Anchoring Expectations of Inflation

57 Pages·2011·0.44 MB·English

Anchoring Expectations of Inflation
Michael MAGILL
University of Southern California,
Martine QUINZII
University of California, Davis
First version: June 2009
This version: May 24, 2011
Abstract. This paper proposes an abstract framework for studying how monetary policy can
anchor agents’expectations of inflation—the prime mandate ofa CentralBank—in a regime where
monetary-fiscalpolicyisRicardiansothatagents’expectationsareindeterminate. Thenewelement
in the analysis lies in the way expectations are modeled as an equilibrium sequence of probability
distributionson afinite intervalofinflationrates. Toanchor agents’expectations theCentralBank
announces a public forecast of inflation and a bond-pricing (term-structure) policy to make the
forecastcredible. Inastationaryenvironmentwecharacterizetheconditionsthattheannouncement
and the term-structure policy must satisfy to be compatible with an equilibrium of the economy
and to be “strongly credible” in sense that the announced forecast is the only inflation process
compatible with the Central Bank’s term-structure policy. When applied to inflation targeting, a
term-structure rule which anchors agents’expectations has the property that the entire yield curve
is a monotone function of the current inflation rate, a natural generalization of the Taylor rule
applied to the entire term structure.
WearegratefultoparticipantsattheConference inhonorofAndreuMas-ColellatBarcelonaUniversity,the
SAET Conference on General Equilibrium at Ichsia, the 6th Cowles Conference in General Equilibrium at
YaleUniversityandtoparticipantsinseminars atColumbiaUniversity,UniversityofLund,CambridgeUni-
versity, the Swiss Central Bank, the Swiss Finance Institute, University of Zurich, University of Mannheim,
University of Rome, University of California,Santa Cruz, The Institute for Advanced Studies, Vienna, and
the University of Southern Californiafor helpful comments. All remaining errors are ours.
Theoriginalpapercirculatedunderthe title“ExpectationsofInflation,theTermStructure ofInterest Rates
and Monetary Policy”.
Research support from the Swiss Finance Institute is gratefully acknowledged.

1. Introduction
This paper provides an abstract framework for studying how monetary policy can anchor agents’
expectations of inflation in a regime where monetary-fiscal policy is Ricardian so that agents’
expectationsof inflationarea prioriindeterminate. The new elementin our analysisliesin the way
we model expectations as an equilibrium sequence of probabilitydistributions, and in our focus on
the consistency conditions that must be satisfied between bond prices and agents’ expectations of
inflation.
The framework is inspired by the current practice of inflation targeting. As a result of the
adverse experience of the 1970’s with an approach based on monetary aggregates, the focus of
CentralBanks has shiftedawayfromcontrollingmonetaryaggregatesto usinginterestrate policies
to control inflation, and in the last twenty years, many Central Banks, have come to adopt the
policy of inflation targeting. A stylized way of characterizing this approach to monetary policy—
convenient for the analysis that follows—maybe expressed as follows.1 Price stability—thecentral
focusofmonetarypolicy—isexpressedfirstbyaninflationtargetπ?,typicallytheannualpercentage
changein aconsumer priceindex (usuallyofthe orderof2%-3%),and thisisviewed astheaverage
rate around which inflation should fluctuate. It is explicitly recognized that due to fundamentals
or possibly due to agents’ expectations regarding future inflation, the inflation rate is inevitably
a random variable. Thus at any given date t the inflation rate π will typically differ from the
t
target rate π 6= π?. The second component of a Central Bank’s inflation targeting policy consists
t
of a periodic official announcement of its best prediction of the sequence of probability distributions
of inflation over a specified horizon T into the future such that the mean of this sequence of
distributions (approximately) reaches the target π? in a specified horizon T˜ < T (where T˜ and T
are typically of the order of two and three years respectively). This periodic announcement of its
inflation forecast is an integral part of an inflation targeting policy which is typically expressed by
a so-called fan chart showing the iso-probability contours for future inflation (see Figure 1 for the
May2009CPI fan chart ofthe Bank of England). The monetary policywhich should lead inflation
to follow the announced process is typically taken to be the short-term (overnight) interest rate,
chosen and announced by the Central Bank at these regular intervals.
The abstract model that we present incorporates many of the features of inflation targeting
described above. In particular it assigns a central role to an announced forecast by the Central
Bank, which takes the form of a sequence of probability distributions of future inflation rates to
1ForacompleteanalysisofinflationtargetingseeWoodford(2007,2010),Walsh(2009)andSvensson(2007,2010).
1
direct agents’ beliefs which—in this Ricardian setting—are otherwise indeterminate. The main
finding is that credibly anchoring agents’ expectations to an announced sequence of probability
distributions is more demanding than anchoring their beliefs to the mean path of this sequence of
distributions, and requires more instruments for monetary policy than just the short-term interest
rate.
Chart 2 CPI inflation projection based on market
interest rate expectations and £125 billion asset
purchases
Percentage increase in prices on a year earlier
6
5
4
3
2
1
+
0
–
1
2
3
2005 06 07 08 09 10 11 12
Figure 1: Fan chart issued by the Bank of England in May 2009. The fan chart shows confidence
intervals around the mode of the distribution of inflation rates at each future date, the darkest
band being a 10% confidence interval around the mode and each band adding 10% probability.
In any setting in which agents can buy and sell goods on markets and can trade securities
there must be a way of “keeping the accounts”. For this it is essential that there be a unit of
account, and monetary models focus on the factors which determine the purchasing power of the
unit of account. Pinning down this purchasing power can be done in many ways and Woodford
(2008) has emphasized the importance of doing this “without money”. However to illustrate our
approach to anchoring expectations in the simplest setting we will find it convenient to use a basic
Lucas-style cash-in-advance model of money with a production technology in which the short-term
2
nominal interest rate influences output. Monetary-fiscal policy is Ricardian, i.e. monetary policy
is primary and fiscal policy adapts to the resulting nominal value of the debt to ensure that the
government’s liabilitiesare always ultimately paid off (in present-value terms). The novelty of the
model is that agents’ expectations of inflation, modeled as a sequence of probability distributions
onafixed support,areendogenous variables: thetaskofthemonetaryauthorityistoinduce agents
to hold a sequence of probabilitydistributionsof its choice on this support, similarto the sequence
of probability distributions in the fan chart that the Central Bank announces to direct agents’
expectations of inflation. We have not attempted to introduce the much more realistic staggered
price setting-module typical of the New-Keynesian (NK) model (Woodford (2003)): this would
lead to an additional interesting feedback between expectations and prices. Adding this module
however substantiallycomplicates the analysis without changing the basic message of our model of
anchoring agents’ expectations.
Themodel iscloseinspirittothemodel ofNakajima-Polemarchakis(2005)2 whoshowedthatif
monetary policy consists solely of fixing the short-term interest rate, and fiscal policy is Ricardian,
then the equilibrium is indeterminate. In such a setting, at each date, the only attribute of the
random inflation at the next period which is determined in equilibrium is its mean which is tied
down by the Fisher equation relating the short-term nominal interest rate to the real interest rate
and the mean of next period’s inflation. Macroeconomists use the “Taylor principle”—increasing
the interest rate more than proportionately to the deviation of the current inflation from target
inflation—to focus on the unique trajectorywhich stays close to the steady state when the interest
rate is chosen to satisfy such a Taylor rule (King (2000), Woodford (2003)). Whether or not it is
appropriate to exclude all other equilibria from consideration has been the subject of discussion
(Benhabib-Schmitt-Grohe-Uribe (2002), Cochrane (2007)).
Ifwe acceptthe argumentthattheTaylorprinciple tiesdownexpectationsofinflation,then the
NK model asserts that all future endogenous variables are well-defined functions of the exogenous
shocks, which in the model are defined by calibrated processes. In practice the Central Bank
periodically makes its best prediction of future shocks given current information and uses this to
make its forecast of the expected path of inflation and output, and the associated policy path
of future short-term interest rates, which it then makes public. If the announcement is credible,
these expected short rates announced at date t, should coincide with those implicit in the prices
of bonds of longer maturities traded on the market at the same date. Svensson (2011) compared
2See also Bloise-Dr`eze-Polemarchakis (2005) for an example of indeterminate equilibrium when the government
policy is specified inpart inreal terms, in part in nominal terms.
3
the policy-path for the Swedish Central Bank (Riksbank) announced in September 2010 with the
forwardrates derived fromthe marketprices of long-termbonds atthe same date: these tworates,
three years forward, were approximately 3.8% and 1.8% respectively, implying a spread of 2%
between the policy-based and market-based rates for the three-year forward rate. A similarspread
between the two-year forward rate occurred in April 2009. This is a substantial discrepancy, and
as Svensson himself asserts, raises questions about the credibility of the Riksbank’s control over
agents’ expectations.
Thissuggeststhattheremaybe somemeritinexploringamodel akintotheNK model without
immediatelyinvokingthe Taylorprinciple, so as to focus attentionon alternativewaysof modeling
the idea of “anchoring agents’ expectations of inflation”. The approach that we take to modeling
the inherent indeterminacy in a Ricardian model is to allow for “sunspot beliefs” by permitting
agents’ expectations of inflation a priori to be any probability distribution on a given interval of
inflation rates. The Central Bank’s role of anchoring expectations is achieved by providing the
“co-ordinatingdevice” invoked in sunspot models, the co-ordinationbeing achieved by announcing
a sequence of probability distributionsof future inflationrates, conditional on the current inflation
rate and the current real shock. Such an announcement may or may not be viewed as credible
by agents in the private sector. The paper lays out the conditions that must be satisfied by the
announcement if it is to be viewed as credible and is to be adopted as the belief of the agents in
the private sector.
The first requirement for credibility is that the short-term interest rate policy of the Central
Bank be compatible with the announced inflation process and the real side of the economy, a
compatibility expressed by the Fisher equation. However if the Central Bank only determines the
short-term interest rate it can only ensure that its interest rate policy is compatible with the mean
of the inflationprocess, so thatmany beliefs differentfrom the one announced by the CentralBank
are also compatible with its interest rate policy. Since the prices of bonds of longer maturities
express agents’expectations of inflationoverthese future timeintervals,itseems naturaltoreverse
the usual logic of bond pricing—by which expectations determine prices—and let the choice of
bond prices tie down agents’ expectations of inflation. To this end we view the Central Bank as
seeking to control the prices of bonds of longer maturities in the same way that it seeks to control
the short-term interest rate.
We assume a stationary framework and, in the style of a statistician,represent the probability
distribution announced by the CB as a histogram on S subintervals of the support of the agents’
expectations. We show that if the CB determines the prices of S bonds of maturities 1,...,S—
4
which we call a term-structure policy—then the expectations of the agents in the private sector
can be anchored to the announced probability distribution. By this we mean that the announced
probabilitydistributionof inflationratesis the onlyone compatiblewiththe chosen term-structure
policy.
Although the CB has considerable flexibility in the choice of its announced inflation process,
there are limits to this flexibility, if it is to make the announcement credible by a term-structure
policy. The compatibility condition expressed by the Fisher equation requires that the announced
inflation process be compatible with the zero lowerbound (ZLB) on the nominal interest rate: this
eliminates expectations which put too much weight on deflation. Under a Markov assumption,
we characterize the expectations matrices which are compatible with an equilibrium with a non-
negative nominal interest rate (Theorem 8). When applied to inflation targeting, this result shows
that the ZLB places restrictions on the inflation targets that it is possible to achieve: the target
must be sufficiently high to accommodate deviations below the target which are still consistent
with non-negative interest rates. In particular we show that the deflation rate which corresponds
to the Friedman rule cannot be taken as a target when beliefs are stochastic.
Second in order to eliminate other possible beliefs of the agents, the expectations of inflation
must be uniquely associated with a term-structure rule—a property which we callstrong credibility
of monetary policy. We characterize the conditions which an expectations matrix must satisfy
in order to be associated with a term-structure rule which makes it strongly credible (Theorem
11). The main attribute of such matrices is that they translate into yield curves for the bonds of
different maturities which must differ sufficiently when the current inflation rate changes. When
appliedtoinflationtargeting,itfollowsthatanimmediatereturntotargetcannotbe madestrongly
credible, since it would imply a flat yield curve invariant to the current inflation rate for bonds of
all maturities greater than one. Expectations which are stochastically mean reverting to a target
generate a family of yield curves such that the entire yield curve increases with the level of current
inflation—a generalized Taylor rule applied to the entire term structure—and in order that the
yield curves be sufficiently different as a function of current inflation, the mean reversion rate to
the target must not be too rapid.
In Section 2 we present the model and show how the infinite-horizon equilibrium can be trans-
formed into a much simpler “reduced-form ” equilibrium with a minimumnumber of endogenously
determined variables: from these variables the complete “extensive-form” equilibrium can always
be recovered (Theorem 3). This is an important step in the analysis since an extensive form
equilibriumis much toocomplex an object to be amenable to direct analysis. In Section 3 we char-
5
acterize expectations matrices which can be made credible by a term-structure rule in a Markov
representative-agent economy without real shocks to productivity. Section 4 shows how the anal-
ysis can be extended to the case where agents are heterogeneous and the economy is subject to
real shocks. The characterization of equilibrium depends not only on current inflation but also
on the real shock and the distribution of income among the agents. Perhaps surprisingly we show
that under a natural assumption on the real shock process, agents’ expectations of inflation can be
anchored with a monetary policy acting on the same number of bonds as in the simple economy
of Section 3. This result (Theorem 15) permits us to show that in a stationary setting the pure
inflation targeting model of Section 3 can be extended to a model of flexible inflation targeting in
which the interest-rate policy is adapted to the real side as well as inflation. Section 5 addresses
some of the practical aspects of implementing a term-structure policy.
2. Monetary Economy
Consider a monetary economy in discrete time over an infinite horizon with a finite number of
agents, in which money serves not only as a unit of account but also as the medium of exchange.
The objects of trade are goods and securities and the purchase (payment) for either must be
made using money. There are two sets of agents: H, the finite set of households in the private
sector and a government, the monetary-fiscal authority. The monetaryauthority issues money and
nominal (government) bonds of different maturities and seeks to control agents’ expectations of
inflation by announcing an inflation forecast and a bond-pricing (interest rate) policy which makes
the forecast credible. The fiscal authority imposes taxes on the agents commensurate with the
monetary policy sothat the government’sdebt does not growtoofast: in shortthe monetary-fiscal
policy is Ricardian.
There are two sources of uncertainty in the economy: the first is nominal, the second real.
The nominal uncertainty arises from the beliefs of the agents regarding the possible future course
of inflation. The standard approach to modeling such beliefs in general equilibrium theory is to
assume that there is an exogenously given “randomizing device” (sunspots) defined on a known
state space with fixed probabilities, which serves to “co-ordinate” agents’ expectations (Cass-Shell
(1983)). Thedefectofthisapproachisthatsuchrandomizingdevicesaredifficult(ifnotimpossible)
to identify in practice, thus severely curtailing the applicability of the approach. In this paper we
suggestanalternativewayofmodeling“sunspotequilibria”whichretainsthebasicideathatbeliefs
can be self-fulfilling and influence the equilibrium. The agents’ beliefs are modeled as probability
6
distributions over future inflation rates and the “co-ordinating device” is a publicly announced
forecast of future inflation made by the Central Bank to which it seeks to anchor (co-ordinate) the
beliefs of the agents in the private sector.
Uncertainty and Event-Tree. The primitive assumption regarding agents’ expectations is
that there is a finite interval of the real line which serves as the support for the agents’ beliefs
regarding inflation at every date. A key objective of monetary policy is to anchor agents’ expecta-
tions, modeled as a probabilitydistribution on this support. To simplifythe analysis, a probability
distribution is represented by a histogram on S subintervals of the support, each subinterval being
identified by a particular inflation rate in the subinterval (typicallythe midpoint). The real uncer-
tainty arises from the fluctuations in the productivities of the agents in their production activities.
To simultaneouslymodel the nominal and real uncertainties, let S be an index set with S elements
so that Π = {π ,s ∈ S} denotes the representative inflation rates, and let G = {1,...,G} denote
s
the index set for the possible real shocks. Unlike the probabilities of the inflation rates which are
endogenously determined, the probabilities of the real shocks are taken as exogenously given. A
partial history of inflation shocks (s ,...,s ) and real shocks (g ,...,g ) up to date t leads to a
0 t 0 t
typical date-event or node ξ = ((s ,g ),...,(s ,g )) ∈ St × Gt = |D where |D is the set of all
t 0 0 t t t t
possible nodes at date t.
The union of all the partial histories defines the event-tree
∞
|D= |D
t
t=0
[
consisting of all possible date-events ξ into the indefinite future. Any date-event ξ has a date t(ξ),
a unique predecessor ξ− at date t(ξ)−1 and a set of immediate successors ξ+ at date t(ξ)+1. If
ξ = ((s ,g ),...,(s ,g )) then
0 0 t t
ξ− = ((s ,g ),...,(s ,g )), ξ+ = {((s ,g ), ...,(s ,g ),(s,g))| (s,g)∈ S ×G}
0 0 t−1 t−1 0 0 t t
For any node ξ ∈ |D, let |D(ξ) denote the sub-tree originating at ξ and let |D (ξ) denote the nodes
T
of the subtree |D(ξ) at date T.
Agents’ characteristics and actions. Each agent (household) h ∈ H has beliefs over the
event-tree:3 let Bh denote the probability that agent h assigns to passing through node ξ. Then
ξ
3We first present the model withpossibly different expectations for the agents tohighlight the co-ordinating role
of the forecast announced by the monetary authority.
7
Bh = 1 for all t ≥ 0 and Bh = Bh for all ξ ∈ |D. The basic object of interest for an
ξ ξ ξ0
ξX∈|Dt ξX0∈ξ+
agent is his consumption stream of the (single) commodity over the event-tree. However to earn
the right to such a stream the agent will need to work over his lifetime, i.e. to offer labor services
to firms who convert them into output and pay the agent for the labor services. To incorporate
production into the model in the simplest way which at the same time captures the idea that
production is influenced by the nominal interest rate set by the monetary authorities, we use the
device introduced by Lucas-Stokey (1985) and also used by Polemarchakis-Nakajima (2005). At
each date-event ξ ∈ |D an agent has an endowment eh of ‘time’which can be used either for leisure
(`h) or to produce labor services (Lh) with eh = `h+Lh, ξ ∈ |D. The agent sells the labor services
ξ ξ ξ ξ
Lh to a firm which uses them to produce y = ahLh units of output, the productivity of the agent’s
ξ ξ ξ ξ
laborservicesdepending ontherealandnotontheinflationshock.4 Ifch istheagent’sconsumption
ξ
of the good at node ξ, then the pair xh = (ch,`h) generates the flow utility uh(ch,`h) where the
ξ ξ ξ ξ ξ
functions uh,h∈ H, satisfy the following conditions:
Assumption U (Preferences) For each h ∈ H the function uh :|R ×[0,eh]→ |R has the following
+
properties
uh > 0, uh > 0
1. increasing and differentiably strictly concave: c `
uh < 0, uh < 0, uh uh −(uh)2 > 0
cc `` cc `` c`
2. supermodular: uh ≥ 0
c`
uh(c,`)→ ∞ as c → 0, ∀ `∈ (0, eh)
3. Inada conditions: c
uh(c,`)→ ∞ as ` → 0, ∀ c> 0
`
uh(c,`)→ 0 as c→ ∞, ∀ `∈ (0, eh)
4. asymptotic satiation: c
uh(c,`)→ 0 as `→ eh, ∀ c> 0
`
The consumption-leisure choice5 xh = (ch,`h) generates the lifetime expected utility
ξ ξ ξ∈|D
Uh(xh) = δt(ξ)Bhuh(xh), 0< δ < 1 (1)
ξ ξ
ξX∈|D
In addition to the consumption-leisure decision xh the agent needs to make a portfolio decision
ξ
at each date-event which enables him to finance this consumption stream. To model money us-
ing Clower’sidea that only money buys goods or securities we adopt the standard cash-in-advance
4Theformalassumptions willbemadeinSection4whereweassume aMarkov structurefortheshocks: ifatdate
t(ξ)the current shock is (π ,g) then ah =ah.
s ξ g
5Throughoutthepaperweuseboldfacetodenoteavectordefinedoverthewholeevent-tree|D(e.g. xh =(xhξ)ξ∈|D)
or a vector defined over the set of agents (e.g. x =(xh) ).
ξ ξ h∈H
8