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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 849, December 2005 --- Screen Reader Version*

Interest Rate Rules, Endogenous Cycles and Chaotic Dynamics in Open Economies^*

Marco Airaudo^† and Luis-Felipe Zanna^‡

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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:

In this paper we present an extensive analysis of the consequences for global equilibrium determinacy of implementing active interest rate rules (i.e. monetary rules where the nominal interest rate responds more than proportionally to changes in inflation) in flexible-price open economies. We show that conditions under which these rules generate aggregate instability by inducing cyclical and chaotic equilibrium dynamics depend on particular characteristics of open economies such as the degree of (trade) openness and the degree of exchange rate pass-through implied by the presence of non-traded distribution costs. For instance, we find that a forward-looking rule is more prone to induce endogenous cyclical and chaotic dynamics the more open the economy and the higher the degree of exchange rate pass-through. The existence of these dynamics and their dependence on the degree of openness are in general robust to different timings of the rule (forward-looking versus contemporaneous rules), to the use of alternative measures of inflation in the rule (CPI versus Core inflation), as well as to changes in the timing of real money balances in liquidity services ("cash-when-I-am-done" timing versus "cash-in-advance" timing).

Keywords: Small open economy, interest rate rules, Taylor Rules, multiple equilibria, chaos and endogenous fluctuations

JEL classification: E32, E52, F41

1  Introduction

In recent years there has been a revival of theoretical and empirical literature aimed at understanding the macroeconomic consequences of implementing diverse monetary rules in the Small Open Economy (SOE).¹ In this literature the study of interest rate rules whose interest rate response coefficient to inflation is greater than one, generally referred to as Taylor rules or active rules, has received particular attention.² To some extent the importance given to these rules in the SOE literature is just a consequence of some of the benefits that the closed economy literature has claimed for them. For instance, Bernanke and Woodford (1997) and Clarida, Galí and Gertler (2000), among others, have argued that active rules are desirable because they guarantee a unique Rational Expectations Equilibrium (REE) whereas rules whose interest rate response coefficient to inflation is less than one, also called passive rules, induce aggregate instability in the economy by generating multiple equilibria. Despite these arguments supporting active rules in closed economies, Benhabib, Schmitt-Grohé and Uribe (2001b, 2002a,b) have pointed out that they are based on results that rely on the type of equilibrium analysis that is adopted. In fact these policy prescriptions are usually derived from a local determinacy of equilibrium analysis, i.e. identifying conditions for rules that guarantee equilibrium uniqueness in an arbitrarily small neighborhood of the target steady state. In contrast by pursuing a global equilibrium analysis in tandem with the observation that nominal interest rates are bounded below by zero, Benhabib et al. have shown that active rules can induce aggregate instability in closed economies through endogenous cycles, chaotic dynamics and liquidity traps.³

What motivates our paper is the fact that the open economy literature on interest rate rules has also restricted its attention to local dynamics and not to global dynamics, often disregarding the zero bound on the nominal interest rate. By doing this, the literature has gained in tractability but has also overlooked a possibly wider set of equilibrium dynamics.

To the best of our knowledge, our work is the first attempt in the open economy literature to understand how interest rate rules may lead to global endogenous fluctuations. We pursue a global and non-linear equilibrium analysis of a traditional flexible-price SOE model with traded and non-traded goods, whose government follows an active forward-looking rule by responding to the expected future CPI-inflation. We show that conditions under which active rules induce aggregate instability by generating cyclical and chaotic dynamics depend on some specific features of an open economy such as the degree of openness of the economy (measured as the share of traded goods in consumption) and the degree of exchange rate pass-through into import prices (implied by the presence of non-traded distribution services). For example, we find that a forward-looking rule is more prone to induce cyclical equilibria and chaotic dynamics the more open the economy and the higher the degree of exchange rate pass-through. If consumption and money are Edgeworth complements in utility these dynamics occur around an extremely low interest rate steady state. On the other hand if consumption and money are substitutes these dynamics appear around the high interest rate target set by the monetary authority.⁴

For a given specification of the rule, we show that the existence of these dynamics and their dependence on the degree of openness are in general robust to different timings of the rule (forward-looking versus contemporaneous rules), to the use of alternative measures of inflation in the rule (CPI versus Core inflation), as well as to changes in the timing of real money balances in liquidity services ("cash-when-I-am-done" timing versus "cash-in-advance" timing.)

Table 1

Country	Degree of Openness (Imports/GDP)	Response Coefficient to Inflation (ρπ)	Type of Interest Rate Rule
France	0.22	1.13†	Forward-Looking
Costa Rica	0.42	1.47*	Forward-Looking
Colombia	0.20	1.31×	Forward-Looking
Chile	0.28	2.10о	Forward-Looking
United Kingdom	0.28	1.84‡	Contemporaneous
Australia	0.19	2.10‡	Contemporaneous
Canada	0.31	2.24‡	Contemporaneous
New Zealand	0.28	2.49‡	Contemporaneous

Note:  Data from IFS were used to calculate the Imports/GDP share, while the significant estimates for the interest rate response coefficient to the CPI-inflation (ρ_π) come from : ^× Bernal (2003),^† Clarida et al. (1998), ^* Corbo (2000), ^‡ Lubik and Schorfheide (2003), and ^о Restrepo (1999). The degree of openness of the economy is the annual average of imports to GDP share for the period of time used for the estimation of ρ_π.

The relevance of our results stems from the fact that they point out the importance of considering particular features of the open economy in the design of monetary policy. Clearly both the degree of openness and the degree of exchange rate pass-through are open economy features that have been neglected by previous closed economy studies. Furthermore, both are characteristics that vary significantly among economies that follow (or followed) active interest rate rules. For instance Table 1 shows the diverse degrees of openness, measured as the share of imports to GDP, for some industrialized and developing economies that have been claimed to follow interest rate rules. In addition Campa and Goldberg (2004) and Frankel, Parsley and Wei (2005), among others, provide empirical evidence suggesting that the degree of exchange rate pass-through into import prices not only varies across industrialized and developing economies; but it has also varied over time within these economies.

This paper is different from closed economy contributions such as Benhabib et al. (2002a) in some key aspects. First, our analysis shows that the assumption of money in the production function used by Benhabib et al. is not necessary to obtain cyclical and chaotic equilibrium dynamics under rules. We introduce money in the utility function and show that the existence of these global dynamics depends on whether consumption and money are Edgeworth complements or substitutes in the utility function. This observation was raised by Benhabib et al. (2001a) in the local determinacy of equilibrium analysis context but to the best of our knowledge it has not been raised in the context of a global equilibrium analysis.

Second, we show that if consumption and money are complements then it is possible to have " non-monotonic liquidity traps" featuring periodic and aperiodic oscillations around an extremely low interest rate steady state that is different from the target steady state. On the contrary if consumption and money are substitutes then cyclical and chaotic dynamics occur only around the target steady state. Although this case is reminiscent of the one in Benhabib et al. (2002a), it also presents a subtle difference. In their closed economy model period-3 cycles and therefore chaos always occur only for sufficiently low coefficients of relative risk aversion. Our results show that these dynamics can basically appear for any coefficient of relative risk aversion greater than one and provided that the economy is sufficiently open. In this sense open economies are more prone than closed economies to display these cyclical dynamics.

Third, we identify necessary and sufficient conditions for the design of active forward looking rules that do not generate cyclical and chaotic equilibria. For some given structural parameters, these conditions generally entail an appropriate choice of the rule's responsiveness to inflation. Although cyclical dynamics can be ruled out, liquidity traps and (hyper) inflationary paths remain viable equilibria. Previous works on monetary economics, however, have proposed solutions on how to deal with these equilibria.⁵

There are previous works in the SOE literature that have tried to identify conditions under which interest rate rules may lead to local multiple equilibria.⁶ For instance, De Fiore and Liu (2003), Linnemann and Schabert (2002), and Zanna (2003), among others, discuss the importance of the degree of openness of the economy in the local equilibrium analysis. The last work also points out the key role played by the degree of the exchange rate pass-through. Although our work is related to these previous studies it is different from them in the type of equilibrium analysis that is pursued. To the extent that our work considers the zero lower bound for the interest rate and pursues a global equilibrium analysis, it is able to focus on a wider set of equilibrium dynamics.

In contrast to the above mentioned works in the SOE literature, this paper does not consider nominal price rigidities. In this sense it is similar to the closed economy works of Benhabib et al. (2002a,b), Carlstrom and Fuerst (2001), and Leeper (1991) among others. In Airaudo and Zanna (2005) we introduce price stickiness and study, through a Hopf bifurcation analysis, how rules can induce cyclical dynamics that never converge to the target steady state. As in the current paper, the existence of equilibrium cycles depends on some open economy features.

The remainder of this paper is organized as follows. In Section 2 we present a flexible-price model with its main assumptions. We define the open economy equilibrium and derive some basic steady state results. In Section 3 we pursue a local and a global equilibrium analyses for an active forward-looking interest rate rule. In Section 4 we study active forward-looking rules that can preclude the existence of cyclical equilibria. In Section 5 we investigate how the degree of exchange rate pass-through can affect the existence of cyclical dynamics for forward-looking rules. We pursue a sensitivity analysis to gauge the robustness of our main results in Section 6. Finally Section 7 concludes.

2  A Flexible-Price Model

2.1  The Household-Firm Unit

Consider a Small Open Economy (SOE) populated by a large number of infinitely lived household-firm units. They are identical. Each unit derives utility from consumption ($c_{t}$), real money balances ($m_{t}^{d}$), and not working ( $1-h_{t}^{T}-h_{t}^{N}$) according to

$$E_{0}\sum_{t=0}^{\infty}\beta^{t}\left\{ \frac{\left[ \left( c_{t}\right) ^{\gamma}\left( m_{t}^{d}\right) ^{1-\gamma}\right] ^{1-\sigma}-1} {1-\sigma}+\psi(1-h_{t}^{T}-h_{t}^{N})\right\}$$ (1)

$$c_{t}=(c_{t}^{T})^{\alpha}(c_{t}^{N})^{\left( 1-\alpha\right) }$$ (2)

where $\beta,$ $\gamma\in(0,1),$ and $\psi,\sigma>0$ but $\sigma\neq1;$⁷ $E_{0}$ is the expectations operator conditional on the set of information available at time 0; $c_{t}^{T}$ and $c_{t}^{N}$ denote the consumption of traded and non-traded goods in period $t$ respectively; $m_{t}^{d}=\frac{M_{t}^{d}}{p_{t}}$ are real money balances (domestic currency money balances deflated by the Consumer Price Index, CPI, $p_{t},$ to be defined below); $h_{t}^{T}$ and $\ h_{t}^{N}$ stand for labor supplied to the production of traded and non-traded goods respectively and $\alpha\in(0,1)$ is the share of traded goods in the consumption aggregator (2). We interpret this share as a measure of the degree of (trade) openness of the economy. As $\alpha$ goes to zero, domestic agents do not value internationally traded goods for consumption. Then the economy is fundamentally closed. Whereas if $\alpha$ goes to one, non-traded goods are negligible in consumption. We refer to this case as the completely open economy.

Although we use specific functional forms, they are general enough to convey the main message of this paper. They will allow us to show analytically how cyclical dynamics induced by the interest rate rule depend on the degree of openness $\alpha$.⁸ They also allow us to study how these dynamics are affected by whether consumption, $c_{t},$ and money, $m_{t},$ are either Edgeworth substitutes or complements. By defining $U=\frac{\left( c_{t}^{\gamma}m_{t}^{1-\gamma}\right) ^{1-\sigma}-1}{1-\sigma}$ and noticing that the sign of the cross partial derivative $U_{cm}$ satisfies $sign\left\{ U_{cm}\right\} =sign\left\{ \left( 1-\sigma\right) \right\} ,$ then we can distinguish between the case of Edgeworth substitutes when $U_{cm}<0$ ($\sigma>1)$ or the case of complements when $U_{cm}>0$ ($\sigma<1)$. Moreover, given that $\gamma\in(0,1)$ and that the coefficient of relative risk aversion (CRRA) can be expressed as $\widetilde{\sigma}\equiv -\frac{U_{cc}c}{U_{c}}=1-\gamma\left( 1-\sigma\right) ,$ then $\sigma \gtreqqless1$ implies $\widetilde{\sigma}\gtreqqless1.$ As a result of this we will refer to $\sigma$ as the " risk aversion parameter."

The representative unit produces traded and non-traded goods by employing labor according to the technologies

$$y_{t}^{T}=z_{t}^{T}\left( h_{t}^{T}\right) ^{\theta_{T}}\qquad and\qquad y_{t}^{N}=z_{t}^{N}\left( h_{t}^{N}\right) ^{\theta_{N}},$$ (3)

where $\theta_{T},\theta_{N}\in\left( 0,1\right)$ and $z_{t}^{T}$ and $z_{t}^{N}$ are productivity shocks following stationary AR $\left( 1\right)$ stochastic processes. We assume that these shocks are the sole source of fundamental uncertainty.

As standard in the literature, we assume that the Law of One Price holds for traded goods and normalize the foreign price of the traded good to one.⁹ Hence $P_{t}^{T}=\mathcal{E}_{t}$, where $P_{t}^{T}$ is the domestic currency price of traded goods and $\mathcal{E}_{t}$ is the nominal exchange rate. This simplification together with (2) can be used to derive the Consumer Price Index (CPI)

$$p_{t}\equiv\frac{\left( \mathcal{E}_{t}\right) ^{\alpha}\left( P_{t} ^{N}\right) ^{1-\alpha}}{\alpha^{\alpha}(1-\alpha)^{1-\alpha}}.$$ (4)

Using equation (4) and defining the gross nominal devaluation rate as $\epsilon_{t}\equiv\mathcal{E}_{t}/\mathcal{E}_{t-1}$ and the gross non-traded goods inflation rate as $\pi_{t}^{N}\equiv P_{t}^{N}/P_{t-1}^{N},$ we derive the gross CPI-inflation rate

$$\pi_{t}=\epsilon_{t}^{\alpha}(\pi_{t}^{N})^{(1-\alpha)}$$ (5)

where $\pi_{t}\equiv\frac{p_{t}}{p_{t-1}}.$ It is just a weighted average of different goods inflations whose weights are related to the degree of openness, $\alpha$. The real exchange rate ($e_{t}$) is defined as the ratio of the price of traded goods (nominal exchange rate) and the price of non-traded goods

$$e_{t}\equiv\mathcal{E}_{t}/P_{t}^{N}.$$ (6)

Then the gross real exchange rate depreciation, $\frac{e_{t}}{e_{t-1}},$ can be written as

$$\frac{e_{t}}{e_{t-1}}=\frac{\epsilon_{t}}{\pi_{t}^{N}}.$$ (7)

As has become very common in the open economy literature such as Clarida et al. (2001) and Galí and Monacelli (2004) among others, we assume that the household-firm units have access to a complete set of internationally traded claims. In each period $t\geq0$ the agents can purchase two types of financial assets: fiat money $M_{t}^{d}$ and nominal state contingent claims, $D_{t+1}.$ The latter pay one unit of (foreign) currency for a specific realization of the fundamental shocks in $t+1.$ Although the existence of complete markets is a very strong assumption, it is well known that they can be approximated by a set of non-state contingent instruments featuring a wide range of maturities and indexations.¹⁰ In this paper the assumption of complete markets serves the sole purpose of ruling out the unit root problem of the small open economy, allowing us to pursue a meaningful local determinacy of equilibrium analysis. In this way, we can compare the results from the global equilibrium analysis to the ones from the local equilibrium analysis.¹¹Nevertheless our global results on the existence of cyclical and chaotic equilibrium dynamics will still hold if instead we assume incomplete markets, as we will show in Section 6.

Under complete markets the representative agent's flow constraint for each period can be written as

$$M_{t}^{d}+E_{t}Q_{t,t+1}D_{t+1}\leq W_{t}+\mathcal{E}_{t}y_{t}^{T}+P_{t} ^{N}y_{t}^{N}-\mathcal{E}_{t}\tau_{t}-\mathcal{E}_{t}c_{t}^{T}-P_{t}^{N} c_{t}^{N}$$ (8)

where $E_{t}Q_{t,t+1}D_{t+1}$ denotes the cost of all contingent claims bought at the beginning of period $t$ and $Q_{t,t+1}$ refers to the period-$t$ price of a claim to one unit of currency delivered in a particular state of period $t+1,$ divided by the probability of occurrence of that state and conditional of information available in period $t$. Constraint (8) says that the total end-of-period nominal value of the financial assets can be worth no more than the value of the financial wealth brought into the period, $W_{t}$, plus non-financial income during the period net of the value of taxes, $\mathcal{E}_{t}\tau_{t\text{ }}$, and the value of consumption spending.

To derive the period-by-period budget constraint, we use the definition of the total beginning-of-period wealth, in the following period, $W_{t+1}=M_{t} ^{d}+D_{t+1}$ and the fact that the period-$t$ price of a claim that pays one unit of currency in every state in period $t+1$ is equal to the inverse of the risk-free gross nominal interest rate; that is $E_{t}Q_{t,t+1}=\frac{1}{R_{t} }.$ From this, the definition of $W_{t+1}$ and (8) we obtain

$$E_{t}Q_{t,t+1}W_{t+1}\leq W_{t}+\mathcal{E}_{t}y_{t}^{T}+P_{t}^{N}y_{t} ^{N}-\mathcal{E}_{t}\tau_{t}-\frac{R_{t}-1}{R_{t}}M_{t}^{d}-\mathcal{E} _{t}c_{t}^{T}-P_{t}^{N}c_{t}^{N}.$$ (9)

The representative unit is also subject to a Non-Ponzi game condition

$$\underset{j\rightarrow\infty}{\lim}E_{t}q_{t+j}W_{t+j}\geq0$$ (10)

at all dates and under all contingencies where $q_{t}$ represents the period-zero price of one unit of currency to be delivered in a particular state of period $t$ divided by the probability of occurrence of that state and given information available at time $0.$ It satisfies $q_{t}=Q_{1} Q_{2}.....Q_{t}$ with $q_{0}\equiv1$.

The problem of the representative household-firm unit reduces to choosing the sequences $\{c_{t}^{T},$ $c_{t}^{N},$ $h_{t}^{T},$ $h_{t}^{N},$ $M_{t}^{d},$ $W_{t+1}\}_{t=0}^{\infty}$ in order to maximize (1) subject to (2), (3), (9) and (10), given $W_{0}$ and the time paths of $R_{t}$, $\ \mathcal{E}_{t}$, $P_{t}^{N},$ $Q_{t+1}$ and $\tau_{t}.$ Note that since the utility function specified in (1) implies that the preferences of the agent display non-satiation then both constraints (9) and (10) hold with equality.

The first order conditions correspond to (9) and (10) with equality and

$$\alpha\gamma\left( c_{t}^{T}\right) ^{\alpha\gamma(1-\sigma)-1}\left( c_{t}^{N}\right) ^{\left( 1-\alpha\right) \gamma(1-\sigma)}\left( m_{t}^{d}\right) ^{\left( 1-\gamma\right) (1-\sigma)}=\lambda _{t}$$ (11)

$$\frac{\alpha c_{t}^{N}}{(1-\alpha)c_{t}^{T}}=e_{t}$$ (12)

$$\frac{\lambda_{t}}{e_{t}}\theta_{N}\left( h_{t}^{N}\right) ^{(\theta_{N} -1)}=\psi=\lambda_{t}\theta_{T}\left( h_{t}^{T}\right) ^{(\theta_{T} -1)}$$ (13)

$$m_{t}^{d}=\left( \frac{1-\gamma}{\gamma}\right) \left( \frac{1-\alpha }{\alpha}\right) ^{1-\alpha}\left( \frac{R_{t}}{R_{t}-1}\right) c_{t} ^{T}e_{t}^{1-\alpha}$$ (14)

$$\frac{\lambda_{t}}{\mathcal{E}_{t}}Q_{t,t+1}=\beta\frac{\lambda_{t+1} }{\mathcal{E}_{t+1}}$$ (15)

where $\lambda_{t}/\mathcal{E}_{t}$ is the Lagrange multiplier of the flow budget constraint.

The interpretation of the first order conditions is straightforward. Equation (11) is the usual intertemporal envelope condition that makes the marginal utility of consumption of traded goods equal to the marginal utility of wealth measure in terms of traded goods ( $\lambda_{t}$). Condition (12) implies that the marginal rate of substitution between traded and non-traded goods must be equal to the real exchange rate. Condition (13) equalizes the value of the marginal products of labor in both sectors. Equation (14) represents the demand for real money balances. And finally condition (15) describes a standard pricing equation for one-step-ahead nominal contingent claims for each period $t$ and for each possible state of nature.

2.2  The Government

The government issues two nominal liabilities: money, $M_{t}^{s}$, and a one period risk-free domestic bond, $B_{t}^{s},$ that pays a gross risk-free nominal interest rate $R_{t}$. We assume that it cannot issue or hold state contingent claims. It also levies taxes, $\tau_{t},$ pays interest on its debt, ( $R_{t}-1)B_{t}^{s}$, and receives revenues from seigniorage$.$ Then the government's budget constraint can be written as $L_{t}^{s}=R_{t-1}L_{t-1}^{s}-(R_{t-1}-1)M_{t-1}^{s}-\mathcal{E}_{t}\tau_{t},$ where $L_{t}^{s}=M_{t}^{s}+B_{t}^{s}.$

We proceed to describe the fiscal and monetary policies. The former corresponds to a generic Ricardian policy: the government picks the path of the lump-sum transfers, $\tau_{t},$ in order to satisfy the intertemporal version of its budget constraint in conjunction with the transversality condition $\underset{t\rightarrow\infty}{\lim}\frac{L_{t}^{s}/\mathcal{E}_{t} }{\underset{k=0}{\overset{t}{\prod}}\left( \frac{R_{k}}{\epsilon_{k+1} }\right) }=0.$ The latter is described as an interest rate feedback rule whereby the government sets the nominal interest rate, $R_{t}$, as a continuous and increasing function of the deviation of the expected future CPI-inflation rate, $E_{t}\left( \pi_{t+1}\right)$, from a target, $\pi^{\ast}.$¹² For analytical and computational purposes, as in Benhabib et al. (2002a) and Christiano and Rostagno (2001), we use the following specific non-linear rule¹³

$$R_{t}=\rho(E_{t}\pi_{t+1})\equiv1+(R^{\ast}-1)\left( \frac{E_{t}\pi_{t+1} }{\pi^{\ast}}\right) ^{\frac{A}{R^{\ast}-1}}$$ (16)

where $R^{\ast}=\pi^{\ast}/\beta$ and $R^{\ast}$ corresponds to the target interest rate. (16) always satisfies the zero bound on the nominal interest rate, i.e. $R_{t}=\rho(E_{t}\pi_{t+1})>1.$ In addition we assume that the government responds aggressively to inflation. This means that at the inflation target, the rule's elasticity to inflation $\xi\equiv\frac {\rho^{\prime}(\pi^{\ast})\pi^{\ast}}{\rho(\pi^{\ast})}=\frac{A}{R^{\ast}}$ is strictly bigger than 1. Following Leeper (1991) we call rules with this property active rules.

Assumption 0: $\xi=\frac{A}{R^{\ast}}>1.$ That is, the rule is active.

2.3  International Capital Markets

Besides complete markets there is free international capital mobility. Then the no-arbitrage condition $Q_{t,t+1}^{w}=Q_{t,t+1} \frac{\mathcal{E}_{t+1}}{\mathcal{E}_{t}}$ holds, where $Q_{t,t+1}^{w}$ refers to the period-$t$ foreign currency price of a claim to one unit of foreign currency delivered in a particular state of period $t+1$ divided by the probability of occurrence of that state and conditional of information available in period $t$.

Furthermore under the assumption of complete markets a condition similar to (15) must hold from the maximization problem of the representative agent in the Rest of The World (ROW). That is, $\frac{\lambda_{t}^{w}} {P_{t}^{Tw}}Q_{t,t+1}^{w}=\frac{\lambda_{t+1}^{w}}{P_{t+1}^{Tw}}\beta^{w}$ where $\lambda_{t}^{w}$ represents the marginal utility of nominal wealth in the ROW, $\beta^{w}$ denotes the subjective discount rate of the ROW and $P_{t}^{Tw}$ is the foreign price of traded goods. Since we normalize this price to one ( $P_{t}^{Tw}=1)$ then assuming that $\beta^{w}=\beta$ leads to $\lambda_{t}^{w}Q_{t,t+1}^{w}=\lambda_{t+1}^{w}\beta.$

Combining this last equation, with condition (15) and the fact that $P_{t}^{T}=\mathcal{E}_{t}$ yields $\frac{\lambda_{t+1}}{\lambda_{t}} =\frac{\lambda_{t+1}^{w}}{\lambda_{t}^{w}},$ which holds at all dates and under all contingencies. This condition implies that the domestic marginal utility of wealth is proportional to its foreign counterpart: $\lambda _{t}=\Lambda\lambda_{t}^{w}$where $\Lambda$ refers to a constant parameter that determines the wealth difference between the SOE and the ROW. From the perspective of a SOE, $\lambda_{t}^{w}$ can be taken as an exogenous variable. For simplicity we assume that $\lambda_{t}^{w}$ is constant and equal to $\lambda^{w}.$ As a result of this $\lambda_{t}$ becomes a constant. Then

$$\lambda_{t}=\lambda=\Lambda\lambda^{w}.$$ (17)

This allows us to write condition (15) as $Q_{t,t+1}=\frac {\mathcal{E}_{t}}{\mathcal{E}_{t+1}}\beta=\frac{\beta}{\epsilon_{t+1}}$ that together with $E_{t}Q_{t,t+1}=\frac{1}{R_{t}}$ yields

$$R_{t}=\beta^{-1}\left[ E_{t}\frac{1}{\epsilon_{t+1}}\right] ^{-1}$$ (18)

which is similar to an uncovered interest parity condition.

2.4  The Definition of Equilibrium

In this paper we will focus on perfect foresight equilibria. In other words, we assume the all the agents in the economy, including the government, forecast correctly all the anticipated variables. Hence for any variable $x_{t}$ we have that $E_{t}x_{t+j}=x_{t+j}$ with $j\geq0$ implying that we can drop the expectation operator in the previous equations. For instance, under perfect foresight, condition (18) becomes

$$R_{t}=\beta^{-1}\epsilon_{t+1}$$ (19)

that corresponds to the typical uncovered interest parity condition as long as $\beta^{-1}$ represents the foreign international interest rate.¹⁴

In order to provide a definition of the equilibrium dynamics subject of our study, we find a reduced non-linear form of the model. To do so we use the definitions (5) and (7) together with conditions (11)-(14), (17), (19), and the market clearing conditions for money and the non-traded good, $M_{t}^{d}=M_{t} ^{s}=M_{t}$ and $y_{t}^{N}=\left( h_{t}^{N}\right) ^{\theta_{N}}=c_{t}^{N},$ to obtain

$$\pi_{t+1}\left( \frac{R_{t+1}}{R_{t+1}-1}\right) ^{\chi}=\left( \frac {R_{t}}{R_{t}-1}\right) ^{\chi}\beta R_{t}$$ (20)

where

$$\chi=\frac{(\sigma-1)(1-\alpha)(1-\gamma)(1-\theta_{N})}{\sigma\lbrack \theta_{N}+\alpha(1-\theta_{N})]+\left( 1-\alpha\right) (1-\theta_{N} )}.$$ (21)

Combining (16) and (20) and dropping the expectation operator yields

$$\left( \frac{R_{t+1}}{R_{t+1}-1}\right) ^{\chi}=\frac{R_{t}}{R^{\ast} }\left( \frac{R^{\ast}-1}{R_{t}-1}\right) ^{\frac{R^{\ast}-1}{A}}\left( \frac{R_{t}}{R_{t}-1}\right) ^{\chi}$$ (22)

which corresponds to the reduced non-linear form of the model that can be used to pursue the local and global determinacy of equilibrium analyses.¹⁵ We use this equation in order to provide a definition of a Perfect Foresight Equilibrium (PFE).

Definition 1   Given the target $R^{\ast}$ and the initial condition $R_{0},$ a Perfect Foresight Equilibrium (PFE) is a deterministic process $\{R_{t}\}_{t=0}^{\infty},$ with $R_{t}>1$ for any $t,$ that satisfies equation (22) if the interest rate rule is forward-looking.

Although Definition 1 is stated exclusively in terms of the nominal interest rate ($R_{t}$), it must be clear that multiple perfect foresight equilibrium solutions to (22) imply real local and/or global indeterminacy of all the endogenous variables.¹⁶ In other words the indeterminacy of the nominal interest rate implies real indeterminacy in our model because of the non-separability in the utility function between money and consumption.¹⁷

In order to pursue the equilibrium analysis we need to identify the steady state(s) of the economy. From (5), (7) and (20) we obtain that at the steady-state(s), $\pi^{Nss} =\epsilon^{ss}=\pi^{ss},$ and $R^{ss}=\pi^{ss}/\beta.$ Using these and the rule (16) we have that

$$\left( R^{\ast}-1\right) ^{\frac{R^{\ast}-1}{A}}R^{ss}=R^{\ast}\left( R^{ss}-1\right) ^{\frac{R^{\ast}-1}{A}}.$$ (23)

Clearly $R^{ss}=R^{\ast}>1$ is a solution to (23), and therefore a feasible steady state. But if the rule is active at $R^{\ast},$ that is if $\xi=\frac{A}{R^{\ast}}>1,$ then another lower steady state $R^{L}\in\left( 1,R^{\ast}\right)$ exists and it is unique. At this low steady state the elasticity of the rule to inflation satisfies $\xi=\frac {A}{R^{L}}<1.$ The following proposition formalizes the existence of the low steady state $R^{L}$.

Proposition 1   If $\frac{A}{R^{\ast}}>1$ (an active rule) and $R^{ss}>1$ (the zero lower bound) then there exists a solution $R^{ss}=R^{L}\in\left( 1,R^{\ast}\right)$ solving (23) besides the trivial solution $R^{ss}=R^{\ast}$.

Proof. See the Appendix.

The existence of two steady states plays a crucial role in the derivation of our results as in the closed economy model of Benhabib et al. (2002a). As a matter of fact, the steady state equation (23) of our SOE is identical to theirs. It is independent of the non-policy structural parameters. Hence no fold bifurcation (i.e. appearance/disappearance of steady states) occurs because of changes in these parameters$.$ What distinguishes our model from theirs are the equilibrium dynamics off the two steady states. This is a consequence of the following two features of our model. First, by introducing traded and non-traded goods we present an economy with two sectors that although homogeneous in terms of price setting behavior (both feature flexible prices), are fundamentally different in terms of the degree of openness to international trade. As we will see below this degree, measured by $\alpha,$ will influence the equilibrium dynamics. Second, by considering money in the non-separable utility function we are able to study how the existence of cyclical dynamics depend on whether money and consumption are either Edgeworth complements ($\sigma<1$) or substitutes ($\sigma>1$).

In the analysis to follow we will study how $\alpha$ and $\sigma$ affect the local and global equilibrium dynamics in our SOE model while keeping constant the other structural parameters ($\beta,$ $\gamma$ and $\theta_{N}$) and the policy parameters ($A$ and $R^{\ast}$). This will allow us to compare economies that implement the same monetary rule but differ in the degree of openness $\alpha$ and the risk aversion parameter $\sigma.$ To accomplish this goal we will proceed in two steps. First we will analyze how these dynamics are affected by the composite parameter $\chi$ defined in (21). Second by taking into account the dependence of $\chi$ on both $\alpha$ and $\sigma$ we will unveil the effect of the degree of openness and the risk aversion parameter on the existence of local and global dynamics (cycles and chaos). In this sense we will regard $\chi$ as a function of $\alpha$ and $\sigma .$¹⁸That is $\chi(\alpha,\sigma).$ For the second step we will use and refer to (21), and to Lemmata 4 and 5 in the Appendix.In turn, these Lemmata and subsequent propositions will use the following definitions

$$\chi_{\max}\equiv\frac{(1-\gamma)(1-\theta_{N})}{\theta_{N}}\in(0,+\infty )\qquad\chi_{\min}\equiv-(1-\gamma)\in(-1,0)$$ (24)

$$\mu(\sigma)=\frac{(\sigma-1)(1-\gamma)(1-\theta_{N})}{\sigma\theta _{N}+(1-\theta_{N})}$$ (25)

where $\chi_{\max}$ and $\chi_{\min}$ are considered scalars and $\mu(\sigma)$ is considered a function of $\sigma.$

Definition 2   Using (24) and (25) define the scalars $\sigma^{i}\equiv\frac{1-\frac{\Upsilon^{i}}{\chi_{\min}}}{1-\frac {\Upsilon^{i}}{\chi_{\max}}}$ and the functions $\alpha^{i}(\sigma)\equiv \frac{1-\frac{\Upsilon^{i}}{\mu(\sigma)}}{1-\frac{\Upsilon^{i}}{\chi_{\min}}}$ for $i=w,k,f,d$ where $\Upsilon^{w}\equiv R^{L}\left( 1-\frac{R^{\ast}-1} {A}\right) -1<0,$ $\Upsilon^{k}\equiv\left( 1-\frac{R^{\ast}}{A}\right) \left( R^{\ast}-1\right) >0,$ $\Upsilon^{f}\equiv\frac{\Upsilon^{w}}{2}$ and $\Upsilon^{d}\equiv\frac{\Upsilon^{k}}{2};$ and the functions $\alpha ^{i}(\sigma)$ are characterized in Lemma 4 when $\sigma>1$ and in Lemma 5 when $\sigma\in(0,1)$.

3  Equilibrium Dynamics under Forward-Looking Rules

Our study of forward-looking rules is motivated by the evidence provided by Clarida et al. (1998) for industrialized economies and by Corbo (2000) for developing economies. Both works suggest that these economies have followed forward-looking rules.

In order to derive analytical results for both the local and the global equilibrium analyses we will assume that the constant parameters $\gamma,$ $\theta_{N},$ $A$ and $R^{\ast}$ satisfy the following assumptions.¹⁹,²⁰

Assumption 1: $\chi_{\max}>\frac{1}{2}(R^{\ast}-1)\left( 1-\frac{R^{\ast}}{A}\right) .$

Assumption 2: $\chi_{\min}<\frac{1-R^{\ast}}{A}.$

Assumption 3: $R^{\ast}-1>A\left( R^{L}-1\right) .$

3.1  The Local Determinacy of Equilibrium Analysis

The local determinacy of equilibrium analysis for forward-looking rules is pursued by log-linearizing equation (22) around the target steady state $R^{\ast}$, yielding

$$\hat{R}_{t+1}=\left[ 1+\frac{\frac{R^{\ast}}{A}-1}{\frac{\chi}{R^{\ast}-1} }\right] \hat{R}_{t}.$$ (26)

Since $R_{t}$ is a non-predetermined variable, studying local determinacy is equivalent to finding conditions that make the linear difference equation (26) explosive. The next Lemma shows how local equilibrium determinacy depends on $\chi$.

Lemma 1   Define $\Upsilon^{d}\equiv\frac{1}{2}(R^{\ast }-1)\left( 1-\frac{R^{\ast}}{A}\right) >0$ and consider $\chi\in \mathbb{R}$. Suppose the government follows an active forward-looking rule then: 1) the equilibrium is locally unique if $\chi<\Upsilon^{d};$ 2) there exist locally multiple equilibria if $\chi>\Upsilon^{d}$.

Proof. See the Appendix.

These simple determinacy of equilibrium conditions for $\chi$ can be reinterpreted in terms of the degree of openness $\alpha$ and the risk aversion parameter $\sigma$ in the following Proposition.

Proposition 2   Consider $\sigma^{d}$ and $\alpha^{d}(\sigma)$ in Definition 2 where $\sigma^{d}>1$ and $\alpha^{d}:\left( 1,+\infty\right) \rightarrow\left( -\infty,1\right)$ . Suppose that the government follows an active forward-looking rule.

1.  There exists a locally unique equilibrium

(a)  if consumption and money are Edgeworth complements, i.e. $\sigma \in\left( 0,1\right) ,$ and for any degree of openness, i.e. $\alpha \in(0,1);$

(b)  if consumption and money are Edgeworth substitutes, i.e. $\sigma>1,$ and the economy is sufficiently open satisfying $\alpha>\alpha_{\min}^{d};$ where $\alpha_{\min}^{d}\equiv\max\left\{ 0,\alpha^{d}(\sigma)\right\}$ is positive and strictly increasing for $\sigma>\sigma^{d},$ but constant and equal to zero for any $\sigma\in\left( 1,\sigma^{d}\right] .$

2.  There exist locally multiple equilibria if consumption and money are Edgeworth substitutes satisfying $\sigma>\sigma^{d}$, and the economy is sufficiently closed satisfying $\alpha\in(0,\alpha^{d}(\sigma)).$

Proof. See the Appendix.

The results of this proposition show the importance of $\alpha$ and $\sigma$ in the local characterization of the equilibrium. In a nutshell, active forward-looking rules guarantee local uniqueness in the following cases: when regardless of the degree of openness the risk aversion parameter $\sigma$ is sufficiently low; and when the economy is sufficiently open for high values of $\sigma$.²¹ It is in this sense that an active rule might be viewed as stabilizing. Local equilibrium determinacy, however, does not guarantee global equilibrium determinacy. To see this we pursue a global characterization of the equilibrium dynamics in the following subsection.

3.2  The Global Determinacy of Equilibrium Analysis

To pursue the global equilibrium analysis we rewrite equation (22) as the forward mapping $R_{t+1}=f\left( R_{t}\right)$ where

$$f\left( R_{t}\right) \equiv\frac{1}{1-J\left( R_{t}\right) ^{\frac{1} {\chi}}}$$ (27)

and

$$J\left( R_{t}\right) \equiv\frac{R^{\ast}}{\left[ R^{\ast}-1\right] ^{\frac{R^{\ast}-1}{A}}}\frac{\left[ R_{t}-1\right] ^{\chi+\frac{R^{\ast} -1}{A}}}{R_{t}^{1+\chi}}.$$ (28)

Then the global analysis corresponds to studying the global PFE dynamics that satisfy $R_{t+1}=f\left( R_{t}\right)$ given an initial condition $R_{0}>1$ and subject to the zero-lower-bound condition $f^{n}\left( R_{0}\right) >1$ for any $n\geq1.$ The types of cyclical and chaotic dynamics we will be referring to are those conforming to the following definitions.

Definition 3   Period-n cycle. A value " $R$" is a point of a period-n cycle if it is a fixed point of the n-th iterate of the mapping $f(.)$, i.e. $R=f^{n}\left( R\right) ,$ but not a fixed point of an iterate of any lower order. If " $R$" is such, we call the sequence $\left\{ R,f\left( R\right) ,f^{2}\left( R\right) ,...,f^{n-1}\left( R\right) \right\}$ a period-n cycle.

Definition 4   Topological chaos. The mapping $f(.)$ is topologically chaotic if there exists a set " S" of uncountable many initial points, belonging to its domain, such that no orbit that starts in " S" will converge to one another or to any existing period orbit.

The global analysis requires a full characterization of $f(.)$ in (27) not only around its stationary solutions, like in the local analysis, but over its entire domain. In this characterization it is useful to take into account that a necessary condition for the existence of cyclical dynamics in continuously differentiable maps is that the mapping $f\left( .\right)$ slopes negatively at either one of the two steady states.²² Lemma 6 in the Appendix investigates the properties of the mapping $f(.)$ showing that they depend critically on $\chi$. Here we only provide a big picture of the analysis. First of all, the Lemma specifies conditions under which $f\left( .\right)$ satisfies the zero-lower-bound requirement. Second, it makes use of the following conditions

$$sign\left\{ f^{\prime}\left( R_{t}\right) \right\} =sign\left\{ \frac{J^{\prime}\left( R_{t}\right) }{\chi}\right\} \qquad sign\left\{ J^{\prime}\left( R_{t}\right) \right\} =sign\left\{ \frac{1+\chi}{1-\frac{R^{\ast}-1}{A}}-R_{t}\right\}$$ (29)

which imply that for $\chi\neq0$ then $R^{J}\equiv\frac{1+\chi}{1-\frac {R^{\ast}-1}{A}}$ is a critical point of $f\left( .\right)$ as long as $R^{J}>1$. With these conditions the Lemma shows that the mapping $R_{t+1}=f\left( R_{t}\right)$ is always single-peaked for $\chi>0$ whereas for $\chi<0$ it is single-troughed only if $\chi+\frac{R^{\ast}-1}{A}>0$.

Figure 1 displays a graphical representation of the cases where the equilibrium mapping $R_{t+1}=f\left( R_{t}\right)$ has a critical point between the two steady states$.$ The right panel considers the case of $\chi<0$ and $\chi+\frac{R^{\ast}-1}{A}>0,$ while the left panel the case of $\chi>0.$ In the left one, $f\left( .\right)$ always satisfies the equilibrium conditions for any $R_{t}\in\left( 1,+\infty\right)$ and crosses the 45 degree line twice at $R^{L}>1$ and $R^{\ast}>R^{L}$ (the two steady states). Furthermore, $\underset{R_{t}\rightarrow1}{\lim}f\left( R_{t}\right) =\underset{R_{t}\rightarrow+\infty}{\lim}f\left( R_{t}\right) =0$ and there is a maximum at $R^{J}\in\left( R^{L},R^{\ast}\right) .$ In the right panel, all equilibrium conditions are satisfied only within a subset $\left( R^{l},R^{u}\right) \subset\left( 1,+\infty\right)$ defined in Lemma 6. Within that set, $f\left( .\right)$ crosses the 45 degree line at $R^{L}>1$ and $R^{\ast}>R^{L},$ as in the previous case, but now $R^{J}\in\left( R^{L},R^{\ast}\right)$ is a minimum and $\underset {R_{t}\rightarrow R^{l+}}{\lim}f\left( R_{t}\right) =\underset {R_{t}\rightarrow R^{u-}}{\lim}f\left( R_{t}\right) =+\infty.$ These are clearly the cases in which we are interested, as they imply a negative derivative of $f(.)$ at either one of the two steady states.

Figure 1

Figure 1:  This Figure shows the mapping R_t+1 = f(R_t) for χ > 0, and χ < 0 but $\chi+\frac{R^{\ast}-1}{A}>0,$ and R_t denotes the nominal interest rate. A formal characterization of this mapping is provided in Lemma 6 in the Appendix.

Figure 1 together with Lemma 6 suggest that depending on the sign of $\chi,$ cycles may appear around either the active steady state or the passive steady state. Hence we proceed to look for flip bifurcation thresholds for $\chi$ i.e. critical values of $\chi$ that determine a change in the stability properties of the steady state where the map $f$ $\left( .\right)$ is negatively sloped. If the steady state is stable, any equilibrium orbit, that starts in a map invariant set centered around this state, will asymptotically converge to the steady state itself, monotonically or spirally. Thus equilibrium cycles are impossible. On the contrary if the steady state is unstable, such orbit will keep oscillating within the map invariant set and either it converge to a stable $n$-period cycle, or not converge at all displaying aperiodic but bounded dynamics (chaotic equilibrium paths). We first consider the case of $\chi\in\left( R^{L}\left( 1-\frac{R^{\ast}-1} {A}\right) -1,0\right)$ and show that endogenous cyclical dynamics of period 2 can occur around the passive steady state.

Lemma 2   Let $\Upsilon^{w}\equiv R^{L}\left( 1-\frac{R^{\ast}-1}{A}\right) -1,$ and define the points $\underline{R}\equiv f\left( R^{J}\right) ,$ i.e. the image of $R^{J}$ (the critical point of $f\left( .\right) ),$ and $\widetilde{R}\equiv f^{-1}\left( R^{\ast }\right) ,$ i.e. the inverse image of the high steady state. Consider $\chi\in\left( \Upsilon^{w},0\right)$ and assume that $f_{\min}\equiv f\left( R^{J}\right) \geq\widetilde{R}.$²³ Then:

1.   $R^{L}\in\left( \underline{R},R^{J}\right)$ and $R^{L}>\widetilde{R};$

2.  the set $\left[ \underline{R},R^{\ast}\right]$ is invariant under the mapping $f\left( .\right)$ and attractive for any $R_{t}\in\left[ \widetilde{R},\underline{R}\right]$ where $\widetilde{R}<\underline{R};$

3.  period-2 cycles within $\left[ \underline{R},R^{\ast}\right]$ and centered around the passive steady state occur when $\chi\in\left( \frac{\Upsilon^{w}}{2},0\right) .$

Proof. See Appendix.

For the case of $\chi\in\left( 0,\left( 1-\frac{R^{\ast}}{A}\right) \left( R^{\ast}-1\right) \right)$ endogenous cyclical dynamics of period 2 exist around the active steady state, instead.

Lemma 3   Let $\Upsilon^{k}\equiv\left( 1-\frac{R^{\ast}}{A}\right) \left( R^{\ast}-1\right) ,$ and define the points $\overline{R}\equiv f\left( R^{J}\right) ,$ i.e. the image of $R^{J}$ (the critical point of $f\left( .\right) )$ and $\widetilde{R}\equiv f^{-1}\left( R^{\ast}\right) ,$ i.e. the inverse image of the high steady state. Consider $\chi\in\left( 0,\Upsilon^{k}\right)$ and assume $f_{\max }\equiv f\left( R^{J}\right) \leq\widetilde{R}.$ Then:

1.   $R^{\ast}\in\left( R^{J},\overline{R}\right)$ and $\widetilde {R}>R^{L};$

2.  the set $\left[ R^{L},\overline{R}\right]$ is invariant under the mapping $f(.)$ and attractive for any $R_{t}\in(\overline{R},\widetilde{R}]$ where $\overline{R}<\widetilde{R};$

3.  period-2 cycles within $\left[ R^{L},\overline{R}\right]$ and centered around the active steady state occur when $\chi\in\left( 0,\frac {1}{2}\Upsilon^{k}\right) .$

Proof. See Appendix.

Similarly to the local determinacy analysis, the conditions for endogenous cycles derived in terms of $\chi$ can be easily translated into conditions described in terms of the degree of openness $\alpha$ and the risk aversion parameter $\sigma.$ The next Proposition accomplishes this goal.

Proposition 3   Suppose that the government follows an active forward-looking rule.

1.  Consider $\sigma^{f}$ and $\alpha^{f}(\sigma)$ in Definition 2 where $\sigma^{f}\in\left( 0,1\right)$ and $\alpha ^{f}:\left( 0,1\right) \rightarrow\left( -\infty,1\right)$ and assume that consumption and money are Edgeworth complements, i.e. $\sigma\in\left( 0,1\right) .$ Then period-2 equilibrium cycles exist around the passive steady state if the economy is sufficiently open satisfying $\alpha >\alpha_{\min}^{f},$ where $\alpha_{\min}^{f}\equiv\max\left\{ 0,\alpha ^{f}(\sigma)\right\}$ is positive and strictly decreasing for $\sigma \in\left( 0,\sigma^{f}\right) ,$ but constant and equal to zero for any $\sigma\in\left[ \sigma^{f},1\right) .$

2.  Consider $\sigma^{d}$ and $\alpha^{d}(\sigma)$ in Definition 2 where $\sigma^{d}>1$ and $\alpha^{d}:\left( 1,+\infty\right) \rightarrow\left( -\infty,1\right)$ and assume that consumption and money are Edgeworth substitutes, i.e. $\sigma>1.$ Then period-2 equilibrium cycles exist around the active steady state if the economy is sufficiently open satisfying $\alpha>\alpha_{\min}^{d},$ where $\alpha_{\min}^{d}\equiv \max\left\{ 0,\alpha^{d}(\sigma)\right\}$ is positive and strictly increasing for $\sigma>\sigma^{d},$ but constant and equal to zero for any $\sigma\in\left( 1,\sigma^{d}\right] .$

Proof. See the Appendix.

Proposition 3 is one of the main contributions of our paper. It states that at either sufficiently low or sufficiently high risk aversion coefficients ($\sigma$), forward looking rules are more prone to induce endogenous cyclical dynamics the more open the economy; while for $\sigma$ sufficiently close to 1, but different from it, forward looking rules will lead to those dynamics regardless of the degree of openness.²⁴

The second point of this Proposition is also useful to make the following interesting argument. The sufficient condition for the existence of period-2 cycles when $\sigma>1$ and the local determinacy condition stated in Point 1b) of Proposition 2 are exactly the same. This is a clear example of why local analysis can be misleading. By log-linearizing around the steady state, local analysis implicitly assumes that any path starting arbitrarily close to it and diverging cannot be part of an equilibrium since it will eventually explode and thus violate some transversality condition. This is not the case here as the global analysis proves that the true non-linear map features a bounded map-invariant and attractive set around the active steady state. It is then possible to have equilibrium paths that starting arbitrarily close to the target steady state will converge to a stable deterministic cycle.

Given the functional form of $f\left( .\right)$ in (27) it is very difficult to derive analytical conditions for $\alpha$ and $\sigma$ under which forward-looking rules induce either cycles of period higher than 2 or chaotic dynamics. Therefore in order to shed some light on the role of both $\alpha$ and $\sigma$ in delivering these dynamics, as well as to find some empirical confirmation of our analytical results, we pursue a simple calibration-simulation exercise.

Table 2:  Parametrization

θN	β	π*	R*	1 - γ	A/R*
0.56	0.99	1.031¼	1.072¼	0.03	2.24

We set the time unit to be a quarter and use Canada as the representative economy. From Mendoza (1995) we borrow the labor income shares for the non-traded sector and set $\theta_{N}=0.56.$ The steady-state inflation, $\pi^{\ast},$ and the steady state nominal interest rate, $R^{\ast}$, are calculated as the average of the CPI-inflation and the Central Bank discount rate between 1983-2002. This yields $\pi^{\ast}=1.031^{\frac{1}{4}}$ and $R^{\ast}=1.072^{\frac{1}{4}}.$ Then the subjective discount rate is determined by $\beta=\pi^{\ast}/R^{\ast}.$ We use the estimate of Lubik and Schorfheide (2003) for the Canadian interest rate response coefficient to inflation which corresponds to $\frac{A}{R^{\ast}}=2.24$. Estimates for the share of expenditures on real money balances, $1-\gamma$, for Canada are not available. For the US, estimates of this parameter vary from 0.0146 to 0.039 depending on the specification of the utility function and method of estimation. We set $1-\gamma$ equal to 0.03 that is in line with the estimates provided by previous works.²⁵ Table 2 gathers the parametrization.

As in the analytical study, in the simulation exercise we vary $\alpha$ and $\sigma$ keeping the remaining parameters as in Table 2$.$ Nevertheless, an estimate of $\alpha$ for Canada can be obtained from the average imports to GDP share during 1983-2002, yielding $\alpha=0.31.$ In contrast, obtaining an estimate of $\sigma$ is more difficult. As explained before, $\sigma$ is related to the CRRA coefficient $\widetilde{\sigma}$ through $\widetilde {\sigma}=1-\gamma\left( 1-\sigma\right)$ which spans over a wide range. The RBC literature usually sets $\widetilde{\sigma}=2$. This value and $1-\gamma=0.03$ imply $\sigma=2.03.$ Since the value of $\sigma$ determines whether consumption and real money balances are either Edgeworth substitutes or complements we will use different values for the CRRA $\widetilde{\sigma}$. For instance, we let $\widetilde{\sigma}\in\left\{ 0.8,1.5,2,2.5\right\}$ which in tandem with $1-\gamma=0.03$ leads to $\sigma\in\left\{ 0.79,1.51,2.03,2.55\right\}$ respectively$.$

Given $\sigma\in\left\{ 0.79,2.03\right\}$ which corresponds to CRRA of $\tilde{\sigma}\in\left\{ 0.8,2\right\}$ we construct Figure 2. It presents the bifurcation (or orbit) diagrams for the degree of openness $\alpha.$ The left panel considers the case when money and consumption are complements by setting $\tilde{\sigma}=0.8.$ The right panel corresponds to the case when they are substitutes as $\tilde{\sigma}=2.$ With $\alpha\in\left( 0,1\right)$ on the horizontal axis and $R_{t}>1$ on the vertical axis, the solid lines in the diagram correspond to stable solutions of period $n.$ The left and right panels of the figure show how by increasing $\alpha$ an active forward-looking rule can drive the economy into period-2 cycles, period-4 cycles,...period-$n$ cycles and eventually chaotic dynamics. Starting from $\alpha=0,$ both panels show that for low degrees of openness the economy, that is described by the mapping $R_{t+1}=f(R_{t})$ in (27), always settles on a stable steady state equilibrium after a long enough series of iterations. It settles on the passive steady state, $R^{L},$ for $\tilde{\sigma}=0.8$ and on the active steady state, $R^{\ast},$ for $\tilde{\sigma}=2$. Once $\alpha$ reaches some threshold a stable period-2 cycle appears, as indicated by the first split into two branches in both panels. As we increase $\alpha$ further in both panels, both branches split again yielding a period-4 stable cycle. A cascade of further period doubling occurs as we keep increasing $\alpha$, yielding cycles of period-8, period-16 and so on. Finally for sufficiently high $\alpha$ values, the rule produces aperiodic chaotic dynamics, i.e. the attractor of the map (27) changes from a finite to an infinite set of points.

Figure 2:  Orbit-Bifurcation Diagrams
Forward-Looking Rules

Figure 2:  Orbit-bifurcation diagrams for the degree of openness, α. R_t denotes the nominal interest rate. The diagrams show the set of limit points as a function of α, under two different coefficients of risk aversion (CRRA) $\tilde{\sigma}=0.8$ and $\tilde{\sigma}=2$, and under an active forward-looking rule. Depending on α, an active forward-looking rule may drive the economy into period-2 cycles, period-4 cycles, ... period-n cycles and even chaotic dynamics.

From Figure 2 we also see that when consumption and money are complements then cyclical and chaotic dynamics occur around the passive steady state; whereas if they are substitutes then these dynamics appear around the target active steady state. Nevertheless for both cases, forward-looking rules are more prone to induce cycles and chaos the more open the economy.

In order to summarize and compare the results of the local and global determinacy of equilibrium analyses we construct Figure 3. It shows the combinations of the degree of openness and the risk aversion parameter, $\alpha$ and $\sigma,$ for which there is local and/or global (in)determinacy. For $\sigma\geq0$ and $\alpha\in\left[ 0,1\right]$ we plot two threshold frontiers: the flip bifurcation frontier for period-2 cycles around the passive steady state, $\alpha^{f}(\sigma)$ and the frontier $\alpha^{d} (\sigma)$ for both local determinacy and period-2 cycles around the active steady state. Regions featuring a locally unique equilibrium are labeled with a "U", while those featuring locally multiple equilibria are labeled with an "M". Clearly, "U" appears everywhere but below the curve $\alpha^{d}(\sigma)$ implying that local determinacy occurs for a wide range of $\left( \alpha,\sigma\right)$ combinations. In fact note how local determinacy coexists with global indeterminacy.

Figure 3

Figure 3:  Equilibrium analysis for an active forward-looking interest rate rule. This figure shows a comparison between the local equilibrium analysis and the global equilibrium analysis as the degree of openness α and the coefficient of risk aversion σ vary. "M" stands for local multiple equilibria and "U" stands for a local unique equilibrium.

It is also interesting to compare our results with the ones in Benhabib et al. (2002a). There are some important differences. First our results derived in a money-in-the-utility-function set-up point out that it is not necessary to assume a productive role for money to obtain cyclical and chaotic equilibria. Second if consumption and money are complements then it is possible to have liquidity traps as in Benhabib et al. (2002a). But some of them may be "non-monotonic" and converge to a cycle around an extremely low interest rate steady state. On the contrary if consumption and money are substitutes then cyclical and chaotic dynamics occur only around the active steady state. Although this case is reminiscent of the one in Benhabib et al. (2002a), it also presents a subtle difference. In their closed economy model period-3 cycles always occur only for sufficiently low $\sigma,$ while our results show that they can basically appear for any $\sigma>1$ provided that there is enough degree of openness in the economy. In this sense and with respect to closed economies, open economies are more prone to display these cyclical dynamics.

4  Stabilizing Endogenous Fluctuations

The rule's elasticity to inflation was treated as given in the previous analysis, since the objective was to compare the performance of a particular rule across economies differing in trade openness and risk aversion. But this parameter is actually a policy choice. Recognizing this poses the following question. Given all the non-policy structural parameters, in particular, given the degree of openness $\alpha$ and the risk aversion parameter $\sigma,$ what elasticity to inflation will eliminate cyclical and chaotic dynamics? To answer this question we can do the following simple exercise.²⁶

The bifurcation thresholds that determine the existence of cyclical dynamics can be implicitly represented by $\chi=\Upsilon$ where $\chi$ depends on $\alpha,$ $\sigma,$ $\gamma,$ and $\theta_{N}$ and $\Upsilon$ depends on $R^{\ast}$ and $\frac{A}{R^{\ast}}.$ Then we can keep $R^{\ast}$ fixed as well as $\alpha,$ $\sigma,$ $\gamma,$ and $\theta_{N}$ (that determine $\chi$) and use $\chi=\Upsilon$ to solve for the bifurcation thresholds in terms of the elasticity $\xi\equiv\frac{A}{R^{\ast}},$ subject to $\xi>1.$ This will help us to find values of $\xi$ that preclude the existence of cycles.

It is simple to show that, for the case of $\chi<0,$ there cannot exist cycles when²⁷

$$\chi\leq R^{L}\left( 1-\frac{R^{\ast}-1}{A}\right) -1$$ (30)

and for the case of $\chi>0$ when

$$\chi\geq\left( 1-\frac{R^{\ast}}{A}\right) \left( R^{\ast}-1\right)$$ (31)

Let's consider the case of $\chi>0.$ We notice that if the interest rate target is set such that $R^{\ast}<1+\chi,$ then inequality (31) always holds for any active interest rate rule. If instead, $R^{\ast}>1+\chi,$ then inequality (31) is equivalent to $\xi\leq\frac{1}{1-\frac{\chi }{R^{\ast}-1}}$, which means that cycles are ruled out if the interest rate rule is not too active. To illustrate this we use the calibration in Table 2 and set $\alpha=0.4$ and CRRA $=2$ (or equivalently $\sigma=2.03).$ Under this parametrization $\chi>0$ and the right panel of Figure 2 suggests that there are period-2 cycles around the active steady state. In order to rule out them the elasticity to inflation should be below 1.35.

In the case of $\chi<0$, inequality (30) can be written as $R^{L} \geq\frac{1+\chi}{1-\Upsilon},$ for $\Upsilon\equiv\frac{R^{\ast}-1}{A} =\frac{R^{\ast}-1}{R^{\ast}\xi}.$ First of all we notice that, since in equilibrium $R^{L}>1$, then the last inequality always holds when $\frac{1+\chi}{1-\Upsilon}\leq1$, i.e. $\chi+\Upsilon\leq0.$ Given the definition of elasticity, the latter can also be written as $\xi\geq\xi^{H}$ where $\xi^{H}\equiv\frac{R^{\ast}-1}{R^{\ast}\left( -\chi\right) }.$ Hence a sufficiently active rule satisfying $\xi>\xi^{H}$ does not allow for equilibrium cycles. Nevertheless, even if we had $\xi<\frac{R^{\ast} -1}{R^{\ast}\left( -\chi\right) },$ still $R^{L}\geq\frac{1+\chi} {1-\Upsilon}$ could hold. From the implicit definition of $R^{L}\,\$in equation (23) and the related proof in Proposition 1, the last inequality is equivalent to