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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 797, April 2004 --- Screen Reader Version*

Indeterminacy with Inflation-Forecast-Based Rules in a Two-Bloc Model^*

Nicoletta Batini ^† (Bank of England), Paul Levine ^‡ (University of Surrey), and
Joseph Pearlman ^§ (London Metropolitan University)

International Finance Discussion Papers numbers 797-807 were presented on November 14-15, 2003 at the second conference sponsored by the International Research Forum on Monetary Policy sponsored by the European Central Bank, the Federal Reserve Board, the Center for German and European Studies at Georgetown University, and the Center for Financial Studies at the Goethe University in Frankfurt.

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. The views in this paper are solely the responsibility of the author and should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

Abstract:

We examine the performance of forward-looking inflation-forecast-based rules in open economies. In a New Keynesian two-bloc model, a methodology first employed by Batini and Pearlman (2002) is used to 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 forecast horizon lies too far into the future. Second, the problem of indeterminacy is intrinsically more serious in the open economy. Third, the problem is compounded further in the open economy when central banks respond to expected consumer, rather than producer price inflation.

Keywords: Taylor rules, inflation-forecast-based rules, indeterminacy, open economy

JEL Classification: E52, E37, E58

Contents

• 1. Introduction
• 2. Recent Related Literature
• 3. The Model
  • 3.1. Households
  • 3.2. Firms
  • 3.3. The Equilibrium and the Trade Balance
  • 3.4. Linearization
  • 3.5. Sum and Difference Systems
  
  
• 4. Designing and Implementing Optimal Policy
  • 4.1. Formulating the Optimal Rule
  • 4.2. Implementing the Optimal Policy and Simple Rules
  
  
• 5. The Stability and Determinacy of IFB Rules
  • 5.1. Conditions for the Uniqueness and Stability
  • 5.2. The Sum System
  • 5.3. The Difference System
  • 5.3.1. No Home Bias and IFB Rules Based on Producer Price Inflation
  • 5.3.2. No Home Bias and IFB Rules Based on Consumer Price Inflation
  • 5.3.3. The Effect of Home Bias
  
  
• 6. Conclusions
• A. The Optimal Rule
• B. A Topological Guide to The Root Locus Technique
• C. Calibration
• Bibliography

NON-TECHNICAL SUMMARY

Under inflation targeting, the task of the central bank is to alter monetary conditions to keep inflation close to a pre-announced target. Since current inflation is usually predetermined by existing price contracts and so cannot be readily affected via monetary impulses, one class of rules widely proposed under inflation targeting are 'inflation-forecast-based' (IFB) rules (Batini and Haldane (1999)). IFB rules are 'simple' rules as in Taylor (1993), but where the policy instrument responds to deviations of expected, rather than current inflation from target and so allow to bypass the policy lags that exist when inflation is sticky.

These rules are of special interest 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. Besides, estimates of IFB-type rules appear to offer a good description of the actual monetary policy in the US and Europe of recent years.

However, IFB rules have been criticized on various grounds. One recurrent crit- icism by much of the literature has to do with the fact that forward-looking Taylor-type rules tend to lead to real indeterminacy. This implies that when a shock displaces the economy from its equilibrium, there are many possible paths for the real variables leading back to equilibrium. The fact that these rules may introduce indeterminacy and generate so called 'sunspot equilibria' is of interest because sunspot fluctuations-i.e. persistent movements in inflation and output that materialize even in the absence of shocks to preferences or technology-are typically welfare reducing and can potentially be quite large. In practice, the problem of real indeterminacy with these rules seems to take 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. Most of the literature in this area assumes that the economy is closed.

In this paper we extend this literature by studying the uniqueness and stability conditions for an equilibrium under IFB rules for various feedback horizons in open economies. In particular, we study determinacy under these rules in a New Keynesian sticky-price two-bloc model similar to Benigno and Benigno (2001) - BB henceforth - and Clarida et al. (2002) - CGG (2002) henceforth. We modify the BB/CGG (2002) model to include habit formation in consumption and price indexing, changes that help to improve the ability of the model to capture the inflation and output dynamics observed in the Euro area and the US. We also generalize the model to allow for the possibility that agents in the two blocs exhibit home bias in consumption patterns. This produces short-run and long-run deviations from consumption-based purchasing power parity, and improves the model's ability to replicate the large and protracted swings in the real euro/dollar rate observed since the launch of the euro. 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.

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. This reaffirms, for the open-economy case, results found in the literature 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 that the probability of indeterminacy is compounded further in the open economy when central banks in the two blocs respond to expected consumer, rather than expected producer price, inflation. Since both the ECB and the Federal Reserve focus primarily on consumer price inflation, and not on producer price inflation, our results on the poor performance of consumer price based rules have important normative implications.

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. Since current inflation is usually predetermined by existing price contracts and so cannot be readily affected via monetary impulses, one class of rules widely proposed under inflation targeting are `inflation-forecast-based' (IFB) rules (Batini and Haldane (1999)). IFB rules are `simple' rules as in Taylor (1993), but where the policy instrument responds to deviations of expected, rather than current inflation from target. 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 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. As shown in Clarida et al. (2000) - CGG (2000) henceforth- and Castelnuovo (2003), 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: we discuss his critique in section 5. Much of the literature, however, focusses on a more specific possible with Taylor-type rules -that of equilibrium indeterminacy when they are forward-looking. 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 now a problem of real indeterminacy emerges 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 (see CGG (2000) and Batini and Pearlman (2002)). But indeterminacy is also possible if the rule is overly aggressive (Bernanke and Woodford (1997); Batini and Pearlman (2002); Giannoni and Woodford (2002)).² Here we extend this literature by studying the uniqueness and stability conditions for an equilibrium under IFB rules for various feedback horizons in open economies. ³

In a New Keynesian closed-economy model, Batini and Pearlman (2002) illustrate analytically that long-horizon IFB rules (with or without additional feedbacks on the output gap) and with interest rate smoothing can lead to indeterminacy.⁴ This paper employs the same root locus methodology to show analytically the feedback parameters/horizon pairs that are associated with unique and stable equilibria in a New Keynesian sticky-price two-bloc model similar to Benigno and Benigno (2001) - BB henceforth- and Clarida et al. (2002) - CGG (2002) henceforth. We modify the BB/CGG (2002) model to include habit formation in consumption and price indexing, changes that help to improve the ability of the model to capture the inflation and output dynamics observed in the Euro area and the US. We also generalize the model to allow for the possibility that agents in the two blocs exhibit home bias in consumption patterns. This produces short-run and long-run deviations from consumption-based purchasing power parity, and improves the model's ability to replicate the large and protracted swings in the real euro/dollar rate observed since the launch of the euro. 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) decomposition 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 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. ⁵ This reaffirms, for the open-economy case, results found in the literature 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 that the probability of indeterminacy is compounded further in the open economy when central banks in the two blocs respond to expected consumer, rather than expected producer price, inflation.

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 compares IFB rules with monetary policy based on explicit optimization and addresses the `Svensson Critique'. Section 5 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 6 offers some concluding remarks and some possible directions for future research.

2  Recent Related Literature

So far, research on monetary policy strategy has identified a series of circumstances under which forward-looking optimal and simple IFB-type rules might result in multiple equilibria or instability. One of the earliest contributions on indeterminacy under inflation-targeting forward-looking rules is Bernanke and Woodford (1997). Assuming that agents form their expectations rationally, they showed that the equilibrium associated with forward-looking optimal inflation-targeting rules under commitment may not be unique when the central bank targets current (exogenously-determined) private-sector forecasts of inflation, either those made explicitly by professional forecasters or those implicit in asset prices. In this sense, their finding squares with the more general one in Sargent and Wallace (1975), who showed that any policy rule responding uniquely to exogenous factors may induce multiple rational-expectations equilibria.

Subsequent work by Svensson and Woodford (2003), again assuming rational expectations and commitment on the side of the central bank, revealed however that forward-looking optimal inflation targeting based instead on endogenously-determined forecasts as opposed to exogenous, private-sector forecasts might not necessarily lead to superior results. As their work emphasizes, the purely forward-looking procedure, often assumed in discussions of inflation forecast targeting, prevents the target variables from depending on past conditions. In other words, the target variables are not `history-dependent'.<sup>6</sup> This feature makes the rules sub-optimal, perhaps seriously so (Currie and Levine (1993)), and can lead to indeterminacy of the equilibrium (Woodford (1999)). Work on simple IFB rules also revealed that with these rules (i) responding to exogenous, private-sector forecasts, (ii) lacking `history dependence', and/or (iii) disregarding the way in which the private sector forms expectations when agents are not fully rational can result in multiple or unstable equilibria (see Svensson and Woodford (2003); and Evans and Honkapoja (2001, 2002)).

Perhaps the best-known theoretical result in the literature on IFB rules is that to avoid indeterminacy 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, CGG (2000) and, in the small-open-economy context, see De Fiore and Liu (2002)). Bullard and Mitra (2001) reaffirmed this result in a closed-economy model where private agents form forecasts using recursive learning algorithms.

Empirically, CGG (2000) found that the Federal Reserve appears to have indeed responded to expected inflation either one-quarter or one-year-ahead. Furthermore, the coefficient for the interest rate response to expected inflation has been considerably greater than 1 during the Volcker-Greenspan era. They also found that the same coefficient was significantly less than 1 in the pre-Volcker era, a possible cause, they argue, of the poor macroeconomic outcomes at the time. Estimates of an IFB rule augmented with an output gap feedback for the euro area by Castelnuovo (2003), using area-wide synthetic data going back to 1980 Q1, suggest that at an aggregate level, European monetary authorities have also responded to expected inflation one-year-ahead with a coefficient well above unity. This result would explain the successful disinflation observed in Europe in the 1980s, and accords with findings in Faust et al. (2001) on estimates of a similar reaction function for the Bundesbank over a slightly shorter period.⁷

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.⁸ 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)-LWW henceforth- Batini and Pearlman (2002) showed analytically that IFB rules may lead to indeterminacy in the standard IS-AS optimizing forward-looking model used, for example, by Woodford (1999). They also showed that this problem is alleviated if: (i) the central bank responds to averages of expected inflation, instead of expected one-period inflation at a specific horizon; (ii) the response is very gradual (i.e., when interest rate smoothing is high); or (iii) if the rule is augmented with a response to the output gap. Below we build on this work to study indeterminacy with IFB rules responding beyond one quarter in the context of a dynamic two-bloc New-Keynesian 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.⁹

3  The Model

Our model is essentially a generalization of CGG (2002) and BB 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 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-sized¹⁰ symmetric blocs with the same household preferences and technologies. 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 competitively 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.¹¹ The decisions of households and firms are as follows:

3.1  Households

A representative household $r$ in the `home' bloc maximizes

$$\mathcal{E}_{0}\sum_{t=0}^{\infty }\beta ^{t}\left[ \frac{ (... ...1-\varphi }-\kappa \frac{N_{t}(r)^{1+\phi }}{ 1+\phi }\right]$$ (1)

where $\mathcal{E}_{t}$ is the expectations operator indicating expectations formed at time $t$, $C_{t}(r)$ is an index of consumption, $N_{t}(r)$ are hours worked, $H_{t}$ represents the habit, or desire not to differ too much from other consumers, and we choose it as $H_{t}=hC_{t-1}$, where $C_{t}$ is the average consumption index and $h \in [0,1)$. When $h=0$, $\sigma >1$ is the risk aversion parameter (or the inverse of the intertemporal elasticity of substitution)¹². $M_{t}(r)$ 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 $C_{t}^{\ast }(r)$, etc.

The representative household $r$ must obey a budget constraint:

$$P_{t}C_{t}(r)+D_{t}(r)+M_{t}(r)=W_{t}(r)N_{t}(r)+(1+i_{t-1})D_{t-1}(r)+M_{t-1}(r)+\Gamma _{t}(r)$$ (2)

where $P_{t}$ is a price index, $D_{t}(r)$ are end-of-period holdings of riskless nominal bonds with nominal interest rate $i_{t}$ over the interval $[t,t+1]$. $W_{t}(r)$ is the wage and $\Gamma _{t}(r)$ are dividends from ownership of firms. In addition, if we assume that households' labour supply is differentiated with elasticity of supply $\eta$, then (as we shall see below) the demand for each consumer's labor is given by

$$N_{t}(r)=\left( \frac{W_{t}(r)}{W_{t}}\right) ^{-\eta }N_{t}$$ (3)

where $W_{t}=\left[\int_0^1 W_t(r)^{1-\eta} dr \right]^{ \frac{1}{1-\eta}}$ is an average wage index and $N_{t}=\int_0^1 N_t(r) dr$ is aggregate employment.

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

$$C_{t}(r)=C_{Ht}(r)^{1-\omega }C_{Ft}(r)^{\omega }$$ (4)

where $\omega \in \lbrack 0,\frac{1}{2}]$ is a parameter that captures the degree of `openness'. If $\omega =0$ we have autarky, while the other extreme of $\omega =\frac{1}{2}$ gives us the case of perfect integration. For $\omega <\frac{1}{2}$ there is some degree of `home bias'.¹³ If $P_{Ht}$, $P_{Ft}$ are the domestic prices of the two types of good, then the optimal intra-temporal decisions are given by standard results:

$$P_{Ht}C_{Ht}(r) = (1-\omega )P_{t}C_{t}(r)$$ (5)

$$P_{Ft}C_{Ft}(r) = \omega P_{t}C_{t}(r)$$ (6)

with the consumer price index $P_{t}$ given by

$$P_{t}=kP_{Ht}^{1-\omega }P_{Ft}^{\omega }$$ (7)

where $k=(1-\omega )^{-(1-\omega )}\omega ^{-\omega }$. Assume that the law of one price holds i.e. prices in home and foreign blocs are linked by $P_{Ht}=S_{t}P_{Ht}^{\ast },P_{Ft}=S_{t}P_{Ft}^{\ast }$ where $P_{Ht}^{\ast }$ and $P_{Ft}^{\ast }$ are the foreign currency prices of the home and foreign-produced goods and $S_{t}$ is the nominal exchange rate. Let $P_{t}^{\ast }=k{P_{Ht}^{\ast }}^{\omega }{P_{Ft}^{\ast }}^{1-\omega }$ be the foreign consumer price index corresponding to (7). Then it follows that the real exchange rate $E_{t}=\frac{S_{t}P_{t}^{\ast }}{P_{t}}$ and the terms of trade $\mathcal{T}=\frac{P_{Ht}}{P_{Ft}}$ are related by the

$$E_{t}\equiv \frac{S_{t}P_{t}^{\ast }}{P_{t}}=\mathcal{T}^{2\omega -1}$$ (8)

Thus (since $2\omega -1\leq 0$), as the real exchange rate appreciates (i.e., $E_{t}$ falls) the terms of trade improve, except at the extreme of perfect integration where $\omega =\frac{1}{2}$. Then $E_t=1$ 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 $C_{t}(r)=C_{t}$, etc) yields standard results:

$$1 = \beta(1+i_t)\left(\frac{C_{t+1}-H_{t+1}}{C_t-H_t}\right)^{-\sigma}\frac{P_t}{P_{t+1}}$$ (9)

$$\left(\frac{M_t}{P_t}\right)^{-\varphi} = \frac{(C_t-H_t)^{-\sigma}}{\chi P_t}\left[ \frac{i_t}{1+i_t} \right]$$ (10)

$$\frac{W_t}{P_t} = \frac{\kappa}{(1-\frac{1}{\eta})}N_t^\phi (C_t-H_t)^\sigma$$ (11)

(9) is the familiar Keynes-Ramsey rule adapted to take into account of the consumption habit. In (10), 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, (10) is completely recursive to the rest of the system describing our macro-model and will be ignored in the rest of the paper. (11) reflects the market power of households arising from their monopolistic supply of a differentiated factor input with elasticity $\eta$. As labour becomes more homogeneous, this elasticity rises and the real wage households can command then falls.

Households can accumulate assets in the form of either home or foreign bonds. Uncovered interest rate parity then gives

$$1+i_t=\frac{S_{t+1}}{S_{t}}(1+i_t^*)$$ (12)

where $i_t^*$ is the interest rate paid on nominal bonds denominated in foreign currency.

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

$$Y_{t}=\left( \int_{0}^{1}Y_{t}(m)^{(\zeta -1)/\zeta }dm\right) ^{\zeta /(\zeta -1)}$$ (13)

where $\zeta$ is the elasticity of substitution. This implies a set of demand equations for each intermediate good $m$ with price $P_{Ht}(m)$ of the form

$$Y_{t}(m)=\left( \frac{P_{Ht}(m)}{P_{Ht}}\right) ^{-\zeta }Y_{t}$$ (14)

where $P_{Ht}=\left[ \int_0^1 P_{Ht}(m)^{1-\zeta} dm \right]^{ \frac{1}{1-\zeta}}$ . $P_{Ht}$ 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:

$$Y_{t}(m)=A_{t}\left( \int_{0}^{1}N_{tm}(r)^{(\eta -1)/\eta }dr\right) ^{\eta /(\eta -1)}$$ (15)

where $N_{tm}(r)$ is the labour input of type $r$ by firm $m$ and $A_{t}$ is an exogenous shock capturing shifts to trend total factor productivity (TFP) in this sector. Minimizing costs $\int_{0}^{1}W_{t}(r)N_{tm}(r)dr$ 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 $W_{t}$ and it follows that

$$Y_{t}=A_{t}N_{t}$$ (16)

For later analysis it is useful to define the real marginal cost as the wage relative to domestic producer price. Using (11) and (16) this can be written as

$$MC_{t} \equiv \frac{W_{t}}{A_{t}P_{Ht}} = \frac{\kappa }{(1-\frac{1}{\eta })A_{t}}\left( \frac{Y_{t}}{A_{t}... ...) ^{\phi }(C_{t}-H_{t})^{\sigma }\left( \frac{P_{Ft}}{P_{Ht}}\right) ^{\omega }$$ (17)

Now we assume that there is a probability of $1-\xi$ at each period that the price of each intermediate good $m$ is set optimally to $P_{Ht}^{0}(m)$. If the price is not re-optimized, then it is indexed to last period's aggregate producer price inflation.¹⁴ With indexation parameter $\gamma \geq 0$, this implies that successive prices with no reoptimization are given by $P_{Ht}^{0}(m),\,\,P_{Ht}^{0}(m)\left( \frac{P_{H,t+1}}{P_{Ht}}\right) ^{\gamma },\,\,P_{Ht}^{0}(m)\left( \frac{P_{H,t+2}}{P_{Ht}}\right) ^{\gamma },... \,\,\,$ . For each intermediate producer $m$ the objective is at time $t$ to choose $\{P_{Ht}(m)\}$ to maximize discounted profits

$$\mathcal{E}_{t}\sum_{k=0}^{\infty }\left(\frac{\xi}{1+i_t} \... ...t+k}}{P_{Ht}}\right) ^{\gamma }-\frac{W_{t+k}}{A_{t}} \right]$$ (18)

given $i_t$ (since firms are atomistic), subject to (14). The solution to this is

$$\mathcal{E}_{t}\sum_{k=0}^{\infty }\left(\frac{ \xi}{1+i_t} ... ...amma }-\frac{1}{ (1-1/\zeta) }\frac{W_{t+k}}{A_{t}}\right] =0$$ (19)

and by the law of large numbers the evolution of the price index is given by

$$P_{H,t+1}^{1-\zeta }=\xi \left( P_{Ht}\left( \frac{P_{Ht}}{P... ... ^{\gamma }\right) ^{1-\zeta }+(1-\xi )(P_{Ht}^{0})^{1-\zeta }$$ (20)

3.3  The Equilibrium and the Trade Balance

In equilibrium, goods markets, money markets and the bond market all clear. Equating the supply and demand of the home consumer good and using (5) and the foreign counterpart of (6) we obtain

$$Y_t = C_{Ht}+C_{Ht}^*=\frac{P}{P_H} \left[(1-\omega) C+ \omega E C^* \right]$$ (21)

Given interest rates $i, i^*$ (expressed later in terms of a IFB rule) the money supply is fixed by the central banks to accommodate money demand. By Walras' Law we can dispense with the bond market equilibrium condition. Then a perfect foresight equilibrium is defined at $t=0$ as sequences $C_t$, $D_t$, $C_{Ht}$, $C_{Ft}$, $P_{Ht}$, $P_{Ft}$, $P_t$, $M_t$, $W_t$, $Y_t$, $N_t$, $P_{Ht}^O$, 12 foreign counterparts $C_t^*$, $D_t^*$, etc, $E_t$, and $S_t$, given past price indices and exogenous TFP processes. These 26 endogenous variables in total are given by 12 equations: (2), (5), (6), (7), (9), (10), (11), (19), (20), (21), and their foreign counterparts, and (8) and (12).

Combining the Keynes-Ramsey equations with the UIP condition we have that

$$\frac{P_{t}^{\ast }}{P_{t}}\left( \frac{C_{t}-hC_{t-1}}{C_{t... ...1}-hC_{t}}{C_{t+1}^{\ast }-hC_{t}^{\ast }}\right) ^{-\sigma }$$ (22)

Let $z_{t}=\frac{S_{t}P_{t}^{\ast }}{P_{t}}\left( \frac{C_{t}-hC_{t-1}}{ C_{t}^{\ast }-hC_{t-1}^{\ast }}\right) ^{-\sigma }$ . Then (22) implies that $z_{t+1}=z_{t}$. We consider a linearization in the vicinity of a symmetric steady state, $\bar{z}=1$. From the transient nature of the shocks it follows that this steady state remains unchanged and hence $z_{t}=1$ in any stable rational expectations equilibrium. Therefore¹⁵

$$\left( \frac{C_{t}-hC_{t-1}}{C_{t}^{\ast }-hC_{t-1}^{\ast }}... ...) ^{-\sigma }=\frac{P_{t}}{S_{t}P_{t}^{\ast }}=\frac{1}{E_{t}}$$ (23)

The model as it stands with habit persistence ($h>0$), $\sigma >1$ and $\omega \in \lbrack 0,\frac{1}{2})$ exhibits net foreign asset dynamics. This can be shown by writing the trade balance $\mbox{TB}_{t}$ in the home bloc as exports minus imports denominated its own currency:

$$\mbox{TB}_{t}=P_{Ht}C_{Ht}^{\ast }-P_{Ft}C_{Ft}=\omega \left... ...t }-P_{t}C_{t}\right) =\omega P_{t}(E_{t}C_{t}^{\ast }-C_{t})$$ (24)

using (5) and (6), the law of one price $P_{Ht}=S_{t}P_{Ht}^{\ast }$, and recalling the definition $E_{t}\equiv \frac{ S_{t}P_{t}^{\ast }}{P_{t}}$. Therefore there are net foreign asset dynamics unless $C_{t}=E_{t}C_{t}^{\ast }$. This is only compatible with (23) if either $\omega =\frac{1}{2}$ (no home bias), in which case $E_{t}=1$, and we start off with balanced trade; or if $\sigma =1$ and $h=0$ (no habit persistence).¹⁶

3.4  Linearization

We linearize around a baseline symmetric steady state in which consumption and prices in the two blocs are equal and constant. Then inflation is zero, $E_{t}=\bar{E}=1$ and hence from (24) trade is balanced. Output is then at its sticky-price, imperfectly competitive natural rate and from the Keynes-Ramsey condition (9) the nominal rate of interest is given by $\bar{\imath}=\frac{1}{\beta }-1$. Now define all lower case variables (including $i_{t}$) as proportional deviations from this baseline steady state¹⁷. Home producer and consumer inflation are defined as $\pi _{Ht}\equiv\frac{P_{Ht}-P_{H,t-1}}{P_{H,t-1} }\simeq p_{Ht}-p_{H,t-1}$ and $\pi _{t}\equiv\frac{P_{t}-P_{t-1}}{P_{t-1} }\simeq p_{t}-p_{t-1}$ respectively. Similarly, define foreign producer inflation and consumer price inflation. Combining (19) and (20), we can eliminate $P_{Ht}^{0}$ to obtain in linearized form

$$\pi _{Ht}=\frac{\beta }{1+\beta \gamma }\mathcal{E}_{t}\pi _... ...1}+\frac{(1-\beta \xi )(1-\xi )}{(1+\beta \gamma )\xi }mc_{t}$$ (25)

The linearized version of the real marginal cost for producers of intermediate goods in the home bloc, (17), is given by

$$mc_{t}=-(1+\phi )a_{t}+\frac{\sigma }{1-h}(c_{t}-hc_{t-1})+\phi y_{t}+\omega (s_{t}+p_{Ft}^{\ast }-p_{Ht})$$ (26)

The first term on the right-hand-side of (26) is a TFP shock. The second term is a risk-sharing effect: a rise in habit-adjusted consumption leads to an increase in the real wage (see (11)) and hence the marginal cost. The last term is a terms of trade effect, which implies that marginal costs falls if the terms of trade, $p_{Ht}-s_{t}-p_{Ft}^{\ast }$ in linearized form, rises.

Linearizing the remaining equations (8), (9), (12), (21) and (23) yields¹⁸

$$\pi_t-\pi^*_t = 2\omega(s_t-s_{t-1})+(1-2\omega)(\pi_{Ht}-\pi^*_{Ft})$$ (27)

$$c_t-\frac{h}{1+h}c_{t-1} = \frac{1}{1+h}\mathcal{E}_t c_{t+1}-\frac{1-h}{(1+h)\sigma}(i_t-\mathcal{E}_t \pi_{t+1})$$ (28)

$$\mathcal{E}_t\Delta s_{t+1} = i_t-i^*_t$$ (29)

$$y_t = (1-\omega)c_t+\gamma c^*_t-2\omega(1-\omega)(p_{Ht}-s_t-p^*_{Ft})$$ (30)

$$\sigma(c^*_t-c_t-h(c^*_{t-1}-c_{t-1})) = -e_t=(1-2\omega)(p_{Ht}-s_t-p^*_{Ft})$$ (31)

Note that (30) and its foreign counterpart imply that $y_{t}+y_{t}^{\ast }=c_{t}+c_{t}^{\ast }$. Also note that for the case when there is no home bias, $\omega =1/2$. Then (27) reduces to relative purchasing power parity for consumer price inflation.

Turning to spillover effects in our linearized form of the model, consider the case of no home bias. Then from (30) and (26) we obtain

$$mc_{t}=\frac{\sigma }{2(1-h)}\left[ y_{t}-hy_{t-1}+y_{t}^{\a... ...ht] +\phi y_{t}+\frac{1}{2}\left[ y_{t}-y_{t}^{\ast } \right]$$ (32)

It follows that the elasticity of marginal cost for intermediate goods home producers with respect to domestic and foreign current output, given output at time $t-1$, are given by $\kappa \equiv \frac{\partial mc_{t}}{\partial y_{t}}$ and $\kappa _{0}\equiv \frac{\partial mc_{t}}{\partial y_{t}^{\ast }}$ where

$$\kappa = \frac{\sigma}{2(1-h)}+\frac{1}{2}+\phi\,; \quad \quad \kappa_0=\frac{\sigma}{2(1-h)}-\frac{1}{2}$$ (33)

(33) indicates that the risk-sharing effect exceeds the terms of trade effect and there is positive spillover from output onto the marginal cost of the second bloc--implying a negative spillover on output--iff $\frac{ \sigma }{1-h}>1$ in the short-run (i.e., given output in period $t-1$). ¹⁹Iff $\frac{\sigma }{1-h}=1$, the risk-sharing and terms of trade effect cancel and there are no spillover effects. Empirical estimates discussed in Appendix C suggest that $\sigma >1,$ so under this calibration in our model spillover effects on output are negative. The effect of introducing habit is to enhance the risk-sharing effect and enhance these negative short-run spillovers.

3.5  Sum and Difference Systems

Since the economies are symmetric, the easiest way of analyzing them is to use the sum and difference systems, as introduced by Aoki (1981). We denote all sums of home and foreign variables with the superscript $S$, while we denote differences by $D$. The first thing to note when inspecting the equations above is that the sum system is independent of home bias, and can be written as

$$\pi_t^S = \frac{\beta}{1+\beta\gamma}\mathcal{E}_t\pi_{t+1}^S+\frac{\gamma}{1+\beta\gamma} \pi_{t-1}^S$$

$$+ \frac{(1-\beta\xi)(1-\xi)}{(1+\beta\gamma)\xi}[(\phi+\frac{\sigma}{1-h}) y_t^S-\frac{\sigma h}{1-h}y_{t-1}^S-(1+\phi)a_t^S]$$ (34)

$$y^S_t = \frac{h}{1+h}y^S_{t-1}+\frac{1}{1+h}\mathcal{E}_ty^S_{t+1}-\frac{1-h}{(1+h)\sigma} (i_t^S-\mathcal{E}_t\pi^S_{t+1})$$ (35)

where $\pi^S=\pi_H+\pi_F^*$, $y^S=y+y^*$, and we note that $\pi_H+\pi_F^*=\pi+\pi^*$.

However the difference system does depend on the home bias parameter, $\omega$, Writing $\pi^D=\pi_H-\pi_F^*$, $y^D=y-y^*$, etc., it can be written as

$$\pi^D_t = \frac{\beta}{1+\beta\gamma}\mathcal{E}_t\pi_{t+1}^D+\frac{\gamma}{1+\beta\gamma} \pi_{t-1}^D+\frac{(1-\beta\xi)(1-\xi)}{(1+\beta\gamma)\xi}mc^D_t$$ (36)

$$mc_t^D = -(1+\phi)a^D_t+\frac{\sigma}{1-h}(c^D_t-hc^D_{t-1})+\phi y^D_t+2{\bf\omega} (s_t+p^*_{Ft}-p_{Ht})$$ (37)

$$c^D_t = hc^D_{t-1}+\frac{(2\omega-1)}{\sigma}(p_{Ht}-s_t-p^*_{Ft})$$ (38)

$$y^D_t = (1-2\omega)c^D_t-4\omega(1-\omega)(p_{Ht}-s_t-p^*_{Ft})$$ (39)

$$\mathcal{E}_t\Delta s_{t+1} = i^D_t$$ (40)

For the case of no home consumption bias ( $\omega =\frac{1}{2}$) taking first differences of (39) and using (40) we have

$$\mathcal{E}_{t}y_{t+1}^{D}-y_{t}^{D}=i_{t}^{D}-\mathcal{E}_{t}\pi _{t+1}^{D}$$ (41)

In addition, when there is no home bias, the remainder of the difference system reduces to

$$\pi _{t}^{D}=\frac{\beta }{1+\beta \gamma }\mathcal{E}_{t}\p... ...-\xi )}{ (1+\beta \gamma )\xi }(1+\phi )(y_{t}^{D}-a_{t}^{D})$$ (42)

Note, as with other models of the same New Keynesian genre, there is a long-run inflation-unemployment trade-off.²⁰

The sum and difference systems can now be set up in state-space form given the nominal interest rate rule. This Aoki decomposition enables us to decompose the open economy into two decoupled dynamic systems; the sum system, that captures the properties of a closed world economy, and a difference system that instead portrays the open-economy case. In principle, we could close the model with a number of different Taylor-type rules and also, given a policymaker's objective function, with optimal rules for coordinated or independent policies. Here we choose to focus uniquely on IFB rules that feedback exclusively on expected inflation. Before doing so, in the next section we first offer answers to the more general question of why is it interesting to look at simple rules. We also discuss why, within the broader class of simple rules, we consider non-optimal simple rules rather than simple rules which are optimal within the constraints defining their Taylor form of simplicity.

4  Designing and Implementing Optimal Policy

4.1  Formulating the Optimal Rule

The analysis of IFB rules set out in the next section contributes to a large literature on monetary policy rules that focusses primarily on the properties of these non-optimizing simple rules, thereby neglecting the possibility that central banks set monetary conditions by means of some explicit optimizing procedure. This approach has been criticized by Svensson (2001, 2003), and in this section we attempt to address his critique. We start with a commonly used objective function at time $t=0$ for the home bloc of the form

$$\Omega _{0}=-\mathcal{E}_{0}\sum_{t=0}^{\infty }\beta ^{t}[\pi _{t}^{2}+\alpha _{y}(y_{t}-k)^{2}+\alpha _{i}i_{t}^{2}]$$ (43)

with an analogous expression for the foreign bloc. The term k indicates an ambitious output target which captures the distortion in our model arising from imperfect competition. The last term captures the policymaker's concern for deviations of the nominal interest rate from the natural rate $\bar{\imath}=\frac{1}{\beta }-1$. Recently, this `pragmatic' approach to rationalizing policy objectives has been replaced by a completely coherent approach that bases the policymaker's objectives on those of the representative household. CGG (2002) and BB pursue the latter course, and the form (43) with $k=0$ corresponds loosely to their quadratic approximation to the utility of the representative household. ²¹

Our linearized model can be expressed in state-space form as

$$\left[ \begin{array}{l} z_{t+1} \ \mathcal{E}_t x_{ t+1} \... ...hsf{B}\left[ \begin{array}{l} i_t \ i_t^* \end{array}\right]$$ (44)

where $\mathsf{z}_t=[k, a_{t-1}, a_{t-1}^*, y_{t-1}, y_{t-1}^*, \pi_{H(t-1)}, \pi_{F (t-1)}^*]$ is a vector of predetermined variables and $\mathsf{x}_t=[y_{t}, y_{t}^*, \pi_{Ht}, \pi_{F t}^*]$ is a vector of non-predetermined variables. A, and B are matrices with time-invariant coefficients. In our deterministic perfect foresight model, transient shocks are unanticipated and follow an exogenous process such as $a_t=\varrho a_{t-1}; a^*_t=\varrho^* a_{t-1}^*$, $\varrho, \varrho \in [0,1)$, with $a_0$ and $a_0^*$ given. Remaining predetermined variables begin at their steady state values; i.e., $y_0=y_0^*=0$, etc.²² Then the optimization problem for a world social planner is to maximize $a\Omega_O +(1-a) \Omega_0^*$ subject to (44) (where $\Omega_0^*$ is analogously defined for the foreign bloc) is set out in general form in Appendix A.²³The optimal rule given by this problem takes the `history-dependent' form

$$\left[ \begin{array}{l} i_t \ i^*_t \end{array}\right]=D_1z_t+D_2C_{21} \sum_{\tau=1}^{t}(C_{22})^{\tau-1}z_{t-\tau}$$ (45)

say, where $D_1, D_2$, $C_{21}$ and $C_{22}$ are segments of partitioned matrices defined in Appendix A. (45) can be decomposed into two parts. By including a relationship $k_{t+1}=k_t=k$ (a constant) in the state-space representation we can extract an open-loop component of policy that results from the inclusion of an ambitious output target in (43). (45) then decomposes into open-loop trajectories for the nominal interest rate plus a feedback that depends on the initial TFP shocks $a_0$ and $a_0^*$.

4.2  Implementing the Optimal Policy and Simple Rules

The optimal cooperative policy then consists of trajectories for nominal interest rates that would be followed in the absence of initial shocks to TFP (or, in a stochastic setting, in the absence of random shocks) and a reaction function consisting of a feedback on the lagged predetermined variables with geometrically declining weights with lags extending back to time $t=0$, the time of the formulation and announcement of the policy. Together these components constitute an explicit instrument rule. ²⁴As is well-known, there are two fundamental problems with implementing such a rule. First it is time-inconsistent: having announced the policy at time $t=0$, at any time $t>0$ there emerges an incentive for the social planner to redesign both open-loop and feedback components of policy. Second, the cooperative policy is not a Nash equilibrium so there exists at any time, including $t=0$, an incentive to renege and adopt a policy that is the best response to that of the other bloc.

One way of implementing the optimal policy that addresses both the time-inconsistency and cooperation problems is to design objective functions for the two blocs that do not coincide with the true welfare. The aim of the exercise is to choose this design, or `regime', so that if the two blocs independently optimize in a discretionary fashion, then in a non-cooperative time-consistent equilibrium the optimal policy will be implemented. Thus BB, in addressing the cooperation problem, force the central banks to be `inward-looking' in the sense that their loss function only includes domestic target (e.g., producer inflation rather than consumer inflation which implies an exchange rate target). Svensson and Woodford (2003) adopt this modified loss function approach to the time-inconsistency problem for a closed economy. The idea of modifying loss functions so that players in a game have the `wrong' welfare criteria is, of course, not new and is the basis of Rogoff-delegation and Walsh contracts. To a greater or lesser extent all these solutions are susceptible to the critique by McCallum (1995) of Walsh contracts, that they do not solve either the credibility or the coordination problem, but ``merely relocate'' them to demonstrating the commitment of the policymakers to their modified loss functions.

A second way of implementing optimal policy is to build up a reputation for commitment to both the second bloc and to the private sector. In a more realistic incomplete information setting where policymakers' objectives are not known to the public, but policy rules can be observed, the public can learn about the rule by observing the relevant data and applying standard econometric techniques. In principle this should be possible for rules of the form (45), but the New Keynesian features of the model (namely output and inflation persistence) make it particularly complex. This highlights the importance of rules being simple in the sense that the instrument is constrained to feed back on a limited number of variables and their lags such as in a Taylor rule, or their forecasts as in IFB rules.

As well as being more easily verifiable, simple rules may have other advantages. As shown in Currie and Levine (1993) and Tetlow and von zur Muehlen (2001), it is easier to learn about simple rules that (by definition) feed back on a limited selection of easily verifiable macro-variables, than to learn about complex optimal rules such as (45) . Taking this ability to learn into account, simple rules may then outperform their optimal counterparts. Finally it has been suggested that simple rules may be robust with respect to modelling errors (LWW, Taylor (1999)).

Simple rules can be designed to approximate the optimal rule by choosing the feedback parameters so as to maximize an objective function of the form (43). However the simplicity constraint means that the optimal simple rule is not certainty equivalent, unlike the optimal rule unconstrained to be simple. This means that if at time $t=0$ we designed a optimal simple rule of a particular form for our model above, optimal feedback parameters would depend on the transient shocks to TFPs, $a_{0}$ and $a_{0}^{\ast }$ and, in a stochastic setting, on the variance-covariance matrix of white noise disturbances in the stochastic process defining these shocks.²⁵ Then rules that perform well, in the sense of achieving a welfare outcome close to that of the optimal rule, under one assumed set of initial displacements and covariance matrix may well lack robustness in that they may perform badly under a different set of assumptions. However some structures of simple rule may be more robust than others.²⁶

Defining what we mean by the optimal simple rule is then problematic. The literature on determinacy, to which our paper contributes, has a more modest objective of providing guidelines to policymakers in the form of simple criteria for avoiding very bad outcomes that lead to multiple equilibria or explosive behaviour. In our set-up, these guidelines focus on the choice of feedback, interest rate smoothing and feedback horizon parameters. In the following section we pursue this research objective by looking at how such guidelines are affected when we proceed from the closed to the open economy and by the degree of openness in the latter.

5  The Stability and Determinacy of IFB Rules

This section studies two particular forms of simple rule, IFB rules either of the form

$$i_t=\rho i_{t-1}+\theta(1-\rho)\mathcal{E}_t \pi_{t+j}$$ (46)

where $j \ge 0$ is the forecast horizon, which is a feedback on consumer price inflation, or of the form

$$i_t=\rho i_{t-1}+\theta(1-\rho)\mathcal{E}_t \pi_{H t+j}$$ (47)

which is a feedback on producer price inflation. Both rules are in deviation form about some long-run zero-inflation steady state and could represent the feedback component of monetary policy that complements a (possibly optimal) open-loop trajectory designed as in the previous section. We assume that the foreign bloc has a similar rule with the same parameters and forecast horizon.

With rules (46) and (47), policymakers set the nominal interest rate so as to respond to deviations of the inflation term from target. In addition, policymakers smooth rates, in line with the idea that central banks adjust the short-term nominal interest rate only partially towards the long-run inflation target, which is set to zero for simplicity in our set-up.²⁷ The parameter $\rho \in \lbrack 0,1)$ measures the degree of interest rate smoothing. $j$ is the feedback horizon of the central bank. When $j=0$, the central bank feeds back from current dated variables only. When $j>0$, the central bank feeds back instead from deviations of forecasts of variables from target. This is a proxy for actual policy in inflation targeting countries that apparently respond to deviations of current inflation from its short or medium forecast (see Batini and Nelson (2001)). Finally, $\theta >0$ is the feedback parameter: the larger is $\theta$, the faster is the pace at which the central bank acts to eliminate the gap between expected inflation and its target value. We now show that, for given degrees of interest rate smoothing $\rho$, the stabilizing characteristics of these rules depend both on the magnitude of $\theta$ and the length of the feedback horizon $j$ .

5.1  Conditions for the Uniqueness and Stability

To understand better how the precise combination of the pair $(j,\theta )$, IFB rules can lead the economy into instability or indeterminacy consider the model economy (44) with interest rate rules of the form (46) or (47) with $j=0,1$. Shocks to TFP are exogenous stable processes and play no part in the stability analysis. Furthermore we are only concerned with the feedback component of policy. We therefore set $k=a_{t}=a_{t}^{\ast }=0$ in (44). Write the IFB rules in the form²⁸

$$\left[ \begin{array}{l} i_{t} \ i_{t}^{\ast } \end{array}\... ...array}{l} \mathsf{z}_{t} \ \mathsf{x}_{t} \end{array}\right]$$ (48)

where $\mathsf{z}_{t}=[y_{t-1},y_{t-1}^{\ast },\pi _{H(t-1)},\pi _{F(t-1)}^{\ast },i_{t-1},i_{t-1}^{\ast }]$ and $\mathsf{x} _{t}=[y_{t},y_{t}^{\ast },\pi _{Ht},\pi _{Ft}^{\ast }]$ to give the system (44) under control as

$$\left[ \begin{array}{l} \mathsf{z}_{t+1} \ \mathcal{E}_{t}... ...array}{l} \mathsf{z}_{t} \ \mathsf{x}_{t} \end{array}\right]$$ (49)

The condition for a stable and unique equilibrium depends on the magnitude of the eigenvalues of the matrix $A+BD$. If the number of eigenvalues outside the unit circle is equal to the number of non-predetermined variables, the system has a unique equilibrium which is also stable with saddle-path $x_t=-Nz_t$ where $N=N(D)$. (See Blanchard and Kahn (1980); Currie and Levine (1993)). In our model under control, with $j=0,1$, there are 4 non-predetermined variables in total, 2 each for the sum and difference systems and 6 predetermined variables in total, 3 each for the sum and difference systems. Instability occurs when the number of eigenvalues of $A+BD$ outside the unit circle is larger than the number of non-predetermined variables. This implies that when the economy is pushed off its steady state following a shock, it cannot ever converge back to it, but rather finishes up with explosive inflation dynamics (hyperinflation or hyperdeflation).

By contrast, indeterminacy occurs when the number of eigenvalues of $A+BD$ outside the unit circle is smaller than the number of non-predetermined variables. Put simply, this implies that when a shock displaces the economy from its steady state, there are many possible paths leading back to equilibrium, i.e. there are multiple well-behaved rational expectations solutions to the model economy. With forward-looking rules this can happen when policymakers respond to private sector's inflation expectations and these in turn are driven by non-fundamental exogenous random shocks (i.e. not based on preferences or technology), usually referred to as `sunspots'. If policymakers set the coefficients of the rule so that this accommodates such expectations, the latter become self-fulfilling. Then the rule is unable to uniquely pin down the behavior of one or more real and/or nominal variables, making many different paths compatible with equilibrium (see Kerr and King (1996); Chari et al. (1998); CGG (2000); Carlstrom and Fuerst (1999) and Carlstrom and Fuerst (2000); Svensson and Woodford (1999); and Woodford (2000)). The fact that the rule itself may introduce indeterminacy and generate so called `sunspot equilibria' is of interest because sunspot fluctuations - i.e., persistent movements in inflation and output that materialize even in the absence of shocks to preferences or technology - are typically welfare-reducing and can potentially be quite large.

In order to gain insight into the stabilizing properties of IFB rules, following Batini and Pearlman (2002) we analyze their performance by using root locus analysis, a method that we borrow from the control engineering literature. Appendix B outlines how this method works. Use of this method allows us to identify analytically the range of stabilizing parameters $(j,\theta )$ in our sticky-price/sticky-inflation models before indeterminacy sets in. The method produces geometrical representations that show how system eigenvalues change as a function of the change in any parameter in the system. In our particular case we are interested in detecting how the characteristic roots of the model economy evolve as we vary the inflation feedback parameter $\theta$, for given forecast horizons $j$ in the policy rule. As the conditions for stability and determinacy of the model hinge on the value of these roots, from these diagrams we can infer which regions of the $\left( j,\theta \right)$ parameter space are associated with unique and well-behaved REE. Since we condition on increasingly distant forecast horizons in the policy rule, the method entails deriving a separate diagram for each value of $j$. However, in the majority of cases a clear pattern emerges quickly, so in what follows we only draw these diagrams at most for $j$ = 0, 1,...,4.

In the following subsections, we use the Aoki method to analyze separately the sum and difference systems of two symmetric blocs pursuing symmetric IFB rules of the form (46) or (47). The results for the sum system can be thought of as applying to a closed economy. For open economies both sum and difference systems must be saddle-path stable for a stable and unique equilibrium. As previously mentioned, the central banks'choice of responding to consumer or price inflation as well as the existence of a home bias in consumption patterns are all irrelevant in the case of the sum system. In the case of the difference system this is no longer true, and so we investigate changes to these assumptions separately for that case.

5.2  The Sum System

The sum form of the IFB rule is given by

$$i_{t}^{S}=\rho i_{t-1}^{S}+\theta (1-\rho )\mathcal{E}_{t}\pi _{t+j}^{S}$$ (50)

Let $z$ be the forward operator. Taking z-transforms of (34) and (50), the characteristic equation for the sum system is given by:

$$(z-\rho )[(z-1)(z-h)(\beta z-1)(z-\gamma )-\frac{\lambda }{\mu }z^{2}(\phi z+\mu (z-h))]$$

$$+ \frac{\lambda \theta }{\mu }(1-\rho )(\phi z+\mu (z-h))z^{j+2}=0$$ (51)

where we have defined

$$\lambda \equiv \frac{(1-\beta \xi )(1-\xi )}{\xi }\,;\quad \mu \equiv \frac{ \sigma }{1-h}$$ (52)

Equation (51) shows that the minimal state-space form of the sum system has dimension $\max{(5, j+3)}$. Recalling that there are 3 predetermined variables in each of the sum and difference systems, it follows that the saddle-path condition for a unique stable rational expectations solution in the general version of our model is that the number of stable roots (i.e., roots inside the unit circle of the complex plane) is 3 and the number of unstable roots is $\max{(2, j)}$.

To identify values of $(j,\theta )$ that involve exactly three roots of equation (51) we use the root locus technique. In particular, this technique can help us uncover how the range of values of $\theta$ that are consistent with determinacy changes as the feedback horizon $j$ changes. The root locus technique provides topological proofs of our main results (Appendix B describes this technique in detail). The technique involves starting from a polynomial equation and using a set of topological theorems to track the equation's roots as parameters in the system vary. The locus describing the evolution of the roots when parameters change is called the `root locus'. In our analysis here, the polynomial equation is the characteristic equation (51), and we use the technique to graph the locus of $\left( \theta ,z\right)$ pairs that traces how the roots change as $\theta$ varies between 0 and $\infty$. Other parameters in the system, including the feedback horizon parameter $j$ in the IFB rule, are kept constant. So to plot root loci for different feedback horizon we have to generate separate charts, each conditioning on a different horizon assumption. Each chart shows the complex plane (indicated by the solid thin line),<sup>29</sup> the unit circle (indicated by the dashed line), and the root locus tracking zeroes of equation (51) as $\theta$ varies between 0 and $\infty$ (indicated by the solid bold line). The arrows indicate the direction of the arms of the root locus as $\theta$ increases. Throughout we experiment with both a `higher' and a `lower' $\frac {\lambda }{\mu }$, as defined in (52). The economic interpretation of these cases is as follows: from the definitions in (52), the high $\frac {\lambda }{\mu }$ case corresponds to low $\xi$