Discouraging Deviant Behavior in Monetary
Economics⇤
Lawrence Christiano† and Yuta Takahashi‡
April 29, 2019

Abstract
We consider a model in which monetary policy is governed by a Taylor rule. The model
has a unique equilibrium near the steady state, but also has other equilibria. The introductionof a particular escape clause into monetary policy works like an out-of-equilibrium version of
the Taylor principle to exclude the other equilibria. We reconcile our finding about the escapeclause with the sharply diﬀerent conclusion reached in Cochrane (2011). Atkeson et al. (2010)
study a diﬀerent version of the escape clause policy, but that version is fragile in that it lacks
a crucial robustness property.

We are grateful for conversations with Marco Bassetto, V.V. Chari, John Cochrane, Martin Eichenbaum, Patrick
Kehoe, Emi Nakamura, Armen Nurbekyan, Matt Rognlie, Jón Steinsson and John Taylor. Although discussions
with these people helped us greatly to clarify our thoughts, they do not necessarily agree with the arguments
described in this paper.
†
Department of Economics, Northwestern University, l-christiano@northwestern.edu‡
Institute of Economic Research, Hitotsubashi University, yuta.takahashi@r.hit-u.ac.jp
⇤

1

1

Introduction

A key objective of government policy design is to guarantee that a desired equilibrium occurs andbad equilibria are ruled out (the implementation problem). A classic example is Diamond andDybvig (1983), who describe a scenario in which an unregulated banking system is fragile and has
good and bad equilibria (i.e., no-run and run) and deposit insurance guarantees that only the goodequilibrium occurs.
This paper is about the implementation problem in monetary models, which often exhibit
multiple equilibria. An important example is the standard New Keynesian model, which assumes
that fiscal policy is passive and monetary policy is set by a Taylor rule. By a Taylor rule, we mean
an interest rate rule that satisfies the Taylor Principle (i.e., has a big coeﬃcient on inflation). An
influential sequence of papers shows that such monetary models have two steady states. Analysis
of simpler models in which an analytic characterization of the global set of equilibria is possible,
suggests that the equilibrium multiplicity problem may be even greater. In those models, it has
been shown that there are deflation equilibria, hyperinflation equilibria, equilibria in which inflation
exhibits cycles and even chaos (see Benhabib, et al. 2001b; 2001a; 2002a; 2002b) (BSGU).
The literature on the standard New Keynesian model has generally ignored the equilibrium
multiplicity issue simply by focusing on the unique equilibrium that is local to the interior nonstochastic steady state equilibrium. We refer to this equilibrium as the desired equilibrium because
in many models that equilibrium is first best or nearly so. Critics correctly argue that until themultiplicity issue has a convincing resolution, the New Keynesian model cannot be thought of asa model that determines the price level or anything else. Moreover, the bad equilibria cannot
simply be waved aside or ignored on the grounds that we don’t see them in the data. Unless thoseequilibria are ruled out, the fact that we don’t see the non-desired equilibria in the data is evidence
against the model.1
In this paper we examine one argument that has been advanced to justify focusing on the1See, for example, Cochrane (2011).

2

desired equilibrium.2 The argument that we consider, which was suggested by Taylor (1996, p.
37), Benhabib et al. (2002a) and Christiano and Rostagno (2001), is studied in a particular model.3
Ideally, that model would be the New Keynesian model, but a global analysis of equilibrium inthat model appears to be infeasible at this time. We analyze a model which has the flavor of the
New Keynesian model, but is tractable. In addition, our conclusions contradict those reached in
Cochrane (2011) and Atkeson et al. (2010). So, to highlight the reason for the diﬀerence in results,
we work with a model that is comparable to theirs.
The proposal that we study is motivated by the observation in our model that the undesired
equilibria associated with a Taylor rule in eﬀect require the complicity of the government. For
example, the exploding hyperinflation equilibria that are possible under the Taylor rule can only
happen if accommodated by exploding money growth. Our interpretation of why we don’t see
this type of thing in the data is that people believe the government wouldn’t allow it. That is
we consider the proposition that the government follows the Taylor rule in normal times, but iscommitted to a particular escape clause. In particular, it follow the Taylor rule as long as the
economy remains in a neighborhood of the desired equilibrium and commits to implementing aparticular escape clause in the event that a non-desired equilibrium appears to form. The escapeclause specifies that if inflation moves outside a particular monitoring range, then the government
deviates from the Taylor rule in favor of some other policy that directly moves inflation back into
the monitoring range. The policy to which the government deviates under the escape clause might
not be optimal in normal times, but the mere existence of the escape clause prevents undesired
equilibria from forming in the first place. A policy in which the government has one set of rules innormal times, but is prepared to deviate to another set of rules under exigent circumstances is not
unprecedented. For example, many governments which respect individual liberty in normal times
have the authority to impose martial law, an entirely diﬀerent regime, in the event of widespreaddisorder.42Perhaps the best known approach for addressing multiplicity is one based on learning. See Christiano et al.
(2018a) and the extensive literature they cite.
3
The papers cited in the text are related to the work of Obstfeld and Rogoﬀ (1983) (see Obstfeld and Rogoﬀ(2017)).
4
Other examples include the ‘unusual and exigent circumstances’ clause, section 13.3, in the Federal Reserve
Act (see https://www.federalreserve.gov/aboutthefed/section13.htm) used to rationalize the use of unconventional monetary policy in the wake of the 2008 Financial Crisis. Another example is the exigent circumstances3

Our discussion pushes back against the conclusions reached in Cochrane (2011) and in Atkeson
et al. (2010) about the type of escape clause policy considered here. Cochrane (2011) agrees thatthe escape clause rules out the non-desired equilibria. However, he argues that it does so by agovernment commitment to do something infeasible (blow up the economy) in case the undesired
allocations occur. Turning back to the bank-run example mentioned above. Suppose that in fact
the government does not have the resources to provide liquidity in case of a bank run. In thatcase, the government commitment to rescue the banks in the event of a run is no feasible. In
eﬀect, the government’s policy eliminates the bad equilibrium by doing something impossible or,in the language of Cochrane (2011), it commits to blow up the economy. Obviously, this is not aninteresting way to rule out a run equilibrium. Agents understanding that the government cannot
bail out banks in the event of a run will ignore the existence of deposit guarantees.
A problem with Cochrane (2011)’s argument is that he makes it within the framework of astandard equilibrium concept. That concept does not allow for oﬀ-equilibrium events such as thenon-desired allocations. So, it is silent about the economic reason that such allocations are notchosen in equilibrium. We follow Bassetto (2005) and Atkeson et al. (2010) by defining the conceptof a strategy equilibrium, which makes it possible to study how it is that the escape clause excludes
non-desired allocations.5 The strategy equilibrium extends the standard concept of equilibrium by
opening up oﬀ-equilibrium paths at each date. The question of why non-desired allocations are
not equilibria is answered by asking the agents what it is about the escape clause that prevents an
inflation from occurring. Following Atkeson et al. (2010), we construct oﬀ-equilibrium paths byadopting the Dixit-Stiglitz production framework. Each intermediate good price setter chooses its
price simultaneously and without coordinating. To decide what price to set, a price setter must
form a belief about what prices the other agents choose and they must contemplate the associatedcontinuation equilibrium for the economy. Thus, we can think of agents’ choice of price as theirbest response to what others do. In a competitive equilibrium, they select a belief about what
under which the Fourth Amendment to the US Constitution’s prohibition against a Warrantless search and seizure
may be ignored (see https://www.law.cornell.edu/wex/fourth_amendment).
5
Our equilibrium concept coincides with the ‘sophisticated equilibrium’ concept in Atkeson et al. (2010). We
give ours a diﬀerent name because the objects in our equilibrium are sequences rather than functions. Our choice ofequilibrium concept is practical in our setting because we generally work with non-stochastic versions of our model.
By working with sequences, we are able to minimize notation.

4

others do that corresponds to the fixed point of this best response function. For this view about
belief formation to be interesting, we require that for each logically possible belief about what
others do, there is a well defined continuation equilibrium. In addition, we require that there
exists a fixed point. If either of these two requirements are not satisfied, then the problem of
forming a belief about inflation is not well defined and we say that a strategy equilibrium does
not exist. Note that even though the individual agents are atomistic, the way they arrive at theirbelief about current inflation requires strategic thinking. Hence, the reason for the name of ourequilibrium concept.
The escape clause rules out inflation above the monitoring range because such an inflation
rate is not a fixed point of the best response function. A firm that conjectures high inflation
understands that the government will respond by raising the nominal interest rate sharply (thatreflects the Taylor Principle) and lowering future inflation (that reflects the switch to low money
growth). The resulting high real interest rate produces a recession in the model, which reduces
marginal cost. Firms’ best response is to post low prices, with the consequence that high inflation
does not occur. In short, with the escape clause the government asserts that it is ready to engineera Volcker-type recession in case inflation is high, and the private economy responds by setting
prices so that high inflation does not happen. So, expectations of high inflation are not a fixed
point and this is why they are not an equilibrium.6 Deviations from the equilibrium path arediscouraged without blow-up-the-economy threats.
Why do we reach a conclusion so diﬀerent from Cochrane (2011)’s? The answer lies in his
assumption of an endowment economy. That assumption cuts the heart out of the mechanism
by which the escape clause does its work in our model. The fall in output and rise in the realinterest rate that occurs in our model is impossible in an endowment economy. We reproduce
Cochrane (2011)’s blow-up-the-world result in the endowment economy version of our model byshowing that continuation equilibria do not exist for non-fixed point inflation conjectures (that is,that economy does not have a strategy equilibrium). So, we agree that the escape clause works
by a blow-up-the-economy threat in Cochrane (2011)’s model. But, his result does not generalize
to a production economy. We conclude that Cochrane (2011)’s analysis says nothing about the6

We can say that expectations of high inflation are not self-confirming.

5

properties of standard macroeconomic models.
We then consider Atkeson et al. (2010). Their primary economic conclusion is that a key tenetof the New Keynesian canon - that the Taylor principle is a necessary ingredient of good monetary
policy - is false. They endorse the escape clause, but propose shrinking the monitoring range for
inflation to a singleton that only includes the desired inflation rate. The desired equilibrium isindeed uniquely implemented by their policy and, as they emphasize, the size of the coeﬃcient on
inflation in the Taylor rule plays no role in ensuring that inflation remains at its desired level. We
make two observations on Atkeson et al. (2010). First, the way that the escape clause works is verymuch in the spirit of the Taylor Principle. The idea behind the Taylor Principle is that with the
big coeﬃcient on inflation, a rise in inflation produces a rise in the real interest rate and, by slowing
down the economy, that brings inflation back down to its desired level (see Taylor (1999, p.325)’s
discussion of ‘leaning against the wind’). That stabilizing force works reasonably well in a New
Keynesian model in a neighborhood of the model steady state, but it does not exclude equilibria
with inflation far from steady state. As explained above, the escape clause excludes those equilibria
by a mechanism very similar to the way the Taylor Principle works near steady state.7 So, Atkeson
et al. (2010)’s conclusion that the Taylor Principle is not necessary for unique implementation of
the close-to-steady-state equilibrium is misleading, at least when viewed through the eyes of our7

How the Taylor Principle works near steady state is well understood. We briefly review that here for completeness. Consider the standard New Keynesian model without capital, linearized around the first-best equilibrium.
The IS curve is xt = Et xt+1 [rt Et ⇡t+1 rt⇤ ] , where xt denotes the log diﬀerence between output in the equilibrium with the Taylor rule and first-best output. The Taylor rule is rt = ⇡t , where > 1. The Phillips curve is⇡t = Et ⇡t+1 + xt , where  > 0 and 2 (0, 1) . Finally, rt⇤ denotes the natural rate of interest, rt⇤ = Et at+1 at
and at denotes a shock to technology. Suppose that the shock has the representation, at = ⇢ at 1 + "t , whereat = at at 1 , ⇢ 2 [0, 1) and "t is an iid shock. It is easy to verify that the locally unique equilibrium has the
form:
rt

Et ⇡t+1 =

at ,

xt =

(1
(

⇢)
⇢)

at ,

⇡t =

⇢

at

h
i 1
where ⌘ ⇢ (1 ( ⇢)(1⇢) ⇢) + 1. For suﬃciently large, is close to ⇢ and rt Et ⇡t+1 ' rt⇤ , ⇡t ' 0 and xt ' 0.
So, a big value of stabilizes the equilibrium around first best and this is accomplished by the operation of the
Taylor principle (i.e., the real rate of interest increases when inflation is high because > 0 and > ⇢). It is easy
to verify that this result also holds when the technology process is replaced by at = ⇢at 1 + "t or when the shockis instead a stationary disturbance to labor supply.

6

model.8
Second, we show that the Atkeson et al. (2010) analysis is fragile. If a vanishingly small numberof price setters make vanishingly small mistakes (i.e., trembles) the monitoring range would beviolated by accident and not because an undesired equilibrium is forming. Yet, the Atkeson etal. (2010) policy would respond by shifting to the money growth regime. This regime-shift could
induce a substantial drop in welfare if there are shocks to money demand.
Atkeson et al. (2010) report that the good performance of their monetary policy is robust to
trembles, in contrast to the conclusion reached here. Their conclusion reflects that they linearize
the map from individual intermediate good prices to the aggregate price index (see Atkeson etal. (2010, p. 53)). By the law of large numbers, zero-mean trembles are wiped out in that linear
representation. But, our analysis works with the actual nonlinear mapping, in which trembles domatter for Jensen’s inequality reasons. This is why we conclude that Atkeson et al. (2010)’s findingthat their policy uniquely supports the desired equilibrium is not robust to trembles. Our policy,
which adopts a wide monitoring range for inflation and the Taylor principle, uniquely implements
the desired equilibrium and is robust to trembles.
The first section below describes the basic ideas using a simple, static, non-monetary example.
Section 3 sets up our monetary model and shows that a model with a Taylor rule has a continuumof equilibria. We include this well-known result here so that our analysis is self-contained.9 Section
4 shows how the introduction of a monitoring range for inflation and an escape clause renders theequilibrium unique. We explain Cochrane (2011)’s blowing-up-the-world critique of this uniquenessresult. Section 5 defines a strategy equilibrium. We establish that our model has a strategy
equilibrium, and that the desired equilibrium is uniquely implemented by the escape clause. We
describe the key steps in the proof, but we move details to Appendix B. Section 6 reconciles our
findings about the escape clause with Cochrane (2011)’s critique. Section 7 addresses Atkeson
et al. (2010)’s conclusion that the Taylor Principle is not necessary to uniquely implement thedesired equilibrium. We also explain the lack of robustness of their version of the escape clause to8

We do not mean to suggest that the Taylor principle is desirable in all possible models. For example, in
Christiano et al. (2010b) and Christiano (2016) it is shown that if the working capital channel is strong enough,then the Taylor principle could be destabilizing. Christiano et al. (2010a) explain how the Taylor principle could
inadvertently trigger a stock market boom in response to news about the future.
9
See BSGU, as well as Woodford (2003).

7

trembles. Finally, section 8 oﬀers a brief conclusion.

2

A Simple Example

In this section we use a standard, static Laﬀer curve model to illustrate the type of argument
we make in this paper.10 As explained in the introduction, Cochrane (2011)’s conclusion thatthe escape clause eliminates undesired equilibria by a blow-up-the-economy threat is an artifactof his assumption that consumption is exogenous and does not hold when we make consumption
endogenous by introducing production. A similar result holds in the Laﬀer curve model, though
in the Laﬀer curve model the endogeneity/exogeneity issue has to do with endogeneity/exogeneityof government spending. In the standard Laﬀer curve model the required level of government
spending is exogenous, analogous to the exogeneity of consumption in Cochrane (2011)’s endowment economy. In this model there are two equilibria: the desired equilibrium (i.e., the Ramsey
equilibrium) and the undesired one. The objective of the government is to design a policy whichrules out the undesired equilibrium and ensures that the desired equilibrium occurs. We showthat under the standard Laﬀer curve model, a Cochrane (2011)-type result holds. The only way
to rule out the undesired equilibrium is with a blow-up-the-economy threat. But, if government
spending is endogenous then it is possible to rule our the undesired equilibrium without blowing
up the economy.

2.1

The Two Competitive Equilibria in the Laﬀer Curve Model

There is a representative household whose problem is:

max
c,l 0

subject to
10

c

1 2
l
2

c  w (1The model is a variant of the model analyzed in Bassetto (2005).

8⌧ ) l,

where c, l, w, ⌧ denote consumption, labor, the wage rate and the labor tax rate, respectively. The
solution to the household problem isl = (1⌧ ) w.

(2.1)

The government has an exogenous spending requirement, G⇤ 2 (0, 1/4), and its budget constraint
is expressed as follows:G⇤ = ⌧ wL,
where L denotes per capita employment.11 The production function of the representative firm is
y = l, so that competition implies w = 1. Let Y denote the per capita average value of y, so that
we have the following aggregate production relationship: Y = L. Clearing in the goods market
requires:
Y = C + G⇤ ,where C denotes the per capita average value of c.
We define:
Definition 1. A competitive equilibrium given G⇤ is a vector (⌧, c, C, l, L, Y ) such that:
1. given L and ⌧ , the household’s problem is solved and the government budget constraint issatisfied;
2. the following conditions are satisfied: l = L, c = C; goods market clearing; and the aggregate
production relation.
To characterize the set of competitive equilibria, combine l = L and the household’s optimality
condition, (2.1), to obtain L = 1⌧. After substituting into the government’s budget constraint,

we find:G⇤ = ⌧ (111⌧) .

It is more natural to express a government’s budget constraint as a weak inequality,G⇤  ⌧ wL.

That we express that constraint as a strict equality reflects our implicit assumption that the government values
private agent utility. As a result, it would never set ⌧ so that revenues exceeds G⇤ .

9

The graph of the function of ⌧ on the right of this expression is the standard Laﬀer curve and theset of equilibria corresponds to the two values of ⌧ , ⌧1 < ⌧2 , where the Laﬀer curve equals G⇤ .
Corresponding to these two equilibria, there are two values of aggregate employment, Li = 1⌧i ,for i = 1, 2. It is easily verified that the equilibrium associated with L1 is welfare-superior to theequilibrium associated with L2 . We refer to the first equilibrium as the desired equilibrium and to
the latter as the undesired equilibrium.

2.2

The Implementation Problem in the Laﬀer Curve Model

Consider the problem of a government which has the ability to commit in a pre-period to its tax
policy. The government’s problem, the implementation problem, is to design its policy so that theL1 equilibrium occurs and the L2 equilibrium is ruled out. That this is a challenging problem anddoes not necessarily have a solution is explained eloquently in Bassetto (2005).12 To understandthe challenge, suppose that the government commits to setting the tax rate unconditionally tothe particular value, ⌧ = ⌧1 . We then ask the question, “How does the ⌧1 policy rule out the L2
equilibrium?” This is the analog of Cochrane’s question in the monetary economy about how theescape clause rules non-desired equilibria. The answer here is that the L2 equilibrium is ruled out
by the government’s commitment to do something physically impossible in case L2 occurs. In particular, given the resources available to the government, ⌧1 L2 , it commits to transferring G⇤ > ⌧1 L2
resources to producers in exchange for goods. Of course, one might imagine that the government
could raise taxes or reduce spending in the event that L2 is realized, but these possibilities havebeen explicitly ruled out by the commitment made in the pre-period. Cochrane (2011) refers to
a commitment to doing something impossible as blowing-up-the-economy.13 Obviously, a policy
which achieves unique implementation by blowing up the economy if an undesired equilibrium
occurs is not interesting. This kind of policy would simply leave households puzzled and uncertain
about what to do or believe.
A way forward is suggested in Bassetto (2005). In particular, we suppose that in the pre-period
12

See also Bassetto (2002).
Atkeson et al. (2010) refer to this type of solution to the implementation problem implementation via nonexistence.
13

10

the government commits to a strategy that specifies government policies as a reaction to private
sector behavior. In our example the strategies, denoted by ⌧ (L) , react to L where L 2 L and
L contains at least the competitive equilibria, L1 and L2 . To formalize the notion that ⌧ (L) is areaction to L, we introduce two sub-periods, the morning and the afternoon. In the morning, labormarkets meet and clear, so L is determined. In the afternoon, L is taken as given and consumption
and the tax rate, ⌧, are determined in a continuation equilibrium:
Definition 2. A continuation equilibrium conditional on L and the policy, ⌧ (L) , is a triple(c, C, Y ) such that c = C = (1

⌧ (L)) L, Y = L = C + G⇤ , and G⇤ = ⌧ (L) L.

The continuation equilibrium conditions include those associated with the afternoon decisions:the household and government budget constraints. The equilibrium conditions that not included
are those associated with the labor market equilibrium that occurs in the morning (e.g., (2.1)).
Definition 3. A strategy equilibrium associated with ⌧ (·) and G⇤ is a competitive equilibriumsuch that there is a well-defined continuation equilibrium corresponding to any value of L 2 L.
In our example there is exactly one policy strategy with a well-defined continuation equilibrium,for each L 2 L :⌧ (L) =

G⇤.
L

This policy does not solve the implementation problem because each of the two competitive equilibria in section 2.1 is a strategy equilibrium given ⌧ (·).
This model confronts us with a conundrum. Either we have two equilibria and we fail to solve
the implementation problem. Or, the desired equilibrium is the unique equilibrium, but this isaccomplished by blowing up the economy in the event the undesirable equilibrium forms.
There is a parallel between this economy, in which government spending is exogenous, andCochrane (2011)’s endowment economy in which private consumption is exogenous. In both cases,
it appears that the only strategy which uniquely implements the desired equilibrium is one that is
supported by a blow-up-the-economy threat.

11

2.3

A Slight Modification to the Laﬀer Curve Model

We now consider a minor change in the model economy which has the eﬀect of endogenizing
government spending, G. Thus, suppose that G⇤ is simply a desired level of government spending,but the government could tolerate lower or higher government spending, G.14 With this change,
the policy strategy must also include a strategy for G. Consider, for example,⌧ (L) , G (L) =8>>< G⇤ , G⇤
L

L2 < L  1>>:⌧1 , ⌧1 L L  L2

⌧1 , G (L) = ⌧1 L, for all L 2 L = [0, 1) .

It is easy to verify that the desired equilibrium, L1 , is the only strategy equilibrium given ⌧ (·) andG (·). Consider again Cochrane’s question, “how does the policy strategy rule out the undesired
equilibrium?” The answer is that with ⌧ (L) = ⌧1 each household chooses l = L1 , so that there is
only one value of employment that clears the labor market, L = L1 . The undesired equilibrium isevidently not ruled out by a blow-up-the-economy threat because there is a well defined continuation equilibrium for each L 2 L including L = L2 . So, with the above policy there is a uniquestrategy equilibrium and the implementation problem is solved in a satisfactory way.
In general one gains more insight into the answer to Cochrane’s question by transforming theequilibrium into the fixed point of best response functions. Consider an individual household
contemplating how hard to work in the morning. The household uses the previously announced
government policy, ⌧ (L) , to form a belief about the labor tax rate. This in turn requires forming
a belief about how hard other households work, namely, L. Conditional on a belief about L thehousehold chooses to work l (L) = 1⌧ (L) . Under the assumption that all households behave inthe same way, we obtain what we call an aggregate best response function,
L0 = F (L) .
14

For example, suppose we add v (G) to the household utility function, where(1
G = G⇤
v (G) =
 otherwise,for  big enough.

12

That is, if everyone believes L then the aggregate per capita employment choice is F (L). A Nashequilibrium is a fixed point of F.
Figure 2.1 displays the best response functions corresponding to two diﬀerent policy strategies
in the endogenous G version of the Laﬀer curve model. Consider the policy considered in Section
2.2, G (L) = G⇤ ,⌧ (L) = G⇤ /L, L 2 L, where L = [G⇤ , 1) . This strategy creates a strategic
complementarity, which is captured by the fact that the best response function is increasing for
L 2 L. Because of its bow-shape, the best response function crosses the 45 degree line twice. The
two fixed points correspond to the two strategy equilibria in Section 2.2. So, as argued above, theimplementation problem is not solved by this policy.
Figure 2.1 also shows the best response in the model, when policy is given by G (L) = ⌧1 L
and ⌧ (L) = ⌧1 . Notice the dramatic change in the shape of the aggregate best response function.
The policy steers the economy away from the undesired equilibrium and to the desired equilibrium
because l = L1 is the dominant strategy for all L 2 L. The answer to Cochrane’s question, “howdid the policy rule out the undesired equilibrium?” is “if everyone thought that everyone else
would work L2 , then everyone would expect lower government spending G = ⌧1 L2 and respondby working L1 Everyone knowing this fact would never believe L2 in the first place.” Notice howreasonable the logic is and the absence of any bizarre threats to blow up the economy.15
Again, there is a parallel between the analysis here and what follows. With G endogenous we
can uniquely implement the desired equilibrium without a blow-up-the-economy threat in the same
way that the endogeneity of consumption in our production economy allows the escape clause touniquely implement the desired monetary equilibrium.
15

There is a strong resemblance to the bank run analysis of Diamond and Dybvig (1983). Absent government
policy intervention there are two equilibria: run and no-run. With deposit insurance, everyone’s dominant strategy
is not to run, even if everyone else is running on the bank. Everyone knowing that everyone else thinks in the sameway, believes there will be no run.

13

Figure 2.1: Best Response Functions
L0
Best response when ⌧ = ⌧1 , G (L) = ⌧1 L

L245

Best response whenG⇤
⌧ (L) =, G (L) = G⇤
L
L2
L1

L

Note: we set G⇤ = 3/16, so ⌧1 = 1/4 and ⌧2 = 3/4. Also, L1 = 3/4 and L2 = 1/4.

3

Model With Taylor Rule and No Escape Clause

Our monetary model is designed to maximize comparability with the models in 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).16 For
now, the model is deterministic, although in later sections we also allow for an exogenous shock.

3.1

Households

The economy is composed of a large number of identical households. The representative household
solves the following problem:
16

The model was originally constructed by Lucas (1990) and Fuerst (1992). See also Christiano (1990), Christiano
and Eichenbaum (1992), Christiano and Eichenbaum (1995) and Christiano et al. (1997).

14

max 1{ct ,lt ,mt ,dt }t=0

1
X
t=0

t

"

c1t

lt1+

1

1+

#

,

s.t. mt = R̄t (Xt + dt ) + mt 1
Pt c t  m t 1

> 1,

(3.1)

>0

d t + Wt l t + T t

Pt c t ,d t + Wt l t

(3.2)
(3.3)

m 1 given,
where ct and lt denote consumption and employment, respectively. Also, mt denotes the quantityof currency held by the household at the beginning of period t+1. The right side of (3.2) represents
the household’s period t sources of currency. The first source of currency takes the form of interest
and principle received on bank deposits, Xt + dt , at the end of period t after goods markets haveclosed. Here, Xt denotes a government transfer of currency directly into the household’s bankdeposit at the start of period t. Also, dt denotes currency deposited by the household into a bankat the start of period t. The currency that the household carries over from period t
by mt 1 , so that mt 1

1 is denoted

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 in advance constraint, (3.3). Finally, after the goods market closes,
the household receives Tt , which denotes firm profits net of government taxes.
According to (3.2) and (3.3), the trade oﬀ between consumption and leisure oﬀered to thehousehold 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 /¯⇡t+1 .
In the appendix we describe standard conditions which ensure boundedness of the household’sintertemporal consumption opportunity set. It is easily verified that under these conditions, thefirst order conditions associated with household employment, deposits and the cash constraint are,15

respectively,Wt= ct l t ,
Pt
ct

(3.4)

= ct+10 = R̄t

R̄t
,
⇡
¯t+1

(3.5)

1 (mt 1d t + Wt l t

Pt c t ) .

(3.6)

In the Kuhn-Tucker condition, (3.6), each expression in braces is non-negative. The transversality
condition is:(3.7)

lim qT mT = 0,

T !1

where
qT =

8
>
>
<1

T =0

>
Q
>
: Ts=01 R̄s

1

T

.

1

In the appendix we show that, under the boundedness conditions, the first order conditions andtransversality condition are necessary and suﬃcient for household optimization.

3.2

Production

A final output good is produced by a competitive, representative firm using the following production function:

Yt =

Z 1
0

"
" 1

" 1
"

Yi,t di," > 1.

(3.8)

Let pi,t denote the price of intermediate good, Yi,t , i 2 [0, 1]. The firm takes the price of output,
Pt , and the prices of the inputs, pi,t , i 2 [0, 1] as given. The first order conditions associated with
the ith firm’s profit maximization problem is:

Yi,t = Yt

✓

16

pi,t
Pt

◆ "

,

(3.9)

for i 2 [0, 1] . The first order conditions, together with the production function, impose a restriction
across the aggregate price index and the price of intermediate goods:

Pt =

Z 1
0

p1i,t " di

1
1 "

.

(3.10)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 fromthe bank and must repay R̄t Wt li,t after the goods market closes. At that time, the government
provides a subsidy, ⌧t , to the firm so that its post-subsidy end-of-period wage bill is (1⌧t ) R̄t Wt li,t .

The ith firm maximizes end-of-period profits by setting its price as a markup over marginal cost,(1⌧t ) R̄t Wt :pi,t =""

1(1⌧t ) R̄t Wt = Wt ,(3.11)

for each i. The second equality in (3.11) reflects our assumption that ⌧t 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). Because marginal cost is the same for all i, allintermediate good firms set the same price, pi,t . As a result, by (3.10) we have:

Pt = W t .

3.3(3.12)

Government Policy

The government’s monetary transfer, Xt , to households at the start of t satisfies

Xt = (µ̄t1) M̄t 1 ,17µ̄t = M̄t /M̄t 1 .

(3.13)

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 }1
t=0 , so that, in equilibrium,
⇢
⇣⇡
¯t ⌘
Pt+1
R̄t = max 1, R̄⇤ ⇤, ⇡¯t+1 ⌘, R̄⇤ ⌘ ⇡
¯⇤/ ,
⇡
¯
Pt
where ⇡¯⇤

1 and R̄⇤ are the desired inflation and interest rate. Here, we assume that

(3.14)> 1.

In addition, the government levies a lump sum tax, Ttg , on households to finance the subsidyin (3.11) to firms. Thus,
Ttg = ⌧t R̄t Wt lt ,where⌧t = 1"

1"

R̄t 1 .

The government does not purchase any goods, or issue debt.17

3.4

Market Clearing and Equilibrium

The goods, labor, money and loan market clearing conditions are:c t = Yt ,Z 1li,t = lt ,M̄t = mt ,Xt + d t = W t l t ,(3.15)

0for t

0. Also, household profits net of taxes are:

Tt =

Z

i

⇥

pt,i yt,iR̄t Wt (1⇤⌧t ) lt,i diTtg .

To ensure that households are identical, we require m 1 = M̄ 1 .
Let⇣⌘āt = lt , {li,t }i2[0,1] , {pi,t }i2[0,1] , ct , ⇡¯t , R̄t , Wt , µ̄t , M̄t , mt , dt .

(3.16)We define a competitive equilibrium as follows:18
17

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).
18
To shorten the definition, we refer to our equilibrium concept as a competitive equilibrium. We do this eventhough some of the agents in our model are not competitive.

18

Definition 4. A competitive equilibrium under the Taylor rule is a sequence, (āt )1
t=0 , that satisfies,for t0, (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(3.11); condition (ii) corresponds to (3.8)-(3.10); condition (iii) corresponds to (3.4)-(3.7); condition(iv) corresponds to (3.13)-(3.14); and condition (v) corresponds to (3.15). We also work with aconcept of competitive equilibrium under constant money growth, in which (3.13)-(3.14) are replaced
by a constant value for µ̄t .

3.5

Properties of the Taylor Rule Equilibrium

We now obtain a dynamic equation that can be used to identify all the equilibria in our model.
We will see that, in equilibrium, our model coincides with the model Cochrane (2011). Our central
observation on Cochrane (2011) follows from the fact that the two economies do not coincide outof equilibrium.
Combining (3.12) and (3.15), we obtain:
1 = ct l t = ct + ,so that for all t0,
ct = 1.

(3.17)As a result, the intertemporal Euler equation reduces to the Fisher equation:

Rt = ⇡t+1 ,(3.18)

where Rt = ln R̄t /R̄⇤ , ⇡t+1 = ln (¯⇡t+1 /µ̄⇤ ) . We can also express the Taylor rule in scaled andlogged form:
Rt = max Rl , ⇡t ,19

(3.19)

⇣⌘⇤
where Rl ⌘ ln 1/R . It is also useful to define the scaled and logged money growth rate:µt ⌘ ln

✓

µ̄t
µ̄⇤

◆

(3.20),

where µ̄⇤ is the money growth level under the desired equilibrium, µ̄⇤ = ⇡¯ ⇤ . Combining (3.18) and(3.19) we obtain:(3.21)⇡t+1 = max Rl , ⇡t .

From the loan market clearing condition, (3.15), the Kuhn-Tucker condition (3.6) in equilibrium
is expressed as follows:19
0 = R̄t

1

M̄t

Pt c t ,M̄t

Pt c t .

(3.22)

Also, the transversality condition, taking into account the equilibrium conditions, (3.15) and (3.17),
corresponds to:
lim

T !1

T M̄T

PT

= 0.

(3.23)

Equation (3.21) is useful for studying the equilibria in the model for the following reason:
Proposition 1. For any sequence, (⇡t )1
t=0 that satisfies the diﬀerence equation, (3.21), it is possible
to construct the other equilibrium variables in such a way that all the conditions for a competitive
equilibrium are satisfied.

For the proof, see Appendix A.1.
19

We stress that this is an ‘equilibrium condition’ and not, for example, something that any particular agent in
the economy actually looks at. It is derived from (3.6) which contains the currency in the hands of the householdat the start of period t, mt 1 dt + Wt lt . The latter object is in the control of the individual household before theconsumption market opens, but in equilibrium it is equal to M̄t , which is something out of the control of households.

20

Figure 3.1: Fisher Equation and Taylor Rule
⇡t+1
l ,

⇡ t+

ln

✓

⇡
¯⇤

◆

Rl

1

=

m

ax

⇡u

⇡l

R

⇡t

45

⇡t

Rl

Figure 3.1, a variant on the well-known figure in Benhabib et al. (2001b, Fig. 1), graphs(3.21).20 From this figure it is easy to see that the model has many equilibria, each indexed by thevalue of ⇡0 . The desired equilibrium corresponds to ⇡t = 0 for t

0. Consider the two inflation

rates marked in Figure 3.1 by ⇡l and ⇡u . We refer to this interval as the inflation monitoring range.
Note that, due to the high value of , there is exactly one equilibrium in which inflation is always
in the monitoring range, ⇡t 2 [⇡l , ⇡u ] for t

0. The allocations in that equilibrium correspond to

the desired equilibrium. This observation plays an important role in the analysis below.

4

Model With Escape Clause

The fact that the unique equilibrium with ⇡t 2 [⇡l , ⇡u ] is the desired equilibrium for all t

0 is animportant motivation for the following policy:
Definition 5. Taylor rule with an escape clause: if t = 0, or if ⇡s 2 [⇡l , ⇡u ] for s = 0, ..., t 1,
t > 0, then follow the Taylor rule, (3.19), with > 1 in period t. Otherwise,noset (scaled) money
1
growth, µt , to a constant, µ 2 [µl , µu ]. Here, µu = ⇡u and µl = max ⇡l , log µ̄⇤ . Also,1
20

Rl < ⇡l  0  ⇡u < 1.

For an extended discussion see, for example, Woodford (2003, Chapter 2, section 4, p. 123). This figure plays
an important role throughout our analysis and so we include it here for completeness.

21

The lower bound on the range, [µl , µu ] , is designed to guarantee that unscaled money growthunder the escape clause, µ̄, is no less than unity. This helps to ensure uniqueness of equilibrium
under constant money growth.21 The lower bound on the monitoring range excludes inflation rates
that imply the zero lower bound on the interest rate is binding. This assumption is simply madefor analytic convenience.
A competitive equilibrium under a Taylor rule with an escape clause is unique and it is thedesired equilibrium. This result is established in three steps. First, we establish that the constantmoney growth rate under the escape clause has a unique equilibrium:
Lemma 1. Suppose M̄ 1 is given and monetary policy sets M̄t = µ̄M̄t 1 for t
There exists a unique competitive equilibrium with the properties:
R̄t =

1

µ̄ > 1,ct = 1,⇡¯t+1 = µ̄,

for

t

0,

0, where µ̄1.

and P0 = M̄ 1 µ̄.

For the proof, see Appendix A.2.
If Lemma 1 were not true and an equilibrium did not exist, it would be impossible to meaningfully ask what would happen if ⇡t 2
/ [⇡l , ⇡u ] , because agents would not know how to form
expectations about t + 1.22 The second step shows that there is no competitive equilibrium inwhich ⇡t 2
/ [⇡l , ⇡u ] :Lemma 2. Consider the case in which monetary policy is the Taylor rule with the escape clause.
An equilibrium has the following property: ⇡t 2 [⇡l , ⇡u ] for t

0.

This Lemma lies at the heart of our equilibrium uniqueness result. So, we include the proof
here:
Proof. Suppose not, and that there exists an equilibrium with ⇡T 2
/ [⇡l , ⇡u ] where T

0 is thefirst date in which the monitoring range is violated. Consider the case, ⇡T > ⇡u . The Taylor rule21
To see how multiplicity of equilibria can arise when money growth lies between and unity, see Albanesi et al.
(2003, Proposition 4).
22
For our analysis it is convenient that we have a unique equilibrium under constant money growth, but we havenot investigated whether uniqueness is necessary for our conclusions.

22

implies RT = ⇡T > ⇡u . Using Lemma 1, it is easy to verify that ⇡T +1 = µ  ⇡u , so that23
RT

(4.1)

⇡T +1 > 0,

violating the Fisher equation, (3.18). This contradicts the assumption of equilibrium.
Next, consider the case ⇡T < ⇡l . Then, RT = max Rl , ⇡T
RT < ⇡l . Also, it is easy to verify ⇡T +1µ⇡l , so that24

RT⇡T +1 < 0,

 0. But, Rl < ⇡l < ⇡l , so

(4.2)

violating the Fisher equation, (3.18). This contradicts the assumption of equilibrium. This establishes the result sought.
The third step is the main proposition:
Proposition 2. Suppose monetary policy is governed by the Taylor rule with an escape clause.
The only equilibrium is the desired equilibrium.
The result is not surprising, given Lemma 2 and Figure 3.1. The former says that there is noequilibrium with ⇡t 2
/ [⇡l , ⇡u ] . The latter indicates that the only equilibrium with ⇡t 2 [⇡l , ⇡u ] is⇡t = 0 for all t. For a formal proof, see Appendix A.3.
We have established that with the escape clause, the equilibrium with the Taylor rule is unique.
In this way, the problem of implementing the desired equilibrium is solved. Next, we turn to thequestion of whether that solution is an interesting one.
23

The (unsurprising) result, ⇡T +1 = µ, follows from four observations. Observation #1 (see Definition 5) isthat the escape 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 (3.18), R̄T ⌘ R̄⇤ exp (RT ); our assumption, R̄⇤ > 1; and our assumptions , ⇡T > 0 which imply
RT = ⇡T > 0. Observation #4 is that observation #3, (3.22) and (3.17) imply PT = M̄T . The result follows fromobservations #2 and #4.
24
The result, ⇡T +1 µ, is straightforward. As in Footnote (23), Lemma 1 implies PT +1 = µM̄T . From (3.17) wehave cT = 1 so that the cash constraint, (3.22), implies M̄T PT . The result follows.

23

5Unique Implementation of the Desired Equilibrium Without Blowing up the WorldWe know from Proposition 2 that the escape clause rules out non-desired equilibria. In this section
we describe an equilibrium concept that allows us to understand how it rules out equilibria. The
standard concept of equilibrium permits us to ask the agents in the model a narrow range ofquestions: ‘why did you choose to work this level of hours rather than that level?’, or ‘why did you
set a low price rather than a high price?’. Within the standard concept of equilibrium agents takethe broader environment, the aggregate price level, aggregate output, etc., as given. We cannot
ask them ‘big picture’ questions such as, ‘what is it about government policy that makes you think
hyperinflation will not occur?’. This type of question is necessarily about what would happen if theeconomy went oﬀ the equilibrium path. Attempts have been made to study what might happen
oﬀ the equilibrium path simply by studying the equations that hold in a competitive equilibrium(see Cochrane (2011)). However, this sort of inference can go seriously awry. As we show below,
our model oﬀers a stark illustration of this point.
The alternative concept of equilibrium that we use here was first advocated in macroeconomics
by Bassetto (2005). Bassetto (2005)’s framework applies in a situation like ours, in which thegovernment has full commitment to implement its policy rule and we want to understand theeconomic reason that that policy rule excludes equilibria. Atkeson et al. (2010) systemize theapproach by exploiting conceptual similarities between the framework used here and one developedin Chari and Kehoe (1990) to think about equilibrium when the government lacks the ability tocommit.
To integrate oﬀ-equilibrium allocations into the analysis, we must, in eﬀect, build exit-ramps
from the equilibrium path. In principle, that could be done in a variety of ways. For the purposeof comparison, we adopt the approach in Atkeson et al. (2010). This involves taking a closer lookat the intermediate good firms’ pricing decision. We do this in the first subsection below. Section
5.2 discusses the formation of firm beliefs, a feature of our model that plays a central role in
the analysis. Section 5.3 discusses the definition of a strategy equilibrium. The last two sections

24

discuss how high and moderate inflation equilibria are ruled out in the strategy equilibrium of ourmodel. Other cases are studied in Appendix B.

5.1

Sequence of Events During the Period

In our discussion of the model in Section 3 we assumed that the ith firm knows the wage, Wt , atthe time it sets its price, pi,t (see (3.11)). But, this assumption is problematic. The nominal wage
rate, Wt , is determined in markets simultaneously with other variables like Pt . We assume that theintermediate good firms set their price simultaneously and without knowing what the other firms
are doing. Under these circumstances, it is logically impossible for the ith firm to actually observe
Pt when it sets its own price, pi,t . The reason is that by (3.10), Pt is the consequence of the prices
set by intermediate good firms. Obviously, firms cannot see the consequences of their actions until
after their actions have been taken.
Motivated by the preceding observation, we assume that pi,t is set before Wt is realized. To
capture this observation, we divide each period t into two parts: morning and afternoon. Intermediate good firms set their prices, {pi,t }i2[0,1] , in the morning. In the afternoon, the intermediate
good firm prices are taken as given and are part of the state of the economy. Conditional on thisstate, a period t continuation equilibrium unfolds. We provide a formal definition of this equilibrium concept in Section 5.3 below. This is a sequence of markets equilibrium that starts in theafternoon of period t and continues into all future periods.

5.2Best Response Function, Fixed Point and Beliefs

We begin this section by focussing on the price-setting problem of an intermediate good firm. The
second subsection below aggregates across the firms.
5.2.1

The Problem of the Individual Intermediate Good Firm

In the morning of period t, the ith intermediate good firm sets its price, pi,t , as follows:epi,t = Wi,t
.

25

(5.1)

Here, the superscript, e, indicates the firm’s conjecture about the value that a variable (in this case,Wt ) will take on in the afternoon, as part of firm i’s conjecture about the period t continuation
equilibrium. As noted in Section (5.3) below, the economic state in this continuation equilibrium
includes, among other things, the prices set by all the other intermediate good firms, {pj,t }j6=i .
Thus, to set its price, the ith firm must form a conjecture about what the other intermediate goodfirms are doing.
We simplify the situation of the ith firm by adopting the following assumption:
Assumption 1. Intermediate good firm i’s conjecture, pej,t j6=i , about what other firms do ischaracterized by symmetry: pej,t = pej0 ,t for all j, j 0 6= i. We denote pe i,t ⌘ pej,t .
Given symmetry a conjecture, pe i,t , maps trivially (see (3.10)) into a conjecture about theaggregate price index that will be realized in the afternoon as part of the time t continuation
equilibrium. So, with a slight abuse of notation, we also denote the conjectured aggregate priceeindex by pe i,t . The ith firm’s conjecture about the nominal wage, Wi,t
, is given byeWi,t= pe i,t✓e
Wi,tpe i,t◆= pe i,t cei,teli,t= pe i,t cei,t

+

.

(5.2)

In addition to (3.10), we have taken into account the intra-temporal equilibrium condition forlabor, (3.4), that must hold in the afternoon. We have also used the fact that ct = lt in the time
t continuation equilibrium. The object, cei,t , is the ith firm’s conjecture about time t aggregate
consumption in the time t continuation equilibrium. That is not a trivial object to compute, sincehouseholds are forward looking and we turn to that issue below.
Substituting (5.2) into (5.1), we obtain an expression for the ith firm’s price decision:pi,t = pe i,t cei,t

+

.

(5.3)

To summarize, the logic underlying (5.3) is as follows. The ith firm forms a conjecture about what
other firms do, pe i,t . Treating pe i,t (and the history of the economy, which we discuss below) as
the conjectured state, the firm then derives its conjecture about the entire time t continuation
equilibrium. The object on the right side of (5.3) represents the time t market-clearing nominal
26

wage rate in that equilibrium. So, (5.3) expresses the ith firm’s price decision as a function of pe i,t
and the relevant past history.
It is convenient to scale (5.3) by Pt 1 µ̄⇤ and take logs:(5.4)

exi,t = ⇡i,t
+ ( + ) ln cei,t ,ewhere xi,t and ⇡i,t
are given by:

xi,t = ln
5.2.2

✓

pi,t
Pt 1 µ̄⇤

◆

e
⇡i,t
= ln

,

✓

pe i,tPt 1 µ̄⇤

◆

(5.5)

.

Aggregating Across Firms

The large amount of symmetry in the Dixit-Stiglitz environment motivates an additional assumption which facilitates aggregation:
Assumption 2. All intermediate firms form the same conjecture, so that pe i,t = pe j,t ⌘ pet for alli, j.
Under Assumption 2, we can delete the i index from the variables in (5.4) and refer to that
expression as the best response of the ‘representative intermediate good firm’.
We now provide a formal definition of a time t continuation equilibrium. We denote a timet

1 history, ht 1 , as follows:

ht 1 =

8>>< m 1 , P 1 , M̄ 1>>:(ht 2 , at 1 )

t=0
t

,

1

where at is the scaled and logged version of āt defined in (3.16):

at =

⇣

lt , {li,t }i2[0,1] , ⇡te , ct , ⇡t , Rt , Wt , µt , M̄t , mt , bt⌘.

Here, {pi,t }i2[0,1] in āt has been replaced by ⇡te . Also, R̄t in āt has been replaced by Rt . Finally,⇡t and µt denote scaled and logged inflation and money growth. Past history, ht 1 , matters in27this model in part because of the nature of monetary policy. For example, if ⇡j 2 [⇡l , ⇡u ] for alljt

1, then monetary policy in period t is the Taylor rule. Otherwise, it is a constant money

growth rule.
We define:
Definition 6. A time t continuation equilibrium conditional on (ht 1 , ⇡te ) is a sequence, (at+s )1
s=0 ,that satisfies all the date t + s, s 0 competitive equilibrium conditions (see Definition 4), with
one exception. The exception is the period t optimality condition for the intermediate good firm,(3.11).
Consistent with our discussion about the trivial relationship between the aggregate price index
and pe i,t , time t inflation in the continuation equilibrium, ⇡t , is simply equal to ⇡te . The reason the
period t optimality condition for the intermediate good price setting decision is not included in atime t continuation equilibrium is that that equilibrium takes pe i,t as given.
By Definition 6, cet is a function of the state, (ht 1 , ⇡te ) . So, we can write (5.4) as follows:
xt = ⇡te + ( + ) ln cet (ht 1 , ⇡te ) .

(5.6)

In practice, it there is only a finite set of types of history, (ht 1 , ⇡te ) . There are two types of ht 1 :one in which the inflation monitoring range has never been violated; and the other in which it was
violated at least once. The number of types of ⇡te depends on the type of ht 1 . If there has been no
violation of the inflation monitoring range in the past, then, there are four types of ⇡te to consider:⇡te > ⇡u , ⇡te 2 [⇡l , ⇡u ] ,1

Rl < ⇡te < ⇡l and ⇡te 

1

Rl . If there has been a violation in the past,then there are two types of ⇡te to consider: those for which the lower bound on the nominal rateof interest is binding and the others.
The object, xt , in (5.6) is the (scaled and logged) price chosen by the ith firm, given its conjecture, ⇡te , and given ht 1 . Because the ith firm thinks (correctly) that everyone else behaves in thesame way, the xt chosen by the ith firm is also its conjecture about what the others do and, hence,maps into a conjecture about the aggregate price index.25 So a firm’s conjecture that inflation
will be ⇡te leads, via a chain of reasoning captured by (5.6), to another conjecture about inflation,
25

The object, xt , is a scaled and logged price, log [pit / (Pt 1 µ̄⇤ )]. So, Pt = Pt 1 µ̄⇤ ext .

28

namely, xt . We assume that a firm can only have one belief about a given variable. For this reason
we suppose that a firm’s belief is a conjecture that is a fixed point of (5.6):
Definition 7. Given history, ht 1 , the ith firm’s belief about period t inflation is a value of ⇡tewith the property that xt = ⇡te in (5.6).
Thus, we distinguish between a conjecture and a special kind of conjecture which we call a belief.
Beliefs are the conjectures that occur in a competitive equilibrium. For example, we see from (3.17)
that in a competitive equilibrium, ct = 1, so that xt = ⇡te , according to (5.6). Conjectures thatare not beliefs allow us to contemplate oﬀ-equilibrium paths. Studying why such conjectures are
not equilibria provides the economic answer to why a particular oﬀ-equilibrium path is not an
equilibrium.

5.3

Strategy Equilibrium

Consider the following definition:
Definition 8. A strategy equilibrium is a competitive equilibrium (see Definition (4)) with theproperty that for each possible history ht 1 : (i) there is a well defined continuation equilibrium
corresponding to any value of ⇡te and (ii) there exists a ⇡te that is a fixed point of (5.4).
When ht 1 is composed of the equilibrium (aj )tj=01 , then condition (i) considers one-period
deviations from the equilibrium. Because we allow for arbitrary ht 1 we also consider the possibilityof multi-period deviations. Condition (i) is important for our purposes, since it is what allows us
to think in an organized way about why certain oﬀ-equilibrium paths are not equilibria.
We give our equilibrium concept a diﬀerent name from the one in Atkeson et al. (2010) only
because the objects in their equilibrium are functions while in our case they are sequences. We
call ours a strategy equilibrium because the intermediate good firms in our model select a belief
based on how they think the economy would react if that belief were true.
Finally, consider the following definition:
Definition 9. A policy uniquely implements an equilibrium if the competitive equilibrium is
unique and the equilibrium is a strategy equilibrium.
We now state the following result:29

Proposition 3. The Taylor rule with the escape strategy uniquely implements the desired equilibrium.
Given that the equilibrium is unique (see Proposition 2) it remains only to show that thedesired equilibrium satisfies (i) and (ii) in Definition 8. This requires evaluating the continuation
equilibrium for all possible (ht 1 , ⇡te ), where all the possibilities are described after equation 5.6.
The two sections below consider the continuation equilibria and best response function for two
interesting possibilities. For the others, see Lemma 5 in Appendix B.26
5.3.1

How is it that High Inflation is Ruled Out in Equilibrium?

Here, we investigate why ⇡t > ⇡u is impossible in a strategy equilibrium, when ht 1 is a history
in which the monitoring range has never been violated. We need to show that ⇡te > ⇡u cannot bea fixed point of (5.6). To do this, we first derive an expression for consumption in the period t
continuation equilibrium.
Period t + 1 is the first period of the constant money growth regime. According to Definition
5 the money growth rate, µ̄, is greater than unity, so that Lemma 1 implies that ct+1 = 1 andM̄t+1 = Pt+1 . Dividing the latter by the binding (because ⇡te > 0) cash constraint in period t,
M̄t = ct Pt , we obtain
µ̄ =

⇡
¯t+1
.
ct

(5.7)

Substituting into the household’s intertemporal Euler equation, (3.5),(cet )=

R̄t.
cet µ̄

Because the inflation monitoring range has not been violated in the past, the Taylor rule, (3.14),
is in operation in the current period. Substituting the Taylor rule into the latter equation andrearranging, we obtain, after taking logs,
ln cet =

1

⇡te

26

µ
1

.

(5.8)

Atkeson et al. (2010) describe two other models for which they obtain unique implementation with the escapeclause. We discuss those models below.

30

Substituting the last expression into (5.4), we obtain:

xt = 1 +( + )
1( + )µ
.
1

⇡te

(5.9)

The value of ⇡te such that xt = ⇡te is:
µ

⇡te =

< ⇡u .

Since the fixed point of the mapping, (5.9), is unique and lies below ⇡u it follows that there is no
fixed point greater than ⇡u . After scaling and logging, (5.7) implies:e⇡t+1= µ + ln cet

=

⇡te

1

µ
1

(5.10).

We summarize these results in the form of a lemma:
Lemma 3. Suppose ⇡te > ⇡u . Then,ln cet =

1

⇡te

µ
1

e
⇡t+1
=

,

1

⇡te

µ
1

,

(5.11)and there is no fixed point of the best response function with ⇡t > ⇡u .
Our analysis allows us to explain why ⇡t > ⇡u is not a competitive equilibrium. When anintermediate good firm contemplates the possibility that inflation will be high because the otherfirms are setting high prices (i.e., ⇡te > ⇡u ), it understands that the current interest rate will behigh because of the Taylor rule and next period’s inflation will be low because of (5.10). With thereal rate interest high, the firm believes aggregate consumption will be low, so that the demand
for labor and the real wage will be low too. With low conjectured marginal cost, the firm would
raise price by less than the amount implicit in the rise in ⇡te (this is captured by the coeﬃcienton ⇡te in (5.9) being less than unity)27 . Understanding that other firms think in the same way,
the firm would not form a belief, ⇡te > ⇡u in the first place. This is the economic reason why
we cannot have a one-period violation of the inflation monitoring range, ⇡t > ⇡u , followed by no27

Recall from (3.1) that>131

further deviations from equilibrium. For multi-period violations, see Appendix B.
Thus, we have an answer to the question, ‘why can there be no hyperinflation?’ If firms
anticipated such a thing they would anticipate a Volcker-style recession with low marginal costs
and they would choose not to set the high prices necessary for high inflation to occur. This explains
the observations in the introduction where we claimed that the escape clause works like the Taylor
principle.
5.3.2

How is Moderate, but Not Desired, Inflation Ruled Out in Equilibrium?

As in the previous section, consider a history, ht 1 , in which the monitoring range has never been
violated. We now consider the case, ⇡t 2 [⇡l , ⇡u ] and explain why it is that ⇡t 6= 0 cannot be an
equilibrium. Consider the following lemma:
Lemma 4. The only fixed point of the best response function, (5.4), for ⇡te 2 [⇡l , ⇡u ] is ⇡te = 0.
Proof. Consider ⇡te 2 [⇡l , ⇡u ]. To evaluate the right side of the best response function, (5.4), we
need to compute the period t afternoon continuation equilibrium. Consider the periods after t
first. According to (2), the unique competitive equilibrium under the Taylor rule with escapeclause has the property, ⇡t+1 = 0 and ct+1 = 1. We then obtain period t consumption in the periodt afternoon continuation equilibrium by substituting into the period t household intertemporal
Euler equation, (3.5):(cet )

=

cet+1

R̄t
R̄t ⇣ ⇡
¯t ⌘
=
=
,
⇡
¯t+1
µ̄⇤
⇡
¯⇤

where the last equality uses the Taylor rule, (3.14). The Taylor rule is in place in period t becauseof our assumption that the escape clause has not been activated in the past. Substituting the logof the latter expression into (5.4) and collecting terms, we obtain:
xt = ⇡te + (+) ln cet =1( + ) ⇡te

The only fixed point for this expression is ⇡te = 0. Thus, ⇡te 6= 0 is not a fixed point, so that thedesired result is established.

32

The intuition here is similar to the intuition underlying Lemma 3. Suppose ⇡u⇡te > 0.

Because inflation is above the desired level, the nominal interest rate is also high because of the
Taylor rule. At the same time, the rate of inflation in the next period is expected to be at its
desired level so that the real interest rate is high. This creates a fall in output, discouraging firms
from raising prices. Knowing this, firms will not form a belief, ⇡te > 0. Now consider the case,⇡l  ⇡te < 0. In this case, the government would generate a boom in output, giving firms an
incentive to raise prices with the consequence that such a belief is not an equilibrium.
The proof of Lemma 4 provides the economic reason why it is that if we have an equilibrium pathwith ⇡t 2 [⇡l , ⇡u ] for all t, there cannot be any deviations from the competitive equilibrium. The
reason is essentially the same as in Section 5.3.1. When price setters contemplate the conjecturethat ⇡t > 0, they anticipate that monetary policy will create a high real rate of interest, which
will slow down the economy and discourage firms from setting the high prices that ⇡t > 0 requires.
It is interesting to note how once again this resembles the operation of the Taylor principle. The
escape clause is responsible for this response to ⇡t > 0 because it is the reason that the uniquecontinuation equilibrium has ⇡t+1 = 0 after a moderate inflation.

6

Where did Cochrane (2011) Go Wrong?

We have explored why it is that non-desired inflation paths fail to be equilibria when monetary
policy is characterized by the Taylor rule with an escape clause. The reasons do not involve anyone
doing something infeasible. In contrast, as explained in Section 4, Cochrane (2011) argues thatthe escape clause trims undesired equilibria by a commitment to do something infeasible in casean undesired equilibrium occurs.
Why does Cochrane (2011) reach such a diﬀerent conclusion than we do? The diﬀerent conclusions may at first seem surprising, since in both cases a competitive equilibrium is characterized by
the same two equations: the Fisher equation, (3.18), and the Taylor rule, (3.19) (see Proposition
1). This illustrates the danger of drawing inferences about oﬀ-equilibrium events from the equilibrium conditions alone. One must work with an equilibrium concept like the strategy equilibrium,
which uses more information about the underlying economy than is revealed by the equilibrium
33

conditions. As it happens, there is an important diﬀerence between our economy and Cochrane(2011)’s.
Cochrane (2011, p. 574) assumes that the underlying economy is an endowment economy. To
see the eﬀect of this assumption, note that our model economy eﬀectively reduces to an endowment
economy if we assume that households supply a fixed amount of labor inelastically. In this case,consumption takes on the same value if we are on the equilibrium path, or we are on the oﬀequilibrium paths that are contemplated in a Strategy Equilibrium (see Definition 8). So, thehousehold’s intertemporal Euler equation, (3.5), reduces to the Fisher equation, (3.18), both on
and oﬀ-equilibrium. It follows that period t continuation equilibria simply do not exist when⇡te 6= 0. Put diﬀerently, in the endowment economy version of our model, the government commits
to do something infeasible (raise the real rate of interest) in case inflation is higher than desired.
Thus, deviations from the equilibrium path are enforced by a threat to blow up the economy, justas Cochrane (2011) suggests.
It follows that the unique competitive equilibrium of the endowment economy version of ourmodel is not a Strategy Equilibrium. But, Cochrane (2011) is wrong to extrapolate from theproperties of his model to those of monetary models in which output is endogenous. In particular,
his analysis oﬀers no reason to think that the escape clause is an uninteresting way to rule out
non-desired equilibria in the kinds of monetary models used in practice, such as the New Keynesianmodel.
Now we formally show that there does not exist a strategy equilibrium in the endowment versionof our model. The equilibrium conditions need to be modified; namely, the same equilibrium
conditions except for the intra-temporal optimality condition, (3.4), hold and drop lt from the listof date t equilibrium variables, āt , (3.16). For simplicity, we assume that lt = 1 for all t

0.

We show that if ⇡0e > ⇡0u , there does not exist a continuation equilibrium conditional on(h 1 , ⇡0e ) . The argument here is similar to the proof in Lemma 2. Period 1 is the first periodof the constant money growth regime. According to Definition 5 the money growth rate, µ̄, isgreater than unity, so that Lemma 1 implies that c1 = 1 and M̄1 = P1 . Moreover, labor supply
is exogenously given so that ct = 1. Dividing the latter by the binding (because ⇡0e > 0) cash

34

constraint in period t, M̄0 = P0 , we obtain(6.1)µ̄ = ⇡¯1 .
Substituting into the household’s intertemporal Euler equation, (3.5),

(c0 )

R̄0
,
µ̄

=1=

(6.2)

which boilds down to
R̄0 =

1

µ̄.

By scaling and logging both sides, we get

R0 = µ 2 [⇡l , ⇡u ]and at the same time, the interest rate should be high due to the Taylor principle, R0 = ⇡0 > ⇡tu .
Therefore there does not exist a continuation equilibrium for high inflation rate, ⇡te > ⇡tu . Since
there does not exist a continuation equilibrium for (h 1 , ⇡te ), there doest not exist a strategy
equilibrium for this economy.

7

The Atkeson et al. (2010) Argument

Atkeson et al. (2010) direct attention to a policy with , ⇡l = ⇡u = 0, so that the monitoring range
is a singleton composed just of the desired rate of inflation. Our model also supports their findingthat the desired equilibrium is the unique equilibrium outcome under their policy. However, thefirst section below shows that this result is fragile and not robust to small trembles. In the second
section below we explain why the fragility may have important welfare consequences.
Finally, in the introduction we expressed skepticism about Atkeson et al. (2010)’s suggestionthat their policy reveals the Taylor principle is not a necessary part of good monetary policy.
Under that policy, to keep the economy in the desired equilibrium,35> 1 is not necessary since theescape clause (assuming no trembles) can do that job all by itself. It is true that in the context
of the Taylor rule, the parameter setting,> 1, is often referred to as the Taylor Principle, andin that sense Atkeson et al. (2010) are right. But, in Section 6 we showed that in our model theescape clause is a commitment to raise the interest rate when inflation is high as a strategy forkeeping inflation close to its desired level. As discussed in the introduction, this is basically adescription of the Taylor Principle. So, replacing a Taylor rule with

> 1 with an escape clauseis tantamount to replacing one way of implementing the Taylor Principle with another.

7.1

Fragility of the Atkeson et al. (2010) Monetary Policy

We consider the possibility that a tiny subset (mass) of firms make a tiny mistake implementing
their price decision. When this happens, then inflation drops out of the monitoring range, triggeringa regime shift in money policy towards constant money growth.
We illustrate these points here with the use of trembles. Suppose that people form their beliefs
as they do in previous sections, without imagining the possibility of trembles. They choose theirbelief as a solution to a fixed point problem, as discussed in Section 5.2. However, when the timecomes for firms to post their price, a small number make a small mistake (the firm manager’s
hand ‘trembles’ as he/she writes the price on the door). In particular, let pet denote the ith firms
expectation of how other firms post their price for all i 2 [0, 1]:
price in absence of tremble

pi,t =z }| {pet (cet ) +

(7.1)

i,t .

This value of pet has the fixed point property that, given the continuation equilibrium conditionalon pet , firms set the same price. Suppose there is a tremble in the form of the additional variable,
i,t , on the right side of (7.1).

That tremble is a unit mean random variable which is drawn

independently by each firm. The distribution ofit

has two parameters, Jt , t 2 [0, 1]. With

probability 1

Jt , the ith firm does not tremble at all, so thatith firm draws

i,t from a uniform distribution with support, [1Jt andi,t ⌘ 1.

With probability Jt thet , 1 + t ].

We can allow botht to be arbitrarily close to zero so that the subset of firms that tremble is very small and36those that do tremble, do so by a small amount.
According to (3.10), the actual price index will be determined as follows :Pt =Z 1 ⇣
0pet (cet ) +i,t= pet (cet ) + exp (t )
where

"

exp (t ) = 1

⌘1 "

1
1 "

di

2 "

Jt + Jt(1 + t )(1
2 t (2 ")

(7.2)

t)

2 "

#11"

.

As expected,
t ! 0 as Jt ! 0 for fixed t ,t ! 0 as t ! 0 for fixed Jt .

Divide both sides of (7.2) by µ̄⇤ Pt 1 and take logs, to obtain:⇡t = ⇡te + ( + ) ln (cet ) + t .

(7.3)

Suppose the economy has, up to the present time, not experienced a tremble and that it has beenin the desired equilibrium. As a result, firms set ⇡te = 0, cet = 1 and they believe (they thinkt = 0) ⇡t = ⇡te = 0. But, this is not what will actually occur, if Jt , t > 0. In this case, thetremble pushes the economy to switch to the money growth regime because ⇡t = t > 0.

7.2

The Fragility has Real Consequences if there are High-Frequency
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 ruleor the money growth rule. But, if we assume money demand shocks are realized in the afternoon,
then there is a substantial loss in jumping from the Taylor rule to the money growth rule. The
assumption that the money demand shocks are realized in the afternoon is a way to capture thenotion that money demand shocks operate at a higher frequency than price changes. The model is37

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, ⌫t , by inserting them into the cash constraint:Pt ct  (mt 1 + Wt lt

Dt ) ⌫ t .

We assume these shocks are iid, have unit mean and are drawn from a uniform distribution with
continuous support. It is easy to show that under the Taylor rule, lt , ct behave as they do in thedesired equilibrium in the absence of money demand shocks. Under the Taylor rule, the money
supply moves in such a way that consumption and employment are first-best and are insulated
from the realizations of ⌫t . In the money growth regime, things are diﬀerent. It is easy to see thatin that equilibrium,l t = ct = h

⌫t
E⌫t

+

i +1 .

A shift from the interest rate regime to the money growth regime, says due to trembles, would
have a substantial negative welfare eﬀect.

8

Summary

In the current workhorse model in macroeconomics, the New Keynesian model, it is common to
focus on a locally unique equilibrium around the unique interior steady state. This equilibrium wasreferred to here as the ‘desired equilibrium’. The analysis of the desired equilibrium has contributed
to policy debates about a wide range of topics, from inflation targeting to the government spending
multiplier, to how to deal with financial frictions. In addition, the properties of the desired
equilibrium resemble, in some important ways, the dynamic properties of actual data.28 Yet,
it is known that the model has other equilibria. At the very least, it has another, non-interior,steady state (BSGU). Moreover, analysis of simple models suggests there could be other equilibria
as well.
28
For a recent summary of the role in data and policy analysis of the New Keynesian model that focuses on thelocally unique equilibrium around steady state, see Christiano et al. (2018b)

38

This state of aﬀairs represents a serious challenge, because without a convincing way to ruleout the other equilibria the model does not, in fact, pin down output, employment, inflation, etc.
Various rationales have been oﬀered for focussing on the desired equilibrium. Here, we analyze theescape clause strategy proposed by BSGU and Christiano and Rostagno (2001).
The standard analysis is validated if we suppose that the government commits to a regime
switch to some other policy that returns inflation to its desired level in the event that a nondesired equilibrium appears to be forming. There are at least three ways to understand why thismight be a natural policy. First, governments in fact do switch to very diﬀerent policies whenthings appear to be going seriously awry. The recent recession and financial crisis in the US is one
example. So, the notion that a commitment to a regime switch might coordinate expectations on
a desired equilibrium does not seem far-fetched. Second, the non-desired equilibria in the stand-in
for the New Keynesian model we consider here requires government complicity. A commitment tostop accommodating non-desired equilibria in case they occur seems like an obvious remedy.
We study how the escape clause rules out non-desired equilibria. To do this, we use theequilibrium concept that allows for out-of-equilibrium behavior suggested in Bassetto (2005) andAtkeson et al. (2010). Our model allow us to address critics of the escape policy studied in thispaper. In that policy, the government sets an inflation monitoring range that is relatively wide,
and then commits to switch to a money growth regime in case the economy jumps outside themonitoring range. We preserve the stabilizing properties of the Taylor Principle by keeping a highcoeﬃcient on inflation in the Taylor rule. We show that the escape clause rules out all non-desired
equilibria by what one might call a Global Taylor Principle. If the economy were to move awayfrom the desired equilibrium, say by a rise in inflation above the desired level, then the presenceof the escape clause would cause the real rate of interest to rise. The ensuing recession would
discourage individual firms from setting the high prices that the high inflation requires. As aresult, expectations of inflation above target cannot occur in the first place. We show that the
Global Taylor Principle does for out-of-equilibrium inflation what the Taylor Principle does for
actual inflation in a stochastic version of the desired equilibrium.29
29
How the Taylor Principle works in stochastic models is well understood. For completeness, we include a briefsummary of the results in footnote 7.

39Cochrane (2011) argues that the equilibrium uniqueness accomplished by the escape clauseis not due to the escape clause per se. He concludes that it is instead due to the fact that the
government commits to do something infeasible in the event that a non-desired equilibrium forms.
We agree that if this were the last word on the escape clause, it would render that proposal
absurd since the impact on private sector expectations of such a commitment would be completely
unpredictable. However, we show that Cochrane (2011)’s conclusion is simply an artifact of his
endowment economy assumption, since it is infeasible for the government to create a recession.
So, Cochrane (2011)’s findings provide no information about how the escape clause works in an
economy like the New Keynesian model, where production is endogenous. We suspect that in any
model with a monetary transmission that resembles the Taylor Principle, the escape clause will
exclude equilibria that are outside the neighborhood of the desired equilibrium.
We also push back against the argument in Atkeson et al. (2010) that the Taylor Principle isunnecessary to achieve a unique equilibrium. They reach this conclusion by studying a version of
the escape clause in which the inflation monitoring range is reduced to a singleton that includes
only the desired level of inflation. In this case the usual Taylor Principle, having a high coeﬃcient
on inflation in the Taylor rule, plays no role in stabilizing the economy. Everything is accomplishedby the escape clause itself. According to our analysis, their conclusion is misleading, because in ourmodel the escape clause works exactly like an out-of-equilibrium version of the Taylor Principle(the Global Taylor Principle). Moreover, their policy is not of practical relevance because we show
it is not robust to tiny errors in price setting made by a tiny fraction price setters (trembles). In
the presence of money demand shocks, trembles would set oﬀ the escape clause by accident - notbecause a non-desired equilibrium is forming - and the economy would switch to a money growth
rule. The money growth rule is welfare inferior in the presence of money demand shocks, in ourmodel.

40

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43

A

ProofsA.1

Proposition 1

Proof. For each ⇡0 > 0, there is a sequence of inflation, {⇡t }1
t=1 , that satisfies (3.19) in whichinflation explodes to 1. The interest rate associated with such a sequence is greater than unity foreach t, so that (3.22) is binding, MT /PT = cT = 1, for all T . Trivially, the transversality condition,(3.23), is satisfied. Similarly, a sequence that satisfies (3.21) with ⇡0 < 0 converges to ln ( /¯⇡ ⇤ ),
so that actual gross inflation converges to . To see that this sequence satisfies (3.23) note that ⇡tconverges in finite time to its fixed point, ⇡t = ln ( /¯⇡ ⇤ ). Suppose convergence occurs at t = t̄Then, for T > t̄, RT = 1 so (3.5) implies PT =that MT /PT = Mt̄ /Pt̄ , a constant

T t̄

Pt̄ . Setting MT =

T t̄

0.

Mt̄ for all T > t̄ so

cT for all T > t̄, we have that the cash constraint, (3.22), issatisfied and
Mt̄
MT
=
PT
Pt̄

T

! 0,

so that (3.23) is satisfied too. The uniqueness result is stated in Proposition 1.

A.2

Lemma 1

Proof. First, we show that in any equilibrium, R̄t > 1 for all t. Suppose, to the contrary, thatR̄t = 1 for some t

0. Then it follows that R̄t+1 = 1. To see this, note that ⇡¯t+1 =

Fisher equation so that Pt+1 < Pt . At the same time, M̄t+1
These two observations, together with M̄tby theM̄t because of our assumption, µ̄

1.

Pt y imply that M̄t+1 > Pt+1 y. The complementaryslackness condition then implies that R̄t+1 = 1.
By induction, we conclude that R̄t = 1 implies R̄t+s = 1 and ⇡¯t+s+1 =for s

0. But, thiscontradicts the hypothesis of equilibrium because the transversality condition is violated. To see
this, note that for any fixed t,

lim

T !1

T M̄T

PT

T t

= lim

T !1 PT /Pt| {z }
=1

44

t M̄T

Pt

t

=

lim M̄T > 0.
Pt T !1Thus, we conclude that in any equilibrium, R̄t > 1 for all t
Next, we show that with R̄t > 1, for all t

0.

0, the equilibrium conditions uniquely determine all

variables. (3.22) implies that the cash constraint binds, M̄tfor all t

Pt y = 0, for each t

0, so ⇡
¯t+1 = µ̄

0. Also, the Fisher equation implies

R̄t =

1

⇡
¯t+1 =

1µ̄ > 1.

The cash constraint and R̄0 > 1 implies

P0 y = M̄0 =) P0 =M̄ 1 M̄0M̄ 1=µ̄.
y M̄ 1y

We have established: Suppose M̄ 1 is given and monetary policy sets M̄t = µ̄M̄t 1 for tµ̄0, where

1. There exists a unique equilibrium with the properties:

R̄t =

1

µ̄ > 1,

for t⇡¯t+1 = µ̄,0,µ̄
and P0 = M̄ 1 ,ywhich is desired.

A.3

Proposition 2

Proof. Suppose, to the contrary, that ⇡0 6= 0. From Lemma 2 equilibrium has the property that themonitoring range is never violated, i.e., ⇡t 2 [⇡l , ⇡u ]. The Taylor rule, (3.19), the Fisher equation,l(3.18), and ⇡l > R , implies that in equilibrium:⇡t+1 = max Rl , ⇡t = ⇡t .
Evidently, ⇡0 6= 0 implies ⇡t 2
/ [⇡l , ⇡u ] for some t, given
conclude that ⇡0 = 0, establishing the proposition.

45> 1. This contradicts Lemma 2. We

B

Existence of a continuation equilibrium for arbitrary history

To show that the unique competitive equilibrium is a strategy equilibrium, we need to establishthat there exists a continuation equilibrium for arbitrary history. In the paper, we have shownthat a continuation equilibrium exists if the Taylor rule is operative at date t and the inflation rate
is weakly greater than ⇡l . (See Lemma 3 and Lemma 4). We cover the other cases in Lemma 5.
Lemma 5. For any (ht 1 , ⇡te ) , there exists a continuation equilibrium. In particular;
1
1. If the Taylor rule is operative at t and
Rl < ⇡te < ⇡l ,30 the continuation consumption,
employment and money demand at date t are the same Lemma 3.

2. If the Taylor rule is operative at t and ⇡te 
ment and money demand at date t are1

Rl ,31 the continuation consumption, employ-ln ct = ln lt =

Rl
1

µ

(B.1)
(B.2)

µt = ⇡te + ln ct .
Notice that the cash constraint at date t is binding.
3. If the money growth rule is operative and ⇡te 2 D (ht 1 ) , where⇢D (ht 1 ) = ⇡te : (1

) µ + ln

Mt 1
Pt 1

⇡te + µ > Rl ,(B.3)

then the continuation consumption, employment and interest rate at date t areln ct = ln lt = µ + ln
Rt = (1

Mt 1
Pt 1

(B.5)

) ln ct + µ.

4. If the money growth rule is operative and ⇡t 2
/ D (ht 1 ) , thenln ct = ln lt =

1

✓

(B.4)

⇡te



Rl

µ + (µMt 1⇡te ) + ln
Pt 1◆

Rt = Rl .
30
31

The existence of a continuation equilibrium for ⇡te >The existence of a continuation equilibrium for ⇡te >

46

1
1

Rl is already established in Lemma 3 and 4.
Rl is already established in Lemma 3 and 4.

(B.6)
(B.7)

Notice that we have already established existence of a continuation equilibrium if ⇡teTaylor rule is operative at time t.

⇡l and theProof. We construct a continuation equilibrium for each (ht 1 , ⇡te ) . From We know that there
exists a continuation equilibrium after t + 1 from Lemma 1 and Proposition 2. So, we need toconstruct date-t variables so that all the date-t afternoon equilibrium conditions, (3.5), (3.22), andare satisfied.
It is easy to show that the proof in Lemma 3 is applied for showing case (1). Thus we omit
the proof here and left it for readers. Consider case (2); we consider a history in which the Taylor
rule is operative at date t and ⇡te 

1

Rl . Suppose that ⇡te 

1

Rl . The monitoring range isviolated so that the continuation equilibrium from t + 1 is unique, and consumption and price are

ct+1 = 1,⇡t+1 = µ + ln

Mt
.
Pt

(B.8)

By construction of (B.2), the cash constraint (3.22) is satisfied as equality. binding cash constraint,
mt = Pt ct . Also, the Euler equation (3.5) is also satisfied. To show it,ln ct+1 + Rt
| {z } |{z}
=0

=Rl
=Rl

=Rl

⇡t+1 + ln ct|{z}=µ+ln ct(µ + ln ct ) + ln ctµ+(|1) ln ct = 0.
{z
}

=µ Rl

Therefore all the afternoon equilibrium conditions are satisfied by (B.1) and (B.2).
Second, consider case (3). Again we need to show that the Euler equation (3.5) and cash
constraint (3.22) are satisfied with (B.4) and (B.5) conditional on the continuation equilibriumfrom t + 1. In particular, consumption and inflation rate is

ct+1 = 1,⇡t+1 = µ + ln

Mt
.
Pt

(B.9)

Cash constraint is trivially satisfied by construction. Put diﬀerently, the real balance Mt /Pt is
47

equal to consumption ct . The Euler equation is satisfied since

ln ct+1 +| {z }
=0

= (1

=(1

) ln ct + µ

Rt
|{z}

⇡t+1 + ln ct|{z}) ln ct +µµ=µ+ln ct

ln ct + ln ct = 0.

Also, notice that the nominal rate is greater than the lower bound Rl :Rt = (1) µ + ln

Mt 1
Pt 1

(B.10)⇡te + µ > Rlsince ⇡te 2 D (ht 1 ) .
Finally consider case (4). Note that in this case the interest rate implied by (B.10) is weakly
less than Rl . Again from t + 1, there exists a continuation equilibrium withct+1 = 1,⇡t+1 = µ + ln

Mt
= µ + (µPt⇡te ) + ln

Mt 1
.
Pt 1

(B.11)For the last equation, we use the fact that the monetary policy at date h is a constant money
growth policy. Using (B.6), (B.7), and (B.11), we can show that the Euler equation is satisfied:ln ct+1 + Rt
| {z } |{z}
=0

=Rl



µ + (µ

=Rl

⇡t+1
|{z}

+ ln ctM=µ+(µ ⇡te )+ln P t

Mt 1⇡te ) + ln
Pt 1

1
t 1

✓

Rl



µ + (µMt 1⇡te ) + ln
Pt 1

Mt 1
Pt 1

ln ct .

The cash constraint is also satisfied. First notice thatMt

Pt ct () µ⇡te + ln

48

◆

= 0.

It is easy to show that
Mt 1
ln ct
Pt 1
✓

◆
Mt 1 1Mt 1ee= (µ ⇡t ) + ln+
Rl
µ + (µ ⇡t ) + ln
Pt 1
Pt 1
⇢

1
Mt 1
=
(Rl µ) (1) (µ ⇡te ) + ln
.
Pt 1
µ

⇡te + ln

(B.12)

Since ⇡te 2/ D (ht 1 ) , we have
Rl

(1

() Rl

µ



Mt 1
) µ + ln⇡te + µ
Pt 1

Mt 1
(1
) µ + ln
⇡te
Pt 1

0.

(B.13)

Combining (B.13) with (B.12), we getµ⇡te + ln

Mt 1
Pt 1

ln ct

0,

which is equivalent to cash constraint at date t. Thus (B.6) and (B.7) satisfy the date-t afternoon
equilibrium conditions from (ht 1 , ⇡te ) .

49