Indeterminacy with Inflation-Forecast-Based Rules in the Large Open Economy

Indeterminacy with Inflation-Forecast-Based
Rules in the Large Open Economy∗
Nicoletta Batini
Paul Levine
International Monetary Fund
University of Surrey
Joseph Pearlman
London Metropolitan University
November 21, 2005
Abstract
We examine the performance of inflation-forecast-based (IFB) nominal interest rates rules
in a two-bloc model. Using a fairly standard dynamic general equilibrium open economy model,
and a methodology used by Batini and Pearlman (2002), we obtain analytically the feedback
parameters/horizon pairs associated with unique and stable equilibria. Three key findings
emerge: first, indeterminacy occurs for any value of the feedback parameter on inflation if the
IFB rule is too forward-looking. Second, the problem of indeterminacy is intrinsically more
serious in the large open economy. Third, the problem is compounded further when central
banks respond to expected consumer, rather than producer price inflation.
JEL Classification: E52, E37, E58
Keywords: Nominal interest rate rules, inflation-forecast-based rules, indeterminacy, open
economy
∗ Research on this paper was carried out while Levine and Pearlman were visiting the European Central Bank
(ECB) July-August 2003 as part of their Research Visitors Programme. Thanks are owing to the ECB for this
hospitality and to numerous resident and visiting researchers for stimulating discussions. Views expressed in this
paper do not reflect those of the IMF or the ECB. An earlier version of this paper was presented at the International
Research Forum on Monetary Policy in Washington, DC, November 14-15, 2003. Useful comments from participants
at this Conference and from participants at seminars in Autonoma University of Barcelona, Pompeu Fabra, the
University of Frankfurt, the ECB, the Bank of England, the Bank of Canada and the Swiss National Bank are
gratefully acknowledged.

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Contents
1 Introduction
1
2
Recent Related Literature
3
3
The Model
4
3.1
Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3.2
Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.3
The Equilibrium and the Trade Balance . . . . . . . . . . . . . . . . . . . . . . . .
9
3.4
Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3.5
Sum and Difference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
4 The Stability and Determinacy of IFB Rules
13
4.1
Conditions for Uniqueness and Stability . . . . . . . . . . . . . . . . . . . . . . . .
14
4.2
The Sum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
4.3
The Difference System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
4.3.1
No Home Bias and IFB Rules Based on Producer Price Inflation . . . . . .
22
4.3.2
No Home Bias and IFB Rules Based on Consumer Price Inflation . . . . . .
24
4.3.3
The Effect of Home Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
5 Conclusions
28
A A Topological Guide to The Root Locus Technique
29

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1
Introduction
Under inflation targeting, the task of the central bank is to alter monetary conditions to keep
inflation close to a pre-announced target. One class of rules widely proposed under inflation tar-
geting are ‘inflation-forecast-based’ (IFB) rules (Batini and Haldane (1999)). IFB rules are ‘simple’
rules as in Taylor (1993), but where the policy instrument, the nominal interest rate, responds to
deviations of expected, rather than current inflation from target. The horizon in the rule is a
policy parameter, alongside the feedback parameters. In most applications, the inflation forecasts
underlying IFB rules are taken to be the endogenous rational-expectations forecasts conditional on
an intertemporal equilibrium of the model. These rules are of specific interest because as shown a
number of studies discussed in the next section estimates of IFB-type rules appear to be a good
fit to the actual monetary policy in the US and Europe of recent years.1
However, IFB rules have been criticized on various grounds. Svensson (2001, 2003) criticizes
Taylor-type rules in general and argues for policy based on explicit maximization procedures.2
Much of the literature, however, focuses on a more specific possible problem with Taylor-type rules
– that of equilibrium indeterminacy when they are forward-looking. Although empirical evidence
suggests that this has not been a feature of the monetary rule in the US been in recent years
(see next section), a different ill-designed rule or parametrization of the historical rule may cause
indeterminacy, and this is what we explore in this paper.
Nominal indeterminacy arising from an interest rate rule was first shown by Sargent and Wallace
(1975) in a flexible price model. In sticky-price ‘New Keynesian’ models this nominal indeterminacy
disappears because the previous period’s price level serves as a nominal anchor. But a more recent
literature now reveals the possibility of real indeterminacy with IFB rules taking two forms: if the
response of interest rates to a rise in expected inflation is insufficient, then real interest rates fall
thus raising demand and confirming any exogenous expected inflation. But indeterminacy is also
possible if the rule is overly aggressive.3 Here we examine these issues for the large open economy
1They are also of interest for small open economies because similar reaction functions are used in the Quarterly
Projection Model of the Bank of Canada (see Coletti et al. (1996)), and in the Forecasting and Policy System of
the Reserve Bank of New Zealand (see Black et al. (1997)) – two prominent inflation targeting central banks.
2We discuss his critique in a longer working paper version of this paper, Batini et al. (2004a), BLP henceforth.
3Both types of real indeterminacy can be illustrated in a very simple closed economy model: consider a special
case of ‘Phillips Curve’ set out in this paper, πt = Et(πt+1) + ayt, where πt denotes inflation and yt is the deviation
of output from its equilibrium level. Close the model with an ad hoc ‘IS’ curve yt = −b(it − Et(πt+1)) where it
is the nominal interest rate which is set according to an IFB-Taylor rule it = θEt(πt+1) + µyt. Substituting out
for yt and it we arrive at Et(πt+1) =
1+bµ
π
1+bµ
t which has a unique rational expectations solution πt = 0
−ab(θ−1)
iff
1+bµ
> 1 and a stable trajectory, tending to zero inflation in the long run, consistent with any initial
1+bµ−ab(θ−1)
1

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by studying the uniqueness and stability conditions for an equilibrium under IFB rules for various
feedback horizons in a two-bloc world, paying particular attention to possible implications for the
US/euro area region.
This paper employs the same root locus methodology employed by Batini and Pearlman (2002)
–henceforth BP– in the closed economy context to identify analytically the feedback parame-
ters/horizon pairs that are associated with unique and stable equilibria. We employ a standard
dynamic general equilibrium sticky-price model of two interacting large open economies, with some
modifications described below. Whilst recent papers have examined IFB rules for the small open
economy, analyzing a two-bloc model is particularly interesting because it allows us to explore the
implications for rational-expectations equilibria of concurrent monetary policy strategies of the
European Central Bank (ECB) and the Federal Reserve. In addition, by assuming that the two
blocs are identical in both fundamental parameters and in policy, we can use the Aoki (1981) de-
composition of the model into sum and differences forms; we can then examine whether findings in
the literature on the stability and uniqueness of equilibria based on a closed economy assumption
translate to the large open economy case.
Three key findings emerge from this paper. First, we find that indeterminacy occurs for any
value of the feedback parameter on inflation in the forward-looking rule if the forecast horizon lies
too far into the future.4 This reaffirms, for the large open economy case, results found by BP for
the closed economy case. Second, we find that the problem of indeterminacy is intrinsically more
serious in an open than in a closed economy. Third, we find the problem is compounded further in
the large open economy when central banks in the two blocs respond to expected consumer, rather
than expected producer price, inflation.
Our discussion of related literature below suggests that whether or not indeterminacy is actu-
ally observed is an open question. If we work on the assumption that central banks would like
to design rules that avoid this problem, the importance of our results lies in the nature of the
constraint involved by the desire for determinancy. In a related paper that examines IFB rules in
inflation rate otherwise– that is there is indeterminacy if θ < 1 or θ > 1 + 2(1+bµ) . In the latter case, overly
ab
aggressive feedback produces cycles of positive and negative inflation. Thus the inclusion of a feedback on output
reduces the region of indeterminacy. Empirical estimates of µ appear to be small, as discussed in section 2. So, in
our subsequent analysis, we focus exclusively on ‘pure’ IFB rules, i.e. rules without an output gap term.
4 The fact that forward-looking behavior is a source of indeterminacy can again be illustrated using the simple
model of the previous footnote. Consider a rule involving a feedback on current inflation and the current output
gap: it = θπt + µyt. Then re-working the analysis we arrive at Et(πt+1) = 1+bµ+aθ(1+(b+1)µ) which has a unique
1+bµ+a(1+(b+1)µ)
RE solution πt iff θ > 1. For this current-looking rule there is no upper-bound on θ: all values above 1 ensure
determinacy.
2

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an estimated closed economy using US data, Batini et al. (2004b) find that this constraint becomes
increasing tight as the forecast horizon increases with the consequence that optimized rules yield-
ing determinancy are increasingly cautious and have poor stabilization properties, especially when
robustness considerations are taken on board. Our results here indicate that this poor performance
of IFB rules in the closed economy may become even worse in the open economy.
The plan of the paper is as follows. Section 2 offers an overview of the main related papers.
Section 3 sets out our two-bloc model. Section 4 uses the root locus analysis technique to investigate
the stability and uniqueness conditions for IFB rules based on producer price or consumer price
inflation, allowing for the possibility of home consumption bias. Section 5 offers some concluding
remarks.
2
Recent Related Literature
Perhaps the best-known theoretical result in the literature on IFB rules is that to avoid indeter-
minacy the monetary authority must respond aggressively, that is with a coefficient above unity,
but not excessively large, to expected inflation in the closed economy context (see, among others,
Bernanke and Woodford (1997), Clarida et al. (2000), BP, Giannoni and Woodford (2002)) and,
in the small open economy context, see De Fiore and Liu (2002) and Zanna (2003)). Bullard
and Mitra (2001), Bullard and Schalling (2005) reworked this result in closed economy and large
open economy models respectively, where private agents form forecasts using recursive learning
algorithms. The possibility of indeterminacy arising from excessively forward-looking IFB rules
has been explored only for the closed economy by BP.
Empirically, both the Federal Reserve in the post-Volker era and European monetary authorities
post 1980 appear to have indeed responded to expected inflation with a coefficient greater than
unity (see Clarida et al. (2000); Castelnuovo (2003); Faust et al. (2001)).5 The empirical relevance
of monetary rules actually resulting in indeterminancy finds some support in the the work of Lubik
and Schorfheide (2004). In this study a Taylor rule on current inflation and the output gap was
estimated by Bayesian methods as part of a closed economy dynamic stochastic general equilibrium
model fitted to US data. They allow for the possibility of indeterminacy in the form of too little
5Although empirical evidence seems to lend support to the idea that the US and European central banks follow
IFB-type rules, the Lucas Critique suggests that there is a logical distinction between observing that a simple
reduced-form relationship holds between variables and assuming that such a relation holds as a structural equation.
For example, Tetlow (2000) demonstrates that a Taylor rule may seem to explain US monetary policy even if
monetary policy is set optimally, conditioning on literally hundreds of state variables.
3

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feedback of inflation and sunspots equilibria. They propose a test for determinacy6and find that
the Volcker-Greenspan regime is consistent with determinacy, but the pre-Volcker regime is not.
The case for an aggressive rule however has been questioned by a number of recent theoretical
studies. First, the result depends entirely on: (a) the way in which money is assumed to enter
preferences and technology; and (b) how flexible prices are. In the closed economy context, both
Carlstrom and Fuerst (2000) and Benhabib et al. (2001) showed, for example, that with sticky
prices the result is overturned when money enters the utility function either as in Sidrauski-Brock
or via more realistic cash-in-advance timing assumptions.7 With these assumptions, if the monetary
authority responds aggressively to future expected inflation it makes indeterminacy more likely,
whereas if it does so to past inflation it makes determinacy less likely.
Second, the result rests on the assumption that, in its attempt to look forward, the central
bank responds only to next quarter’s inflation forecast, not to forecasts at later quarters. However,
real-world procedures typically involve stabilizing inflation in the medium-run, one to two years
out. It follows that the above result may not translate into sound policy prescriptions for inflation
targeters. Complementing numerical results by Levin et al. (2001), BP showed analytically that
IFB rules may lead to indeterminacy in a standard dynamic closed economy general equilibrium
used, for example, by Woodford (1999). Below we build on this work to study indeterminacy
with IFB rules responding beyond one quarter in a dynamic two-bloc general equilibrium model.
In doing so we consider the impact of various degrees of openness and price flexibility on our
indeterminacy results, but stick to the conventional timing used in most open economy optimizing
agents models whereby real money entering the utility function refers to end-of-period balances.
3
The Model
Our model is essentially a generalization of the model use by Clarida et al. (2002), Benigno and
Benigno (2004) and Pappa (2004)8 to incorporate a bias for consumption of home-produced goods,
habit formation in consumption, and Calvo price setting with indexing of prices for those firms
who, in a particular period, do not re-optimize their prices. The latter two aspects of the model
6However Beyer and Farmer (2003) have questioned whether determinate and indeterminate models can be
disentangled in this way, so the jury is still out on whether we have observed indeterminancy.
7 De Fiore and Liu (2002) assume this latter type of cash-in-advance assumption and show, in the context of
a small open economy model, that indeterminacy results are sensitive to the various assumptions on the timing of
transactions.
8Unlike these authors who study optimal rules with and without coordination, our focus is on suboptimal simple
IFB rules.
4

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follow Christiano et al. (2001) and, as with these authors, our motivation is an empirical one: to
generate sufficient inertia in the model so as to enable it, in calibrated form, to reproduce commonly
observed output, inflation and nominal interest rate responses to exogenous shocks.
There are two equally-sized9 symmetric blocs with the same household preferences and tech-
nologies. In each bloc there is one traded risk-free nominal bond denominated in the home bloc’s
currency. The exchange rate is perfectly flexible. A final homogeneous good is produced competi-
tively in each bloc using a CES technology consisting of a continuum of differentiated non-traded
goods. Intermediate goods producers and household suppliers of labor have monopolistic power.
Nominal prices of intermediate goods, expressed in the currency of producers, are sticky.
The monetary policy of the central banks in the two blocs takes the same form; namely, that of
an IFB nominal interest rate rule with identical parameters. The money supply accommodates the
demand for money given the setting of the nominal interest rate according to such a rule. Since
the paper is exclusively concerned with the possible indeterminacy or instability of IFB rules, we
confine ourselves to a perfect foresight equilibrium in a deterministic environment with monetary
policy responding to unanticipated transient exogenous TFP shocks. The decisions of households
and firms are as follows:
3.1
Households
A representative household r in the ‘home’ bloc maximizes
1−ϕ
M
∞

t (r)
(C
P
N

E
t(r) − Ht)1−σ
t
t(r)1+φ
0
βt 
+ χ
(1)
1
1
1 + φ 
t=0

− σ
− ϕ
− κ

where Et is the expectations operator indicating expectations formed at time t, Ct(r) is an index
of consumption, Nt(r) are hours worked, Ht represents the habit, or desire not to differ too much
from other consumers, and we choose it as Ht = hCt−1, where Ct is the average consumption
index and h ∈ [0,1). When h = 0, σ > 1 is the risk aversion parameter (or the inverse of the
intertemporal elasticity of substitution)10. Mt(r) are end-of-period nominal money balances. An
analogous symmetric intertemporal utility is defined for the ‘foreign’ representative household and
the corresponding variables (such as consumption) are denoted by C∗t(r), etc.
9The population in each bloc is normalized at unity. It is straightforward to allow for different sized blocs; then
the Aoki decomposition, aggregates must be population-weighted and differences expressed in per capita terms.
10When h = 0, σ is merely an index of the curvature of the utility function.
5

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The representative household r must obey a budget constraint:
PtCt(r) + Dt(r) + Mt(r) = Wt(r)Nt(r) + (1 + it−1)Dt−1(r) + Mt−1(r) + Γt(r)
(2)
where Pt is a price index, Dt(r) are end-of-period holdings of riskless nominal bonds with nominal
interest rate it over the interval [t, t+1]. Wt(r) is the wage and Γt(r) are dividends from ownership
of firms. In addition, if we assume that households’ labour supply is differentiated with elasticity
of supply η, then (as we shall see below) the demand for each consumer’s labor by a particular
firm producing good m is given by
W
−η
N
t(r)
t(m, r) =
N
W
t(m)
(3)
t
1
1
where W
−η
t =
1 W
is an average wage index and N
N
0
t(r)1−η dr
t(m) =
1
0
t(m, r)dr is aggregate
employment by firm m.
We assume that the consumption index depends on the consumption of a single type of final
good in each of two identically sized blocs, and is given by
Ct(r) = CHt(r)1−ωCFt(r)ω
(4)
where ω ∈ [0, 1] is a parameter that captures the degree of ‘openness’. If ω = 0 we have autarky,
2
while the other extreme of ω = 1 gives us the case of perfect integration. For ω < 1 there is some
2
2
degree of ‘home bias’.11 If PHt, PFt are the domestic prices of the two types of good, then the
optimal intra-temporal decisions are given by standard results:
PHtCHt(r) = (1 − ω)PtCt(r); PFtCFt(r) = ωPtCt(r)
(5)
with the consumer price index Pt given by
Pt = kP 1−ωP ω
Ht
F t
(6)
where k = (1 − ω)−(1−ω)ω−ω. Assume that there is complete exchange rate pass-through onto
the import and export prices of final goods. Then the law of one price holds; i.e. prices in home
11The effect of home bias in open economies is also studied in Corsetti et al. (2002) and De Fiore and Liu (2002).
6

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and foreign blocs are linked by PHt = StP ∗ , P
where P ∗ and P ∗ are the foreign
Ht
F t = StP ∗
F t
Ht
F t
currency prices of the home and foreign-produced goods and St is the nominal exchange rate. Let
P ∗
ω
1−ω
t = kP ∗
P ∗
be the foreign consumer price index corresponding to (6). Then it follows that
Ht
F t
the real exchange rate Et = StP ∗t and the terms of trade
are related by the relationship
Pt
Tt = PHt
PF t
S
E
tP ∗
t
t ≡
=
P
T 2ω−1
t
(7)
t
Thus (since 2ω − 1 ≤ 0), as the real exchange rate appreciates (i.e., Et falls) the terms of trade
improve, except at the extreme of perfect integration where ω = 1 . Then E
2
t = 1 and the law of
one price applies to the aggregate price indices.
In a perfect foresight equilibrium, maximizing (1) subject to (2) and (3) and imposing symmetry
on households (so that Ct(r) = Ct, etc) yields standard results:
C
−σ
P
1
= β(1 + i
t+1 − Ht+1
t
t)
(8)
Ct − Ht
Pt+1
M
−ϕ
t
(C
i
=
t − Ht)−σ
t
(9)
Pt
χPt
1 + it
Wt
κ
=
N φ
P
t (Ct − Ht)σ
(10)
t
(1 − 1)
η
(8) is the familiar Keynes-Ramsey rule adapted to take into account of the consumption habit.
In (9), the demand for money balances depends positively on consumption relative to habit and
negatively on the nominal interest rate. Given the central bank’s setting of the latter, (9) is
completely recursive to the rest of the system describing our macro-model and will be ignored in
the rest of the paper. (10) reflects the market power of households arising from their monopolistic
supply of a differentiated factor input with elasticity η.
Households can accumulate assets in the form of either home or foreign bonds. Uncovered
interest rate parity then gives
S
1 + i
t+1
t =
(1 + i∗
S
t )
(11)
t
where i∗t is the interest rate paid on nominal bonds denominated in foreign currency.
7

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3.2
Firms
Competitive final goods firms use a continuum of non-traded intermediate goods according to a
constant returns CES technology to produce aggregate output
1
ζ/(ζ−1)
Yt =
Yt(m)(ζ−1)/ζdm
(12)
0
where ζ is the elasticity of substitution. This implies a set of demand equations for each interme-
diate good m with price PHt(m) of the form
P
−ζ
Y
Ht(m)
t(m) =
Y
P
t
(13)
Ht
1
1
where P
−ζ
Ht =
1 P
. P
0
Ht(m)1−ζ dm
Ht is an aggregate intermediate price index, but since final
goods firms are competitive and the only inputs are intermediate goods, it is also the domestic
price level.
In the intermediate goods sector each good m is produced by a single firm m using only
differentiated labour with another constant returns CES technology:
1
η/(η−1)
Yt(m) = At
Nt(m, r)(η−1)/ηdr
(14)
0
where Nt(m, r) is the labour input of type r by firm m and At is an exogenous shock capturing
shifts to trend total factor productivity (TFP) in this sector. Minimizing costs 1 W
0
t(r)Nt(m, r)dr
and aggregating over firms leads to the demand for labor as shown in (3). In a equilibrium of equal
households and firms, all wages adjust to the same level Wt and it follows that Yt = nf Yt(m) =
nf AtNt(m) = AtNt where nf is the number of firms and Nt is aggregate employment.
For later analysis it is useful to define the real marginal cost as the wage relative to domestic
producer price. Using (10) and Yt = AtNt this can be written as
W
κ
Y
φ
ω
M C
t
t
t ≡
=
(C
(15)
A
t − Ht)σ PFt
tPHt
(1 − 1)A A
P
η
t
t
Ht
Now we assume that there is a probability of 1−ξ at each period that the price of each intermedi-
ate good m is set optimally to P 0 (m). If the price is not re-optimized, then it is indexed to last pe-
Ht
8

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