The work in recent decades on how agents manage to coordinate their beliefs is loosely divided by Binmore (1987) into two approaches: the eductive approachand evolutive approach. The former imagines that agents undergo a private, mental reasoning process
in a pre-period to arrive at a set of beliefs. The eductive approach was illustrated in the example
above. The second, evolutive, approach assumes that agents form their beliefs based on observing
data as they accumulate over time. No doubt, the correct approach is a combination of the two.27
More recently, Evans et al. (2018) make a convincing case that realistic models cannot completely
avoid the evolutive approach.
As noted above, the standard rational expectations approach in macroeconomics sidesteps the coordination question altogether by simply assuming that coordination occurs (e.g., Guesnerie (2002)’s‘clairvoyance’).28 As we explain below, tractability drives us to use a combination of the eductive
approach and clairvoyance. We take a step away from full rational expectations towards eduction(rather than the evolutive approach) because it is closest in spirit to the rational expectations framework used in the analysis of monetary policy rules.
2.2.4

Escape Clauses

Finally, our analysis is related to the general topic of escape clauses. Escape clauses allow governments
to deviate from normal policy (which might work well in normal times) to an alternative policy in casethings go wrong. An escape clause may work well by anchoring the normal equilibrium, even if thepolicy under the escape clause is not very efficient. For example, in the last section of the paper we
introduce money demand shocks, so that a money growth rule is not efficient for the classic reasons
spelled out in Poole (1970). Still the commitment to deviate to that money growth rule in the eventthat an inflation monitoring range is violated (as long as it is designed properly), may ensure thatthe conditions that trigger the escape clause never happen in the first place, as in the Diamond andDybvig (1983) analysis of deposit insurance. Another example is the ‘suspension clause’ in the UnitedStates Constitution (Article I, Section 9, Clause 2), which gives the United States Congress the rightto suspend habeas corpus ‘...in cases of rebellion or invasion [when] the public safety may requireit.’ An example that applies to monetary policy is the ‘unusual and exigent circumstances’ clause,
Section 13.3, in the Federal Reserve Act, used to rationalize the use of unconventional monetary
policy in the wake of the 2008 Financial Crisis and during the current COVID-19 crisis. The policy
to which the government deviates under the escape clause might not be optimal in normal times, but
27

For example, Binmore (1987, p. 185) states, “It is not denied that the middle ground between these extremes ismore interesting than either extreme.”
28
This approach is also dominant in practical applications of game theory. It is common to assume that agents
simply know (without further explanation) that everyone else will play the Nash strategy.

9the mere existence of the escape clause may reduce chance that undesired equilibria form in the firstplace.

3

Competitive Equilibrium

Our monetary model is designed to allow us to make the points summarized in the previous section,and to facilitate comparability with the related monetary policy literature (see especially Cochrane(2011) and Atkeson et al. (2010)). Versions of the model have been studied extensively in a series of
papers and the variant we consider is closest to the model used in Christiano and Rostagno (2001).29

3.1

Households

The economy is composed of a large number of identical households. The representative household
solves the following problem:max{ct ,lt ,mt ,dt }∞
t=0

∞
X
t=0

βt

"

#
1+ψ
c1−γ
l
t
− t
,
1−γ 1+ψ

γ > 0, γ 6= 1, ψ ≥ 0s.t. mt = R̄t (Xt + dt ) + mt−1 − dt + Wt lt − Pt ct + Tt ,Pt ct ≤ mt−1 − dt + Wt lt

(5)
(6)
(7)

m−1 given,
where ct ≥ 0 and lt ≥ 0 denote consumption and employment, respectively. Also, mt−1 denotes thequantity of currency held by the household at the beginning of period t. The right side of equation(6) represents the household’s period t sources of currency. The first source of currency takes theform of interest and principle, R̄t (Xt + dt ) , received on bank deposits, Xt + dt , at the end of periodt after goods markets have closed. Here, Xt denotes a government transfer of currency directly into
the household’s bank deposit at the start of period t. Also, 0 ≤ dt denotes currency deposited by
the household into a bank at the start of period t and R̄t denotes the gross nominal rate of interest.
The currency that the household carries over from period t − 1 is denoted by mt−1 , so that mt−1 − dt
represents the currency that the household chooses to keep on hand at the start of period t. Other
currency available to the household at the start of period t is the wage bill, Wt lt , which it receives
before work begins. The household spends currency, Pt ct , in the goods market, subject to the cash
constraint, (7). Finally, after the goods market closes, the household receives Tt , which denotes firm
profits net of government taxes.30
29

The model was originally constructed by Fuerst (1992), which in turn was inspired by Lucas (1990). See also
Christiano (1990), Christiano and Eichenbaum (1992), Chari et al. (1995), Christiano and Eichenbaum (1995) andChristiano et al. (1997).
30
Nothing substantial depends on our assumption, γ 6= 1. We discuss this case later.

10

According to equation (6) and equation (7), the trade off between consumption and leisure offeredto the household by the market is a function of the real wage, Wt /Pt . Similarly, the tradeoff between
consumption in different periods is determined by the real rate of interest, R̄t /π̄t+1 .
It is easily verified that under these conditions, the first order conditions associated with household
employment, deposits and the cash constraint are, respectively,Wt= cγt ltψ ,
Pt

(8)

R̄t
,
π̄t+1

0 = R̄t − 1 (mt−1 − dt + Wt lt − Pt ct ) .

c−γ= βc−γ
t
t+1

(9)
(10)

In equation (10), each expression in braces is non-negative. The transversality condition is:lim qj mj = 0,

(11)

j→∞

−1
Q
j−1
R̄
. In Online Appendix A.5, we show that, under thewhere qj is 1 for j = 0 ands=0 s
boundedness conditions, the first order conditions and transversality condition are necessary andsufficient for household optimization.

3.2

Production

A final output good is produced by a competitive, representative firm using the CES production
function with the elasticity of substitution κ > 1 :
Yt =

Z 1

(κ−1)/κ
diYi,t0κ/(κ−1).

Let pi,t denote the price of intermediate good, Yi,t , i ∈ [0, 1]. The firm takes the price of output, Pt ,and the prices of the inputs, pi,t , i ∈ [0, 1] as given. The first order conditions associated with the ith
firm’s profit maximization problem is:
Yi,t = Yt



Pt
pi,t

κ

.

(12)

The first order conditions, together with the production function, impose a restriction across theaggregate price index and the price of intermediate goods:
Pt =

Z 1

1−κ
pi,t
di

0

11

1
 1−κ

.

(13)

The ith intermediate good, Yi,t , is produced by a monopolist with the following production function,
Yi,t = li,t . Here, li,t denotes the labor input employed by the ith firm.
We assume that the ith intermediate good firm must pay its period t wage bill, Wt li,t , at the
beginning of the period, before it has obtained its period t receipts. The firm borrows Wt li,t from thebank and must repay R̄t Wt li,t after the goods market closes. At that time, the government provides
a subsidy, τt , to the firm so that its post-subsidy end-of-period wage bill is (1 − τt ) R̄t Wt li,t . Since
the firm wants to borrow an infinite amount when R̄t < 1, equilibrium requires R̄t ≥ 1. The ith firm
maximizes end-of-period profits by setting its price as a markup over marginal cost, (1 − τt ) R̄t Wt :pi,t =

κ(1 − τt ) R̄t Wt = Wt ,
κ−1

(14)

for each i. The second equality in equation (14) reflects our assumption that τt is selected to neutralize
the interest rate and monopoly power distortions in the model. We neutralize the monopoly power
distortion to streamline notation and we neutralize the interest distortion to ensure that Cochrane(2011)’s model is a special case of ours (see below).31 Because the marginal cost for i is the same forall the other firms, all intermediate good firms set the same price, pi,t . As a result, by equation (13)the intermediate good firm equilibrium condition is:
P t = Wt .

3.3

(15)

Government Policy

The government’s monetary transfer, Xt , to households at the start of t satisfies
Xt = (µ̄t − 1) M̄t−1 ,µ̄t = M̄t /M̄t−1 .

(16)Here, M̄t denotes the end-of-period t per capita stock of money and µ̄t denotes the gross money
growth rate. Monetary policy selects a sequence, {µ̄t }∞t=0 , so that, in equilibrium, π̄ φ 
Pt+1tR̄t = max 1, R̄∗ ∗, π̄t+1 ≡, R̄∗ ≡ π̄ ∗ /β,
(17)
π̄
Pt

where π̄ ∗ ≥ 1 and R̄∗ are the desired inflation and interest rate. Here, we assume that φ > 1. In
addition, when the lower bound on R̄t is binding, we assume that the government sets M̄t = Pt ct ,where ct denotes aggregate consumption.
In addition, the government levies a lump sum tax, Ttg , on households to finance the subsidy inequation (14) to firms. Thus, Ttg = τt R̄t Wt lt ,where τt = 1 − (κ − 1) /κR̄t−1 . The government does notpurchase any goods, or issue debt.32
31

See Christiano and Rostagno (2001) for an extended analysis of the case, (1 − τt ) κ/ (κ − 1) = 1.
A number of interesting issues concerning fiscal policy are left out of the analysis. For example, a property of
our model is that a non-negative money growth rate rules out a zero interest rate equilibrium. In the presence ofgovernment debt, this result is not necessarily true. For further discussion and a defense of the position taken here,
see Christiano and Rostagno (2001, Section 2.4).
32

12

3.4

Market Clearing and Equilibrium

The goods, labor, money and loan market clearing conditions are:ct = Y t ,Z 1

li,t = lt ,M̄t = mt ,Xt + dt = Wt lt ,

(18)

0

for t ≥ 0, respectively. Also, household profits net of taxes are Tt =Ttg . Equations (12) and (14) imply, using equation (18):


R 
p
Y
−
R̄
W
(1
−
τ
)
l
di−
i,t
i,t
t
t
t
i,t
i

ct = lt .

(19)

To ensure that households are identical, we require m−1 = M̄−1 .
Let the time t variables be denoted by:āt = lt , ct , Pt , R̄t , Wt , µ̄t , M̄t , mt , dt .

(20)

To simplify notation, we delete reference to pi,t and li,t because these are simply equal to Pt and lt ,respectively, for each i ∈ [0, 1]. We define a competitive equilibrium that starts at date 0 as follows:33
Definition 1. A competitive equilibrium under the Taylor rule is a sequence, (āt )∞t=0 , that satisfies,
for t ≥ 0, (i) intermediate good firm optimality; (ii) final good firm optimality; (iii) household
optimization, conditional on m−1 ; (iv) government policy; and (v) market clearing.
For purposes of equilibrium characterization, it is useful to note that condition (i) corresponds
to equation (14); condition (ii) corresponds to equation (12) and equation (13); condition (iii) corresponds to equation(8) - equation (11); condition (iv) corresponds to equation (16) - equation (17);and condition (v) corresponds to equation (18).
We define competitive equilibrium under alternative monetary policies by obvious adjustments to
Definition 1. For example, we can define a competitive equilibrium under constant money growth, by
replacing equation (16) - equation (17) with a constant value for µ̄t . Similarly, we can define a competitive equilibrium under the Taylor strategy. The Taylor strategy was discussed in the introduction,
and will be defined formally in Section 3.6 below.

3.5

Properties of Competitive Equilibrium Under the Taylor Rule

We now obtain a dynamic equation that can be used to identify all the equilibria in our model. It
turns out that the analysis of equilibrium is greatly simplified by scaling and logging the variables.
Later, when we discuss how rational agents coordinate their expectations, the log transform will also
be very convenient.34
33

To shorten the definition, we refer to our equilibrium concept as a competitive equilibrium. We do this even thoughsome of the agents in our model are not competitive.
34
See the discussion of Proposition 9 below.

13

Combining equation (15) and equation (18), we conclude that for all t ≥ 0, ct = lt = 1. As aresult, the intertemporal Euler equation reduces to the Fisher equation:
Rt = πt+1 ,(21)

where Rt = ln R̄t /R̄∗ , πt+1 = ln (π̄t+1 /π̄ ∗ ) . Also, µ̄∗ = π̄ ∗ is the money growth level in the desired
equilibrium. We can also express the Taylor rule in scaled and logged form:



∗



Rt = max {Rl , φπt } , M̄t = Pt ct , if Rt = Rl ,(22)

where Rl ≡ ln 1/R and ct denotes aggregate consumption. It is also useful to define the scaled
and logged money growth rate:
 
µ̄t
.
(23)
µt ≡ lnµ̄∗
Combining equation (21) and equation (22) we obtain the following first order difference equation:
πt+1 = max {Rl , φπt } .

(24)Combining the loan market clearing condition, equation (18), the household Kuhn-Tucker condition,
equation (10), and monetary policy in the zero lower bound, we have the equilibrium cash condition:M̄t = Pt ct .

(25)

Also, the transversality condition, taking into account the equilibrium conditions, (18) and the constant consumption level, corresponds to:lim β T

T →∞M̄T −1
= 0.
PT

(26)

Equation (24) is useful for studying the equilibria in the model for the following reason:
Proposition 1. For any sequence, (πt )∞t=0 that satisfies the difference equation, (24), it is possible
to construct the other equilibrium variables in such a way that all the conditions for a competitive
equilibrium are satisfied.
That a model which has an equilibrium characterization of the form in equation (24) has multiple
equilibria, is well-known. For completeness, we include a proof in Online Appendix A.4.35
Figure 2, a variant on the well-known figure in Benhabib et al. (2001b, Fig. 1), depicts equation(24). From this figure it is easy to see that the model has many equilibria, each indexed by thevalue of π0 . The desired equilibrium corresponds to πt = 0 for t ≥ 0. Consider the two inflation
35

Other work that studies models with the equilibrium characterization, equation (24), include Benhabib et al.
(2001a), Woodford (2003, chapter 2) and Cochrane (2011). The model underlying these characterizations is anendowment economy, while ours is not. This distinction plays an important role in our analysis.

14rates marked in Figure 2 by πl and πu . We refer to this interval as the inflation monitoring range.
Note that, due to the high value of φ, there is exactly one equilibrium, the desired equilibrium, inwhich inflation is always in the monitoring range, πt ∈ [πl , πu ] for t ≥ 0. This observation plays animportant role in the analysis below.
Figure 2: Fisher Equation and Taylor Rule
πt+1
 R,
ax
l

π t+

ln



β
π̄ ∗



Rl
φ

1

=

m



φπ

t

◦

45

πu

πl

πt

Rl

3.6

Taylor Strategy

The fact that the unique equilibrium with πt ∈ [πl , πu ] is the desired equilibrium for all t ≥ 0 is an
important motivation for the following policy:
Definition 2. Taylor strategy: if t = 0, or if πs ∈ [πl , πu ] for s ≤ t − 1, for t > 0, then follow the
Taylor rule, (22), with φ > 1 in period t. Otherwise, monetary policy sets (scaled and logged) money
growth to µt = µ + ρ ln ct for t ≥ 0, where µ ∈ [πl , πu ]. Here, πl and πu are two constants satisfying
max



1
Rl
, ln ∗φµ̄< πl ≤ 0 ≤ πu < ∞.

(27)

Note our money growth rule responds to (aggregate) consumption. The parameter, ρ, is irrelevant
for establishing uniqueness of competitive equilibrium. It is important later when we consider offequilibrium paths (see Section 5 below). We will see in Section 4.2 that the value of ρ plays a rolein guaranteeing unique implementation of the desired equilibrium. The restriction on µ guarantees
that if the escape clause is activated, then inflation will be brought to the interior of the monitoring
range.
A competitive equilibrium under a Taylor strategy is unique and it is the desired equilibrium. This
result is established in three steps. First, we establish that the economy has a unique equilibrium
under our money growth rule:Lemma 1. Suppose M̄−1 is given and monetary policy sets M̄t = µ̄cρt M̄t−1 for t ≥ 0, where µ̄ > 1.
There exists a unique competitive equilibrium with the properties:R̄t = β −1 µ̄ > 1,

ct = 1,π̄t+1 = µ̄,
15for t ≥ 0,

and P0 = M̄−1 µ̄.

For the proof, see Online Appendix A.5.
If Lemma 1 were not true and an equilibrium did not exist, it would be impossible to meaningfully
ask what would happen if πt ∈/ [πl , πu ] , because agents would not know how to form expectations
36
about t + 1. The second step shows that there is no competitive equilibrium in which πt ∈/ [πl , πu ] :Lemma 2. Consider the case in which monetary policy is the Taylor strategy defined in Definition
2. An equilibrium has the following property: πt ∈ [πl , πu ] for t ≥ 0.
The intuition for the proof is straightforward given Figure 2. A formal proof appears in Online
Appendix A.5.
The basic result of this section is:
Proposition 2. Suppose monetary policy is governed by the Taylor strategy. The only equilibriumis the desired equilibrium.
Proof. Suppose, to the contrary, that π0 6= 0. From Lemma 2 equilibrium has the property that themonitoring range is never violated, i.e., πt ∈ [πl , πu ]. The Taylor rule, (22), the Fisher equation, (21),l/ [πl , πu ] for some t,
and πl > Rφ , imply that in equilibrium: πt+1 = φπt . Evidently, π0 6= 0 implies πt ∈given φ > 1. This contradicts Lemma 2. We conclude that π0 = 0, establishing the proposition.
The result follows almost immediately from Lemma 2 and Figure 2. The former says that there
is no equilibrium with πt ∈/ [πl , πu ] . The latter indicates that the only equilibrium with πt ∈ [πl , πu ]is πt = 0 for all t ≥ 0.
Proposition 2 in effect assumes φ > 1 because that is included in our definition of the monetary
policy rule (see Definition 2). It is interesting to note that φ > 1 is not necessary for a uniqueequilibrium in the extreme case, πl = πu = µ = 0. In this case, uniqueness is guaranteed for any
φ 6= 0 (this result was shown in Atkeson et al. (2010)). To verify this, simply retrace the proof
of Lemma 2, which is the heart of the proof of Proposition 2. We discuss the implications of this
proposition for policy and for the Taylor principle in Section 6 below. We refer to this policy as
follows:
Definition 3. The zero monitoring range strategy is a version of the policy in Definition 2 with
πl = πu = µ = 0 and φ 6= 0.
Following is a formal statement of the uniqueness result:
Proposition 3. Suppose monetary policy is the zero monitoring range policy. The only equilibriumis the desired equilibrium.
36

For our analysis it is convenient that we have a unique equilibrium under the money rule, but we have notinvestigated whether uniqueness is necessary for our conclusions.

16

4

The Market as a GameThis section transforms the market economy of the previous section into a large, non-cooperative
game among the atomistic intermediate good producers. In period t, the ith intermediate good firm’s
payoff is a function of its own action (its price), the history of the economy, ht−1 , as well as its
belief about the price set by other intermediate good firms. As in the example in Section 2, the ith
firm’s belief about the price set by other firms matters. This is because what the other firms do
determines the continuation equilibrium of the economy, which in turn affects the ith firm’s payoff.
In contemplating the continuation equilibrium, the intermediate good firms think of households as
optimizers with clairvoyance about market prices, so that their decisions satisfy first order conditions
and clear markets.37 In addition, each firm understands the monetary policy strategy. The firstsection below explains how ht−1 and the ith firm’s belief map - via a continuation equilibrium - into
the ith firm’s optimal choice of its own price. This mapping defines the firm’s best response function.
We show that in contemplating the various continuation equilibria associated with alternative beliefs,
the firm’s best response function depends on the nature of monetary policy. This result may at firstseem surprising, since we do not impose any exogenous price setting or other nominal frictions in themodel. Monetary policy matters when intermediate good firms form expectations because, as they
contemplate different prices set by others, the continuation equilibrium behaves like a sticky pricemodel.
The first subsection below studies the price setting decision of the ith intermediate good producer,
as a function of an arbitrary belief and history. The second and third subsections formally define andthen discuss the strategy equilibrium of the model.

4.1

The Price Decision of the ith Intermediate Good Producer

As explained in Section 3.2, the ith optimizing intermediate good firm wishes to set its price, pi,t , tothe aggregate wage rate. In the first subsection below we describe the conundrum discussed intuitively
in Section 2. The conundrum implies that to make its price decision, the ith firm must form a belief
about how other intermediate good firms are setting their date t price. The second subsection defines
histories in terms of our scaled variables. The third subsection describes the reasoning that allows theith firm to combine its knowledge of the past history of the economy with its belief about others’ prices
to map into a belief about the nominal wage rate, Wt . This mapping, which is used to define the ith
firm’s best response function, is represented analytically in Proposition 4. The key ingredient of thebbest response function is an ht−1 , πi,tcontinuation equilibrium. The fourth and fifth subsections
provide explicit derivations and economic interpretations of the best response functions.
37

Recall the discussion about clairvoyance in Section 2.2.3. By assuming that households have clairvoyance we meanthat they have rational expectations, i.e., their beliefs correspond to the competitive (continuation) equilibrium.

174.1.1The Role of Beliefs in a Firm’s Price Setting DecisionIntermediate good firms set their prices simultaneously and without coordination. Recall from equation (14) that firms would like to set their price, pi,t , equal to the nominal wage rate. For the reasonsdescribed in Section 2, it is not possible for the ith firm to literally observe Wt at the time that it
sets its price. This is because Wt is determined in an equilibrium in which the state is composed of
the observed past history, ht−1 , as well as the aggregate price level, Pt . The aggregate price level isnecessary for wage determination in the labor market because, for example, households care about
Wt /Pt , and not Wt per se. That the ith firm cannot observe Pt at the time it chooses pi,t is obvious
from equation (13), which indicates that Pt is the consequence of the prices set by all intermediate
good firms. The ith firm cannot observe the consequences, Pt , of the other firms’ actions until after
those actions have been taken. It follows that when the ith firm sets its price, it must do so based ona belief about how other firms set their price.
We make the following symmetry assumption about beliefs: the ith firm believes that all otherfirms set the same price, and we denote that price by pbi,t . The symmetry assumption is automaticallysatisfied in a competitive equilibrium. There, each firm’s belief about how others set their price iscorrect and, hence, identical. An implication of our symmetry assumption is that a given belief, pbi,t ,maps trivially into a belief about the aggregate price index via equation (13).
4.1.2

History, ht−1 , and Scaling the Variables

We begin by providing a formal definition of a history, ht−1 . A history consists of events observed
until the end of date t − 1. Formally, P , M̄  t = 0−1−1
ht−1 =.
(h , a ) t ≥ 1
t−2 t−1

The two variables in h−1 are the initial state of the household (e.g., m−1 = M̄−1 ) and the lagged
price level needed to define period 0 inflation in the Taylor rule. Also,at = lt , ct , πt , Rt , wt , µt , M̄t , mt , dt .

(28)

Here, πt corresponds to Pt in āt , which is the unscaled version of at (see equation 20). We obtain πtfrom Pt by scaling the latter with Pt−1 µ̄∗ and logging the result. Similarly, Rt , µt correspond to R̄t , µ̄tin āt , after scaling and logging. Finally,
wt ≡ ln

Wt
Pt−1 µ̄∗

18



.

(29)

Let pbi,t denote the scaled and logged price that the ith firm believes the other firms set:38
pbi,tPt−1 µ̄∗

b
πi,t
≡ ln

!

.

(30)

Let xi,t denote the scaled and logged price set by the ith intermediate good firm:
xi,t = lnpi,tPt−1 µ̄∗



.

(31)


b
Continuation Equilibria: Definition and Characterization
ht−1 , πi,tbA continuation equilibrium associated with ht−1 , πi,t
is formally defined as follows:bcontinuation equilibrium is a sequence, (ai,t+s )∞Definition 4. A ht−1 , πi,t
s=0 , that satisfies (i) alltime t + s equilibrium conditions, s > 0 and (ii) all time t equilibrium conditions except the period t
optimality condition for the intermediate good firm, equation (15).
4.1.3

We use the i subscript on the period t and future a’s to highlight that these variables correspondto the beliefs of the ith firm. Past a’s (hence, ht−1 ) do not require an i subscript because they are
public information. For a list of all the equilibrium conditions in the model, see the discussion afterbDefinition 1. We obviously must place at least mild regularity restrictions on πi,t
for the analysis
to be interesting. In particular, the notion that the ith firm might entertain the thought that otherfirms set a negative or zero price is uninteresting because no continuation equilibrium could exist inthat case.39 This is true, independent of the nature of monetary policy. So, we exclude any belief as
unreasonable unless it satisfies:
b
πi,t
> −∞.
(32)
b
We place no restrictions on ht−1 or πi,t
, apart from equation (32). For example, we do not requirethat the variables in ht−1 satisfy the date t − 1 and earlier equilibrium conditions. But, we do allowbconfigurations of ht−1 , πi,tthat correspond to the unique competitive equilibrium under the Taylor
strategy.40
b
Most of the discussion below is devoted to the construction of ai,t in the ht−1 , πi,tcontinuation∞bequilibrium. This is because the mapping from ht−1 , πi,tto (ai,t+s )s=1 is straightforward due tob
It is easy to establish that ht−1 , pbi,t and ht−1 , πi,tspan the same information. First, we show that, given ht−1bband pi,t , it is possible to compute πi,t in equation (30). Note that Pt−1 = π̄t−1 × · · · × π̄0 P−1 for t ≥ 0 and that π̄t−s ,1 ≤ s ≤ t, can be recovered from ht−1 because π̄t−s = exp (πt−s ) µ̄∗ (see the discussion after equation (21)). Thisestablishes that Pt−1 can be recovered from ht−1 , so that we have our first result. Second, it trivial to verify that, givenbht−1 and πi,t
, one can derive pbi,t .
39bThis corresponds to πi,tbeing a complex number or −∞, respectively.
40
A history would correspond to an equilibrium if (i) the elements of ht−1 are composed of allocations and pricesb
up to date t − 1 in competitive equilibrium under the Taylor strategy; and (ii) πi,t
= πt , the inflation rate in thisequilibrium. In this case, the continuation equilibrium is simply the date t + s, s ≥ 0 allocations and prices in theunique competitive equilibrium under the Taylor strategy.
38

19

Lemma 1 and Proposition 2. In addition, we have a special interest in a particular element of ai,t ,bnamely the time t wage rate. We denote the mapping, F, from ht−1 , πi,tto the scaled and logged
wage, wi,t , defined in equation (29) byb.
wi,t = F ht−1 , πi,t(33)bthen it sets its price as follows (recall equation (14)):
In this notation, if the ith firm believes πi,tbxi,t = wi,t = F ht−1 , πi,t.

(34)

Thus, in our environment, F can also be interpreted as ith intermediate good firm’s best responsebis the firm’s optimal choice of its own price, conditional on otherfunction. That is, F ht−1 , πi,tbfirms setting their price to πi,t.
Before formally stating the function, F, it is convenient to repeat our restrictions on preferences(see equation (5)) and on the parameters of the monetary policy rule (see Definition 2):
ψ ≥ 0, γ > 0, γ 6= 1,
Rl1
φ > 1, max, ln ∗ < πl ≤ 0 ≤ πu ,
φ
µ̄

(35)
µ ∈ [πl , πh ] .

(36)

Then,
Proposition 4. If monetary policy is governed by the Taylor strategy (i.e., equations (35) and(36) are satisfied) and the money growth rule parameter, ρ, satisfies ρ < min {1, γ}, then a uniquebht−1 , πi,tcontinuation equilibrium exists, with F given by(i) if ht−1 has the property that the inflation monitoring range has never been violated, then F
takes the following form:b

F ht−1 , πi,thi
 1 − φ (γ + ψ) π b
i,t
γ
=


π b − γ+ψ min R , φπ b − µ
l
i,t
i,t
γ−1

b
πi,t
∈ [πl , πu ]

(37)

b
πi,t
∈
/ [πl , πu ] ,(ii) if ht−1 has the property that the inflation monitoring range has been violated at least once, thenF takes the following form:

where





M̄t−1
γ+ψ
γ+ψ
b

1 − 1−ρ πi,t + 1−ρ µ + ln Pt−1

b

h
i
F ht−1 , πi,t=  ρ+ψ π b + γ+ψ µ + ln M̄t−1 + µ − R
l
i,t
ρ−γ
γ−ρ
Pt−1
D (ht−1 ) ≡



1−γ
b
πi,t
:

1−ρ



M̄t−1
b
µ + ln− πi,t
Pt−1

20



b
πi,t
∈ D (ht−1 )

,

(38)

otherwise+ µ > Rl



(39)

b
and D (ht−1 ) corresponds to the values of πi,thaving the property that the nominal interest ratebin the ht−1 , πi,t continuation equilibrium is positive.

The proof of Parts (i) and (ii) appear in Sections 4.1.4 and 4.1.5, respectively.41 The graph of Fin equations (37) and (38), for the indicated parameter values, is displayed in Figure 3. The figure
t−1
Figure 3: Best Response Function, µ = 0, γ > 1, M̄
= 1, ρ = 0
Pt−1(b) Money Growth Regime(a) Taylor Rule Regimexi,tbxi,t = πi,t
+



γ+ψ
1−γ








b
min Rl , φπi,t−µ

xi,t =

ρ+ψ 
ρ − γ πb

i,t −

xi,t
γ+ 
ψ
ρ−γ µ+

ln M̄t−1
Pt− + µ1

φ−1 Rl

πl

πu

◦

l



b
πi,t

b
πi,t

µ + ln45−Rφ (γ + ψ) b
xi,t = 1 −
πi,t
γ

Mt−1
1−ρ
+
(µ − Rl )
Pt−1
1−γ





γ+ψ
γ+ψ
M
b
xi,t = 1 −πi,t+µ + ln t−1
1−ρ
1−ρ
Pt−1

◦

45

Note: panel (a) refers to histories, ht−1 , in which the inflation monitoring range has never been violated. Panel (b) refers to histories,
ht−1 , in which the inflation monitoring range has been violated at least once.

in each panel indicates that the best response function crosses the 45 degree line exactly once, atthe origin.42 This crossing, or fixed point, corresponds to the unique Nash equilibrium: if every firm
had the same Nash belief about what others do, then the actions of all the firms would validatethat belief. The existence and uniqueness of the Nash equilibrium in equation (37) does not dependon the values of the parameters, as long as they satisfy equations (35) and (36). Section 4.2 below
establishes existence and uniqueness of the fixed point of F (ht−1 , ·) for general ht−1 .
4.1.4


b
ht−1 , πi,tContinuation Equilibria: ht−1 in Which the Monitoring Range Has
Never Been Violated

We now consider F for ht−1 in which the Taylor rule is in place in period t (see Figure 3a ). Becauseof our symmetry assumption about beliefs, equation (13) implies that Pi,t = pbi,t in the continuation
equilibrium. It follows that the third entry in ai,t , which is period t scaled and logged inflation, πi,t ,has the property,bπi,t = πi,t.
(40)bHere, πi,t
is defined in equation (30). Turning to the wage rate, note that,

Wi,t = Pi,tWi,t= pbi,t (ci,t )ψ+γ ,
Pi,t

41

(41)

As we note in Section B.1 below, we leave one part of the proof for Online Appendix B.
In the case of Panel (b), the unique Nash equilibrium would be at a point different from the origin if µ 6= 0 orM̄t−1 /Pt−1 6= 1.
42

21

where, ci,t and Wi,t /Pi,t denote time t consumption and the real wage in the continuation equilibriumb
implied by ht−1 , πi,t
. The second equality in equation (41) uses the fact that Pi,t = pbi,t and substitutes out the real wage, using the household intratemporal Euler equation, equation (8). In addition,
equation (41) uses the equilibrium condition that consumption equals labor, equation (19).43
After scaling the variables in equation (41) by Pt−1 µ̄∗ , using equation (30) and taking logs, we
obtainbwi,t = πi,t+ (γ + ψ) ln ci,t .
(42)
Evidently, to compute wi,t we require ci,t .
b
We consider πi,ton two segments of the real line. These segments are motivated by the factbthat πi,t maps directly into period t (scaled and logged) inflation in a continuation equilibrium (seeb∈/ [πl , πu ] , as violation of the inflation monitoring range
equation (40)). We refer to the segment, πi,tb
and refer to the segment, πi,t ∈ [πl , πu ], as moderate inflation.
Violation of the inflation monitoring range
Because the inflation monitoring range is violated in period t, period t + 1 is the first period of
the money growth regime. According to Definition 2 the gross money growth rate, µ̄, is greater thanunity, so that Lemma 1 guarantees existence and uniqueness of (ai,t+s )∞s=1 . To obtain the remaining
elements of ai,t it is useful to note that, according to Lemma 1, ci,t+1 = 1 and M̄i,t+1 = Pi,t+1 . Dividing
the latter by the period t equilibrium cash constraint, (25), M̄i,t = ci,t Pi,t , we obtain44µ̄ =π̄i,t+1
.
ci,t

(43)

Here, π̄i,t+1 denotes the gross inflation rate from t to t + 1 in the continuation equilibrium associatedbwith ht−1 , πi,t
. Substituting equation (43) into the household’s intertemporal Euler equation, (9),
1/(1−γ)
we obtain ci,t = β/µ̄R̄i,t. Because ht−1 has the property that the inflation monitoring range
has not been violated in the past, the Taylor rule, (17), is in operation in period t. Substituting the
Taylor rule into the latter equation and rearranging, we obtain, after scaling and taking logs,
ln ci,t =



1 
b
min Rl , φπi,t
−µ ,
1−γ

(44)

where µ is the scaled and logged value of µ̄. In equation (44) we used the fact that inflation, πi,t , inb
the continuation equilibrium is πi,t
. Substituting from equation (44) into equation (42), we obtain:bwi,t = πi,t
−
43



γ+ψ
γ−1







b
b
min Rl , φπi,t− µ = F ht−1 , πi,t.

(45)

Our symmetry assumption on beliefs plays a role in equation (41) by ruling out a price dispersion term that would
appear otherwise.
44
Note that we assume the cash constraint is satisfied as a strict equality, regardless of whether or not the interest
rate is at the zero lower bound. When the interest rate is positive, equality reflects reflects household optimality,
equation (10). When the interest rate is zero, equality reflects our assumption that in that case, monetary policy sets
the aggregate money stock equal to the current nominal value of aggregate consumption.

22


b
This completes our discussion of the mapping from ht−1 , πi,tto the continuation equilibrium forthe case in which the inflation monitoring range is violated for the first time in period t. Equations
b(44) and (45) provide explicit expressions for ci,t and wi,t in the ht−1 , πi,tcontinuation equilibrium.
b
The other variables in the ht−1 , πi,t continuation equilibrium are straightforward to derive.
There is a point of contact in our environment between models in which (i) prices are flexible andmonetary policy is neutral and (ii) sticky price models. In the competitive equilibrium of our model,
real variables are determined independent of monetary policy. By contrast, from the perspective of anb’s, when γ > 1 the world behaves in period t likeintermediate good agent contemplating different πi,ta simple sticky price New (or, old) Keynesian model, in which monetary policy is not neutral. To see
this, consider the ith intermediate good firm contemplating a lower (scaled and logged) price set bybbother firms, πi,t
, in the range where the zero lower bound on the interest rate is not binding, πi,t
> Rl /φb(see Figure 3a). The firm in effect contemplates how the ht−1 , πi,t continuation equilibrium changes
with an exogenous decrease in the aggregate price level, ignoring the equilibrium condition, (15).45
There are two channels by which the nominal wage in the continuation equilibrium changes in responseb. The first channel is an inflation channel. In the continuation equilibrium inflation,
to a change in πi,tb
πi,t , falls by the same amount as πi,tbecause of equation (13) and our symmetry assumption on beliefs.
The inflation channel is captured by the first term on the right of the equality in equation (45). It
reflects that, holding the real wage constant, the nominal wage falls by the same amount, in log
terms, as inflation. The interest rate channel reflects that a cut in πi,t results in a fall in the nominalinterest rate, via the Taylor rule. Other things the same, this cut in the interest rate has a direct
positive impact on consumption via the Euler equation, (9). This direct impact on consumption of
the interest rate cut is amplified by general equilibrium effects when γ > 1. In particular, the initialrise in consumption raises πi,t+1 (see equation (43)). This reduces the real interest rate, triggering
a secondary positive effect on consumption, which raises πi,t+1 further, and so on. With γ > 1 eachsuccessive general equilibrium jump in consumption is smaller and the sum of the effects is finite.
bThus, a fall in πi,ttriggers a jump in ci,t .46 The resulting rise in the demand for labor produces a risein the equilibrium (scaled and logged) real wage (see the second term after first equality in equationb(45)). Because the interest rate effect on the real wage dominates the inflation effect of a fall in πi,t,ththe nominal wage rate rises. The i firm best-responds by raising the price that it sets. This is why
the ith firm’s best response function has a negative slope when γ > 1.47b
Now suppose that the firm contemplates a reduction in πi,twhen the zero lower bound on thebinterest rate is binding, φπi,t < Rl . With the nominal interest rate at its lower bound, interest rate45b
In this section, we allow for arbitrary πi,tand we temporarily ignore any restrictions that optimizing behavior bybother agents may imply for πi,t . We consider such restrictions in Section 5 below.
46
The sum of the direct and indirect effects referred to in the text can be expressed as a geometric sum involving
1/γ. It can be verified that the term, 1/ (1 − γ) in equation (44) can be interpreted as reflecting the application of thegeometric sum formula.
47
It is easy to verify that when γ > 1, 1 − φ (γ + ψ) / (γ − 1) < 0. Later, we explain why the model is not interesting
if 0 < γ ≤ 1. Throughout the analysis, we assume ψ ≥ 0. We also assume φ > 1 except when we consider consider the
special case, πl = πu = µ = 0,as in Proposition 3.

23channel described in the previous paragraph is shut down. With only the inflation channel operative,b
the best response function is parallel to the 45◦ line (see equation (45)). Reductions in πi,twhen theinterest rate is at its lower bound have no impact on real wage, real interest rate, consumption andemployment.
bcontinuation equilibrium
We have carefully discussed the case, γ > 1. When γ < 1 the ht−1 , πi,t
is still well defined and the best response function continues to have a unique fixed point at zero.
However, the global behavior of the best response function is qualitatively very different. For example,
bbwhen γ < 1.48 In Section 5 we will argue that γ < 1 isis strictly increasing in πi,tF ht−1 , πi,teconomically uninteresting in this model.
Moderate inflation
The analysis of moderate inflation in period t is qualitatively similar to the case just consideredwhen γ > 1 (again, see Figure 3a). The difference is that here, period t+1 inflation in the continuation
equilibrium is perfectly anchored at its desired level of zero (thus, equation (45) does not enter the
analysis). The reason is that the Taylor rule will be in operation in t + 1, which implies from
Proposition 2 that πi,t+1 = 0 and ci,t+1 = 1. With πi,t+1 fixed in this way, the best response function
does not exhibit the sensitivity to whether γ is greater or less than unity that we see in equation 5.
Substituting πi,t+1 = 0 and ci,t+1 = 1 into the period t household intertemporal Euler equation,(9), we obtain (after scaling and logging),b− γ ln ci,t = Ri,t = φπi,t,(46)

after making use of the Taylor rule, (17). Substituting into equation (42) and collecting terms, we
obtain:φbb
wi,t = 1 − (γ + ψ) πi,t
= F ht−1 , πi,t
.
(47)
γ

b
For the range of πi,t’s considered here, the zero lower bound is not encountered. Apart from that,the analysis is qualitatively the same as in the version of previous discussion with γ > 1. Thereis a quantitative difference because period t + 1 inflation is anchored at its desired level when themonitoring range is not violated in the current period. This weakens the interest rare channel by
eliminating what we called the general equilibrium effects associated with a change in consumption.
Still, we can see in equation (47) that the interest rate channel still dominates the inflation channel,because φ (γ + ψ) /γ > 1. This completes our discussion of the mapping from ht−1 , πtb to thecontinuation equilibrium for the moderate inflation case.
48

One way to understand the dramatic impact of γ is to recall our geometric sum argument in the case, γ > 1.
The standard geometric sum formula underlies the coefficient, (1/γ) / (1 − 1/γ), in equation (45). The geometric sum
formula ‘works’ whether the infinite sum converges or not, but it switches sign from positive to negative when it ismisused. In fact, the infinite sum formula was only used in the text to convey intuition. The mathematical computationof the continuation equilibrium did not use that formula. In Section 5.5 we argue that the model is not economicallyinteresting when γ < 1 (see also Proposition 11, which rules γ < 1 out).

24

4.1.5bht−1 , πi,tContinuation Equilibria: ht−1 in Which the Monitoring Range Has Been
Violated

We now consider histories, ht−1 , in which the economy is in the money growth regime from time t andbcontinuation equilibrium
into the future (see Figure 3b). The values of Pi,t and πi,t in the ht−1 , πi,tfollow immediately from equation (13) and the symmetry assumption on beliefs. Lemma 1 describes
b49
the part of the ht−1 , πi,tcontinuation equilibrium beginning in t + 1, (ai,t+s )∞In particular,
s=1 .
ci,t+1 = 1 and Pi,t+1 = M̄i,t µ̄,using the fact that the cash constraint is binding in t + 1 and using themonetary policy rule (see Lemma 1). Then,ln ci,t+1 = 0, Pi,t+1 = M̄i,t µ̄, πi,t+1 = µ + ln

M̄i,t
M̄i,t
M̄t−1
b
, ln
= µ + ρ ln ci,t − πi,t
+ ln
,
Pi,t
Pi,t
Pt−1

(48)

where the fourth term is the period t money rule after taking logs and rearranging. We do notinclude the i subscript on lagged real balances because they are contained in ht−1 , which is public
information.
b
We suppose that the zero lower bound on the interest rate in the ht−1 , πi,tcontinuation equilibrium is strictly non-binding, i.e., Ri,t > Rl . Then, the period t cash constraint (10) is binding, which,when combined with the loan market clearing condition (see equation (18)), implies ln ci,t = M̄i,t /Pi,t .
Substituting out for M̄i,t /Pi,t using the monetary policy rule and rearranging, we obtain:


M̄t−1
1
b
− πi,t .
µ + lnln ci,t =
1−ρ
Pt−1

(49)

Using (49) to substitute out for ln ci,t in equation (42), we conclude that the (scaled and logged) wage
is given by



γ+ψ
M̄t−1
b
b
b
F ht−1 , πi,t = πi,t +− πi,t .
µ + ln
(50)
1−ρ
Pt−1

b
We now derive a necessary and sufficient condition on πi,tto guarantee that the zero lower bound on
the interest rate is not violated. The household intertemporal Euler equation (9) implies, using (48)and (49),



M̄t−1
1−γ
b
µ + ln− πi,t + µ.
Ri,t =1−ρ
Pt−1This expression shows that Ri,t is strictly greater than Rl if, and only if, πi,t ∈ D (ht−1 ) as defined inequation (39).
Now consider the case in which the zero lower bound binds, namely, πi,t ∈/ D (ht−1 ) . The Euler
equation (9) in the zero lower bound is, after rearranging and using equation (48),49

Period t + 1 corresponds to period 0 in the lemma.

25



M̄t−1
1
b
µ − πi,t + ln
+ µ − Rl .
ln ci,t =
γ−ρ
Pt−1

(51)


b
continuation equilibrium is given by
Using equation (42), the time t wage in the ht−1 , πi,tFbht−1 , πi,t



=



ρ+ψ
ρ−γ





M̄t−1
µ + ln+ µ − Rl .
γ−ρ
Pt−1

γ+ψ
b
πi,t
+

(52)

In Online Appendix B.1, we show that the household cash constraint is satisfied in period t in thebht−1 , πi,tcontinuation equilibrium constructed above.
As in Section 4.1.4, the intuition behind F is very simple, and we describe it in loose terms here.
To simplify the discussion, we assume that ρ = 0.50 The ‘model’ used by the ith intermediate goodb
, resembles afirm to construct the continuation equilibrium corresponding to a particular belief, πi,tstandard sticky price model. The model is essentially static, because the firm assumes that starting inperiod t + 1 all variables jump to their competitive equilibrium continuation values. We first considerb∈ D (ht−1 ) .
the case in which the zero lower bound on the interest rate is not binding, i.e., πi,tSuppose the ith intermediate good firm contemplates a reduction in the (scaled and logged) price setbby other firms, πi,t
. As in Section 4.1.4, this variation in beliefs operates through two channels. The
first channel, the inflation channel, drives the nominal wage down one-for-one in log terms (see thefirst term after the equality in equation (50)). The second channel operates through the interest rate.
With the time t money stock a function of ht−1 , a fall in the time t aggregate price level raises time
t real balances. The household would not be content with its previous deposit decision because thatnow entails carrying non-interest bearing cash in excess of what is needed for consumption. So, thebby increasing
intermediate good producer imagines the household would respond to the drop in πi,tdeposits, di,t , in equation (7). This increased supply in the loan market drives the interest rate down
and encourages households to raise consumption. Another factor that raises consumption derives
from the fact that the period t + 1 price level is determined by ht−1 when ρ = 0 (see equation (48)).51
So, the fall in the period t price level raises πi,t+1 ,thus reducing the real interest rate even more.
The resulting sharp increase in the demand for labor raises the real wage. The interest rate channelbis stronger than the inflation channel when γ > 1, so a fall in πi,traises the nominal wage in thecontinuation equilibrium (again, see equation (50)). The ith firm best-responds by raising its ownbprice level. Further reductions in πi,teventually make the zero lower bound on the interest rate bind.
b
The positive impact of such reductions in πi,ton consumption are now somewhat smaller becausebonly the inflation channel is operative and the interest rate channel is gone. Thus, reductions in πi,tin the region of the zero lower bound raise consumption and the wage, but not as strongly as when
50

This parameter will play an important role in the analysis in the next section, and will be discussed there.
According to equation (48), Pi,t+1 is a function of M̄i,t . When ρ = 0 then M̄i,t = µ̄M̄t−1 (see Lemma 1). The
object, M̄t−1 , can be recovered from ht−1 .
51

26

b
πi,t
∈ D (ht−1 ).52
Although we think of the money growth rule and the interest rate rule as part of one regime, it
is interesting how similarly the economy behaves under each rule. In each case, the mapping frombeliefs about inflation to actual price decisions is heavily influenced by an interest rate channel. In
this respect, our model provides an example of an argument developed in Taylor (1999). He showsthat monetary arrangements that are apparently very different can all have the same property thata rise in the nominal interest rate acts as device to moderate (or, even reverse) a rise in inflation.
In his discussion, Taylor (1999) includes not only the types of interest rate and money growth rules
discussed here, but also the operation of the specie-flow mechanism under the Gold standard. We
return to this theme in Section 6.

4.2

Strategy Equilibrium

It is convenient to define the following equilibrium concept:
Definition 5. An ht−1 continuation equilibrium is a sequence, (at+s )∞s=0 , that satisfies all time t + s
equilibrium conditions, s ≥ 0.
We now focus on Nash equilibria, in which all intermediate good firms have the same beliefs. So,
we can drop the subscript, i. A Nash equilibrium belief is a πtb with the property, πtb = F ht−1 , πtb .
We denote such a πtb by π b (ht−1 ). In principle, π b (ht−1 ) can be empty or it can be a set-valued
function of ht−1 . In the next proposition, we show that π b (ht−1 ) is single-valued for all ht−1 . In
an addition, the following result establishes properties of an ht−1 continuation equilibrium, and its
relation to an ht−1 , πtb continuation equilibrium:
Proposition 5. If γ > 0, γ 6= 1, and ρ < min {γ, 1}, our model has the following properties:(i) for every history, ht−1 , there exists a unique continuation equilibrium,(ii) an ht−1 continuation equilibrium is an ht−1 , πtb continuation equilibrium iff πtb ∈ π b (ht−1 ),(iii) the mapping, π b (ht−1 ) , is single-valued for each ht−1 ,(iv) for histories, ht−1 , in which the inflation monitoring range has never been violated, the uniquefixed point of F is πtb = 0.
52

The best response function has slope, −ψ/γ, when the zero lower bound is binding and it has slope 1 − (γ + ψ) ,when it is not binding. To see that it is flatter when the zero lower bound is binding, we show that −ψ/γ > 1−(γ + ψ) .
To see that this isthe case, add ψ/γ to both sides and collect terms in γ +ψ, so that the previous inequality is equivalent

to 0 > − (ψ + γ) 1 − γ1 , which is true when γ > 1. The same result is true for the values of ρ allowed in the monetary
growth rule (see Definition 2). We explain the economic rationale the restriction on ρ in section 5.5.

27

For the proof, see Online Appendix B.2.
The above properties of continuation equilibria are crucial for an equilibrium concept that plays
an important role in our analysis:
Definition 6. A competitive equilibrium is a Strategy Equilibrium if the model has the following two
properties: (i) for each possible history, ht−1 , there is a well-defined ht−1 continuation equilibrium;
and (ii) for each possible ht−1 , πtb with πtb > −∞, there is a well defined ht−1 , πtb continuation
equilibrium.
Our Strategy Equilibrium is substantively not different from the ‘sophisticated equilibrium’ concept used in Atkeson et al. (2010). It has a different name because the equilibrium objects in Atkeson
et al. (2010) are functions while they are sequences here. We use the name, ‘strategy equilibrium’
even though all the agents in our model are atomistic and they know their decisions have no impact
on economic aggregates. Still, the reasoning of our intermediate good firms has a strategic aspect toit. They contemplate different beliefs about the actions of others and the impact that others’ actions
have on economic aggregates. For an equilibrium to be a strategy equilibrium, it is necessary thatthe continuation equilibrium associated with each belief is well-defined.
It is easily established Definition 6 is satisfied by our model:
Proposition 6. The Taylor rule equilibrium with an escape clause is a strategy equilibrium for γ > 0,γ 6= 1 and ρ < min {γ, 1} .
Proof. The result is immediate by Proposition 4, (i) and Proposition 5, (i).
It is also interesting to construct the best response function, F, under the under the zero monitoring range policy in Definition 3, with πl = πu = µ = 0 and φ 6= 0. In this case, we know from
Proposition 3 that the desired equilibrium is the only competitive equilibrium. Consider the case,ht−1 , in which the inflation monitoring range has never been violated. The best response function,b
F, in this case is the second expression in equation (37) for all πi,t
. In the case of ht−1 in which there
has been a violation of the monitoring range, the function, F, is given in equations (38) and (39).
This establishes that the equilibrium for this monetary policy is a strategy equilibrium. Formally,
Proposition 7. Suppose (i) monetary policy is governed by the zero monitoring range strategy, and(ii) γ > 0, γ 6= 1 and ρ < min {γ, 1}. Then, the unique equilibrium is a strategy equilibrium.
This proposition was established in Atkeson et al. (2010). The economic intuition underlying
F in the case of the policy considered here is essentially the same as it is for the case, πl < πudiscussed in Sections 4.1.4 and 4.1.5. This is obviously true for histories, ht−1 , in which there has
been a past deviation, because the escape clause is irrelevant in this type of history. It is also truefor histories, ht−1 , in which there never has been a deviation. For those histories, the best response
function is the function graphed in Figure 3a, where the middle segment is shrunk to a singleton at
the origin. Whether that middle segment is simply a point at the origin, or has positive length, as
28

in Figure 3a, has no significant impact on the underlying economic intuition a described above. So,
the policy referred to in Proposition 7 is yet one more example of Taylor (1999)’s observation thatseemingly different policy arrangements could nevertheless assign an important role to the interest
rate for stabilizing inflation. This observation may at first seem surprising, since Proposition 7 allows
for φ > 0 but arbitrarily small. In Section 5, we argue that φ < 0 is not economically interesting.

4.3

Observations About the Strategy Equilibrium

The requirement that an equilibrium be a strategy equilibrium places restrictions on the model
parameters (see Propositions 6 and 7). We illustrate this with two examples in which the parameter
restrictions are not satisfied and the equilibrium fails to be a strategy equilibrium. Our first example
considers the log specification of utility, γ = 1. Our second example considers a particular set of
restrictions which in effect convert our economy into an endowment economy. Consistent with theresults in Cochrane (2011), the unique competitive equilibrium corresponding to the Taylor strategy
is not a strategy equilibrium in the endowment economy case. The reason is that the Taylor strategy
implements the desired equilibrium by changing the real interest rate in the event that the economy
leaves the desired allocations. In an endowment economy such a policy is not a strategy equilibrium,
because there exists no monetary policy that can change the real interest rate. Of course, this logic
does not apply to our model because it is a production economy in which it is possible for monetary
policy to change the real interest rate.
First, consider the following proposition:
Proposition 8. Suppose γ = 1. The competitive equilibrium associated with the Taylor strategy is
not a strategy equilibrium.
Proof. It suffices to show that there exists an ht−1 , πtb for which there does not exist a continuation
equilibrium. Thus, suppose ht−1 is a history in which the monitoring range has never been violated.
Consider πtb ∈/ [πl , πu ] . In period t + 1 the escape clause will be invoked and the economy will switch
to the money growth regime. By Lemma 1, we haveln ct+1 = 0.

(53)

The cash constraint is binding in periods t and t + 1, so equation (43) is satisfied andπt+1 = ln ct + µ,

(54)

after scaling and logging. Taking into account that in the continuation equilibrium, πt = πtb ∈/ [πl , πu ],it follows that in the scaled and logged Taylor rule,
Rt = φπt ∈/ [πl , πu ] ,
29

(55)

because φ > 1. After scaling and logging the household intertemporal Euler equation, (9),ln ct = ln ct+1 − [Rt − πt+1 ] .

(56)

Substituting equations (53), (54), into equation (56) we obtain
Rt = µ ∈ [πl , πu ] ,(57)which contradicts equation (55).
We infer from this example that the Taylor strategy would not be an effective way to exclude undesired equilibria if γ = 1. A different policy would be necessary to implement the desired equilibriumin this case.
Another closely related example is the version of the model in which γ > 0 and in which households
supply labor inelastically, lt = 1. In this case, the model economy is effectively an endowment economy,
as in Cochrane (2011). By essentially the same logic as in the γ = 1 case, the Taylor strategy is
not a strategy equilibrium.53 So, using the escape clause to eliminate undesired equilibria in an
endowment economy is, once again, uninteresting.54 If the model economy does not have γ = 1
or is not an endowment economy, then equations (55) and (56) do not imply a contradiction. A
suitable adjustment in ln ct can reconcile the two equations. Indeed, Proposition 6 implies that suchan adjustment would occur in our model (which is not an endowment economy) with γ 6= 1. The
economics of the adjustment in ln ct will be discussed in the next section.

5

Rationalizable Implementation

This section derives sufficient conditions and a necessary condition for policy to uniquely implement
the desired equilibrium. We begin, in the first subsection, by providing a standard definition ofrationalizability and relating it to the concept of iterated deletion discussed in Section 2. The second
subsection motivates a refinement of the concept of rationalizability, robust rationalizability, which
we use in our analysis. The third subsection provides sufficient conditions for rationalizability, as
well as a necessary condition. These conditions are the formal representation of the ‘leaning against
the wind, but not too aggressively’ principle discussed in Section 2. To establish our conditions forrationalizability, we must generalize existing results in the literature (see Desgranges (2014)). To
our knowledge, the literature does not address models with best response functions like ours. The
53

The proof by contradiction in the endowment economy requires a minor adjustment to the argument in the text.
The argument leading to equation (55) is unchanged. Equation (57) also holds, but in the endowment economy it isderived from the fact, ln ct = ln lt = 0, for all t. In particular, πt+1 = µ by equation (54) and Rt = πt+1 by equation(56) implies equation (57).
54
As noted in the introduction, this observation was first made in Cochrane (2011). The endowment economy
assumption is made in Cochrane (2011, p. 574) and our proof by contradiction argument (see Footnote 53) coincides
with the argument in Cochrane (2011, p. 584).

30

best response function in our model lacks differentiability, continuity and has a segment which hasslope unity (see Figure 3). The fourth and fifth subsections make use of our rationalizability result toestablish sufficient conditions for unique implementation of the desired equilibrium. We also provide
a necessary condition. We obtain these results for the Taylor strategy, defined in Definition 2, andalso for the zero monitoring range strategy defined in Definition 3.
Our analysis in this paper follows Atkeson et al. (2010) by focusing on one-period deviations
from an arbitrary history, ht−1 . We have extended the analysis to multi-period deviations.55 For
space reasons, that analysis is reported in Online Appendix C.6. Conceptually, the extension tomultiperiod deviations is straightforward. However, because of the segmented structure of our best
response function, the type of global analysis that we do in the body of the paper is less tractable. So,
Online Appendix C.6 proceeds in the style of Evans et al. (2018) by doing the multi-period analysis
locally to the unique Nash equilibrium of the relevant game, for arbitrary ht−1 . We obtain a setof necessary and sufficient conditions for the local, multi-period analog of unique implementation.
Those conditions are slightly weaker than the conditions in the one-period setting.56 We do so forboth monetary policies studied in the paper: the Taylor strategy and the zero monitoring range
strategy.

5.1

Rationalizability

In our environment, intermediate good firms choose an action which is a best response to a belief
about the price set by other firms.57 We are interested in determining what belief would be reasonable,or rationalizable, for an individual firm under CK. To this end, it is useful to first define the set of
possible beliefs, Π0 = (−∞, ∞). These are the beliefs for which there exists a well defined continuation
equilibrium (see the discussion before equation (32)). We also define a justified belief:bb∈ Π0 and (ii)
, about the actions of others is justified if (i) πi,tDefinition 7. An agent’s belief, πi,tb
πi,tis itself a best response to some belief in Π0 by everyone else.

Rationalizability has the following recursive definition:bbDefinition 8. We say that a belief, πi,t
∈ Π0 , is rationalizable if (i) πi,t
is justified and (ii) the beliefbto which πi,t is a best response is also justified.

It is well known that there are two ways to determine the rationalizability of a given belief, chainof justification and iterated deletion. Both play an important role in our analysis, and so we definethem formally here.
55

We do not appeal to the ‘one-shot-deviation principle’ in game theory (see e.g., Fudenberg and Tirole (Section 4.2,
1991)) because our setting is different from the environment in which that principle is derived. In that environment
there is a large agent who makes a decision which affects the evolution of the economy (see, e.g., Chari and Kehoe(1990)). In our environment all agents are atomistic, so that their own decisions are inconsequential.
56
See equations (63) and (67) below.
57
We continue to maintain our symmetry assumption on beliefs.

31

bGiven the recursive nature of Definition 8, a belief, πi,t
, is rationalizable if it is possible to constructba chain of justification for πi,t . To define this chain we first construct a sequence of sets. Thebfirst element in the sequence is the singleton, πi,t
. The lth element in the sequence, for l > 1, isb:58
F −l ht−1 , πi,t


 b
b
b
, ... .
(58)
, ..., F −l ht−1 , πi,t
πi,t , F −1 ht−1 , πi,tb
, is a sequence of beliefs in equation (58) such that each
A chain of justification for the belief, πi,tb
is rationalizable according toelement belongs to Π0 . In there exists such a sequence, then πi,t−lbDefinition 8. Rationalizability can fail if F
ht−1 , πi,t ∩ Π0 is empty for some l. This constructivebmethod for verifying that πi,t
is rationalizable is called the chain of justification method.
The iterated deletion method is the one used in the example in the Section 2. It identifies theentire set of rationalizable beliefs. As such, it represents a second method for determining whether
a given belief is rationalizable. Suppose that the firm starts out with the set of possible beliefs, Π0 .
Then, F (ht−1 , Π0 ) denotes the set of best responses associated with all possible beliefs, Π0 . Consider
an element, xi ∈ Π0 such that xi ∈/ F (ht−1 , Π0 ) . The ith firm knows that there is no circumstancein which it would choose xi and it realizes that others would never choose xi either. So, the ith firm
would drop xi from the set of possible beliefs, Π0 . That is, the ith individual would restrict its beliefs
T
about what others do to Π1 = Π0 F (Π0 ).59
T
This process can be repeated, leading to a sequence, Πk+1 = Πk F (ht−1 , Πk ), for k ≥ 0. Noticethat by construction, the sequence is non-increasing, Πk+1 ⊆ Πk for k ≥ 0. Consider the set of beliefs,b
πi,t∈ Π0 , such that they remain undeleted even as k → ∞:∗

Π (Π0 ; F ) =

∞
\

Πk .

(59)

k=0

We specify Π0 and F as arguments of Π∗ because we will have occasion to evaluate the operator, Π∗ ,for sets other than, Π0 , and functions other than F . We do not include the argument, ht−1 , in Πkand Π∗ in order to simplify notation.
As is well known, the set, Π∗ (Π0 ; F ), coincides with the set of rationalizable beliefs in the senseof Definition 8. For completeness, we present a proof in Online Appendix C.1.60
For each possible ht−1 we envision that every intermediate good firm constructs a set of rationalizable beliefs using either the chain of justification or the iterated deletion method, or even a
combination of the two. This private mental process used by firms to restrict their beliefs (perhaps
to a singleton) is called eduction (see Binmore (1987, page 184)).


b
b
−l
b
is defined similarly for l ≥ 2.
Here, x ∈ F −1 ht−1 , πi,tif πi,t
=
F
(h
,
x).
Also,
F
h
,
π
t−1
t−1
i,t
T
59
The reason we define Π1 = Π0 F (Π0 ) and not simply Π1 = F (Π0 ) is because our analysis must be robust to
situations in which F (Π0 ) ( Π0 , in which case F 2 (Π0 ) is not well defined.
60
For a textbook treatment, see Mas-Colell et al. (1995, Section 8.C).
58

32

5.2

Rationalizability with Refinement

Our paper is about designing policy so that agents’ expectations are uniquely anchored to the desired
equilibrium. As it turns out, a technicality prevents achieving this kind of uniqueness in the model
described up to now. The technicality is the assumption implicit in our analysis that intermediate
good agents can discriminate between an exact zero and an infinitesimally small number. Intuitively,
our refinement assumes that there exists an arbitrarily small positive number, such that agentscannot distinguish between it and zero. With this refinement, there is still a nontrivial policy design
problem, but failure of expectations to be anchored only occurs for economically interesting reasons.
The version of rationalizability which incorporates our refinement, robust rationalizability, is defined
below.
To understand the motivation for our refinement, let ht−1 be a history in which the inflationb
monitoring range has never been violated. Consider the rationalizability of a particular belief, πi,t<
Rl /φ. This belief lies in the region bounded on the right by the kink point in the best responsebfunction where the zero lower bound on the interest rate is binding (see Figure 3a). For any such πi,t,there is a unique chain of justification belonging to the set of sequences in equation (58). That chain
is constructed by iterating on the inverse of the segment of F that is parallel to the 45◦ line and has
the property that the zero lower bound is binding.61 If we exponentiate each element in that chainof justification, the resulting sequence corresponds to a sequence of beliefs for the ith intermediate
good firm about the (scaled) prices set by others. The limiting belief in that sequence is not justified,because it corresponds to a belief that others set their price to zero. However, technically that limitbis never actually reached. For this reason, all beliefs, πi,t
< Rl /φ, are in fact rationalizable (seeDefinition 8). We find this result economically uninteresting. Although the limiting belief is neverreached, the sequence of beliefs in the chain of justification does enter any arbitrarily small interval ofb< Rl /φ, rests critically on the assumption thatzero in finite steps. So, rationalizability of beliefs, πi,tagents have the capacity to differentiate infinitesimally small numbers from an exact zero. We drop
this assumption and replace it by the assumption that intermediate good firms have an ε cognitive
impairment. In particular, we assume that an intermediate good firm mentally records a number
smaller than ε > 0 as an exact zero, where ε can be arbitrarily small.62b, generally contains two elements. However, candidate chains ofAt least for small values of l, F −l ht−1 , πi,tjustification constructed using the greater of the two elements inevitably reach a point where no inverse exists. Suchb
candidates are therefore not part of a chain of justification for πi,t.
62
For a recent discussion of limitations on the ability of people to distinguish numbers that are very close to eachother, see Woodford (2019). For an early discussion in a decision-theoretic context, see Luce (1956). Interestingly
computers have the same cognitive ‘problem’ because they can only register a finite set of numbers. For example,
MATLAB registers positive numbers equal to 10−324 or less, as an exact zero. Computers also have a symmetricproblem with large numbers (in MATLAB, numbers larger than 10309 are registered as +∞). So, in principle we
could adopt a refinement which posits a cognitive impairment in thinking about large numbers. For example, we could
suppose that there exists an arbitrarily small δ, with δ > 0, such that x > 1/δ cannot be distinguished from 1/δ.
Online Appendix C.3 studies such a refinement and its implications for on- and off- equilibrium paths in our model.
In any case, our analysis in the body of this paper does not require any further refinement beyond the one discussed
in the text. That is because, for model parameterizations that satisfy necessary and sufficient conditions for uniqueimplementation, our environment places a natural upper bound on beliefs about inflation (see the discussion after
61

33

Under the assumption that an intermediate good firm is ε impaired, the firm (mistakenly) concludes that the chain of beliefs discussed in the previous paragraph does in fact reach zero.63 With
this refinement, the best response function, F, changes to F ε :



b
b
b
≡ max F ht−1 , πi,t
, ln ε , for πi,t
≥ ln ε,F ε ht−1 , πi,t

(60)


b
b
is not defined.64 This is because an ε impaired firm
< ln ε, F ε ht−1 , πi,tfor any ht−1 . When πi,tb< ln ε perceives that other firms set their price to zero. The firm’s decision problem
contemplating πi,t
is not well-defined for such a belief because there does not exist a continuation equilibrium.65 Thebrationale for the max operator in equation (60) is as follows. Recall that F ht−1 , πi,tcorresponds tob.
the (logged and scaled) nominal wage rate in the continuation equilibrium associated with ht−1 , πi,tb< ln ε. The firm would in this case perceive the nominal wage to be zero.
Suppose that F ht−1 , πi,tWith a zero marginal cost of production the profit maximizing firm is led to set its price to the lowest
positive value. In the absence of the cognitive impairment there is no solution to this problem.66
However, with our cognitive impairment the firm would choose to set its price to the unique smallest
price that it can perceive, which is greater than zero. That price is ln ε. This explains why xi = ln εbwhen F ht−1 , πi,t
< ln ε.67b, about actions by others which has the property
Let Πε0 denote the set of ε possible beliefs, πi,tthat a continuation equilibrium exists when intermediate good firms are ε impaired. That is:Πε0 = [ln ε, ∞),b

0 < ε < eπ (ht−1 ) .

(61)

We think of ε > 0 as being arbitrarily small. Here, π b (ht−1 ) is the unique fixed point of F (ht−1 , ·) ,which is the version of the best response function in which agents have no cognitive impairment (seeProposition 5). Similarly, we define ε justifiability and ε rationalizability analogously to Definitions
7 and 8, respectively.
We now provide the formal definition of robust rationalizability. This definition formalizes theintuition described above, which is that our refinement does not commit to any particular value ofε, other than that ε is small and positive. We begin by considering a specific value of ε satisfying
Proposition 11 below). As a result, extending our refinement to large numbers would have no impact on the analysisof rationalizability.
63
The situation is reminiscent of the literature on the amount of time, T, required for the neoclassical model toconverge to its steady state. The mathematically correct answer is that convergence never occurs (i.e., there is no
finite T large enough so that the capital stock is mathematically equal to its steady state value). Practical analysts
understand this, but they also understand that the amount of time needed to get within any ε > 0 region of the steadystate is finite. So, in practice, one defines ‘time to arrive in steady state’ as the amount of time, Tε , needed to enter
some specified ε neighborhood of steady state. Our refinement assumes that the agents in our model adopt this typeof logic when examining a sequence of prices that converges to (but does not actually reach) zero.
64
There are alternative representations of ε impairment. For example, we could suppose that for ε > 0 but small,
agents perceive 0 ≤ x ≤ ε as x = ε. This does not change our results.
65
Recall the discussion before equation (32).
66
Recall equation (12).
67
Our conclusion implicitly makes use of the quasi-concavity of the intermediate good firm’s objective function.

34

b

0 < ε < eπ (ht−1 ) . Let Π∗ (Πε0 , F ε ) denote the associated set of rationalizable beliefs.68 For notational
simplicity, we denote the correspondence from ε to Π∗ (Πε0 , F ε ) by Π∗ (ε) . The set, Π∗ (0) , is the setof rationalizable beliefs in the sense of Definition 8. It is easy to verify that Π∗ (ε) is weakly increasing
as ε declines for all ε ≥ 0.69 It follows that the limit, Π∗,r , of Π∗ (ε) as ε goes to zero is given by:Π∗,r =[Π∗ (ε) .

(62)

π b (ht−1 )0<ε≤e

We suppress the dependence of Π∗,r on ht−1 to simplify notation. Rationalizability under our refinement is defined as follows:bb∈ Π∗,r , where Π∗,r is defined, is robustly rationalizable if πi,tDefinition 9. We say that a belief, πi,tin equation (62).

The fact that Π∗ (ε) is lower- but not upper- hemicontinuous at ε = 0 provides an alternative
perspective on our refinement and explains our use of the adjective, robust, in Definition 9. Considerb∈ Π∗,r . It follows from the weakly increasing property of Π∗ (ε) ,a robustly rationalizable belief, πi,tbbbis rationalizable∈ Π∗,r , has the robustness property that πi,t∈ Π∗ (0). Thus, a belief, πi,tthat πi,twhether or not the ith agent has an ε cognitive impairment, as long as ε is small enough. By contrast,bbconsider a belief having the properties, πi,t∈ Π∗ (0) ,πi,t∈/ Π∗,r . Such a belief is rationalizable whenfirms have no cognitive impairment, ε = 0. But, it is not rationalizable for any cognitive impairment,
ε > 0, no matter how small ε is. In this sense, rationalizability of such a belief lacks robustness.

5.3

Sufficient and Necessary Conditions for Rationalizability

This section provides a result which plays a central role in the next section, where we establish
conditions for unique rationalizability. It is convenient to temporarily use a simpler notation which
abstracts from our specific economic setting. Accordingly, suppose the best response of a player ina large game is given by G. Let L and l denote the actions, respectively, of the others and of theindividual. So, l = G (L) . Suppose that the graph of the function is, with one exception, strictlyinterior to the butterfly-shaped, shaded area in Figure 4a. The exception is the point market witha dot, where the boundaries of the butterfly meet. The boundaries are defined by the dashed lines
with angles 45◦ and 135◦ . The domain, Ω, of the function G is defined by projecting the butterfly
onto the horizontal axis. The dot in the figure corresponds to a Nash equilibrium, a value of L suchthat L∗ = G (L∗ ).
The proposition below provides conditions under which the sequence of sets, Gk (Ω), k = 1, 2, ...
converges to the singleton, L∗ :See equation (59) for the definition of the operator, Π∗ .
εε00A simple way to see the weakly increasing result is by induction. Trivially, x ∈ Π0 implies x ∈ Π0 for ε < ε.
0
Similarly, F (Πε0 ) ⊆ F Πε0 . The result follows by repeating this type of argument and using the definition of Π∗ inequation (59).
68

69

35

G(L)

G(L)
135 ◦

135 ◦

◦

45

L∗

◦

45

L∗

G̃ (L)L∗L

L∗(a) The Butterfly

L(b) Butterfly and Necessity

Proposition 9. Suppose that a function G satisfies the following properties:(i) the domain of G, Ω, is a bounded set on the real line containing L∗ ,(ii) L∗ is a fixed point for G,∗|(iii) supL∈Ω\{L∗ } |G(L)−L< 1.
|L−L∗ |

Then we have Π∗ (Ω; G) = limk→∞ Gk (Ω) = {L∗ }.
Here, Π∗ (Ω; G) is defined in equation (59). The role of the boundedness condition in (i) will be
explained below. It is easy to see that conditions (ii) and (iii) imply that the fixed point of G is
unique. Condition (iii) is the mathematical statement that the function, G, lies inside the butterfly.70
We use the ‘sup’ operator in condition (iii) because the ‘max’ operator may not be well defined withthe type of best response function, F , in our model, because it is not a continuous function.
A formal proof of the proposition appears in Appendix A. In the special case that Ω is a boundedinterval, the intuition is straightforward. In this case, condition (iii) implies Gk (Ω) ⊆ Gk−1 (Ω) fork = 2, 3, ... . It is easy to verify that because this sequence of sets is weakly shrinking, it is guaranteedthat Π∗ (Ω; G) = limk→∞ Gk (Ω) . That the limit is L∗ can be verified using the kind of logic used in
the discussion of Figure 1a in Section 2.
The sufficient conditions of Proposition 9 are somewhat more general than they might at first
appear. In particular, these conditions may not be satisfied for the natural economic representationof the model variables, but they may be satisfied after a suitable transformation. In this case, theAn alternative, equivalent, version of condition (iii) is: ∃ρ < 1 such that for all L ∈ Π1 \ {L∗ } , |F (L) − L∗ | <ρ |L − L∗ |. To see this, suppose first that the previous condition holds. Then |F (L) − L∗ | / |L − L∗ | < ρ. Taking thesupremum, we have supL∈Π1 \{L∗ } |F (L) − L∗ | / |L − L∗ | ≤ ρ < 1. So, (iii) in Proposition 9 holds. Second, supposethat condition (3) holds. Then let δ = supL∈Π1 \{L∗ } |F (L) − L∗ | / |L − L∗ | and ρ = (1 + δ) /2 > δ. It follows that
|F (L) − L∗ | / |L − L∗ | ≤ δ < ρ for all L ∈ Π1 . Therefore |F (L) − L∗ | ≤ δ |L − L∗ | < ρ |L − L∗ | for all L ∈ Π1 \ {L∗ } .
The two results establish our equivalence result.
70

36

conclusion of Proposition 9 also applies to the original variables.71 In the case of our monetary model,
we have been working with the equilibrium conditions expressed in log form. This transformation
was convenient in our analysis of the competitive equilibrium of the model. As we shall see below,
the piecewise linearity of F in our model when the variables are expressed in log form makes it easy
to apply the sufficient conditions of Proposition 9.
To see that the conditions are sufficient but not necessary, consider the piecewise-linear function in
Figure 4b (see the solid line). That function does not lie in the butterfly, but it can easily be verified
that Π∗ (Ω; G) = {L∗ } anyway. For the purpose of our analysis, there does exist one particularly
useful necessary condition:
Proposition 10. If G (·) has a linear segment passing through L∗ then unique rationalizability,Π∗ (Ω; G) = {L∗ } , implies that the slope of the linear segment has slope less than unity in absolutevalue.
The function, G̃ (L), in Figure 4b does not satisfy the necessary condition of Proposition 10 andso unique implementation does not occur. It is easy to verify that values of L close enough to L∗ are
all rationalizable.

5.4

Implementation

We seek restrictions on the monetary policy parameters which guarantee that private agents uniquely
coordinate (or, anchor) their beliefs on the desired equilibrium. If the restrictions are satisfied, thenmonetary policy achieves unique implementation. We begin with the following definition:
Definition 10. A competitive equilibrium satisfies unique implementation if the model has thefollowing two properties: (i) it is a strategy equilibrium and (ii) for each ht−1 there exists a uniqueb.
robustly rationalizable belief, πi,tIn the first subsection we discuss unique implementation under the Taylor strategy (see Definition
2). The second subsection considers the zero monitoring range policy in Definition 3.
5.4.1

Taylor Strategy

We now state the following result:
Proposition 11. Consider the model in which monetary policy is governed by the Taylor strategy(see Definition 2) and the money growth rule parameter, ρ, satisfies ρ < min {1, γ}. A sufficient
Here is a simple illustration. Let l = G (L) = L−1/2 , with domain Ω = [Ll , Lu ] , where 0 < Ll < Lu . This function,
G, has a unique fixed point at L = 1. In addition, it is easy to verify that Gk (Ω) → {1}, as k → ∞. However, G
does not satisfy condition (iii) of Proposition 9. To see this, note that as L → 0, G (L) → ∞, thus breaking out of thebutterfly in Figure 3. However, if l and L are replaced by their logs, then the system is linear, ln l = −1/2 × ln L, withdomain, Ω = [ln Ll , ln Lu ] . This system does satisfy the sufficient conditions of 9.
71

37

condition for unique implementation of the unique equilibrium of that model is:
1−φ

γ+ψ
< 1,
γ−1

1 − (γ + ψ) < ρ < 1 −

γ+ψ
.
2

(63)

A necessary condition for unique implementation is:
1−φ

γ+ψ
< 1,
γ

ρ<1−

γ+ψ
.
2

(64)

Before sketching the proof of Proposition 10, we discuss the sufficient conditions in equation (63)and the necessary conditions in equation (64). We stress three observations about the sufficientconditions. First, there is no restriction on the boundaries of the inflation monitoring range πl andπu , beyond the (modest) restrictions stated in Definition 2. Second, depending on the values of theparameters associated with the private sector, γ and ψ, it is possible that there is no value for themonetary policy parameter, φ, that satisfies the sufficient conditions for unique implementation. Notethat φ > 1 implies that the first expression in equation (63) is equivalent to:72
φ<2

γ−1
,
γ+ψ

(65)

which cannot be satisfied if γ < 1. Equation (65) indicates that the sufficient conditions can also
be violated in case γ > 1 and ψ is large enough. The third interesting feature of the conditions,(63), is the restriction on ρ. Condition (65) implies, via the second condition in equation (63),that ρ < 0. Thus, a version of the Taylor strategy which switches to a constant money growth rule
(i.e., ρ = 0) when the escape clause is activated does not satisfy the sufficient conditions for uniqueimplementation. In the next section we provide an example with ρ = 0 in which the Taylor strategy
fails to anchor expectations.
Now, consider the two necessary conditions in equation (64). The first and second of these
conditions correspond to Proposition 10 applied to the Taylor rule and the money growth rule,
respectively, in the Taylor strategy. Given that φ, γ, ψ are positive, the first condition in equation 64
is equivalent to
2
.
(66)
φ<
1 + ψ/γNote that ψ and γ play similar, though opposite, roles in determining the upper bound on φ. With anincrease in ψ, or a decrease in γ, a given value of φ corresponds to a more aggressive response in the
Taylor rule to inflation. To see this, suppose a firm contemplates a higher belief about the prices set
by others. In the continuation equilibrium, this implies a higher level of the interest rate, dependent
on the given value of φ. For a fixed value of γ, this implies a particular drop in consumption (seeTo see this, note that the first condition in equation (63) implies −1 < 1 − γ+ψ
γ−1 φ < 1. Subtracting 1 from eachterm and multiplying the result by −1, we obtain 2 > γ+ψ
γ−1 φ > 0. Given φ > 1 > 0, the second inequality in thelast expression implies γ > 1, so that the first condition in equation (63) coincides with equation (65).
72

38

equation (46)). This drop in consumption produces a greater fall in marginal cost, the higher isψ (see equation (47)). Similarly, for a fixed value of ψ, a lower value of γ increases intertemporal
substitution in consumption. This leads to a larger drop in consumption and reduces marginal costby more (see equations (46) and (47)). This is why a higher value of ψ/γ increases the likelihoodthat a given φ leans against the wind too aggressively.
The upper bound in equation (66) may appear surprisingly low, given the values of ψ and γ often
used in practice. There is reason to think that in the type of models that work well empirically, thisupper bound might be higher. For example, Online Appendix E presents a version of our model with
Calvo-style price-setting frictions. In that model, only a subset of the intermediate good firms adjust
their price in a given period. When the number of such firms is small, then the upper bound on φ ishigher. The intuition for this resembles the intuition for γ and ψ just described. In particular, whena price setter considers higher prices set by other price setters, the impact on aggregate inflation(hence, on policy) is smaller, the smaller is the number of price setters. With fewer price setters(more stickiness in prices), policy is less aggressive for given value of φ, so that an upper bound likethe one in equation (66) is higher.73
To prove Proposition 11, we first consider the type of history, ht−1 , in which the inflation monitoring range has never been violated. Because φ in the monetary policy rule is positive, the conditions(65) in Proposition 11 require γ > 1. Proposition 6 then implies that the equilibrium is a strategy
equilibrium, so that part (i) of Definition 10 is satisfied. The key result used to establish part (ii) of
Definition 10 is Proposition 9, with G (·) = F ε (ht−1 , ·) ,for a specific, small value of ε > 0. To meet
condition (i) of Proposition 9, we must identify a suitable bounded set, Ω. We define Ω as the resultof the first iteration in the iterated deletion method: Ω = Πε0 ∩ F (ht−1 , Πε0 ) .74 That Ω is bounded
below follows from the lower bound on Πε0 (see equation (61)). The set, Ω, is also bounded above,
given the parameter restrictions, (63). This is because there is a maximal value of F (ht−1 , ·) (see,for example, Figure 4). It is straightforward to verify that the parameter restrictions also imply thatF ε (ht−1 , ·) lies inside the butterfly in Figure 4a. That is, conditions (ii) and (iii) of Proposition 9 aresatisfied, where the unique fixed point of F ε (ht−1 , ·) is L∗ = π b (ht−1 ) . It follows from Proposition

9 that Π∗ (Ω; F ε ) = π b (ht−1 ) for each ε > 0. Since Π∗ (ε) = Π∗ (Ω; F ε ) it follows that the set of
robustly rationalizable beliefs (see Definition 9), Π∗,r , is composed of the singleton, π b (ht−1 ) . It
follows that part (ii) of Definition 10 is satisfied. We conclude that π b (ht−1 ) is the only robustly
rationalizable belief.
Now consider the type of history, ht−1 , in which the inflation monitoring range has been violatedb
at least once. Lemma 18 in Online Appendix C.4 establishes that equation (63) implies F ht−1 , πi,t
is bounded above and satisfies the conditions in Proposition 9. Thus, unique rationalizability isestablished for this type of history, so that Proposition 11 holds.
73

See equation (151) in Online Appendix E.
See the discussion in the paragraph before equation (59) for a demonstration that a rational firm would ignore any
belief that is in Πε0 , but not in Ω.
74

39

Figure 4: Best Response Function, F ε , with µ = 0

xi,t

13

5◦

ln ε φ−1 Rl

πu

πl

π max

b
πi,t

◦

45



b
b
The graph displays F ε ht−1 , πi,tfor πi,t∈ Ω, where Ω is the horizontal segment from ln ε to π max and π max =


ε
ε
b
is constructed using F and ε according to equation
maxx∈Πε0 F (ht−1 , x) and Π0 is defined in equation (61). Also, F ε ht−1 , πi,tNotes:(60). The shaded area corresponds the shaded area in Figure 4a. Also, ht−1 corresponds to a history in which the inflation monitoring
range has never been violated.

5.4.2

Zero Monitoring Range Policy

It is also of interest to consider the zero monitoring range policy in Definition 3, with πl = πu = µ = 0
and φ 6= 0. Consider the case, ht−1 , in which the inflation monitoring range has never been violated.
The best response function for this type of history is the function graphed in Figure 4, with the middle
segment shrunk to a singleton at the origin. Alternatively, it is the second element of F in equationb(37) for all πi,t
. So, for this type of ht−1 , |1 − (γ + ψ) φ/ (γ − 1)| < 1 guarantees that the best response
function lies in the butterfly. This in turn is sufficient to guarantee unique implementation in thesense of Definition 10. In the case of ht−1 in which there has been a violation of the monitoring range,
the result in Proposition 11 applies. We summarize these observations in the form of a proposition
as follows:
Proposition 12. Suppose monetary policy is the zero monitoring range policy (see Definition 3).
Unique equilibrium of the model satisfies unique implementation if and only if|1 − (γ + ψ) φ/ (γ − 1)| < 1,

ρ<1−

γ+ψ
.
2

(67)

then unique equilibrium of the model satisfies unique implementation.
Note that unique implementation requires γ 6= 1 and φ 6= 0. Interestingly, negative or small
positive values of φ are consistent with unique implementation. The logic behind unique implementation of the above policy is essentially the same as it is for the policy considered in Proposition
11. The policy has the effect that if a particular intermediate good producer believes others set highprices, then - via the interest rate channel - the continuation equilibrium is such that that agent’s
best response is to post low prices. What is crucial for unique implementation is that variations in40

beliefs generate relatively small responses. Formally, the responses must be small enough that thebest response function lies inside the butterfly. This result is guaranteed when the model parameterssatisfy equation (67).

5.5Discussion of Proposition 11

We now discuss cases where the necessary and sufficient conditions in Proposition 11 are not satisfied,so that unique implementation fails. We describe a baseline set of parameter values which satisfies theconditions of Proposition 11. We then consider perturbations in which the conditions are not satisfied.
In the perturbations, the competitive equilibrium is still unique and it is a strategy equilibrium. This
discussion goes to the core message of the paper: the finding that an equilibrium is unique and that
it is a strategy equilibrium is not sufficient for unique implementation. For a policy to satisfy uniqueimplementation requires stronger restrictions on policy parameters.
Our baseline parameter values are:
ψ = 1/2, γ = 6, φ = 1.5, ρ = −2, µ = 0
The value of ρ implies that if consumption goes up by 1 percent, then monetary policy in the money
growth regime reduces the money stock by 2 percent. Table 1 summarize the restrictions required toguarantee various equilibrium properties. The baseline parameters satisfy all properties in Table 1.
We now perturb the value of each parameter so that unique implementation fails.
Table 1: Parameter Restrictions(a) Taylor strategy
φ

ρ

γ

ψ

Uniqueness of competitive equilibrium (CE)φ>1

No Restriction

γ>0

ψ≥0

CE is a strategy equilibriumNoneρ < min {1, γ}

γ 6= 1None

CE is uniquely implemented

γ−1
φ < 2 γ+ψ

ρ < −ψ

γ >2+ψ

(b) Zero monitoring range policy
φ

ρ

γ

ψ

Uniqueness of competitive equilibrium (CE)φ>0

No Restriction

γ>0

ψ≥0

CE is a strategy equilibriumNoneρ < min {1, γ}

γ 6= 1None

CE is uniquely implemented

0 < γ+ψγ−1 φ < 2

ρ < 1 − γ+ψ2Note: in each table, the first row indicates parameter restrictions used to establish uniqueness of equilibrium for our model (Proposition2); subsequent rows indicate incremental restrictions to obtain the results in the first column (these correspond to Propositions 6 and 11).

41

Consider φ first. Given our baseline model parameters, the upper bound on φ is 1.53. The
impact on the best response function of a φ larger than this upper bound can be seen in Figure 5a.
Note that this change causes the best response function to have a slope that is steeper −1. With ab
is associated with a bigger increase in Ri,t in the continuation
higher value of φ an increase in πi,tequilibrium, which in turn results in a bigger drop in ci,t and, hence, the wage. The result is thatthe ith firm’s best response function no longer lies interior to the butterfly, and so it violates our
sufficient condition for unique robust rationalizability (see Proposition 9). Intuitively, a high valueof φ has the consequence that the ith firm’s action is more sensitive to variations in its beliefs, aphenomenon that makes unique (robust) rationalizability less likely. Consistent with our discussion
in Section 2, when φ is too large, the ‘leaning against the wind, but not too aggressively’ principlebis violated. To see that unique rationalizability does not hold, consider a belief, πi,t6= 0, that lies
inside the monitoring range (see Figure 5a). Because the best response function has a slope steeper
than −1, the inverse of F has a slope less than unity in absolute value. Thus, a chain of justificationb
which converges into the Nash equilibrium (i.e., the origin in
can be constructed for the posited πi,t
the figure). Since the Nash equilibrium is justified, the entire chain of beliefs is justified and so thebis rationalizable.75 We conclude that when φ lies above its upper bound, then rationality
posited πi,tand CK are not sufficient for agents to coordinate on the unique competitive equilibrium. When φis too large, inflation expectations are not anchored.
Next we consider a perturbation in γ. Figure 5b displays what happens if γ < 1 but close
to unity. There is a segment of the best response function that lies interior to the butterfly, but
the best response dramatically leaves the butterfly outside the inflation monitoring interval, [πl , πu ].
The reason for this very sharp change in the best response function is the presence of 1 − γ in the
denominator of F outside the monitoring range. The switch in the value of γ relative to unity reflects
the switch in the sign of the response of equilibrium consumption to a change in the nominal interestbrate. With γ < 1 a rise in πi,tdrives Ri,t up, but the real rate down, and so induces households
to increase consumption (see equation (44)). This in turn raises the wage rate and induces the ithbintermediate good firm to increase its price by even more than the rise in πi,t(see equation (45)).
So, the Taylor strategy violates the leaning against the wind principle discussed in Section 2 andunique rationalizability fails. I also fails because there is a second Nash equilibrium.76 The failure of75

Note that the inverse of F actually has two elements. In the text we focused on the upper element, while the otherelement has to do with the region to the left of the zero lower bound, Rl /φ. For rationalizability, one need only find
one chain of justification (see Definition 8), so that nothing was lost in the text when we ignored the lower element in
the inverse of F .
76
This example is reminiscent of the example in the introduction to Kocherlakota (2018), which motivates hisanalysis. Essentially, his example is version of the example in our Figure 1a, in which the best response function cuts
the 45◦ line from below. Suppose we apply our cognitive refinement for a specific ε > 0 but small. In this case, thebest response function becomes horizontal for expected inflation, πib , at or below ln ε (see equation (60) and also thehorizontal segment in Figure 5b). The horizontal segment extends from the best response function to the 45◦ line,
so that the cognitive gives rise to a second Nash equilibrium. If we introduce an analogous cognitive impairment inperceiving large numbers (see Online Appendix C.3), then a similar horizontal segment would appear for sufficiently
high inflation. This change would give rise to a third Nash equilibrium. In this example the cognitive impairment has
the effect of compactifying the space of expected inflation. As in Kocherlakota (2018)’s discussion, this compactification
increases the number of Nash equilibria. This effect on the number of Nash equilibria reflects that a best response42the leaning against the wind principle in this example was discussed informally in Subsection 2.2.2
above.
Figure 5: Best Response Functions with Perturbations on Baseline Parameter Values(a) Best Response Function φ is bigxi,tbxi,t = πi,t+

φ−1 Rl πl



γ+ψ
1−γ





(b) Best Response Function γ < 1 but γ close to unity
xi,t




b
min Rl , φπi,t−µ
φ (γ + ψ) b
xi,t = 1 −
πi,t
γ

b
πi,t

πu

b
πi,t

πu

πl

ln ε
φ−1 Rlφ (γ + ψ) b
xi,t = 1 −
πi,t
γ

◦

45

13 ◦
5

◦

ln ε

45

b
xi,t = πi,t
+



γ+ψ
1−γ





(c) When ρ = 0
xi,t =

ρ+ψ 
ρ − γ πb

i,t −

xi,t
γ+ 
ψ
ρ−γ µ+

ln M̄t−1
Pt− + µ
1

−R

l



b
πi,t

µ + ln

◦

45

Mt−1
1−ρ
+
(µ − Rl )
Pt−1
1−γ





γ+ψ
γ+ψ
M
b
xi,t = 1 −πi,t+µ + ln t−1
1−ρ
1−ρ
Pt−1

13 ◦
5

Next, we turn to rationalizability in histories, ht−1 , in which the inflation monitoring range has
been violated at least once. According to Table 1 the money growth rule must have a ‘leaning against
the wind’ feature, ρ < −ψ. Since ψ ≥ 0, there is no way that ρ = 0 can satisfy the sufficient conditions
for implementation. To see why this is the case consider Figure 5c. The slope of the best response
function, F, reflects the assumptions, ρ = 0 and ψ > 0. The best response of intermediate goodfunction cutting the 45◦ line from below violates the butterfly condition, (iii) in Proposition 9. For example, when thebest response function cuts the 45◦ from above, so the butterfly condition is satisfied, then it is easily verified that ourrefinement and even its extension to large numbers does not affect the number of Nash equilibria. Kocherlakota (2018)
argues that analysis in models where the number of Nash equilibria is not robust to compactification can be very
misleading. We reach a similar conclusion: analysis of the competitive equilibrium of a model in which the butterfly
condition is not satisfied is misleading because there is no basis for the notion that the equilibrium would actuallyoccur. For that equilibrium to occur it is necessary for the butterfly condition to be satisfied.

43

135 ◦




b
min Rl , φπi,t−µproducers is very sensitive to variations in their beliefs. We discussed the case, ρ = 0, in Section 4.1.5.
b
There, we explained that when a firm contemplates a low value of πi,t
, the associated continuation
equilibrium has a low real interest rate, thus stimulating consumption and raising marginal cost. This
is why the best response function has a negative slope. As it turns out, when ρ = 0 that slope is
steeper than −1. This is too steep to allow iterated deletion to do its work and so is not compatiblebwith rationalizability. By making ρ < 0, the problem is fixed. As ci,t rises with a low πi,tthen theperiod t money stock is reduced and this mitigates the decline in the real rate, flattening the best
response function. When ρ < −ψ the best response lies inside the butterfly diagram and we obtain
unique implementation.

6

The Taylor Strategy Versus Two Alternatives

We have identified parameter restrictions, equation (63), under which three apparently different
monetary policy regimes uniquely implement77 desired inflation: (i) the Taylor strategy - φ > 1 andπl ≤ πu ; (ii) the zero monitoring range strategy - φ 6= 0 and πl = πu = µ = 0;78 and (iii) a puremoney growth rule (see Lemma 1). We explain how these three different policy strategies illustrate theobservation in Taylor (1999) that many apparently different policy arrangements, including the Gold
Standard and money growth rules, can be thought of as interest rate rules which stabilize inflation byraising the interest rate in case inflation rises.79 According to Taylor (1999), the important questionis which interest rate policy works best. In the deterministic model described thus far, each of ourthree policies performs equally well because it supports the same desired allocations.
In this section we introduce two modifications which, while preserving much of the tractability ofthe existing model, allows us to also incorporate considerations that make the model more realistic.
With these changes, the model suggests that the best policy among those considered here is the Taylor
strategy, with πl < πu . We make the argument in two steps. First, we point out that zero monitoring
range policy, (ii), has a knife edge property: remaining in the Taylor rule regime is not robust to
arbitrarily small trembles by an arbitrarily small number of intermediate good firms. Without further
changes, this is of no importance in our existing model because the monetary growth rule to which
the economy switches under trembles also uniquely implements the desired equilibrium. Our second
change is to introduce money demand shocks which, since at least the classic paper of Poole (1970),
are an important motivation for interest rate rules. The idea is that the interest rate rule implicit
in a money growth rule implies excessive, welfare-reducing volatility in the interest rate when there
77

Here, by implementation we mean Definition 10.
Notice unique implementation is consistent with φ < 0 under policy (ii). However, in this case unique implementation requires γ < 1 (see equation (63)). Perhaps surprisingly, the Taylor logic works in this case as well, though
it does so via the real interest rate, not the nominal rate. This can be seen by retracing the logic in the subsection,‘Violation of the inflation monitoring range’ in Section 4.1.4, for the case, φ < 0 and γ < 1.
79
Recall the discussion of the interest rate channel in the case of the Taylor strategy (see the discussion after equation(45)); in the case of the zero monitoring rate strategy (see the discussion after Proposition 7); and in the case of thepure money growth rule (see the discussion after equation (52)).
78

44

are substantial shocks to money demand. As we show below, our model with money demand shocks
captures the Poole (1970) idea. A policy of the type, (i), is welfare superior to a money growth rule
like (iii) or to (ii) when there is the possibility of a tremble.

6.1

TremblesWe consider a history, ht−1 , in which no deviation from the inflation monitoring range has occurred.
We imagine that in period t a tiny subset (mass) of firms make a tiny mistake when they set theirprices. When such mistakes happen, then inflation jumps out of the monitoring range under policy(ii), triggering a regime shift to the money growth rule.
Formally, suppose that intermediate good firms form their beliefs as they do in previous sections,
without imagining the possibility of trembles. They arrive at a unique set of beliefs by iterated
deletion, as in Section 5.80 However, when the time comes for firms to post their period t price,a small fraction of them make a tiny mistake. Say, the hand of each of these firms trembles as it
endeavors to write its price the menu. As in Section 4.1.1, let pbi,t denote the ith firm’s expectation of
how other firms post their price. Then the optimal price chosen by ith firm ispi,t = pbi,t (ci,t )γ+ψ υi,t ,(68)

where ci,t denotes consumption in the continuation equilibrium associated with pbi,t . Suppose pbt isthe unique belief that survives iterated deletion, so that all firms, i ∈ [0, 1], adopt it. Then, eachfirm (not being aware that a tremble might occur) resolves to set its own price to pbt . This being the
Nash belief, it follows that ci,t = 1 in the continuation equilibrium imagined by each of the i firms.81
Next, suppose that there is a tremble in the form of the additional variable, υi,t , on the right side ofequation (68). That tremble is a unit mean random variable which is drawn independently by eachfirm. The distribution of υi,t has two parameters, Jt , δt ∈ [0, 1]. With probability 1 − Jt , the ith firm
does not tremble at all, so that υi,t ≡ 1. With probability Jt the ith firm draws υi,t from a uniform
distribution with support, [1 − δt , 1 + δt ]. We can allow both Jt > 0 and δt > 0 to be arbitrarily close
to zero.

According to equation (13), the actual price index in the ht−1 , pbt continuation equilibrium isdetermined as follows:
Pt =

Z 1
0

where

1−ε
pbt υi,t
di

"

1
 1−ε

= pbt exp (κt )(1 + δt )2−ε − (1 − δt )2−ε
exp (κt ) = 1 − Jt + Jt
2δt (2 − ε)
80

(69)
1
# 1−ε

.

We assume that the parameter restrictions required for Proposition 11 are satisfied.
As was discussed earlier, ci,t = 1 in the continuation equilibrium associated with the Nash equilibrium belief. See
equation (46)8145

As expected, κt → 0 as Jt → 0 or δt → 0. Divide both sides of equation (69) by µ̄∗ Pt−1 take logs andmake use of the definition of ln pbt / (µ̄∗ Pt−1 ) in equation (30) to obtain:πt = πtb + κt ,(70)

where πtb = 0, the uniquely rationalizable belief.82 Since realized inflation at date t is different fromzero, πt = κt 6= 0. Since the inflation monitoring range is violated, the tremble pushes the economy
into the money growth regime.83

6.2

Money Demand Shocks

In the model as it is set up now, the regime shift does not matter from a welfare standpoint. The
same desired equilibrium outcomes occur whether the economy is following the Taylor rule or themoney growth rule. But, if we assume money demand shocks are realized after firms set their prices,then there is a substantial loss in switching from the Taylor rule to the money growth rule. The
assumption that the money demand shocks are realized after firms set their prices is a way to capture
the notion that money demand shocks operate at a higher frequency than price changes. The model
is a variant of the sticky price model in Christiano et al. (1997), where time t prices are predeterminedwhen time t shocks are realized.
We introduce money demand shocks, νt , by inserting them into the cash constraint, equation (7):Pt ct ≤ (mt−1 + Wt lt − dt ) exp (νt ) .
We assume νt is a mean-zero stochastic process, distributed independently over time. With thischange in the model, it is no longer analytically tractable. As a result, we study a log-linear approximation of the model in a neighborhood of the desired equilibrium. We redo all the analysis inprevious sections, using local versions of the concepts of equilibrium, rationalizability and implementation studied in the previous sections. The results are similar and are provided in Online Appendix
D. We establish the following:
Proposition 13. Consider the version of the model with trembles (see Section 6.1) and with sufficiently small shocks, νt . Welfare under the zero monitoring range strategy (Definition 3) is lower
than welfare under the Taylor strategy with πl < πu .

As discussed in the introduction to this section, this result is in effect a formalization of the finding
in Poole (1970).
82

See (iv), Proposition 5.
Atkeson et al. (2010) find that their model is robust to trembles. That is because in their model the mapping fromintermediate good prices to the aggregate price index is linear.
83

46

7

Concluding Remarks

This paper analyzes a monetary policy strategy in which an interest rate rule is adopted in normal
times and switches to a monetary growth rule if inflation gets out of hand. We present a modeland describe sufficient conditions, as well as some necessary conditions, under which the inflation
expectation of each agent is firmly anchored at its desired level. The concept of implementation that
we use takes into account that agents take an active role in forming their expectations.
To derive sufficient conditions for expectations to be anchored requires that our analysis be global
and so tractability required that we limit ourselves to a simple model. In our model we identified
a principle - leaning against wind, but not too aggressively - which guarantees that expectations
are anchored. We conjecture that this principle generalizes to the type of models used in empiricaland policy analysis. Our analysis of the necessary conditions for expectations to be anchored in a
simple New Keynesian model with price-setting frictions lends support to our conjecture.84 Since themethods used to identify necessary conditions are based on linear approximations in a neighborhoodof steady state, they can be generalized to the kinds of models used in practice.

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49

A

Appendix: Proof for Proposition 9

In order to prove Proposition 9, we use the following lemma.
Lemma 3. Suppose that F satisfies the conditions in Proposition 9. Then for any set C ⊂ X1 ,sup |F (x)| < sup |x| .
x∈C

(71)

x∈C

(x)|
Proof. Let δ = supx∈X1 \{0} |F|x|< 1. Then for all x ∈ C,|F (x)| ≤ δ |x| . Taking the supremum of
the right-hand-side, we have for all x ∈ C,|F (x)| ≤ δ supx∈C |x| . Now taking the supremum of theleft-hand-side, we havesup |F (x)| ≤ δ sup |x| < sup |x| .
x∈C

x∈C

x∈C

The last inequality holds because δ < 1. We establish equation (71).
The following is a proof for Proposition 9.
T
Proof. Since 0 ∈ Xn for all n ≥ 1, 0 ∈ limn→∞ Xn = ∞n=1 Xn . So it suffices to show that there∞
exists a decreasing sequence of {Yn }n=1 such that Xn ⊂ Yn , and Yn → {0} . We define recursively
Yn as follows. Set Y1 as follows: Y1 = [−α1 , α1 ] , where α1 = supx∈Y1 |x|. For all n ≥ 2, set
Yn = [−αn , αn ] ,where
αn = sup |f (x)|(72)
x∈Yn−1

Then it is trivial that Xn ⊂ Yn for all n. Also notice that αn is also expressed as follows:
αn = sup {|x| ; x ∈ Yn } = sup {|f (x)| ; x ∈ Yn−1 } .

(73)

We show that αn is strictly decreasing and converging to 0.
Suppose that αn is not strictly decreasing. So there exists k such that 0 < αk ≤ αk+1 . Fromequation (72), αk ≤ supx∈[−αk ,αk ] |f (x)| . Equation (71) in Lemma (71) implies that
αk ≤

sup
x∈[−αk ,αk ]|f (x)| <supx∈[−αk ,αk ]|x| = αk ,which is a contradiction. Therefore αk > αk+1 .
A decreasing sequence bounded by below has a limit αk ↓ α∗ . Now we show that α∗ = 0. Supposethat α∗ > 0. Then again equation (71) in Lemma (71) implies that
α∗ =

sup
x∈[−α∗ ,α∗ ]|f (x)| <supx∈[−α∗ ,α∗ ]|x| = α∗ ,which is a contradiction again. Thus, α∗ is 0. Thus we conclude that Xn ⊂ Yn ↓ {0} so that
Xn → {0} as n → ∞.
50

BAppendix: Proof of Proposition 11

Proof. That the equilibrium is a strategy equilibrium follows from the fact that conditions (i) and (ii)of the Definition of a strategy equilibrium (see Definition 6) are guaranteed by Proposition 5. Thisestablishes part (i) of Definition 10. It remains to establish part (ii) of that definition. We restrict
the domain of F (ht−1 , ·) to Πε1 . This domain satisfies condition (i) of Proposition 9. By (iii) of
Proposition 5, the fixed point, π (ht−1 ), of F (ht−1 , ·) exists and is unique, so that (ii) of Proposition 9
is satisfied. Online Appendix C.4 formally establishes that conditions (63) guarantees that condition(iii) of Proposition 9 is satisfied for any ht−1 . It is easy to see this informally for histories, ht−1 , inwhich there has never been a violation of the inflation monitoring range. In this case, the graph of F
is strictly interior to the butterfly graph except at the unique Nash equilibrium (see Figure 4). Withthe three conditions of Proposition 9 satisfied, we conclude that F k (ht−1 , Πε1 ) → π (ht−1 ) as k → ∞.

T T∞
k
ε
The object on the right of (iii) in Proposition 16, Πε1
k=1 F (ht−1 , Π1 ) ,is the set containing thesingleton π (ht−1 ). Condition (iii) of that proposition then implies that Π∗ (Πε1 ) = {π (ht−1 )} for eachπ (ht−1 ) > ε > −∞. It follows trivially that[−∞<ε≤π(ht−1 )Π∗ (Πε1 ) = {π (ht−1 )} .

Also, π (ht−1 ) is rationalizable because it is a Nash equilibrium, π (ht−1 ) = F k (ht−1 , π (ht−1 )). So,
conditions (i) and (ii) of Definition 10 are verified. This establishes that conditions (i) and (ii) of
Definition 10 are satisfied. The result follows.

51

Online Appendix to ‘Anchoring Inflation Expectations’Lawrence J. Christiano and Yuta Takahashi

For convenience, the title of each section below has the form, ‘Appendix: XX’, where XX corresponds to the title of a manuscript section.

A

Appendix: Competitive Equilibrium

In this section we establish results for Section 3 in the paper. The first three subsections below derivevarious properties of the competitive equilibrium. In the first subsection we describe restrictions onmarket prices that are necessary for an equilibrium to exist. Moreover, for the household problem
to be well defined in an equilibrium, it is necessary that prices have the property that all budgetfeasible choices of the household result in finite discounted utility. The second subsection takes theserestrictions as given and establishes conditions on household choices that are necessary and sufficientfor their optimization problem.

A.1

Equilibrium Restrictions

We derive six properties of the set of equilibrium prices. To describe the first property, for convenience,
we repeat the representative household’s objective and constraints:

lim

T →∞

T
X

βt

t=0

"

#
lt1+ψ
ct1−γ
−
,
1−γ 1+ψ

γ > 0, γ 6= 1, ψ > 0

s.t. mt ≤ R̄t (Xt + dt ) + mt−1 − dt + Wt lt − Pt ct + Tt ,Pt ct ≤ mt−1 − dt + Wt lt

(74)
(75)
(76)

m−1 given,∞
A budget-feasible sequence, (ct , lt , mt , dt )∞
,
has
the
property
that,
given
P
,
R̄
,
W
,
T
,
X
and
t
t
t
t
t
t=0
t=0
m−1 , it satisfies the flow budget constraints (75), the cash constraints (76), and the following nonnegativity constraints
0 ≤ (ct , lt , mt , dt ) , lt ≤ N.
(77)
Here, N > 0 denotes the household’s endowment of time.
We define a feasible sequence, (ct , lt , mt , dt )∞t=0 , as a budget-feasible sequence for which the limit ofpartial sums in equation (74) is finite. The household problem is to maximize equation (74) over the1

set of feasible sequences. Our first property of prices is that the feasible set of sequences is non-empty.
If this were not so, then there would not be an equilibrium.
We briefly discuss why we restrict ourselves to a subset of the set of budget-feasible sequences. In
the case, γ > 1 restricting ourselves to the set of feasible sequences is done without loss of generality.
In this case, the elements in the partial sum in equation (74) are all negative. As a result, thesequence of partial sums either converges to a finite, non-positive number or to −∞. We can ignore
sequences which generate −∞ utility because the household would always prefer one of the feasible
sequences. In case of γ < 1 the limit of partial sums in equation (74) is bounded above and below.
The reason is that we only consider equilibrium prices. In an equilibrium it must be that ct = lt > 0
and lt ≤ N , so that period utility is bounded above and below.
Our second property of prices is that Pt > 0 for all t. If in any period Pt ≤ 0, then no feasible
sequence is optimal for the household. For any sequence of consumptions, higher consumption would
also be feasible and it would generate higher utility. Third, consider the nominal interest rate. In
many models, R̄t < 1 cannot be an equilibrium because households could in that case be able toturn the loan market into a money pump, allowing unbounded consumption in t + 1. In our model
households cannot borrow, so their optimization problem may be well-defined for R̄t < 1. However,as discussed in Section 3.2, the fact that firms borrow implies R̄t ≥ 1 in any equilibrium. If theR̄t < 1 in some period, then it is the intermediate good firm that can turn the loan market into amoney pump in that period. With R̄t < 1 the firm problem has no solution. For any high level ofborrowing it would earn even more profits by borrowing more. After paying off the extra principle
and interest on any borrowing, it would have extra money left over which it could send as additional
profits to its owner. Fourth, Wt > 0 is necessary for an equilibrium. If the condition were not met,then for any level of employment the firm could earn more profits by hiring more workers.
Fifth, it will be useful for us to use the concept of a date 0 price of a time t good, qt , defined in
the following way:
1
t=0
qt = Q.
(78)
 t R̄−1 t ≥ 1
s=1 s
Because R̄t > 1 it follows that qt > 0 for all t. Taking into account equation (77) we also knowthat qt mt ≥ 0 for each t. Sixth, firm optimization implies Pt = Wt and the resource constraint andtechnology imply ct = lt . Summarizing, we have
Wt , Pt > 0, Pt = Wt , ct = lt , R̄t ≥ 1, qt mt ≥ 0,(79)

for each t.

A.2

Restatement of the Household Problem

We find it convenient to reformulate the household problem in the canonical form used by Stokey etal. (1989). This allows us, in the next section, to easily apply the arguments in Stokey et al. (1989)
2

and Kamihigashi (2002) to obtain the necessary and sufficient conditions for household optimality.
Of course, we solve the household problem under the assumption that prices correspond to those inan equilibrium.
We impose (obviously, without loss of generality) the flow budget constraint, (6), as a strict
equality. We use that equation to solve for lt :
mt − R̄t (Xt + dt ) + mt−1 − dt + Tt − Pt ct
,
lt =
Wt

(80)

for all t ≥ 0. We denote the date t choice variables by the 3 × 1 column vector xt :xt = (ct , mt , dt )0 ,We denote the period utility function in the reformulated system by the function, F :mt − R̄t (Xt + dt ) − mt−1 + dt − Tt + Pt ct
F (xt−1 , xt ) = u ct ,
.
Wt

(81)

Note that F is concave and differentiable in (xt−1 , xt ) since u is has these properties in c and l.
Obviously, the function, F (xt−1 , xt ) , is not only determined by the household choice variables(xt−1 , xt ), but also by the other variables, Pt , R̄t , Wt , Tt , Xt and m−1 , which are beyond the controlof the household. To keep the notation simple, we do not make the dependence of F on the latter
variables explicit.
We now consider the various inequality constraints in the model. By equation (80), the upperand lower bound constraints on labor imply:0 ≤ mt − R̄t (Xt + dt ) + mt−1 − dt + Tt − Pt ct ≤ Wt N,(82)

for all t ≥ 0. These inequalities are not binding on the households at equilibrium prices. Equilibrium
conditions guarantee that the first inequality is strict at equilibrium prices. To see this, recall fromequation (79) that, in an equilibrium, Pt = Wt . So, equality of the household marginal rate of
substitution with the real wage implies cγt ltψ = Wt /Pt = 1. From this it follows that ct and lt arestrictly greater than zero. The second inequality in equation (82) is strict because we do not restrict
the magnitude of N.
We express the household’s cash constraint (after substituting out for Wt lt from equation 80) as
well as the restrictions, mt ≥ 0 and dt ≥ 0,respectively, as follows:At xt ≥ bt .

3

(83)

Here, At and bt are sequences taken as given by the household:
0 1 −R̄t
R̄t Xt + TtAt =  0 1 0  , bt = 
0
.
0 0 1
0
In any equilibria, bt becomes positive, so that we assume that bt ≥ 0. We do not include the restriction,
ct ≥ 0, in equation (83) because we showed above that it is non-binding at equilibrium prices. Anotherinequality in the model is the last condition in equation (79):qt Ãt xt ≥ 0, ÃT =

h

0 1 0

i

.

(84)

The date 0 problem of the household given equilibrium prices is:max∞{xt }t=0 ∈C

TXlimT →∞

β t F (xt−1 , xt ) ,

(85)

t=0

subject to the restriction in equation (83). Here, C denotes the set of sequences for which the limit ofpartial sums is finite. We assume the set, C, is non-empty for otherwise there would be no equilibrium.
The set, C, is a function of the equilibrium prices, but to keep the notation simple we do not makethat dependence explicit.

A.3

Necessary and Sufficient Conditions for Household Optimality

In this section we discuss the necessary and sufficient conditions for household optimization at equilibrium prices. That the first order conditions (see equations (8), (9) and (87)) are part of the necessary
and sufficient conditions is easy to verify and we do not discuss that here. Instead, we focus on thesufficiency and necessity of a transversality condition.
The first order conditions are:F2 (xt−1 , xt ) + µt × At + βF1 (xt , xt+1 ) = 0,(86)diag (µt ) (At xt − bt ) = 0,(87)

µt ≥ 0,(88)

plus equation (83). Here, F1 and F2 denote 1 × 3 row vectors of derivatives with respect to thefirst and second arguments, respectively. Also, µt is the 1 × 3 row vector of multipliers on the cash
constraint, At xt ≥ bt . diag (µt ) is the 3 × 3 square matrix whose diagonal elements are µt .
The transversality condition is:lim β T F1 (xT −1 , xT ) xT −1 = 0.

T →∞

4

(89)

To interpret (89) in terms of our underlying model, note
β T F1 (xT −1 , xT ) xT −1 = β T−ul,Tc−γ R̄T
mT −1 = 0
qT −1 mT −1 ≥ 0.
WT
P0 R0

(90)

We let ul,t and uc,t denote the partial derivatives of household utility with respect to lt and ct ,respectively. The first equality in equation (90) uses the definition of F in equation (81). The
second equality uses the labor first order condition, ul,T = −uc,T WT /PT , the intertemporal first ordercondition,
Pt−1
uc,t
×
= R̄t−1 ,βPtuc,t−1and the definition of qt in equation (78) for t = 1, 2, ..., T . The inequality in equation 90 follows fromequation (79). According to equation (90), equation (89) requires that the present value of money
should be zero in the limit as T → ∞.
A.3.1

Sufficiency of Transversality Condition

According to the following proposition, if a budget-feasible sequence, {xt }∞t=0 , satisfies (86), (87) and(88) for t ≥ 0 and the transversality condition, (89), then that sequence solves the household problem
in that there is no other feasible sequence that generates higher utility.
∞
Proposition 14. Suppose prices correspond to a competitive equilibrium and let {x∗t }∞t=0 and {µt }t=0denote a sequence of decisions and multipliers associated with the household problem. If the decisions
and multipliers satisfy (86), (87), (88), and (89), then the sequence, {x∗t }∞t=0 , solves the householdproblem, (85).

Proof. The rest of the proof style follows that of Theorem 4.15 in Stokey et al. (1989). Let {xt }∞t=0
denote an arbitrary feasible sequence. Define
DT =

T
X
t=0




β t F x∗t−1 , x∗t − F (xt−1 , xt ) .

We wish to show thatlim DT ≥ 0.

T →∞First, concavity of F implies:
F (xt−1 , xt ) ≤ F1 x∗t−1 , x∗t
Then,
DT ≥

T
X
t=0





xt−1 − x∗t−1 + F2 x∗t−1 , x∗t (xt − x∗t ) .






β t F1 x∗t−1 , x∗t x∗t−1 − xt−1 + F2 x∗t−1 , x∗t (x∗t − xt ) .
5

Collecting terms
DT ≥ F1 x∗−1 , x∗0x∗−1 − x−1



+

T −1
X
t=0





β t F2 x∗t−1 , x∗t + βF1 x∗t , x∗t+1 (x∗t − xt ) + β T F2 x∗T −1 , x∗T (x∗T − xT ) .

or, after using equation (86):DT ≥ F1x∗−1 , x∗0x∗−1 − x−1



+

T −1
X
t=0


β t [−µt At (x∗t − xt )] + β T F2 x∗T −1 , x∗T (x∗T − xT ) .

It is easy to show that
T −1
X
t=0

β

t

[−µt At (x∗t − xt )] =

T −1
X
t=0

β t [(−µt (At x∗t − bt ) + µt (At xt − bt ))] .

By the complementary slackness condition, (87), µt (At x∗t − bt ) = 0 for all t. Also since {xt }∞t=0 is∗budget-feasible, At xt − bt ≥ 0 for all t ≥ 0. Thus, combining with the fact that µt ≥ 0 and x−1 = x−1 ,we getDT ≥ β T F2 x∗T −1 , x∗T (x∗T − xT ) .

Again using equation (86) again, we haveDT ≥ β T F2 x∗T −1 , x∗T (x∗T − xT )
= β T −βF1 x∗T , x∗T +1 − µT AT (x∗T − xT )≥ −β T +1 F1 x∗T , x∗T +1 x∗T + β T +1 F1 x∗T , x∗T +1 xT .

The transversality condition, (89), and feasibility of xt imply that, respectively,lim β T F1 x∗T −1 , x∗T x∗T = 0,

T →∞We conclude thatlim β T +1 F1 x∗T , x∗T +1 xT ≥ 0.

T →∞lim DT ≥ 0.

T →∞DT is the difference between two partial sums and the regularity conditions in (i) guarantee thatthose partial sums converge. This establishes our result.
A.3.2

Necessity of Transversality Condition

We turn to the necessity of 89. We establish the result using the argument in Kamihigashi (2002).
We show that his argument applies almost without change, even though he works with a version of
the Stokey et al. (1989) model, while ours is a monetary model with a cash constraint.
6

Proposition 15. Suppose prices correspond to a competitive equilibrium and {x∗t }∞t=0 is a solution∗ ∞to the household problem, (85). Then, {xt }t=0 satisfies (89).
∗ ∞Proof. Consider a class of perturbations, {xt (λ, T )}∞t=0 , on the optimal path, {xt }t=0 :xt (λ, T ) =


x ∗
t

λx∗
t

t≤T

,

t>T

where γ ≤ λ < 1. Also, γ is a scalar, 0 < γ < 1, having the property that {xt (λ, T )} is a feasible
interior sequence with finite discounted utility value for each λ ∈ [γ, 1). Feasibility is trivial andimposes no restriction on γ, since At x∗t ≥ bt implies At xt (λ, T ) ≥ bt for all t ≥ 0, and each λ ∈ [γ, 1).
Similarly, the natural debt limit, (84), also places no restriction on γ. To see this, note thatlim qt Ãt xt (λ, T ) = λ lim qt Ãt x∗t ≥ bt .
T →∞

T →∞

That we can always choose a value of γ (perhaps very near to unity) so that {xt (λ, T )}∞t=0 has finite
discounted utility follows from continuity of F.
Since {x∗t } is optimal, it follows that:
∞>

∞
X
t=0

t

βF

x∗t−1 , x∗t



≥

∞
X

β t F (xt−1 (λ, T ) , xt (λ, T )) .

(91)

t=0

Kamihigashi (2002)’s argument exploits a property of concavity. In particular, the fact that the slopeof a concave function, f : [γ, 1] → R, is declining implies
f (1) − f (λ)
f (1) − f (γ)
≤
,
1−λ
1−γ

(92)

for λ ∈ [γ, 1) (see Kamihigashi (2002), Lemma 3). Rearranging 91, we obtainβ

T +1 F





∞
∗
∗
∗
∗
X
x∗T , λx∗T +1 − F x∗T , x∗T +1
t+1 F xt , xt+1 − F λxt , λxt+1
≤
β
1−λ
1−λ
t=T +1


∞
∗
∗
∗
∗
X
t+1 F xt , xt+1 − F γxt , γxt+1
≤
,
β
1
−
γ
t=T +17(93)

where the second inequality is an application of (92). Notex∗T , λx∗T +1 − F x∗T , x∗T +1
β
1−λ

λ↑1
T +1
∗
∗
→−β
F2 xT , xT +1 x∗T +1
=β T +2 F1 x∗T +1 , x∗T +2 x∗T +1 + β T +1 µT +1 AT +1 x∗T +1 − bT +1 + β T +1 µT +1 bT +1
=β T +2 F1 x∗T +1 , x∗T +2 x∗T +1 + β T +1 µT +1 bT +1 ≥ 0.
T +1 F

Here, the limit result follows from differentiability of F , the equality uses (90) and the weak inequality
reflects feasibility.
Driving λ → 1 in (93) and using the latter result, we obtain,
0≤β

T +2

F1
|

x∗T +1 , x∗T +2
{z

≥0



x∗T +1 +β T +1 µT +1 bT +1 ≤
}

|

{z

}

≥0

∞
X

t=T +1

β

t+1 F



x∗t , x∗t+1 − F γx∗t , γx∗t+1.
1−γ

As T → ∞, the term on the right converges to zero since the both infinite sums are finite whensummed over all t ≥ 0. This establishes (89).

A.4

Proposition 1

Proof. For each π0 > 0, there is a sequence of inflation, {πt }∞t=1 , that satisfies (22) in which inflation
explodes to ∞. The interest rate associated with such a sequence is greater than unity for each t,so that (25) is binding, M̄T /PT = cT = 1, for all T . Trivially, the transversality condition, (26), issatisfied.
1
M̄T −1
= βT
→ 0.
βT
PT
πT
Similarly, a sequence that satisfies (24) with π0 < 0 converges to ln (β/π̄ ∗ ), so that actual grossinflation converges to β. To see that this sequence satisfies (26) note that πt converges in finite time
to its fixed point, πt = ln (β/π̄ ∗ ). Suppose convergence occurs at t = t̄ ≥ 0. Then, for T > t̄, RT = 1
so (9) implies PT = β T −t̄ Pt̄ . Setting M̄T = β T −t̄ M̄t̄ for all T > t̄ so that M̄T /PT = M̄t̄ /Pt̄ , a constant≥ cT for all T > t̄, we have that the cash constraint, (25), is satisfied and
lim β

T →∞

T M̄T −1PT

β T −t t M̄T −1
βt
= lim
β
=
lim M̄T −1 > 0
T →∞ PT /Pt
Pt
Pt T →∞| {z }=1

so that (26) is satisfied too. The uniqueness result is stated in Proposition 1.
Appendix:8

A.5

Taylor Strategy

Following is a proof of Lemma 1. For convenience, we restate Proposition 1 here:
Lemma 4. Suppose M̄−1 is given and monetary policy sets M̄t = µ̄cρt M̄t−1 for t ≥ 0,where µ > 1.
There exists a unique competitive equilibrium with the properties:R̄t = β −1 µ̄ > 1,

for t ≥ 0,

ct = 1,and P0 = M̄−1 µ̄.

Proof. First, equations (8), (15) and (19) imply ct = 1, so that monetary policy sets money growthto µ̄. Second, we show that in any equilibrium, R̄t > 1 for all t. Suppose, to the contrary, that R̄t = 1
for some t ≥ 0. Then it follows that R̄t+1 = 1. To see this, note that π̄t+1 = β by equation (9) sothat Pt+1 < Pt . Note also that M̄t+1 /M̄t = µ̄ > 1. Combining equations (10), (16), (18) and ct = 1
we have M̄t ≥ Pt . The results, Pt+1 < Pt and M̄t+1 /M̄t > 1 then imply (using ct+1 = 1) M̄t+1 > Pt+1 .
The complementary slackness condition, (10), then implies that R̄t+1 = 1. By induction, we conclude
that R̄t = 1 implies R̄t+s = 1 and π̄t+s+1 = β for s ≥ 0. Finally,qT = qt ,for all T ≥ t (see the equation after (11)). Note that for any fixed t,
qT M̄T = qt M̄T → ∞,because qt is fixed and M̄T is increasing in T . This contradicts the transversality condition (11), so
we conclude that in any equilibrium, R̄t > 1 for all t ≥ 0.
Third, we show that with R̄t > 1, for all t ≥ 0, the equilibrium conditions uniquely determine
all variables. Equation (25) implies that the cash constraint binds, M̄t − Pt = 0, for each t ≥ 0, so
π̄t+1 = µ̄ for all t ≥ 0. Fourth, the Fisher equation, (9), implies
R̄t = β −1 π̄t+1 =

µ̄
> 1.
β

The cash constraint, c0 = 1 and R̄0 > 1 implies
P0 = M̄0 =⇒ P0 = M̄−1

M̄0
= M̄−1 µ̄.
M̄−1

We have established the results desired.
Following is a proof of Lemma 2:
Proof. Suppose not, so that there exists an equilibrium with πT ∈/ [πl , πu ] where T ≥ 0 is the firstdate in which the monitoring range is violated. Consider the case, πT > πu . The Taylor rule implies
9RT = φπT > πu . Using Lemma 1, it is easy to verify that πT +1 = µ ≤ πu , so that85
RT − πT +1 > 0,(94)

violating the Fisher equation, (21). This contradicts the assumption of equilibrium.

Next, consider the case πT < πl . Then, RT = max Rl , φπT ≤ 0. But, Rl < φπl < πl , so RT < πl .
Also, it is easy to verify πT +1 ≥ µ ≥ πl , so that86
RT − πT +1 < 0,(95)

violating the Fisher equation, (21). This contradicts the assumption of equilibrium. This establishes
the result sought.

B

Appendix: The Market as a Game

B.1

Proposition 4bcontinuation equilibrium in Proposition
Here, we we consider the cash constraint in the ht−1 , πi,t(4). The proposition considers two types of histories, ht−1 : those in which the inflation monitoring
range has never been violated (see Part (i)) and those in which the inflation monitoring range has
been violated at least once (Part (ii)). The cash constraint is always satisfied as an equality in Part (i)bcontinuationby the design of our monetary policy rule. However, our construction of the ht−1 , πi,tequilibrium (see Section 4.1.5) in the second type of history ignored the time t cash constraint. This
is not an issue when the equilibrium interest rate is positive. But, needs to be verified in the casethat the interest rate is zero. We do that here.
b∈/ D (ht−1 ), or, more
According to Proposition (4), the zero lower bound is binding for πi,texplicitly,
1−γAi,t + µ ≤ Rl ,
(96)
1−ρ
where

Ai,t ≡ µ + lnM̄t−1b
− πi,t.
Pt−1

For convenience, we repeat the relevant equilibrium conditions derived in the text. Equation (48)combines the period t + 1 equilibrium allocations as well as the period t and t + 1 money growth rule85

The (unsurprising) result, πT +1 = µ, follows from four observations. Observation #1 (see Definition 2) is that theescape clause is invoked in period T + 1. Observation #2 is that observation #1 and Lemma 1 imply PT +1 = µM̄T .
Observation #3 is that the unscaled interest rate, R̄T , exceeds unity. The latter can be seen from: the definition
after (21), R̄T ≡ R̄∗ exp (RT ); our assumption, R̄∗ > 1; and our assumptions φ, πT > 0 which imply RT = φπT > 0.
Observation #4 is that observation #3, (25) and the equilibrium condition, ct = 1, imply PT = M̄T . The result follows
from observations #2 and #4.
86
The result, πT +1 ≥ µ, is straightforward. As in Footnote (85), Lemma 1 implies PT +1 = µM̄T . Combining
equation (15) and equation (18), ct = 1 for all t ≥ 0. So the cash constraint, (25), implies M̄T ≥ PT . The result
follows.

10

to obtain:
πi,t+1 = µ + ρ ln ci,t + Ai,t .

(97)

The household Euler equation in the zero lower bound, making use of the fact, ln ci,t+1 = 0, is
πi,t+1 = Rl + γ ln ci,t ,(98)

where Rl denotes the scaled and logged interest rate at the zero lower bound (see Section 3.5).
Combining the loan market clearing condition, Wt lt = Xt + dt , (see equation (18)) with the household
cash constraint, equation (7), we obtain:
ln

M̄i,t
≥ ln ci,t
Pi,t

(99)

Finally, the money growth rule is, expressed in terms of real balances,
ln

M̄i,t
= ρ ln ci,t + At .
Pi,t

(100)

Equations (97), (98) and (100) are the three equations used in the text to solve for the threebwhich have theunknowns, M̄i,t , ci,t , πi,t+1 . These are solved given Ai,t , Pi,t implied by ht−1 , πi,tproperty that the equilibrium nominal rate of interest is at its lower bound (i.e., Ai,t satisfies equation(96). Here, we verify that in the equilibrium constructed in in the text, the household cash constraint,(99) is satisfied when ρ < min {γ, 1}.
Combining equation (97) and (98), we can solve for ln ci,t :ln ci,t =µ − Rl + Ai,t
.
γ−ρ

(101)

Also, after substituting out for ρ ln ci,t + At in equation (97) from equation (100) we obtain πi,t+1 =M̄i,tµ + ln Pi,t
, which, when combined with (98) yields
γ ln ci,t = lnM̄i,t
+ µ − Rl ≥ ln ci,t + µ − Rl ,Pi,twhere the weak inequality uses (99). Collecting terms, we have(γ − 1) ln ci,t ≥ µ − Rl .

(102)

Substituting out for ln ci,t in (102) using (101), we obtain,(γ − 1)µ − Rl + Ai,t
≥ µ − Rl .
γ−ρ

11

(103)

This establishes that the equilibrium conditions, (97), (98), (97) and the cash constraint, (99) imply
equation (103). It is easily verified that if equation (103) and the equilibrium conditions, (97), (98),(97), are satisfied, then the cash constraint, (99) is satisfied. We conclude that equation (103) isequivalent to the cash constraint, given our three equilibrium conditions. For this reason, in thisappendix we refer to (103) as the cash constraint.
With this characterization result in hand, we ask whether the cash constraint, (103), is satisfied
at the equilibrium values of ci,t and πi,t+1 derived in the text, for Ai,t such that the nominal interest
rate is at its lower bound (i.e., that satisfy (96)), given our restrictions on γ and ρ.
Consider first the case, γ > 1 and ρ < 1. The interest rate being at its lower bound places thefollowing bound on Ai,t (see equation (96)):
Ai,t ≥1−ρ
(Rl − µ) .
1−γ

(104)

The coefficient on Ai,t in the cash constraint, (103), is positive so the term on the left of that inequality
is minimized by minimizing the value of Ai,t . Putting Ai,t at the minimum consistent with the interest
being at its zero lower bound (see equation (104)), we obtain1−ρ
µ − Rl + 1−γ(Rl − µ)
µ − Rl + Ai,t≥ (γ − 1)= µ − Rl .
(γ − 1)γ−ρ
γ−ρ

We conclude that the cash constraint, (103), is satisfied.
Now consider the case, γ < 1 and ρ < γ. The bound on Ai,t implied by the interest rate being at
its lower bound is (e.g., equation (96)) is
Ai,t ≤1−ρ
(Rl − µ) .
1−γ

(105)

The coefficient on Ai,t in the cash constraint, (103), is negative, so the term on the left of that
inequality is minimized by maximizing the value of Ai,t . Putting it at the maximum consistent with
the interest rate being at its lower bound is (see equation (105)), we obtain:1−ρ
µ − Rl + 1−γ(Rl − µ)
µ − Rl + Ai,t(γ − 1)≥ (γ − 1)= µ − Rl .
γ−ργ−ρWe conclude that the cash constraint, (103), is satisfied. This establish the result we set out to prove.

B.2

Proof of Proposition (5)Result (i) follows from Lemma 1 and Proposition 2. Now consider (ii). Suppose we have a πtb that has
the fixed point property in (ii). The only thing that distinguishes an ht−1 continuation equilibrium and
an ht−1 , πtb continuation equilibrium (see Definition 4) is that the latter does not necessarily satisfy
the time t intermediate good equilibrium condition, Pt = Wt (see equation (15)). This difference can12be described in terms of F . Suppose πtb is an arbitrary belief about how other firms set their (scaledand logged, as in equation (30)) prices. In the continuation equilibrium associated with ht−1 , πtb ,the (scaled and logged) aggregate price index is simply πtb itself. The (scaled and logged) wage ratein the ht−1 , πtb continuation equilibrium is F ht−1 , πtb (see equation (33)). So, a fixed point of F
corresponds to a sequence, (at+j )∞j=0 , that satisfies all equilibrium conditions in periods t,t + 1, ....
Hence the sequence is an ht−1 continuation equilibrium. Now consider the converse in part (ii). Let(at+j )∞
j=0 denote an ht−1 continuation equilibrium. By definition, all period t, t + 1, ... equilibrium
conditions (including equation (15)) are satisfied. Let πt denote period t inflation in this equilibriumand set πtb = πt . Because equation (15) is satisfied, it follows that πtb = F ht−1 , πtb . This establishes(ii).
Consider (iii). First, we establish existence of a fixed point for F, for any ht−1 . According topart (i) there is a unique ht−1 continuation equilibrium. By the converse proof of part (ii) above,it then follows that there exists a fixed point for F. To see that this fixed point is unique, suppose
it is not. If there were two fixed points of F for some history, ht−1 , each corresponds to a distinct
ht−1 continuation equilibrium. This contradicts (i). Finally, consider (iv). Condition (iii) implies

there exists a unique fixed point of F, and condition (ii) implies that the ht−1 , πtb continuation
equilibrium associated with that fixed point coincides with the ht−1 continuation equilibrium. Since,by Proposition 2, inflation in the ht−1 continuation equilibrium is πt = 0, it follows that the same istrue in the ht−1 , πtb continuation equilibrium. But, πtb = πt in the latter equilibrium, establishing
(iv).

C

Appendix: Rationalizable Implementation

C.1

Iterated Deletion and Rationalizability

Let L denote an action taken by others and let l = F (L) denote the individual’s best response to
L. In terms of this notation, Definition 7 is expressed as follows. Consider the following sequence of
sets:
 −k
∞
F (L) k=0 .
(106)Here, F −1 (L) denotes the set of beliefs, Ω, for which L is a best response. That is, F (Ω) = L. Also,F −2 (L) ≡ F −1 (Ω) .
The set, F −2 (L) , is the set of beliefs which justify the actions corresponding to each element in Ω. The
sets, F −n (L) , for n > 2 are defined similarly. Finally, F −0 (L) ≡ {L} . If, for all n, F −n (L) 6= {∅},
then there exists at least one sequence of numbers in equation (106) that is a chain of justified beliefs
that supports L. If so, then according to Definition 7, we say that L is rationalizable.
Under the method of iterated deletion, the agent’s initial candidate set of beliefs corresponds to13

the set of justified beliefs, which we denote by Π1 . Then F (Π1 ) denotes the set of best responses
associated with some justified belief. Suppose there is a value of l ∈ Π1 which is not a best responseto any L ∈ Π1 , so that there no circumstances in which the individual would choose l. Under CK,
the individual would also expect that others would never choose such an action, and so it would bedeleted from the set of candidate beliefs. Specifically, the individual would restrict its beliefs aboutTwhat others do to Π2 = Π1 F (Π1 ). 87
T
This process can be repeated, leading to a sequence, Πk+1 = Πk F (Πk ), for k = 1, 2, ... . Noticethat by construction, the sequence is non-decreasing, Πk ⊆ Πk−1 for k = 2, ... . Consider the set ofelements, L ∈ Π1 , such that they remain undeleted even as k → ∞:Π∗ (Π1 ) =

∞
\

Πk .

(107)

k=1

So, the set, Π∗ (Π1 ) , is composed of the beliefs in Π1 that survive iterated deletion. We will also
have occasion to apply Π∗ to sets different from the set of justified beliefs, Π1 . When the argumentof Π∗ (·) is Π1 , we economize on notation by dropping the argument. Following is a characterizationof the relationship between Π∗ and the set of rationalizable beliefs:
Proposition 16. The following hold:(i) L is rationalizable if and only if L ∈ Π∗ ;(ii) if there exists a set Π having the property, F (Π) = Π, then each element in Π is rationalizable;(iii) Π∗ (Π) ⊆ Π


k
F
(Π)
for any set, Π, that is a subset of the domain of F . 88
k=1

T T∞

The proof is provided in the next section.
Result (i) is useful because sometimes it is more convenient to study rationalizability by chainof justification and at other times iterated deletion is more convenient. Part (ii) of Proposition 16
provides another characterization of rationalizability that is useful for us below. Part (iii) will bevery useful for us below. It implies that if the object on the right of ‘⊆’ is a singleton when Π = Π1 ,then Π∗ (Π1 ) is a singleton too, and we have (by part (i)), unique rationalizability. 89

C.2

Proof of Proposition 16

Proof. (i) (Sufficiency) Suppose, to the contrary, that L ∈/ Π∗ but L is rationalizable according to
Definition 8. By rationalizability, L ∈ Π1 . The fact, L ∈/ Π∗ , implies by equation (107), that there
exists a k such that L ∈/ Πk but L ∈ Πk−1 . Since L is rationalizable, F −k (L) is not empty and is a
T
If C is a set, then g (C) = {y : y = g (x) , for x ∈ C} . Also, A B means {y : y ∈ A, y ∈ B}.
88
Here F 2 (A) ≡ F (F (A)) , and F k (A) is recursively as follows: F k (A) = F F k−1 (A) for all k ≥ 2.
89
If F (Π) ⊆ Π then, the ‘⊆0 can be replaced by an equality.

87

14

subset of Π1 . It then follows that L ∈ F k (Π1 ) . In particular, there exists a sequence of {L0s }k−1s=1 suchthatL = F L0k−1 , L0s = F L0s−1 , L01 ∈ Π1 ,for all s ≤ k − 1. Note that Ls ∈ Πs for all s ≤ k − 1 by construction. So, L ∈ Πk , a contradiction.
(Necessity) Suppose that L ∈ Π∗ , but is not rationalizable. If follows by Definition 7 that there
does not exist a chain of justified beliefs that supports L. This means that there exists a k ≥ 1 suchthat F −k (L) = {∅} and F −s (L) 6= {∅} for all s ≤ k − 1. It follows that F k−1 (L) ∈/ Π1 and, if k > 1,s
F (L) ∈ Π1 for all s ≤ k − 1. So, L ∈/ Πk , a contradiction.
(ii) Consider an arbitrary L1 ∈ Π, where F (Π) = Π. There exists a L2 ∈ Π which justifies L1 . By
induction, we can construct a sequence, {Lk }∞k=1 , recursively. Therefore L1 is rationalizable.
(iii) The claim immediately holds since Πk+1 ⊂ F (Πk ) for all k ≥ 1.

C.3

Impact of Refinement on Competitive Equilibria

In the body of the paper we show that our refinement has the effect of trimming economicallyuninteresting off-equilibrium paths under the Taylor rule with an escape clause. Could it be thatthe refinement trims all undesired competitive equilibria, eliminating the need for the escape clausein the first place? Here, we show that the answer is ‘no’. The set of equilibria with the Taylor ruleand no escape clause is essentially the same with or without our refinement. Section 3.5 indexes theequilibria by the initial gross rate of inflation, π̄0 > 0.90 The refinement eliminates a minuscule set of
equilibria by trimming those with π̄0 < π̄ ∗ ε for arbitrarily small ε > 0.91
But, what if we extend our refinement so that intermediate good firms have a similar cognitive
impairment which prevents them from being able to distinguish large numbers? Suppose we adopt
a cognitive impairment which results in a cap on the price level. Such an impairment would not beinteresting because it would rule out all equilibria in which inflation is positive, such as our desired
equilibrium. An alternative approach results in an arbitrarily high, but finite, cap, π̄ cap , on inflation.
We explore this approach in detail and conclude that this refinement does not trim a substantial
number of equilibria and thus is not a substitute for the escape clause.
One might suppose that a cap on inflation would trim competitive equilibria in which the initialinflation rate, π̄0 , is larger than its desired value, π̄ ∗ .92 A rationale for that conjecture might be theargument in Section 3.5 which shows that, absent any refinement and absent the escape clause, allequilibria with π̄0 > π̄ ∗ lead to exploding inflation rates. We argue that such a conjecture is wrong.
In fact there are many equilibria with π̄0 > π̄ ∗ . The only equilibria that are ruled out are the ones
with π̄0 > π̄ cap . This does not significantly trim the number of equilibria since π̄ cap can be arbitrarily
large.
90

Obviously, there can be no equilibrium with π̄0 ≤ 0, since π̄0 = P0 /P−1 , where Pt denotes the price of consumption
in period t.
91
Recall, π̄ ∗ denotes the gross inflation rate in the desired equilibrium.
92
A cap on inflation, π̄ cap , would have no impact on our analysis of rationalizability in off-equilibrium paths because
we find that there is a natural upper bound on inflation on those paths.

15

With the refinement and no escape clause there remains an equilibrium corresponding to π̄ ∗ <π̄0 ≤ π̄ cap . Each equilibrium exhibits ‘exploding inflation’ in the sense that in each equilibrium
inflation converges to the very high number, π̄ cap . The key reason that inflation does not literally
explode to infinity is that when the cognitive impairment becomes binding, the Euler equation ceases
to be the Fisher equation. We proceed now to make these observations formal.
In our analysis, only the intermediate good firms have a cognitive impairment. All other agentssolve their equilibrium conditions exactly. A cognitive impairment may prevent the intermediate good
firm from setting its period t price equal to the actual period t wage rate.93 We assume that there isan arbitrarily small value of δ, where δ lies in the interval, (0, 1) , such that when the intermediate
good firm is confronted with a wage rate, Wt , with the property, Wt > (Pt−1 π̄ ∗ ) /δ, then it actuallyperceives the wage to be (Pt−1 π̄ ∗ ) /δ. Absent our refinement, optimization leads firms to set Pt = Wt(see equation (15)). With our refinement, the equation characterizing firm optimization, (15), is
replaced by the following complementary slackness-type condition:(Wt − Pt ) (ln (1/δ) − πt ) = 0, Wt ≥ Pt , ln (1/δ) ≥ πt .

(108)Here, recall that πt is the logged and scaled inflation rate, πt = ln (Pt / (Pt−1 π̄ ∗ )). In terms of scaled
and logged inflation, the inflation cap is the third term in equation (108).
The representative household’s Euler equation for employment, equation (8), together with theresource constraint, ct = lt , corresponds toWt= cγ+ψ
, γ > 0, ψ ≥ 0.
t
Pt

(109)

Given Pt > 0, equations (108) and (109) are equivalent to:1(ct − 1) ln − πtδ

= 0, ct ≥ 1, ln (1/δ) ≥ πt .

(110)

The intertemporal Euler equation, equation (9), combined with the Taylor rule, equation (17), is:ln (ct+1 /ct ) =1(φπt − πt+1 ) .
γ

(111)

We now consider the set of equilibria for the model. As in Section 3 equilibria can be indexed by
π0 . Obviously, the equilibria associated with π0 ≤ 0 are the same with or without the refinement.
Also, equilibria associated with π0 > ln (1/δ) no longer exist. This is a small set of equilibria, since
δ > 0 can be arbitrarily small. We now consider the equilibria associated with π0 ∈ (0, ln (1/δ)] .
In particular, we construct a sequence,{ct , πt }∞t=0 , corresponding to each π0 ∈ (0, ln (1/δ)] which
93

Recall, we are studying the competitive equilibrium, in which it is assumed (recall the clairvoyance assumption
discussed at the end of Section 15) that the firm directly observes the actual equilibrium wage rate. The cognitive
impairment results in a possible distinction between the wage rate that is observed and the wage rate that is perceived.

16

satisfies equations (111) and (110) for t ≥ 0.94 After that we show that each of these sequences is anequilibrium because it satisfies the transversality condition of the household, equation (11).
Consider t = 0. Since 0 < π0 ≤ ln (1/δ), equation (110) is satisfied by setting c0 = 1. Computeπt for t = 1, ..., T + 1 using
πt = φt π0 ,where t = T + 1 is the first date when πt > ln (1/δ) .95 Since π0 > 0 and φ > 1, a finite T ≥ 1 exists.
Then, the computed πt , along with ct = 1, for t = 0, ..., T, satisfy equations (111) and (110).
Now consider equation (111) for t = T , setting πT +1 = ln (1/δ):ln (cT +1 ) =1(φπT − ln (1/δ)) > 0,
γ

(112)

so that cT +1 > 1. Then, for t > T + 1, set πt = ln (1/δ) and1ct+1= (1/δ) γ (φ−1) .
ct

(113)

with cap on price level. With {ct , πt }∞t=0 computed in this way, equations (110) and (111) are satisfied
for all t.
To verify the transversality condition, note that for j > 1 equations (112) and (113) imply:
1

cT +j = cT +1 (1/δ)(j−1) γ (φ−1) .
In the transversality condition, equation (11), we have that mj = Pj cj and qj is proportional toβ j u0 (cj ) /Pj , where u0 (cj ) denotes the marginal utility of consumption. Note that qj mj is proportionalT0
to β T cT u0 (cT ) = β T c1−γ
T ,thus ensuring that β cT u (cT ) → 0 if γ ≥ 1. We have established thefollowing result:
Proposition 17. Suppose γ ≥ 1 and that the intermediate good firm has the cognitive impairment
defined in this subsection. There exists an equilibrium corresponding to each π0 ∈ (−∞, ln (1/δ)]. The
equilibrium associated with each −∞ < π0 ≤ 0 coincides with the corresponding equilibrium without
the refinement (see Proposition 1). The constructed equilibrium associated with each 0 < π0 ≤ ln (1/δ)
has an inflation rate that converges to ln (1/δ) in finite steps.
With this cognitive impairment there are multiple equilibria, including equilibria with very highinflation, as in the version of the model with no refinement and no escape clause. The nature of theequilibria are someone different, however, even if the number of equilibria is roughly the same. Most
importantly, without the refinement all the equilibria generate the same amount of welfare. With therefinement welfare associated with high inflation is very low. This is because consumption growth is94
95

Equilibria in which π0 > − ln δ are ruled out by (110).
Here, we drop the dependence of T on δ and π0 to simplify notation.

17

eventually greater than unity and eventually becomes extraordinarily high (see equation (113)). Thisis a large deviation from the desired equilibrium, which is first best.

C.4

Rationalizability When The Monitoring Range Has been Violated

Proposition 18. Consider an arbitrary history ht−1 in which the monitoring range has violated. If
condition (63) is satisfied, then there exists a unique robustly rationalizable belief.
bProof. Fix any arbitrary small 0 < ε ≤ eπ (ht−1 ) . We show that for any such ε, the conditions for
Proposition 9 are satisfied for some set. Then Proposition 9 implies that Π∗ (ε) is a singleton for allb
0 < ε ≤ eπ (ht−1 ) . Since the Nash belief, π b (ht−1 ) , is included in Π∗ (ε) , we conclude that there exists
a unique robustly rationalizable belief.
In order to show that the three conditions for Proposition 4 are satisfied, we first show that


b
b
b
b
F ht−1 , πi,t
has the following properties: (i) F ht−1 , πi,t
is increasing in πi,tfor all πi,t
≤ π̄ b (ht−1 )and decreasing otherwise where π̄ b (ht−1 ) is a unique value satisfying the following linear equation,
1−γ
1−ρ



M̄t−1
b
µ + ln− πi,t + µ = Rl ;
Pt−1


b
is less than 1 in its absolute value.
(ii) the slop of F ht−1 , πi,tb
In order to show (i), the conditions (63) imply that γ > 1 and ρ < 1. Therefore if πi,t
≤ π̄ b (ht−1 ) ,

b
b
b
b
then πi,t∈/ D (ht−1 ) . Otherwise, πi,t
∈ D (ht−1 ). Consider πi,t
≤ π̄ b (ht−1 ) . The slope of F ht−1 , πi,tis (ρ + ψ) / (ρ − γ), which is given by (ii) in Proposition 4. Since ρ < γ from the assumption,
ρ − γ < 0. From the last condition in (63), we haveρ+ψ <1−γ+ψγ−ψ+ψ =1−
.
2
2

(114)

2 + ψ < γ.

(115)

The first condition in (63) implies that

So, combining the above two equations (114) and (115), we obtainρ+ψ <1−

2+ψ−ψ
γ−ψ
<1−
= 0.
2
2


b
b
Therefore, (ρ + ψ) / (ρ − γ) is positive. Now consider πi,t
> π̄ b (ht−1 ) . The slope of F ht−1 , πi,t
is


γ+ψ
1 − 1−ρ , which is given by (ii) in Proposition 4. It is easy to show that the slope is negative since
1 − (γ + ψ) < ρ.
Now we prove (ii). It is immediate that (ρ + ψ) / (ρ − γ) < 1 since(ρ + ψ) > (ρ − γ) ,and ρ − γ < 0. It is easy to show that 1 − γ+ψis bigger than −1 if ρ < 1 − γ+ψ
.
1−ρ
2
18

The two properties (i) and (ii) imply that π max = maxπ∈[ln ε,∞) F (ht−1 , π) < ∞. Thus intermediatebfirms can disregard any beliefs πi,t
> π max . Set Ω = [ln ε, π max ]. Then all the conditions in Proposition
9 are satisfied. So we conclude that for all 0 < ε < exp (π (ht−1 )) ,Π∗ (ε) = π (ht−1 ) .
Therefore Πr,∗ = π (ht−1 ) , which is desired.

C.5

Local RationalizabilityIn this section, we derive F (N ) ht−1 , π b for the history in which the monitoring range has been
violated once and show that DF (N ) is a lower triangular matrix whose diagonal elements are 1 −(γ + ψ) / (1 − ρ) .
Consider first ht−1 in which there has been
range. First notice that
 a violation in the monitoringM̄t−1bat the Nash equilibrium belief, π (ht−1 ) = µ + ln Pt−1 , 0, · · · , 0 , the zero lower bound does notbind. This comes from the fact that the interest rates at the equilibrium belief are given by Ri,t+j = µ
for all j ∈ {0, · · · , N − 1} . Therefore, we can choose a small neighborhood of π b (ht−1 ) so that forany π b ∈ Π, the zero lower bound does not bind.
For π b ∈ Π, the money growth rule (see Definition 2) is in place in periods t + j, for all j ≥ 0. Webknow from Lemma 1 that ln ci,t+N = πi,t+N = 0. The period t + j wage is the simple function of πi,t+j
and ln ci,t+j given in equation (42). We obtain ln ci,t+j simply by iterating forward on the equilibrium
t−1. Iterating in this way yields:
cash constraint (25) with the given initial real balance, ln M
Pt−1
"
#
j−1
X


1
M
t−1
b
b
b
b
ln ci,t+j =µ − πi,t+j,+µt+s πi,t, · · · , πi,t+s− πi,t+s
+ ln
1−ρ
Pt−1
s=0
t−1
:
where µt+s (πt , · · · , πt+s ) is recursively defined given the initial real balance ln M
Pt−1

"
#
s−1
X


ρ
M
t−1
b
b
b
b
µt+s (πt , · · · , πt+s ) = µ +µ − πi,t+j+µt+k πi,t, · · · , πi,t+k− πi,t+k
+ ln
,
1−ρ
P
t−1
k=0
with the understanding that the summation is defined as zero if s − 1 < 0. So, we have for all t(N )Fjht−1 , π


b

"
#
j−1
X


M
γ
+
ψ
t−1
b
b
b
b
b
− πi,t+s+ ln= πt+j−1+µ − πi,t+j+µt+s πi,t, · · · , πi,t+s
.
1−ρ
Pt−1
s=0


(N )
It is clear that Fj
ht−1 , π b is a linear function of π b and only depends on the past and current
inflation. So DF (N ) is a lower triangular matrix. It is trivial show that the diagonal elements are
1 − (γ + ψ) / (1 − ρ) . We establish the desired.

19

C.6

Multiple Shots

Our strategy equilibrium concept allows the ith intermediate good firm to entertain a wide range ofbeliefs about what other firms do in period t when it makes its own period t price decision. But,
conditional on a given belief about what other firms do in period t, only one sequence of future
actions by all intermediate good firms (including the ith firm itself) is contemplated. These are theactions in the unique ht−1 , πtb continuation equilibrium. In effect, the ith firm treats the other firms
in period t as players in a large period t game, but all players (including the ith firm itself) are treatedas robots satisfying equilibrium conditions (or, playing Nash) in future periods.
A natural question is whether we obtain the same results when we consider N −period beliefs,
 b b
0
−
→
−b
b
captured by the N × 1 vector, →π bi,t ≡ πi,t, πi,t+1 , ..., πi,t+N−1 . The vector, π i,t , contains the beliefsof the ith intermediate good firm about the prices set by the other intermediate good firms in periods
t + s, s = 0, ..., N − 1, for ∞ > N > 0. In this sense, the analysis up to now has only addressed
N = 1 period beliefs. For the ith firm to determine its best response when N > 1 requires that it
−compute an ht−1 , →π bi,t continuation equilibrium. This is a sequence, (ai,t+s )∞s=0 , that satisfies alltime t + s equilibrium conditions, s ≥ 0, not including the firm optimality conditions, equation (15),in periods t + s, for s = 0, ..., N − 1. The variables in the continuation equilibrium that are of interest
−to the ith firm are (wi,t , ..., wi,t+N −1 ) = F (N ) ht−1 , →π bi,t . The firm’s best response is the analog of−(34), (xi,t , ..., xi,t+N −1 ) = F (N ) ht−1 , →π bi,t , where xi,t+s is the (scaled and logged) price set by thefirm in period t + s. The function, F (N ) inherits the piecewise linear structure that it has for N = 1(see Proposition 4). With N = 1, F (N ) is composed of 5 linear segments. As N increases, the typeof global analysis we were able to do in the N = 1 case becomes unwieldy. Instead, we proceed as in
Evans et al. (2018) by doing a local analysis of implementation.
−−
Let Π (ht−1 , ε) denote an ε-cylinder around the Nash equilibrium, →π b (ht−1 ) , defined by →π b (ht−1 ) =


−
−
−
π b (ht−1 ) . Thus, Π (ht−1 , ε) = →π b (ht−1 ) + →π b ∈ RN : πib ≤ ε, i = 1, ..., N , where
F (N ) ht−1 , →
−
ε > 0, but small.96 We will show that the N × 1 vector-valued function, F (N ) ht−1 , →π bi,t , is linear−for →π bi,t ∈ Π (ht−1 , ε) and has a unique zero for any ht−1 . We will exploit this linearity property toestablish what we call local rationalizability.
−
Consider first ht−1 in which there never has been a violation in the monitoring range. For →π bi,t ∈Π (ht−1 , ε), the Taylor rule is in place in periods t + j, for j ≥ 0. We know from Proposition (2) thatbln ci,t+N = πi,t+N = 0. The period t + j wage is the simple function of πi,t+j
and ln ci,t+j given inequation (42). We obtain ln ci,t+j by iterating backward on the household Euler equation, with thegiven initial conditions for ci,t+N and πi,t+N . Iterating in this way yields:
N −2

 φ b
1X
b
b
φπi,t+k− πi,t+k+1− πi,t+N −1 ,ln ci,t+j = −
γ k=j
γ
96

For ht−1 in which the monitoring range has never been violated, we can choose ε = min {−πl , πu }. Otherwise, we
can choose the ε- cylinder so that the zero lower bound is neverencountered. (See equation (39) for the restrictions
−on πtb when N = 1.) For both types of ht−1 , F (N ) ht−1 , →π bi,t in our model is a linear function.

20

where we define the summation as zero when j > N − 2. We have
(N )
Fj


−
b
−
ht−1 , →π bi,t = πt+j−1



γ+ψ
γ

" N
−2
X

k=j−1



#

b
b
b
+ φπt+N− πt+k+1
φπt+k
−1 ,


−
π bi,t . Evidently, the function is linear
where j = 1 is the top element of the N × 1 vector, F (N ) ht−1 , →−in →π bi,t and F (N ) has the following representation:
−−F (N ) ht−1 , →π bi,t = DF (N ) × →π bi,t ,where DF (N ) denotes an N ×N upper diagonal matrix with 1− γ+ψ
φ along its diagonal. Equationγ(63) implies that 1 − γ+ψ
φ < 1. With its eigenvalues inside the unit circle, DF (N ) is a convergentγk
matrix97 , so that DF (N ) (Π (ht−1 , ε)) → {0} as k → ∞.98 We adopt the idea of local rationalizability in Evans et al. (2018) by assuming that Π (ht−1 , ε) is the set of justified beliefs and that thisis common knowledge.99 In that case, it follows trivially by the convergence property of DF (N ) that→−
π bi,t = 0 is the unique locally rationalizable belief. In this sense, we obtain a local version of uniqueimplementation.
In the case where ht−1 contains at least one violation of the inflation monitoring range, it is also−−possible to construct F (N ) ht−1 , →π bi,t locally for →π bi,t in an ε-cylinder, Π (ht−1 , ε), about the Nash−equilibrium. The Nash equilibrium, →π b (ht−1 ), exists and is unique because of the uniqueness of an
ht−1 continuation equilibrium (see Ii) of Proposition 5). Our linear characterization requires onlythat the cylinder, Π (ht−1 , ε) , be sufficiently ‘narrow’ that it excludes inflation rates low enough to
avoid the zero lower bound on the interest rate. In this case, the N × 1 vector-valued function,


− b
−
−
π b (ht−1 ) . The N × N matrix, DF (N ) ,π i,t − →π bi,t = DF (N ) × →F (N ) , can be written F (N ) ht−1 , →is lower triangular with the scalar, 1 − (γ + ψ) / (1 − ρ), in every entry on its diagonal. So, DF (N )is a convergent matrix if |1 − (γ + ψ) / (1 − ρ)| < 1. This condition is satisfied under the parameter
restrictions, equation 63, sufficient for unique implementation (see Proposition 11).
We summarize the above results as follows. Consider first the analog of unique implementation
for arbitrary N :Definition 11. Let Π (ht−1 , ε) be an ε- cylinder about the unique Nash equilibrium. Suppose it is CK
that intermediate good firms only consider beliefs in the set, Π (ht−1 , ε). A competitive equilibrium
satisfies locally unique N-shot implementation if there exists ε > 0 such that: (i) there is a continuation−−equilibrium for each ht−1 and for each ht−1 , →π bi,t such that →π bi,t ∈ Π (ht−1 , ε) and (ii) for each ht−1 ,−Π∗ (Π (ht−1 , ε)) = →π b (ht−1 ) , where Π∗ is defined in 107.

97
A convergent matrix, T, is a matrix that has the property, limk→∞ T k = 0. A matrix, T, is convergent iff its largest
eigenvalue is less than unity in absolute value (see Meyer (2000, p. 617)).




k

k
98
)
DF (N
(Π) is defined
as a set DF (N ) x; x ∈ Π . DF (N ) (Π) is defined as recursively. DF (N ) (Π) =



 
k−1
DF (N )DF (N )
Π for all k ≥ 2.
99
Evans et al. (2018) refer to their idea as ‘local eductive stability’.

21Condition (i) corresponds to the requirement that the equilibrium locally be a strategy equilibrium−(see Definition 6). Condition (ii) requires that only the Nash equilibrium, →π b (ht−1 ) , survives iterated
deletion of beliefs.
We now state the following result:
Proposition 19. Consider the model in which monetary policy is governed by the Taylor strategy(see Definition 2). Then the competitive equilibrium satisfies locally unique N-shot implementation if
and only if
1−



γ+ψ
γ



φ < 1,

1−



γ+ψ
1−ρ



< 1.

(116)

Suppose monetary policy is governed by the zero monitoring range policy (see Definition 3). Then thenecessary and sufficient conditions for locally unique N-shot implementation are the second conditionin equation (116) and the following:
1−



γ+ψ
γ−1



φ < 1.

The first condition in equation (3) pertains to histories, ht−1 , when the inflation monitoring range
has never been violated. The condition applies to the slope of the best response function in a smallneighborhood around the unique Nash equilibrium. When π l < π u , that slope is different than when
π l = π u (see, for example, Figure 5a). This is the reason for the different sufficiency conditions acrossthe two monetary policies considered in the Proposition 19.

D

Appendix: The Taylor Strategy Versus Two Alternatives

This section describes the stochastic version of the model studied in the main text. The global
equilibrium analysis of this model under the Taylor strategy is more tedious than the analysis of thedeterministic model. While the equilibrium conditions in the deterministic model are linear in thelog of the variables, this is not the case in the stochastic version of the model. So, we work witha version of the equilibrium conditions that have been log-linearized about the unique equilibrium
allocations in the deterministic version of the model. These allocations are also the allocations in thelocally unique stochastic version of the model.100 As in the analysis of Poole (1970), the Taylor rulehas the effect of preventing the money demand shock from having an impact on allocations. Under
the money growth rule, consumption and employment are a function of the realization of the money
demand shock. However, inflation is not a function of that shock (prices are set before the money100

In the deterministic model, the off-equilibrium path in which monetary policy is the money growth rule can have aconstant term, µ, that lies anywhere in a specific interval (see Definition 2). This means that we are somewhat flexible
in what the inflation rate and interest rate are in these off-equilibrium paths. In the stochastic model, we set µ = 0,
so that the inflation rate in the money growth regime is the same as it is in the interest rate regime. This ensures theaccuracy of our log-linear approximation.

22

demand shock is realized) and has the consequence that much of the best response function analysis
in the main text goes through with relatively little modification.
The first section below in effect repeats all the analysis done in the paper for the stochastic
model. However, our various concepts are modified to reflect that our analysis is local to the Taylor
strategy equilibrium in the deterministic model. Thus, Section 3 below defines and analyzes a linear
competitive equilibrium under the Taylor strategy. We establish local uniqueness of that equilibrium.
Section 4 transforms the equilibrium into a game and describes the model best response functions.
We establish that the linear competitive equilibrium is a local strategy equilibrium. Finally, the third
subsection below establishes unique local implementation of the equilibrium. All this analysis pertains
to what we call one-period deviations in the introduction to Section 5. The extension to multi-period(or, multi-shot) deviations discussed in Online Appendix C.6 applies equally to the stochastic model
discussed here. That is because, as we show below, the properties of the best response function, F ν ,in the stochastic model are locally identical to the best response function, F, in the model without
shocks. The main reason for this is that prices are set before the realization of the current period
money demand shock, which is itself independent over time.
Finally, the results for trembles in Section 6.1 applies equally to the stochastic model studied
here. We evaluate welfare before t = 0 when the equilibrium starts up. When a tremble occurs, theaggregate price level is perturbed as in equation (69)). Under the zero monitoring range strategy,πl = πu , so that a tremble, no matter how small, induces a reversion to the constant money growth
regime. In the Taylor strategy with πl < πu , a tremble, if sufficiently small (or, if πu −πl is sufficiently
big), will never trigger the escape clause. Under the Taylor strategy, the equilibrium is first best withct = 1 for all t. Under the money growth regime, consumption fluctuates stochastically in response to
the money demand shock (see equation (122)). As a result, in the version of the model with trembles,
the zero monitoring range strategy is welfare inferior to the Taylor rule strategy with πl < πu . Thisestablishes We state the result for trembles in Proposition 13.

D.1

Money Demand Shocks

We introduce a shock to money demand, νt , that is realized after intermediate good firms set theirprice:101
Pt ct ≤ (mt−1 + Wt lt − dt ) exp (νt ) .
(117)
All other time t variables are determined after the realization of νt . We modify our definition of ahistory, ht−1 , to include the record of all shocks up to, and including, period t − 1. For reasonsof tractability, we limit ourselves to an analysis of the log-linearized equilibrium conditions and we
ignore the non-negativity constraint on the interest rate.102 The equilibrium conditions, after log
linearizing in the neighborhood of the desired equilibrium, are:101

The model is a variant of the sticky price model in Christiano et al. (1997), where time t prices are predeterminedwhen time t shocks are realized.
102
Our style of analysis in this section follows the approach in Atkeson et al. (2010).

23

ln ct = Et ln ct+1 −1(Rt − Et πt+1 )
γ

(118)

0 = Et−1 ln ct

(119)

M̄t
Pt

(120)

ln ct = νt + ln
Rt = φπt .

(121)Here, Et denotes the conditional expectation based on shocks up to period t. The first equation is
the log-linearized version of the household’s intertemporal optimality condition (9).103 The second
equation combines the intermediate good firm’s optimality condition (hence, the presence of Et−1 )with the household’s intra-temporal optimality condition, (8), and the resource constraint, equation(19).104 The third equation is the equilibrium version of the cash constraint, (117), that takes into
account loan market clearing (see equation (18)). In our analysis in this section, we impose (120) as
a strict equality. The last equation is the version of the Taylor rule, (22), which ignores the lower
bound constraint on Rt .
As in Section 3.6, when the government conducts its monetary policy based on the money growth
rule, equation (121) is replaced with the money growth rule in Definition 2.
Monetary policy in the log-linearized economy is defined as a version of Definition 2 in which we
ignore the non-negativity constraint on the interest rate:
Definition 12. The Taylor strategy in the linearized economy is: (i) for histories, ht−1 , in whichπs ∈ [πl , πu ] for each s ≤ t − 1 the period t rule is given by (121) with φ > 1 and (ii) for the otherhistories the money growth rule is the one in Definition 3.6.
We define a linear competitive equilibrium as follows:
Definition 13. A linear competitive equilibrium with an escape clause is an allocation a = (at )∞t=0such that (i) a satisfies equations (118), (119), (120); (ii) monetary policy is given in Definition 12;(iii) there exists a constant C > 0 such that var (πt ) < C for all t ≥ 0.
By var(πt ) we mean the variance of πt conditional on information available in period t = 0.
Condition (iii) in Definition 13 guarantees that the variance of all other real variables is also bounded.
D.1.1

Uniqueness of Competitive Equilibrium with Taylor Rule under Escape Clause

The analysis proceeds as follows. We begin by establishing a version of Lemma 1 which (as before)
helps to guarantee that inflation expectations are anchored in the long run. In particular, under the103

Log-linearization in this equation entails two approximation errors. First, we replace ln Et (1/ct+1 ) by Et ln (1/ct+1 )after taking the log of equation (118). Second, we replace ln Et (1/π̄t ) with Et ln (1/π̄t ). We also scale inflation before
logging, but that involves no approximation error.
104
As in equation (118), the only approximation in equation (119) is that we replace ln Et−1 ct with Et−1 ln ct .

24money growth rule in Lemma 1, there exists a unique linear competitive equilibrium. The equilibrium
allocations are similar those in Lemma 1, with the obvious exception that we have a money demand
shock.
Lemma 5. Suppose M̄−1 is given and the monetary rule is M̄t = cρt µ̄M̄t−1 for t ≥ 0. There exists a
unique linear competitive equilibrium with the properties:
Rt = µ +

ρ−γ
νt ,
1−ρ

ln ct =

1
νt ,
1−ρ

πt+1 = µ +

ρ
νt ,
1−ρ

for t ≥ 0,and

P0 = M̄−1 µ̄. (122)

Recall (see equation (23)) that µ is scaled and logged µ̄. The uniqueness proof is straightforward
and we include it here.105Proof. Combining the cash constraint (120) with equation (119), Et−1 ln M̄t /Pt = 0. Rearranging
the latter and taking into account that Pt is contained in the t − 1 information set, we haveln Pt = Et−1 ln cρt µ̄M̄t−1 = ln µ̄ + ln M̄t−1 .

(123)

The object in parentheses uses monetary policy to substitute out for M̄t . The second equality inequation (123) uses equation (119). First differencing equation (123) and using the monetary policy
rule, we obtain
πt+1 = µ + ρ ln ct(124)
for all t ≥ 0. Consumption is determined by combining equation (120), the monetary policy rule andequation (123):1
M̄t= νt + ρ ln ct =
νt .
(125)
ln ct = νt + ln
1−ρM̄t−1 µ̄
Finally, the expression for Rt in equation (122) is determined by combining equations (119), (118)and (124). Uniqueness follows by construction.
Next, we prove that in any linear competitive equilibrium with an escape clause, the inflation
monitoring range is not violated:
Lemma 6. Suppose monetary policy is given by Definition 12. Any linear competitive equilibrium
has the property that πt ∈ [πl , πu ] for all t ≥ 0.
105

The proof is simpler than the proof of Lemma 1 because that proof respects the lower bound constraint on thenominal interest rate. To establish uniqueness, we have to consider the possibility that the lower bound is binding
and the cash constraint is not satisfied as an equality. To rule out this kind of equilibrium the proof of Lemma 1 uses
the transversality condition of the household (see equation (11)). When we ignore the lower bound on the interest
rate, as we do here, the transversality condition is not needed to establish uniqueness of the equilibrium under themoney growth rule. Interestingly, as long as the money demand shock, has a suitable lower bound, the non-negativity
constraint on the nominal interest rate, Rt ≥ Rl , is satisfied in the unique equilibrium.

25

The proof appears in Online Appendix D.2 and essentially follows the proof of Lemma 3.6. Lemma
6 is crucial for establishing the analog of Proposition 2:
Proposition 20. Suppose monetary policy is given by Definition 12. The only linear competitive equilibrium is the desired equilibrium. Money growth in this equilibrium is M̄t = exp (−νt + νt−1 ) M̄t−1 µ̄.
Given Lemma 6, the proof is straightforward and is given in Online Appendix D.3. That proof
makes use of the variance condition, (ii) in Definition 13.
Comparing ct in the money growth equilibrium (see Lemma 5) with ct in the equilibrium inwhich monetary policy is the interest rate rule in Definition 12, we can see the Poole (1970) result.
Under the money growth rule, consumption fluctuates inefficiently. By contrast, the interest rate
rule supports the desired allocations (i.e., ct = 1 and πt = 0, independent of νt ), which are first-bestin our environment. Not surprisingly, there exists a money growth rule that can achieve the same
result. Indeed, Proposition 20 describes a money growth rule for which the only equilibrium is thedesired allocations. However, that rule requires observing the money demand shocks. The advantageof the interest rate rule is that it does not require observing νt and nevertheless is able to cancel thewelfare-reducing effects of νt on ct .
D.1.2

Local Strategy Equilibrium

Now we consider a local strategy equilibrium, which is the analog of Definition 6.
Definition 14. Let Π (ht−1 , ε) be an ε- neighborhood of the unique Nash equilibrium. A linear
competitive equilibrium is a local strategy equilibrium if there exists ε > 0 such that there is a linearbb∈ Π (ht−1 , ε).
such that πi,tcontinuation equilibrium for each ht−1 and for each ht−1 , πi,tb
As in Section 4, we show that for each ht−1 and πi,t
, there exists a continuation equilibrium. Firstbconsider ht−1 in which the monitoring range has never been violated. We construct the ht−1 , πi,tb
∈ [πl , πu ]. The ith intermediate good firm wishes to set its price tocontinuation equilibrium for πi,tthe expected value of the wage. As before, the realized time t nominal rate can be expressed as thebproduct of the real wage and the aggregate price level. The (scaled) aggregate price level is exp πi,tby our symmetry assumption on beliefs and the price equation, (13). Households observe the realized
time t real wage at the time that they make their ct and lt decisions. It follows that the realizedγ+ψ
period t real wage is ci,t, after using the resource constraint, equation (19). So, the (scaled) priceth
set by the i firm is set as follows:b
exp (xi,t ) = Et−1 exp ((γ + ψ) ln ci,t ) exp πi,t.

bAfter linearizing this expression in terms of xi,t , ln ci,t and πi,taround their values in the desired
equilibrium, we obtain:bxi,t = (γ + ψ) Et−1 ln ci,t + πi,t
.
(126)

26

b
The mapping from πi,tto ci,t is obtained by an argument similar to the one used in Section 4.
According to the monetary policy rule, the Taylor rule (121) is operative at date t. Moreover, theTaylor strategy will be in place in period t + 1 as well. According to Proposition (20), the uniqueequilibrium beginning in t + 1 has the property, ci,t+1 = 1, πi,t+1 = 0 regardless of the realizationof νt and νt+1 . Combining these facts with the intertemporal Euler equation, (118), and the Taylorbb, so that after substitution into
rule, (121), the mapping from πi,tto ln ci,t is simply ln ci,t = − φγ πi,tequation (126) we obtainφ (γ + ψ) bνb
F ht−1 , πi,t = 1 −πi,t .
(127)γHere, F ν denotes the segment of the ith firm’s best response function for the indicated ht−1 and forbπi,t∈ [πl , πu ]. The superscript, ν, indicates that this is the best response function in the linearized
economy. This segment of the best response function coincides with the same segment in the (nonb∈ [πl , πu ]linearized) version of our economy without ν shocks (compare equation (127) with the πi,tbbsegment of F in equation (37)). More broadly, it is easy to verify that F ν ht−1 , πi,t
= F ht−1 , πi,tb
, except when the zero lower bound on the interest rate is binding in the nonfor all ht−1 and all πi,tlinearized model. It is easy to verify that F ν corresponds to the best response function in Proposition
4 in which the min operator is replaced by φRt and the set, D, is replaced by the real line. In termsof Figure 3, F ν is F in which the ‘hook’ to the left of the origin is pushed off to −∞. We summarize
these findings as follows:
Proposition 21. In the linearized economy with monetary policy given in Definition 12, the uniquelinear competitive equilibrium is a local strategy equilibrium with best response function, F ν , for thefollowing two types of history, ht−1 :(i) if ht−1 has the property that the inflation monitoring range has never been violated, then F νtakes the following form:

Fν

h
i
φ
b
b

∈ [πl , πu ]
1
−
(γ
+
ψ)
πi,t
πi,t

γ
b
i
ht−1 , πi,t = h,
 1 − φ γ+ψ π b + γ+ψ µ π b ∈
/
[π
,
π
]
l
u
i,t
i,t
γ−1
γ−1

(128)

(ii) if ht−1 has the property that the inflation monitoring range has been violated at least once, thenF ν takes the following form:Fνbht−1 , πi,t



=



γ+ψ
1−
1−ρ





M̄t−1
µ + ln
,
1−ρ
Pt−1

γ+ψ
b
πi,t
+

(129)

(iii) the set of fixed points of F ν , {x : x = F ν (ht−1 , x)} , coincides with π b (ht−1 ) in part (iii) of
Proposition 5.
We established (i) above. For a proof of (ii), see Section D.4 below. Part (iii) follows from theobservation that F = F ν in a neighborhood of the unique Nash equilibrium of the version of the27model without the money demand shock.
D.1.3

Unique Implementation

The definition of unique implementation suited to the framework in this Online Appendix is:
Definition 15. A linear competitive equilibrium satisfies unique local implementation if the modelhas the following two properties: (i) it is a local strategy equilibrium and (ii) there exits ε such that
for each ht−1 , Π∗ (Π (ht−1 , ε)) = π b (ht−1 ) , where Π (ht−1 , ε) is an ε- neighborhood of the unique
Nash equilibrium, π b (ht−1 ).
This definition of unique local implementation is essentially the same as the concept of ‘locallystrongly eductive stability’ in Evans et al. (2018, p. 831).

D.2

Lemma 6

Proof. Suppose not, so that there exists an equilibrium with πT ∈/ [πl , πu ] where T ≥ 0 is the firstdate in which the monitoring range is violated. The Taylor rule implies RT = φπT ∈/ [πl , πu ] . The
money rule will be followed in period T + 1 and Lemma 5 implies PT +1 = M̄T µ̄. Dividing by PT andusing the period T cash constraint, we obtain πT +1 = µ + ln cT . By the period T Euler equation,(118) we have, using (119) and (121),11
ln cT = − (RT − πT +1 ) = − (φπT − πT +1 ) .
γγ
Substituting πT +1 = µ + ln cT into the above equation,1
ln cT = − (φπT − µ − ln cT ) .
γTaking the expectation as of time T − 1, using (119) and the facts, µ ∈ [πl , πu ] , φ > 1 :πT =µ∈ [πl , πu ] ,φa contradiction. This establishes the result sought.

D.3

Proposition 20

Proof. Suppose, to the contrary, that πt 6= 0 for some t. From Lemma 2 equilibrium has the property
that the monitoring range is never violated, i.e., πt ∈ [πl , πu ]. The Taylor rule, (121), the firstorder conditions by the representative household, (118) and firm optimality, (119), imply that inequilibrium:− γ ln ct = φπt − πt+1 .
(130)
28

Taking the expectation as of date t − 1, we obtain
φπt = Et−1 πt+1 .
We can equivalently write this as
πt = φπt−1 + εt−1 ,where {εt }∞t=0 independent over time, with Et−1 εt = 0 and εt ⊥πt−s , s ≥ 0. Then, after repeatedsubstitution, we obtaintXπt = φt π0 +φs−1 εt−s .
s=1

Then the variance of πt isvar (πt ) =tX

φs−1 var (εt−s ) .

s=1

If var (εs ) 6= 0 for some s, then the variance of πt goes up without bound since φ > 1. So, in any
local equilibrium, εt = 0 for all t ≥ 0.
πt = φt π0 .
(131)Then, if πt 6= 0 the inflation monitoring range will be violated eventually since φ > 1. But, thatwould contradict Lemma 6. So, π0 = 0. Equation (131) implies that πt = 0 for all t = 0. Therefore
the nominal interest rate Rt is zero for all t. Also, equation (130) implies that the consumption ct
is zero for all t ≥ 0. Therefore, the desired allocation is the unique equilibrium under the Taylor
strategy.

D.4

Proof for Proposition 21bbsuch that it lies inside of the monitoring range. Suppose that πi,t∈
/ [πl , πu ] .
Proof. (i) Consider πi,tThen according to Lemma 6, the price level at t + 1 is ln Pt+1 = ln M̄t + ln µ̄. Differencing (123) andusing the cash constraint (120), we obtain

πt+1 = µ + ln ct − νt .

(132)

 b

1
Substituting this expression into the Euler equation (118), we get : ln ct = 1−γ
φπi,t − µ + νt .So
the substituting the equation into equation (126),Fbht−1 , πi,t






γ+ψ b
γ+ψ
= 1−φ
πi,t +
µ.
γ−1
γ−1

29

Now we prove (ii). Combining the monetary policy and the cash constraint (126),


1
M̄t−1
b
ln ct =
νt + µ − πi,t + ln
.
1−ρ
Pt−1


b
for the histories in which theSubstituting this equation into equation (126), we obtain F ht−1 , πi,tmonitoring range has been violated in the past:FEbht−1 , πi,t








γ+ψ b
γ+ψ
M̄t−1
= 1−
π +
µ + ln.
1 − ρ i,t 1 − ρ
Pt−1Appendix: Conclusion

Here, we consider a simple version of the standard New Keynesian model with Calvo-style pricesetting frictions. We start by describing the non-linear equations of the model and we derive thebest response functions for intermediate good firms in the game representation of the model. We
analyze the stochastic version of the model in a small neighborhood of its interior nonstochastic steadystate. We derive an upper bound on φ that is necessary for the desired equilibrium to be uniquely
implementable (the bound is the analog of the first expression in equation (64)). We show that thisupper bound is higher, the smaller is the set of marginal price setters. This substantiates a claim
made after equation (66) and in the conclusion of the main text. Our analysis provides a roadmap toderive analogous necessary conditions in a wide range of specifications of the New Keynesian model.
We establish our results by working with the equilibrium conditions and the best response functions linearized around non-stochastic equilibrium. Our logic is that if an equilibrium fails to be
uniquely rationalizable conditional on beliefs being within an arbitrarily small neighborhood of theequilibrium, then unique rationalizability (hence, unique implementation) fails for the underlying
nonlinear system. Since our analysis is local, we have no need to specify our escape clause and theinflation monitoring range must simply be an interval that contains the target inflation rate in its
interior.106 We must assume (though we do not display) that there exists an escape strategy whichguarantees global uniqueness of the equilibrium that we study.

E.1

Households

The representative household solves:
106

Thus, we do not consider the zero monitoring range policy in Definition (3).

30

max

E0

∞
X

β

t=0

s.t.

t



N 1+ϕ
log Ct − exp (τt ) t
1+ϕPt Ct + Bt+1 ≤ Wt Nt + Rt−1 Bt + Tt ,(133)

where Bt denotes the beginning-of-period t stock of bonds, and Ct , Nt , Wt , Pt , Tt denote consumption,
employment, the nominal wage rate, the nominal price level, and the profits net of governmenttransfers and taxes, respectively. The exogenous disturbance to preferences, τt , had the following law
of motion:τt = λτt−1 + ετt .
The first order optimality conditions are:
1 Rt
1
= βEt
Ct
Ct+1 π̄t+1
Wteτt Ct Ntϕ =.
Pttwhere π̄t ≡ PPt−1
.

E.2

Firms

E.2.1

Aggregate Output

Final goods are produced by a representative, competitive firm using the following production function:
 ε
Z
1

ε−1

ε−1

Yi,tε di

Yt =

.

0

Profit maximization leads to the following demand for Yi,t
Yi,t = Yt



Pt
Pi,t

ε

where Pi,t is the price of Yi,t . Substituting the demand curve back into the production function andrearranging, we obtain the following restriction across prices:
Pt =

Z 1

(1−ε)
Pi,t di

0

31

1
 1−ε

.

(134)

E.2.2

Intermediate Good Firm

The intermediate good,Yi,t , is produced by a monopolist using the following production function:
Yi,t = eat Ni,t ,where∆at = ρ∆at−1 + εat , ∆at ≡ at − at−1 .
The monopolist chooses its price and output subject to the demand curve and the Calvo price friction:
Pi,t =(P̃t
Pi,t−1

with probability 1 − θ.
with probability θ(135)With probability 1 − θ the firm chooses its price, denoted by P̃t , optimally. We denote the realpre-subsidy marginal cost by st :Wt

st = Patt = eτt Ct Ntϕ /eat ,eafter using the household’s intratemporal Euler equation. The post-tax marginal cost is (1 − ν) st ,so that the firm’s objective is:
∞
X

Eti

j=0

β j υt+j [Pi,t+j Yi,t+j − Pt+j (1 − ν) st+j Yi,t+j ] ,where υt+j denotes the Lagrange multiplier on household budget constraint. Here, the superscript, i,on the expectation operator is meant to signal that the expectation is over the idiosyncratic priceshock, in addition to aggregate variables. As is customary with the standard, simple New Keynesianmodel, we set ν to eliminate the monopoly distortion in steady state:ε(1 − ν) = 1.
ε−1
Let Pet denote the price set by the 1 − θ firms who optimize at time t. These firms are referred toas the ‘marginal price setters’. The expected value of future profits are the sum of two parts: futurestates in which price is still Pet , so Pet matters, and future states in which the price is not Pet , so Pet isirrelevant. In this way we can express the intermediate good firm objective as follows
Eti

∞
X
j=0

β j υt+j [Pi,t+j Yi,t+j − Pt+j (1 − ν) st+j Yi,t+j ]

z

}|t

= Et

∞
X
j=0

Z

{
h
i
(βθ)j υt+j P̃t Yi,t+j − Pt+j (1 − ν) st+j Yi,t+j +χt ,32

where Zt is the present value of future profits over all future states in which the firm’s price is P̃t and
Xt is the present value over all other states, so dχt /dPet = 0. Also, the absence of the i superscript
on Et indicates that the expectation only involves aggregate variables, because the expectation overidiosyncratic variables is captured explicitly by the presence of θj in the objective.
Substitute out demand curve in the firm’s objective, and ignoring χt ,
Et

∞
X
j=0

= Et

h
i
(βθ)j υt+j P̃t Yi,t+j − Pt+j (1 − ν) st+j Yi,t+j

∞
X
j=0

h
i
ε
(βθ)j υt+j Yt+j Pt+jP̃t1−ε − Pt+j (1 − ν) st+j P̃t−ε .

Take the FONC with respect to P̃t :
Et

∞
X

(βθ)

j

ε
υt+j Yt+j Pt+j

j=0

so that
Et

∞
X



 −ε
−ε−1
(1 − ε) P̃t+ εPt+j (1 − ν) st+j P̃t
= 0,

ε+1
(βθ)j υt+j Yt+j Pt+j

j=0

"

#
P̃t
− st+j = 0,
Pt+j

using the fact that the subsidy cancels the monopoly markup. Substitute out for the Lagrange
multiplier:= υt+j
"
#
z
}| {
∞
0
X
u
(C
)
P̃
t+j
t
ε+1
Yt+j Pt+j− st+j = 0.
Et
(βθ)j
P
P
t+j
t+j
j=0
and using the assumed log-form of utility, as well as the resource constraint, Yt+j = Ct+j , we obtain:
Et

∞
X
j=0

(βθ)j (Xt,j )−ε [p̃t Xt,j − st+j ] = 0,
p̃t ≡
π̄t ≡

P̃t
,
Pt




1

Pt
, Xt,j = π̄t+j π̄t+j−1 ···π̄t+1
1
Pt−1

Xt,j = Xt+1,j−1

1

π̄t+1

j > 0.

Solving the above expression:Pj−εPt eτt p∗t Nt1+ϕ + Pt Et ∞st+j
j=1 (βθ) (Xt,j )
P̃t =
P∞
j
1−ε
Et j=0 (βθ) (Xt,j )
33

j≥1

(136)

j=0
(137)

p̃t =
say. Consider the numerator term:

Kt = Et

∞
X

P∞

j
−ε
st+j
Kt
j=0 (βθ) (Xt,j )
,
=
P∞
j
1−ε
Ft
Et j=0 (βθ) (Xt,j )

Et

(138)

(βθ)j (Xt,j )−ε st+j

j=0

 =Xt,j , ∵(137) −ε
}|
{
z
∞
X


j−1  1
Xt+1,j−1 = st + βθEt(βθ) 
 st+jπ̄t+1
j=1= st + βθEt



1

π̄t+1

−ε X
∞

−ε
(βθ)j Xt+1,j
st+1+jj=0

law of iterated expectationsz}|{== st + βθEtst + βθEt Et+1



1
π̄t+1

=Kt+1



−ε X
∞

−ε
(βθ)j Xt+1,j
st+1+j

j=0

}|
{
−ε z
∞
X
1
−ε
Et+1
(βθ)j Xt+1,j
st+1+jπ̄t+1

j=0

so, we have,
Kt = st + βθEt



1
π̄t+1

−ε

Kt+1 (1)(139)

Similarly, for Ft :ε−1
Ft = 1 + βθEt π̄t+1
Ft+1 (2).

E.3

(140)

Aggregate Restrictions

Evaluating equation x(134) using the assumption, equation (135):
Pt =



1−ε
(1 − θ) P̃t1−ε + θPt−1

or, after rearranging,
p̃t =

"

ε−1

1 − θ (π̄t )
1−θ

Let
Yt∗ ≡

Z 1
0

34

1
# 1−ε

Yi,t di.

1
 1−ε

.

,

Substituting out from the demand curve for Yi,t , we obtain:Yt∗ = Yt

Z 10where Pt∗ =

R
1

−ε
Pi,t
di
0

 −1
ε

Pi,t
Pt

−ε

di = Yt Ptε (Pt∗ )−ε ,. We conclude
Yt = p∗t eat Nt ,

where
p∗t =



Pt∗
Pt

ε

, Pt∗ =

Z 1
0

−ε
Pi,t
di

− 1ε

.

The variable, p∗t , is the distortion implied by price dispersion originally derived by Yun (1996).
Applying the Calvo result to Pt∗ :h−ε i −1ε∗Pt∗ = (1 − θ) P̃t−ε + θ Pt−1

After rearranging,


−1
π̄tε
−ε
= (1 − θ) p̃t + θ ∗
pt−1
−1
−ε
# 1−ε
"
ε−1
ε
π̄
1 − θ (π̄t )= (1 − θ)+ θ ∗t 
1−θ
pt−1

p∗t ≡

E.4



Pt∗
Pt


ε

(4)

Linearized Equilibrium Conditions

Collecting the 7 (non-linear) equilibrium conditions in our 7 endogenous variables:eτt Ct NtϕAt# 1"(ε−1) 1−ε
K
1
−
θπ̄
t
t
ε−1
Ft+1 (2),Ft = 1 + βθEt π̄t+1
=
(3)
Ft
1−θ

−1
ε
! ε−1
(ε−1)
ε
1 − θπ̄tθπ̄p∗t = (1 − θ)+ ∗ t  (4)
1−θ
pt−1

ε
Kt = st + βθEt π̄t+1
Kt+1 (1), st ≡

1 Rt
1
= βEt
(5),
Ct
Ct+1 π̄t+1
Rt  π̄t φ
=
(7),
R
π̄

Ct = p∗t eat Nt (6)35where the last equation is the monetary policy rule. We now log-linearize these equations about
nonstochastic steady state to obtain the standard three equation NK model.
It is worth noting that when θ = 0, the allocations and prices in the equilibria of the model
considered here coincides with the allocations and prices in the equilibria of the model consideredin the main manuscript (see Section 3). This is so, even though the underlying models are very
different.107 In the policy rule, (7), π̄ denotes the policy maker’s inflation target and R denotes thenominal interest rate in non-stochastic steady state. As in the standard presentation of the threeequation NK model, we set π̄ = 1, the steady state is given by:K=11(1), s = 1, F =(2)
1 − βθ
1 − βθ

K= 1 (3),Fp∗ = 1(141)

1 = β exp (−∆a) R (5)c = N,
where ct = Ct /At and absence of a time subscript indicates non-stochastic steady state.
Let x̂t ≡ (dxt ) /x, where dxt denotes a small deviation of xt from its steady state value. Then,log linearizing equation (139) around steady state,b̄K̂t = ŝt (1 − βθ) + βθ επ t+1 + K̂t+1 .

We conclude (after applying the same manipulations to the other price optimality conditions):b̄t+1 + K̂t+1 (a)K̂t = (1 − βθ) ŝt + βθEt επb̄t+1 + F̂t+1 (b)F̂t = βθEt (ε − 1) π
K̂t = F̂t +θ
b̄t . (c)
π
1−θ

(142)
(143)
(144)

Substitute out for K̂t in equation (142) using equation (144) and then substitute out for F̂t fromequation (143) to obtain:107

In the model in this appendix, we are not specific about what gives rise to money demand. One possibility, whichis consistent with all the equilibrium conditions described here, is that real balances enter additively in the utility
function and money accumulation is explicitly included in the household’s budget constraint, equation (133). At thislevel, the model is obviously quite different from the one in the body of the paper, despite their being observationallyequivalent in terms of equilibrium prices, interest rates, employment and output.

36b̄t+1 + F̂t+1
βθEt (ε − 1) πCollecting terms, we get :θθb̄t = (1 − βθ) ŝt + βθEt επb̄t+1 + F̂t+1 +b̄t+1 .
π
π
+
1−θ
1−θ

(1 − θ) (1 − βθ)b̄t+1ŝt + βEt πθNow, we go for ŝt . First, we rewrite st as follows:b̄t =πexp (τt ) Ct Ntϕst =At 1+ϕexp (τt ) Ct=
∵ Nt = Ct / (p∗t At )(p∗t )ϕ
At
1+ϕ

1
Ct
i
h
= ∗ ϕ(pt )A exp − τt
t

1+ϕ

1
X 1+ϕ ,
(p∗t )ϕ t=

where Xt is the ratio of actual consumption to what is easily verified to be its first-best level, Ct∗ :Ct
Ct
Xt = ∗ =
exp
Ct
At



τt
1+ϕ



= ct exp



τt
1+ϕ



.

(145)

Since π̄ = 1,p̂∗t = θp̂∗t−1 .
So, under the assumption that the economy started up long ago, to a first order approximation(around the efficient steady state), we get p̂∗t = 0. Then,
1 + ϕ 1+ϕX
xt = (1 + ϕ) xtŝt =s (p∗ )ϕ

where xt denotes the output gap, xth≡ X̂t .i The output gap here is the log deviation of actual output
τtrelative to its first-best level, At exp − 1+ϕ. Our calculation takes into account that in non-stochastic
steady state, actual output and its first best level coincide. In this way, we have derived the Phillips
curve:(1 − θ) (1 − βθ)
b̄t =
b̄t+1
π
(1 + ϕ) xt + βEt πθb̄t = π̄tπ̄−π̄ = πt ,Note that ππt =(1 − θ) (1 − βθ)(1 + ϕ) xt + βEt πt+1 .
θ

37

(146)

We now derive the NK IS curve. The ‘interest rate’, Rt∗ , when consumption is at its first best
solves:
h
i
τt
A
exp
−
∗
t
1+ϕ
1
C
i = Rt∗ Et βi.
hh
1 = Rt∗ Et β ∗t = Rt∗ Et β
−τt
t+1
Ct+1
At+1 exp − τ1+ϕ
exp ∆at+1 − τt+1
1+ϕMultiplying the intertemporal Euler equation of the household by At :

Et




1
1
Rt
−β
= 0(5),
ct
ct+1 exp (∆at+1 ) π̄t+1

where ct ≡ Ct /At . Alternatively,


Et 

1
−β
Xt
X

t+1 exp

1−τt∆at+1 − τt+1
1+ϕ





Rt 
= 0(5),π̄t+1

After log-linearizing (5):
τt+1 − τtˆt+1 − ∆at+1 − ∆a −Et 1 − xt − 1 − xt+1 + R̂t − π̄
= 0,
1+ϕ


where the d0 s are dropped from τt because it has the same steady state at each date. Then,



 
τt+1 − τtˆt+1 + ∆at+1 − ∆a −Et xt − xt+1 − R̂t − π̄
= 0.
1+ϕ


(147)

Now, we log-linearize the equation that defines Rt∗ :R̂t∗ = Etτt+1 − τt∆at+1 − ∆a −1+ϕSubstituting this into equation (147)xt = Et xt+1 − R̂t − πt+1 − R̂t∗ .

(148)

The Taylor rule is:

Rt  π̄t φ=,Rπ̄which, after log-linearization about nonstochastic steady state is:R̂t = φπt .

(149)

Equations (146), (148) and (149) represent the three equations of the standard simple NK model.

38

E.5

Best Response Function of an Individual, Marginal Price Setter

We now discuss the decision of the ith individual marginal price setter. This price setter is one of
the measure, 1 − θ, of marginal price setters. Each marginal price setter makes its price decision,P̃i,t , independently and without communicating with the others. Let zi,t denote P̃i,t /Pt−1 . As in thebdiscussion in the paper, the ith firm forms a belief, zi,t, about what the other marginal price settersbb
are doing (after scaling by Pt−1 ). Conditional on zi,t and ht−1 , the firm can compute an ht−1 , zi,tcontinuation equilibrium. The variables, Ki,t and Fi,t in that continuation equilibrium are requiredfor it to evaluate the terms on the right of its optimality condition, equation (138). Let π̄i,t denote thebcontinuation equilibrium. In this notation, the firm’s optimality
value of Pt /Pt−1 in the ht−1 , zi,tcondition, equation (138), is:Ki,tzi,t =
π̄i,t .
Fi,t
Log-linearizing around steady state:ẑi,t = K̂i,t − F̂i,t + πi,t .
Substituting the linearized expressions for K̂t and F̂t , equations (142) and (143), we obtainθ
Et πt+1 + πi,t .
1−θbcontinuation equilibrium. Note
As before, the subscript, i, indicates the ith price setter’s ht−1 , zi,tthat πt+1 does not have a subscript, i. This reflects that we only consider one-shot deviations and thatafter log-linearizing around an undistorted steady state, our linearized model has no state variables.
As a result, xt+j and πt+j are not functions of j, j > 0. Also, we haveẑi,t = (1 − βθ) ŝi,t + βŝi,t = (1 + ϕ) xi,t
xi,t = Et xi,t+1 − R̂i,t − πt+1 − R̂t∗ .
There is no i subscript on R̂t∗ because that is common knowledge. It follows that






θ
Et πt+11−θ= πi,t [1 − (1 − βθ) (1 + ϕ) φ] + (1 − βθ) (1 + ϕ) Et xt+1 + πt+1 + R̂t∗ + βẑi,t = πi,t + (1 − βθ) (1 + ϕ) Et xi,t+1 −R̂i,t − πt+1 − R̂t∗+βθEt πt+1
1−θInflation is related to what the other marginal price setters are doing based on the cross pricerestriction:
1

 1−ε

b 1−ε
+θ
π̄i,t = (1 − θ) zi,t
.

39

b
Log-linearizing this expression around π̄i,t = zi,t
= 1, we obtainb
πi,t = (1 − θ) ẑi,t.

So, we conclude that the best response function denoted by f, has the following form:



θ
b
b
Et πt+1 = f ẑi,t, ξt .
ẑi,t = [1 − (1 − βθ) (1 + ϕ) φ] (1 − θ) ẑi,t+(1 − βθ) (1 + ϕ) Et xt+1 + πt+1 + R̂t∗ + β
1−θ
|
{z
}
=ξt

(150)
Let πt denote the Nash equilibrium of the best response function, equation (150). It is a general resultthat the competitive equilibrium coincides with the Nash equilibrium when the economy is representedb= πtN ash andas a game. So, we can verify our calculations by solving equation (150) for ẑi,t = ẑi,tverifying that πtN ash = πt , where πt satisfies the Phillips curve, equation (146). A simple chain-ofjustification argument verifies the standard result that the Nash equilibrium (e.g., the competitive
equilibrium) is rationalizable. We now inquire whether there exists another rationalizable equilibriumin an arbitrarily small neighborhood, D (πt ; ε) , of πt , where D (πt ; ε) = {x : |πt − x| < ε} and ε > 0
but sufficiently small. We say that the equilibrium satisfies locally unique rationalizability if theset, Π∗ (D (πt ; ε) ; f ) , is a singleton, πt , for all competitive equilibrium values of πt and all t, forε sufficiently small.108 The correspondence, Π∗ , is defined in equation (59). In the present caselocally unique rationalizability coincides with the requirement that the slope of the best responsebfunction, f, in terms of ẑi,tis less than unity, in absolute value. From equation (150), that coefficientis [1 − (1 − βθ) (1 + ϕ) φ] (1 − θ). Trivially, this coefficient is less than unity. So locally uniquerationalizability obtains if and only if[1 − (1 − βθ) (1 + ϕ) φ] (1 − θ) > −1,
which is equivalent to:1(1 − θβ) (1 + ϕ)


1+

1
1−θ



> φ.

(151)

A qualitative feature of this expression is interesting from the perspective of our ‘leaning against thewind, but not too aggressively’, principle. In particular, the upper bound on φ is less restrictive, thebigger is θ. When the individual marginal price setter considers what other do, what matters is theirimpact on the aggregate price index, which is small when their number is small. This is captured by
the expression, (1 − θ) , in front of the inequality in equation (151).

E.6

The Taylor Rule Works Well in Normal Times

Here, we substantiate the claim made in Footnote (??) and repeated in Section 7, that the Taylor rulehas good operating characteristics (‘works well’) in a neighborhood of the unique equilibrium local
108

We do not include ξt in the definition because the Phillips makes ξt a simple function of πt .

40

to the interior steady state. After log-linearization, the equilibrium conditions reduce to three: theIS curve, equation (148); the monetary policy rule, equation (149); and the Phillips curve, equation(146). For simplicity, we display these equations here. The IS curve is:
xt = Et xt+1 − [rt − Et πt+1 − rt∗ ] ,where xt denotes the log-difference between equilibrium and output and output at its first-best level.
Also, rt∗ corresponds to R̂t∗ above andrt∗ = Et at+1 − at ,where at is an exogenous shock with the following law of motion:∆at = ρ∆at−1 + εt ,∆at = at − at−1 , ρ ∈ [0, 1)

The Phillips curve is:
πt = βEt πt+1 + κxt , β ∈ (0, 1) ,where κ > 0. Finally, the Taylor rule is:
rt = φπt ,where φ > 1 and rt corresponds to R̂t above.
It is easy to verify that the locally unique equilibrium has the form:rt − Et πt+1 = ψ∆at ,xt = ψ (1 − βρ) / [κ (φ − ρ)] ∆at ,πt = [ψ/ (φ − ρ)] ∆atwhereψ ≡ ρ [(1 − βρ) (1 − ρ) / (κ (φ − ρ)) + 1]−1 .
Evidently, for φ sufficiently large, ψ is close to ρ and rt −Et πt+1 ' rt∗ , πt ' 0 and xt ' 0. We conclude
that based on the analysis of equilibrium alone, a big value of φ works well in the sense that thepolicy stabilizes the equilibrium around first best. It is easy to verify that this stabilization result
also holds when the technology process is replaced by at = ρat−1 + εt or when the shock is instead astationary disturbance to labor supply.

41