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

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 849 December 2005 Interest Rate Rules, Endogenous Cycles and Chaotic Dynamics in Open Economies Marco Airaudo 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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 www.federalreserve.gov/pubs/ifdp/.

Interest Rate Rules, Endogenous Cycles and Chaotic Dynamics in Open Economies ∗ Marco Airaudo Luis-Felipe Zanna † ‡ First Draft: January 2004 This Draft: December 2005 Abstract Inthispaperwepresentanextensiveanalysisoftheconsequencesforglobalequilibriumdeterminacy 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 conditionsunderwhichtheserulesgenerateaggregateinstabilitybyinducingcyclicalandchaoticequilibrium 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. Forinstance,wefindthataforward-lookingruleismorepronetoinduceendogenouscyclicalandchaotic 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 Classifications: E32, E52, F41 Theauthorswouldliketothankthefollowingpeopleforcommentsandsuggestions: Gian-ItaloBischi,DavidBowman,Chris ∗ Erceg, Jon Faust, Laura Gardini, Dale Henderson, Salvatore Nisticò, and the seminar participants at the Board of Governors, the LUISS University, the University of Urbino, the 2004 Meeting of the Society for Nonlinear Dynamics and Econometrics (SNDE), the 2004 MMF Group Symposium on Banking and Monetary Economics and the 2004 Latin American Meeting of the Econometric Society. The views expressed in this paper are solely the responsibility of the authors and should not be interpretedasreflectingtheview oftheInstituteofEconomicPolicyortheBoardofGovernorsoftheFederalReserveSystem orofany otherperson associated with the FederalReserve System. Institute of Economic Policy I, Humboldt University, Spandauer Strasse 1, Berlin D-10178, Germany. Email: † marco.airaudo@staff.hu-berlin.de Board of Governors of the Federal Reserve System, 20th Street and Constitution Avenue, NW, Washington, D.C., 20551. ‡ Phone: (202)452-2337. Fax: (202)736-5638. E-mail: Luis-Felipe.Zanna@frb.gov.

1 Introduction Inrecentyearstherehasbeenarevivaloftheoreticalandempiricalliteratureaimedatunderstandingthe 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 guaranteeequilibriumuniquenessinanarbitrarilysmallneighborhoodofthetargetsteadystate. Incontrast 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. Tothebestofourknowledge, ourworkisthefirstattemptintheopeneconomyliteraturetounderstand 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 showthatconditionsunderwhichactiverulesinduceaggregateinstabilitybygeneratingcyclicalandchaotic dynamicsdependonsomespecificfeaturesofanopeneconomysuchasthedegreeofopennessoftheeconomy (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 1SeeBall(1999),Clarida,GalíandGertler(1998,2001),GalíandMonacelli(2004),Kollman(2002),LubikandSchorfheide (2003),and Svensson (2000),among others. 2See Taylor(1993). 3Inthiscontextaliquiditytrapisunderstoodasadeceleratinginflationdynamicswheretheeconomyisheadedtoasituation of low and possibly negative inflation and low and possibly zero interest rates and in which monetary policy is ineffective to stop thisprocess. 1

economyandthehigherthedegreeofexchangeratepass-through. IfconsumptionandmoneyareEdgeworth 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 contemporaneousrules),totheuseofalternativemeasuresofinflationintherule(CPIversusCoreinflation), aswellastochangesinthetimingofrealmoneybalancesinliquidityservices(“cash-when-I-am-done”timing versus “cash-in-advance” timing.) Table 1: Country Degree of Openness Response Coefficient Type of Interest (Imports/GDP) to Inflation (ρ ) Rate Rule π 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 2.10 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 ) come from : Bernal (2003), Clarida et al. (1998), π × † Corbo (2000), Lubik and Schorfheide (2003), and 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, 4Theexistenceoftwostationaryequilibriaisaconsequenceofcombiningactiveinterestrateruleswiththezerolowerbound on nominalinterestratesasshown by Benhabib etal. (2001b). 2

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,ouranalysisshowsthattheassumptionofmoneyintheproductionfunctionusedbyBenhabib etal. isnotnecessarytoobtaincyclicalandchaoticequilibriumdynamicsunderrules. Weintroducemoney intheutilityfunctionandshowthattheexistenceoftheseglobaldynamicsdependsonwhetherconsumption 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 “nonmonotonic 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 relativeriskaversiongreaterthanoneandprovidedthattheeconomyissufficientlyopen. Inthissenseopen economies are more prone than closed economies to display these cyclical dynamics. Third,weidentifynecessaryandsufficientconditionsforthedesignofactiveforwardlookingrulesthatdo notgeneratecyclicalandchaoticequilibria. Forsomegivenstructuralparameters,theseconditionsgenerally entailanappropriatechoiceoftherule’sresponsivenesstoinflation. Althoughcyclicaldynamicscanberuled 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 TherearepreviousworksintheSOEliteraturethathavetriedtoidentifyconditionsunderwhichinterest 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 economyinthelocalequilibriumanalysis. Thelastworkalsopointsoutthekeyroleplayedbythedegreeof 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. IncontrasttotheabovementionedworksintheSOEliterature,thispaperdoesnotconsidernominalprice 5See Alstadheim and Henderson (2004), Benhabib et al. (2002b) or Christiano and Rostagno (2001) for the liquidity traps case and Obstfelfand Rogoff (1983)forthe inflationary pathscase. 6Fortwo country modelssee Batini,Levine and Pearlman (2004)among others. 3

rigidities. InthissenseitissimilartotheclosedeconomyworksofBenhabibetal. (2002a,b),Carlstromand 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. InSection4westudyactiveforward-lookingrulesthatcanprecludetheexistenceofcyclicalequilibria. 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 ), real money balances (md), and t t not working (1 hT hN) according to − t − t E ∞ βt (c t )γ md t 1 − γ 1 − σ − 1 +ψ(1 hT hN) (1) 0 h ¡ 1¢ σi − t − t  X t=0  −    c =(cT)α(cN)(1 α) (2) t t t − where β, γ (0,1), and ψ,σ > 0 but σ = 1;7 E is the expectations operator conditional on the set of 0 ∈ 6 information available at time 0; cT and cN denote the consumption of traded and non-traded goods in t t period t respectively; md = M t d are real money balances (domestic currency money balances deflated by the t pt Consumer Price Index, CPI, p , to be defined below); hT and hN stand for labor supplied to the production t t t of traded and non-traded goods respectively and α (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 α goestozero, domesticagentsdonotvalueinternationallytradedgoodsforconsumption. Thentheeconomy isfundamentallyclosed. Whereasifαgoestoone,non-tradedgoodsarenegligibleinconsumption. Werefer 7Thecaseofσ=1correspondstothecaseofseparabilityamongconsumptionandmoneyintheutilityfunction. Itimplies no distortionary effects of transaction money demand. It can be easily shown that no equilibrium cycles occur in this case. Hence,we do notconsideritin the analysis. 4

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 α.8 They also allow us to study how these dynamics are affected by whether consumption, c , and money, m , are either Edgeworth substitutes or complements. By defining t t U = (cγ t m1 t− 1 γ) σ 1 − σ − 1 and noticing that the sign of the cross partial derivative U cm satisfies sign { U cm } = − sign (1 σ) , then we can distinguish between the case of Edgeworth substitutes when U < 0 (σ > 1) cm { − } or the case of complements when U >0 (σ <1). Moreover, given that γ (0,1) and that the coefficient cm ∈ of relative risk aversion (CRRA) can be expressed as σ Uccc =1 γ(1 σ), then σ T1 implies σ T1. ≡− Uc − − As a result of this we will refer to σ as the “risk aversion parameter.” e e The representative unit produces traded and non-traded goods by employing labor according to the technologies yT =zT hT θT and yN =zN hN θN , (3) t t t t t t ¡ ¢ ¡ ¢ whereθ ,θ (0,1)andzT andzN areproductivityshocksfollowingstationaryAR(1)stochasticprocesses. T N ∈ t t We assume that these shocks are the sole source of fundamental uncertainty. Asstandardintheliterature,weassumethattheLawofOnePriceholdsfortradedgoodsandnormalize the foreign price of the traded good to one.9 Hence PT = , where PT is the domestic currency price of t E t t tradedgoodsand isthenominalexchangerate. Thissimplificationtogetherwith(2)canbeusedtoderive t E the Consumer Price Index (CPI) p ( E t )α P t N 1 − α . (4) t ≡ αα(1 α)1 α ¡− ¢− Usingequation(4)anddefiningthegrossnominaldevaluationrateas(cid:18) / andthegrossnon-traded t t t 1 ≡E E− goods inflation rate as πN PN/PN , we derive the gross CPI-inflation rate t ≡ t t − 1 π =(cid:18)α(πN)(1 α) (5) t t t − where π pt . It is just a weighted average of different goods inflations whose weights are related to t ≡ pt 1 − the degree of openness, α. The real exchange rate (e ) is defined as the ratio of the price of traded goods t (nominal exchange rate) and the price of non-traded goods e /PN. (6) t ≡E t t 8WecouldconsideraCESfunctionforaggregateconsumption(2)toemphasizethefactthattheintratemporalelasticityof substitution between the two types of consumption can be different from one. However, this would not affect our conclusions on the role ofopenness,butsimply preventusfrom obtaining analyticalresults. 9In Section 5 we will relax the assumption of the Law of One Price” at the consumption level by introducing non-traded distribution costs. This willallow us to modelan imperfectdegree ofexchange rate pass-through into importprices. 5

Then the gross real exchange rate depreciation, et , can be written as et 1 − e (cid:18) t = t . (7) e πN t 1 t − 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 0 the agents can purchase two types of financial assets: ≥ fiat money Md and nominal state contingent claims, D . The latter pay one unit of (foreign) currency for t t+1 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 instrumentsfeaturingawiderangeofmaturitiesandindexations.10 Inthispapertheassumptionofcomplete marketsservesthesolepurposeofrulingouttheunitrootproblemofthesmallopeneconomy,allowingusto pursueameaningfullocaldeterminacyofequilibriumanalysis. Inthisway, wecancomparetheresultsfrom the global equilibrium analysis to the ones from the local equilibrium analysis.11 Nevertheless 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 Md+E Q D W + yT +PNyN τ cT PNcN (8) t t t,t+1 t+1 ≤ t E t t t t −E t t −E t t − t t whereE Q D denotesthecostofallcontingentclaimsboughtatthebeginningofperiodtandQ t t,t+1 t+1 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 , plus non-financial income during the t period net of the value of taxes, τ , and the value of consumption spending. t t E To derive the period-by-period budget constraint, we use the definition of the total beginning-of-period wealth, in the following period, W =Md+D and the fact that the period-t price of a claim that pays t+1 t t+1 oneunitofcurrencyineverystateinperiodt+1isequaltotheinverseoftherisk-freegrossnominalinterest rate; that is E Q = 1 . From this, the definition of W and (8) we obtain t t,t+1 Rt t+1 R 1 E Q W W + yT +PNyN τ t − Md cT PNcN. (9) t t,t+1 t+1 ≤ t E t t t t −E t t − R t −E t t − t t t 10See Angeletos (2002). 11See Schmitt-Grohé and Uribe (2003). 6

The representative unit is also subject to a Non-Ponzi game condition limE q W 0 (10) t t+j t+j j ≥ →∞ at all dates and under all contingencies where q represents the period-zero price of one unit of currency to t 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 =Q Q .....Q with q 1. t 1 2 t 0 ≡ The problem of the representative household-firm unit reduces to choosing the sequences cT, cN, hT, { t t t hN, Md, W in order to maximize (1) subject to (2), (3), (9) and (10), given W and the time paths t t t+1 } ∞t=0 0 of R , , PN, Q and τ . Note that since the utility function specified in (1) implies that the preferences t E t t t+1 t 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 αγ cT t αγ(1 − σ) − 1 cN t (1 − α)γ(1 − σ) md t (1 − γ)(1 − σ) =λ t (11) ¡ ¢ ¡ ¢ ¡ ¢ αcN t =e (12) (1 α)cT t − t λ e tθ N hN t (θN− 1) =ψ =λ t θ T hT t (θT− 1) (13) t ¡ ¢ ¡ ¢ md t = 1 − γ γ 1 − α α 1 − α R R t 1 cT t e1 t− α (14) µ ¶µ ¶ µ t − ¶ λ λ tQ =β t+1 (15) t,t+1 t t+1 E E where λ / is the Lagrange multiplier of the flow budget constraint. t t E Theinterpretationofthefirstorderconditionsisstraightforward. Equation(11)istheusualintertemporal 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 (λ ). Condition (12) implies that the marginal rate of t 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 forrealmoneybalances. Andfinallycondition(15)describesastandardpricingequationforone-step-ahead nominal contingent claims for each period t and for each possible state of nature. 2.2 The Government Thegovernmentissuestwonominalliabilities: money,Ms,andaoneperiodrisk-freedomesticbond,Bs, t t that pays a gross risk-free nominal interest rate R . We assume that it cannot issue or hold state contingent t claims. It also levies taxes, τ , pays interest on its debt, (R 1)Bs, and receives revenues from seigniorage. t t − t 7

Then the government’s budget constraint can be written as Ls =R Ls (R 1)Ms τ , where t t − 1 t − 1− t − 1 − t − 1−E t t Ls =Ms+Bs. t t t 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, τ , in order to satisfy the intertemporal t versionofitsbudgetconstraintinconjunctionwiththetransversalitycondition lim Ls t / Et =0.Thelatter t t Rk →∞ k=0 (cid:18)k+1 isdescribedasaninterestratefeedbackrulewherebythegovernmentsetsthenomin(cid:84)al(cid:19)inter(cid:20)estrate,R t ,asa continuousandincreasingfunctionofthedeviationoftheexpectedfutureCPI-inflationrate,E (π ),from t t+1 atarget, π .12 Foranalyticalandcomputationalpurposes, asinBenhabibetal. (2002a)andChristianoand ∗ Rostagno (2001), we use the following specific non-linear rule13 A R =ρ(E π ) 1+(R 1) E t π t+1 R∗− 1 (16) t t t+1 ≡ ∗ − π µ ∗ ¶ where R = π /β and R corresponds to the target interest rate. (16) always satisfies the zero bound on ∗ ∗ ∗ the nominal interest rate, i.e. R = ρ(E π ) > 1. In addition we assume that the government responds t t t+1 aggressively to inflation. This means that at the inflation target, the rule’s elasticity to inflation ξ ≡ ρ0(π∗)π∗ = A is strictly bigger than 1. Following Leeper (1991) we call rules with this propertyactive rules. ρ(π ) R ∗ ∗ Assumption 0: ξ = A >1. That is, the rule is active. R ∗ 2.3 International Capital Markets Besides complete markets there is free international capital mobility. Then the no-arbitrage condition Qw t,t+1 = Q t,t+1Et+1 holds, where Qw t,t+1 refers to the period-t foreign currency price of a claim to one unit Et 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, λw t Qw = PTw t,t+1 t λw t+1βw where λw represents the marginal utility of nominal wealth in the ROW, βw denotes the subjective PTw t t+1 discount rate of the ROW and PTw is the foreign price of traded goods. Since we normalize this price to t one (PTw =1) then assuming that βw =β leads to λwQw =λw β. t t t,t+1 t+1 Combining this last equation, with condition (15) and the fact that PT = yields λt+1 = λw t+1, which t E t λt λw t holds at all dates and under all contingencies. This condition implies that the domestic marginal utility of wealth is proportional to its foreign counterpart: λ = Λλwwhere Λ refers to a constant parameter that t t determinesthewealthdifferencebetweentheSOEandtheROW.FromtheperspectiveofaSOE,λw canbe t taken as an exogenous variable. For simplicity we assume that λw is constant and equal to λw. As a result t 12In the sensitivity analysis presented in Section 6 we also study contemporaneousand backward-looking rules. 13Cycles and chaoswould also occurifthe interestrate rule wasa linearfunction ofinflation. 8

of this λ becomes a constant. Then t λ =λ=Λλw. (17) t This allows us to write condition (15) as Q t,t+1 = E E t+ t 1 β = (cid:18)t β +1 that together with E t Q t,t+1 = R 1 t yields R =β 1 E 1 − 1 (18) t − t(cid:18) · t+1¸ 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 we have that E x = x with j 0 implying that we can drop the expectation operator in t t t+j t+j ≥ the previous equations. For instance, under perfect foresight, condition (18) becomes R =β 1(cid:18) (19) t − t+1 that corresponds to the typical uncovered interest parity condition as long as β 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-linearformof themodel. Todosoweusethedefinitions(5)and(7)togetherwithconditions(11)-(14), (17), (19), and the market clearing conditions for money and the non-traded good, Md = Ms = M and t t t yN = hN θN =cN, to obtain t t t R χ R χ ¡ ¢ π t+1 = t βR (20) t+1 R 1 R 1 t µ t+1 − ¶ µ t − ¶ where (σ 1)(1 α)(1 γ)(1 θ ) χ= − − − − N . (21) σ[θ +α(1 θ )]+(1 α)(1 θ ) N N N − − − Combining (16) and (20) and dropping the expectation operator yields R t+1 χ = R t R ∗ − 1 R∗ A− 1 R t χ (22) R 1 R R 1 R 1 µ t+1 − ¶ ∗ µ t − ¶ µ t − ¶ 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 14This holdsby the previousanalysissince EtQw t,t+1 = R 1 t w =β. 15Thisequationabstractsfrompossibleeffectsthatfiscalpoliciescanhaveontheequilibriumdynamicsoftheeconomy. The reason is that we have assumed a Ricardian fiscal policy under which the intertemporal version of the government’s budget constraintand itstransversality condition willbe alwayssatisfied. 9

Perfect Foresight Equilibrium (PFE). Definition 1 Given the target R and the initial condition R , a Perfect Foresight Equilibrium (PFE) is a ∗ 0 deterministic process R , with R > 1 for any t, that satisfies equation (22) if the interest rate rule is { t } ∞t=0 t forward-looking. Although Definition 1 is stated exclusively in terms of the nominal interest rate (R ), it must be clear t 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), πNss =(cid:18)ss =πss, and Rss =πss/β. Using these and the rule (16) we have that (R ∗ 1) R∗ A− 1 Rss =R ∗ (Rss 1) R∗ A− 1 . (23) − − Clearly Rss =R >1 is a solution to (23), and therefore a feasible steady state. But if the rule is active at ∗ R , that is if ξ = A >1, then another lower steady state RL (1,R ) exists and it is unique. At this low ∗ R ∗ ∈ ∗ steady state the elasticity of the rule to inflation satisfies ξ = A <1. The following proposition formalizes RL the existence of the low steady state RL. Proposition 1 If A > 1 (an active rule) and Rss > 1 (the zero lower bound) then there exists a solution R ∗ Rss =RL (1,R ) solving (23) besides the trivial solution Rss =R . ∗ ∗ ∈ Proof. See the Appendix. The existence of two steady states plays a crucial role in the derivation of our results as in the closed economymodelofBenhabibetal. (2002a). Asamatteroffact,thesteadystateequation(23)ofourSOEis 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 ofthefollowingtwofeaturesofourmodel. First, byintroducingtradedandnon-tradedgoodswepresentan economywithtwosectorsthatalthoughhomogeneousintermsofpricesettingbehavior(bothfeatureflexible prices), are fundamentally different in terms of the degree of openness to international trade. As we will see 16By indeterminacy we refer to a situation where one or more real variables are not pinned down by the model. We use the terms indeterminacy and multiple equlibria interchangeably. The same comment applies to determinacy and a unique equilibrium. 17Infact,foranygivenRt,bysimplymanipulatingequations(5),(7),(11)-(14),(17),(19),andthemarketclearingconditions formoneyandthenon-tradedgood,wecanobtainalltheremainingrealendogenousvariables. Forinstancethelaborallocated for the production of the non-traded good and the real exchange rate can be expressed as functions of the gross interest rate only, hN t = hN(Rt) and et = e(Rt), respectively, whereas from the market clearing condition, cN t = hN t θN, we obtain cN t =cN(Rt). (cid:3) (cid:4) 10

below this degree, measured by α, 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 (σ <1) or substitutes (σ >1). Intheanalysistofollowwewillstudyhowαandσaffectthelocalandglobalequilibriumdynamicsinour SOEmodelwhilekeepingconstanttheotherstructuralparameters(β,γ andθ )andthepolicyparameters N (A and R ). This will allow us to compare economies that implement the same monetary rule but differ ∗ in the degree of openness α and the risk aversion parameter σ. To accomplish this goal we will proceed in two steps. First we will analyze how these dynamics are affected by the composite parameter χ defined in (21). Second by taking into account the dependence of χ on both α and σ we will unveil the effect of the degreeofopennessandtheriskaversionparameterontheexistenceoflocalandglobaldynamics(cyclesand chaos). Inthissensewewillregardχasafunctionofαandσ.18 Thatisχ(α,σ).Forthesecondstepwewill 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 (1 γ)(1 θ ) χ − − N (0,+ ) χ (1 γ) ( 1,0) (24) max ≡ θ ∈ ∞ min ≡− − ∈ − N (σ 1)(1 γ)(1 θ ) µ(σ)= − − − N (25) σθ +(1 θ ) N N − where χ and χ are considered scalars and µ(σ) is considered a function of σ. max min 1 Υi 1 Υi Definition 2 Using (24) and (25) define the scalars σi −χmin and the functions αi(σ) −µ(σ) for ≡ 1 Υi ≡ 1 Υi −χmax −χmin i = w,k,f,d where Υw ≡ RL 1 − R∗ A− 1 − 1 < 0, Υk ≡ 1 − R A ∗ (R ∗ − 1) > 0, Υf ≡ Υ 2 w and Υd ≡ Υ 2 k ; and the functions αi(σ) are ch³aracterized´in Lemma 4 whe³n σ >1´and in Lemma 5 when σ (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. Inordertoderiveanalyticalresultsforboththelocalandtheglobalequilibriumanalyseswewillassume that the constant parameters γ, θ , A and R satisfy the following assumptions.19,20 N ∗ 18χalso dependson γ and θN. Howeverwe willnotpursue any bifurcation analysiswith respectto these parameters. 19Assumption1isnecessaryandsufficientfortheexistenceofrangesofαandσwherelocalindeterminacyoccurs,aswellas fortheexistenceofaflipbifurcationfrontierinthecaseofσ>1. Assumption2allowsmonotonicliquiditytrapsforlowvalues of σ. It could therefore be dropped without affecting the possibility of cycles around the passive steady state. Assumption 3 whichbasicallyrequiresenoughseparationbetweenthetwosteadystates-isusefulinprovingtheexistenceofaflipbifurcation frontierinthecaseofσ<1.Extensivealgebra(notreportedhereforreasonsofspace)showsthatAssumption3holdsif,fora given targetR ,the rule is sufficiently active. ∗ 20The calibration exercise that we present below suggests that these assumptions are not unrealistic. For given monetary 11

Assumption 1: χ > 1(R 1) 1 R∗ . max 2 ∗ − − A Assumption 2: χ min < 1 −A R∗. ³ ´ Assumption 3: R 1>A RL 1 . ∗ − − ¡ ¢ 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 , yielding ∗ R∗ 1 Rˆ = 1+ A − Rˆ . (26) t+1 χ t " R 1 # ∗− SinceR isanon-predeterminedvariable,studyinglocaldeterminacyisequivalenttofindingconditionsthat t makethelineardifferenceequation(26)explosive. ThenextLemmashowshowlocalequilibriumdeterminacy depends on χ. Lemma 1 Define Υd ≡ 1 2 (R ∗ − 1) 1 − R A ∗ > 0 and consider χ ∈ R. Suppose the government follows an active forward-looking rule then: 1)³the equil´ibrium is locally unique if χ<Υd; 2) there exist locally multiple equilibria if χ>Υd. Proof. See the Appendix. These simple determinacy of equilibrium conditions for χ can be reinterpreted in terms of the degree of openness α and the risk aversion parameter σ in the following Proposition. Proposition 2 Consider σd and αd(σ) in Definition 2 where σd >1 and αd :(1,+ ) ( ,1). 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. σ (0,1), and for any degree of ∈ openness, i.e. α (0,1); ∈ (b) if consumption and money are Edgeworth substitutes, i.e. σ >1, and the economy is sufficiently open satisfying α > αd ; where αd max 0,αd(σ) is positive and strictly increasing for min min ≡ σ >σd, but constant and equal to zero for any σ© 1,σdª. ∈ ¡ ¤ 2. There exist locally multiple equilibria if consumption and money are Edgeworth substitutes satisfying σ >σd, and the economy is sufficiently closed satisfying α (0,αd(σ)). ∈ policyparametersandforgivenθ,bothAssumptions1and2implyaminimumshareofrealbalancesinutility. Wecaninfact i r n e e w q r u it a e li A ty ss is um ab p o t u io t n 0 1 .0 a 0 s 6, (1 w − hil γ e ) w > es 1 2 e ( t 1 −1 θN θN γ ) (cid:19)= R 0 ∗ A . − 03 1 (cid:20)co ( n A si − ste R n ∗ t ) ly . w Fo it r h th th e e ca li l t i e b r r a a t t u io re n . u S s i e m d il i a n rl T y a , b A le ss 2 u , m t p h t e io r n igh 2 t c h a a n n b d e s w id r e it o t f en th a i s s − 1 − γ> R∗ A− 1,with the righthand side equalto 0.007 in ourcalibration. 12

Proof. See the Appendix. The results of this proposition show the importance of α and σ 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 σ is sufficiently low; and when the economy is sufficiently open for high values of σ.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 Topursuetheglobalequilibriumanalysiswerewriteequation(22)astheforwardmappingR =f(R ) t+1 t where 1 f(R ) (27) t ≡ 1 J(R t )χ 1 − and J(R ) R ∗ [R t − 1]χ+R∗ A− 1 . (28) t ≡ [R ∗− 1] R∗ A− 1 R t 1+χ Then the global analysis corresponds to studying the global PFE dynamics that satisfy R =f(R ) given t+1 t an initial condition R > 1 and subject to the zero-lower-bound condition fn(R ) > 1 for any n 1. 0 0 ≥ 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 = fn(R), but not a fixed point of an iterate of any lower order. If “R” is such, we call the sequence R,f(R),f2(R),...,fn 1(R) a period-n cycle. − © ª 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. Theglobalanalysisrequiresafullcharacterizationoff(.)in(27)notonlyarounditsstationarysolutions, 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(.) 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 χ. Here we only 21In fact this result is more general since a quick inspection of Proposition 2 suggests that if the the economy is very open, an active rule leads to a unique equilibrium regardless ofthe valuesofthe otherstructuralparameters. 22See Lorenz (1993). 13

χ > 0 χ < 0 R R t+1 t+1 f(R) f(R) t t 1 1 1 RL RJ R* R 1 Rl RL RJ R* Ru R t t Figure 1: This Figure shows the mapping R t+1 = f(R t ) for χ > 0, and χ < 0 but χ+ R∗ A− 1 > 0. R t denotes the nominal interest rate. A formal characterization of this mapping is provided in Lemma 6 in the Appendix. provide a big picture of the analysis. First of all, the Lemma specifies conditions under which f(.) satisfies the zero-lower-bound requirement. Second, it makes use of the following conditions J (R ) 1+χ sign f (R ) =sign 0 t and sign J (R ) =sign R (29) { 0 t } ½ χ ¾ { 0 t } (1 − R ∗ A− 1 − t ) which imply that for χ = 0 then RJ 1+χ is a critical point of f(.) as long as RJ > 1. With these 6 ≡ 1 − R∗ A− 1 conditions the Lemma shows that the mapping R =f(R ) is always single-peaked for χ>0 whereas for t+1 t χ<0 it is single-troughed only if χ+ R∗ A− 1 >0. Figure1displaysagraphicalrepresentationofthecaseswheretheequilibriummappingR =f(R )has t+1 t acriticalpointbetweenthetwosteadystates.Therightpanelconsidersthecaseofχ<0andχ+R∗ A− 1 >0, while the left panel the case of χ > 0. In the left one, f(.) always satisfies the equilibrium conditions for any R (1,+ ) and crosses the 45 degree line twice at RL > 1 and R > RL (the two steady states). t ∗ ∈ ∞ Furthermore, lim f(R ) = lim f(R ) = 0 and there is a maximum at RJ RL,R . In the right t t ∗ panel, all equi R li t b → ri 1 um conditi R o t n → s + a ∞ re satisfied only within a subset Rl,Ru (1 ∈ ,+¡ ) defi¢ned in Lemma ⊂ ∞ 6. Within that set, f(.) crosses the 45 degree line at RL > 1 and ¡R ∗ > R¢L, as in the previous case, but now RJ RL,R is a minimum and lim f(R )= lim f(R )=+ . These are clearly the cases in ∗ t t ∈ Rt→ Rl+ Rt→ Ru − ∞ which we a¡re intere¢sted, as they imply a negative derivative of f(.) at either one of the two steady states. Figure1togetherwithLemma6suggestthatdependingonthesignofχ,cyclesmayappeararoundeither the active steady state or the passive steady state. Hence we proceed to look for flip bifurcation thresholds for χ i.e. critical values of χ that determine a change in the stability properties of the steady state where 14

the map f (.) is negatively sloped. If the steady state is stable, any equilibrium orbit, that starts in a map invariantsetcenteredaroundthisstate,willasymptoticallyconvergetothesteadystateitself,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 notconvergeatalldisplayingaperiodicbutboundeddynamics(chaoticequilibriumpaths). Wefirstconsider the case of χ ∈ RL 1 − R∗ A− 1 − 1,0 and show that endogenous cyclical dynamics of period 2 can occur around the pass³ive st³eady state´. ´ Lemma 2 LetΥw ≡ RL 1 − R∗ A− 1 − 1,anddefinethepointsR ≡ f RJ ,i.e. theimageofRJ (thecritical point of f(.)), and R f³ − 1(R ∗ ), i.´e. the inverse image of the high s¡tead¢y state. Consider χ (Υw,0) and ≡ ∈ assume that f f RJ R.23 Then: min ≡e ≥ ¡ ¢ 1. RL R,RJ and RL >eR; ∈ ¡ ¢ 2. the set [R,R ] is invarianet under the mapping f(.) and attractive for any R R,R where R<R; ∗ t ∈ h i 3. period-2 cycles within [R,R ] and centered around the passive steady state occur wehen χ Υwe,0 . ∗ ∈ 2 ¡ ¢ Proof. See Appendix. For the case of χ 0, 1 R∗ (R 1) endogenous cyclical dynamics of period 2 exist around the ∈ − A ∗ − active steady state, inste³ad.³ ´ ´ Lemma 3 LetΥk 1 R∗ (R 1),anddefinethepointsR f RJ ,i.e. theimageofRJ (thecritical ≡ − A ∗ − ≡ point of f(.)) and R ³ f − 1(R´ ∗ ), i.e. the inverse image of the high s¡tead¢y state. Consider χ 0,Υk and ≡ ∈ assume f max f RJ R. Then: ¡ ¢ ≡ e ≤ ¡ ¢ 1. R RJ,R and Re>RL; ∗ ∈ ¡ ¢ 2. the set RL,R is inevariant under the mapping f(.) and attractive for any R (R,R] where R<R; t ∈ £ ¤ 3. period-2 cycles within RL,R and centered around the active steady state occur whene χ 0,1Υk e. ∈ 2 £ ¤ ¡ ¢ Proof. See Appendix. Similarly to the local determinacy analysis, the conditions for endogenous cycles derived in terms of χ can be easily translated into conditions described in terms of the degree of openness α and the risk aversion parameter σ. The next Proposition accomplishes this goal. 23Thisassumptionrulesoutexplosivepathsforinitialconditionsbetweenthetwosteadystates. Itissimilarinflavortothe one used by Boldrin et al. (2001) and Matsuyama (1991). If fmin =f RJ <R there would not be a non-trivial mappinginvariant set. This case displays a different type of multiplicity. It can be shown that there exists a set of points within the (cid:3) (cid:4) (cid:104) set [R,R ] that leave such a set after a finite number of iterations, and settle to an exploding path diverging from the active ∗ steady state (see Matsuyama,1991). 15

Proposition 3 Suppose that the government follows an active forward-looking rule. 1. Consider σf and αf(σ) in Definition 2 where σf (0,1) and αf : (0,1) ( ,1) and assume that ∈ → −∞ consumption and money are Edgeworth complements, i.e. σ (0,1). Then period-2 equilibrium cycles ∈ exist around the passive steady state if the economy is sufficiently open satisfying α > αf , where min αf max 0,αf(σ) is positive and strictly decreasing for σ 0,σf , but constant and equal to min ≡ ∈ zero for any ©σ σf,1ª . ¡ ¢ ∈ £ ¢ 2. Consider σd and αd(σ) in Definition 2 where σd > 1 and αd : (1,+ ) ( ,1) and assume ∞ → −∞ that consumption and money are Edgeworth substitutes, i.e. σ > 1. Then period-2 equilibrium cycles exist around the active steady state if the economy is sufficiently open satisfying α > αd , where min αd max 0,αd(σ) is positive and strictly increasing for σ >σd, but constant and equal to zero for min ≡ any σ 1,σ©d . ª ∈ ¡ ¤ 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 (σ), forward looking rules are more prone to induce endogenous cyclicaldynamicsthemoreopentheeconomy;whileforσsufficientlycloseto1,butdifferentfromit,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 σ > 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 startingarbitrarilyclosetoitanddivergingcannotbepartofanequilibriumsinceitwilleventuallyexplode and thus violate some transversality condition. This is not the case here as the global analysis proves that thetruenon-linearmapfeaturesaboundedmap-invariantandattractivesetaroundtheactivesteadystate. 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(.) in (27) it is very difficult to derive analytical conditions for α and σ underwhichforward-lookingrulesinduceeithercyclesofperiodhigherthan2orchaoticdynamics. Therefore in order to shed some light on the role of both α and σ in delivering these dynamics, as well as to find some empirical confirmation of our analytical results, we pursue a simple calibration-simulation exercise. 24Asmentionedbefore,thediscountinuityofthemodelwithrespecttoσ=1correspondstoautilityfunctionthatisseparable in money and consumption. In this case itis possible to show thatno cyclicaldynamicscan occur. 16

Table 2: Parametrization θ β π R 1 γ A N ∗ ∗ − R ∗ 0.56 0.99 1.0311 4 1.0721 4 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 θ = 0.56. The steady-state N inflation,π ,andthesteadystatenominalinterestrate,R ,arecalculatedastheaverageoftheCPI-inflation ∗ ∗ andtheCentralBankdiscountratebetween1983-2002. Thisyieldsπ ∗ =1.0311 4 andR ∗ =1.0721 4.Thenthe subjective discount rate is determined by β =π /R . We use the estimate of Lubik and Schorfheide (2003) ∗ ∗ for the Canadian interest rate response coefficient to inflation which corresponds to A = 2.24. Estimates R ∗ for the share of expenditures on real money balances, 1 γ, 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 andmethodofestimation. Weset1 γ equalto0.03thatisinlinewiththeestimatesprovidedbyprevious − works.25 Table 2 gathers the parametrization. As in the analytical study, in the simulation exercise we vary α and σ keeping the remaining parameters as in Table 2. Nevertheless, an estimate of α for Canada can be obtained from the average imports to GDP share during 1983-2002, yielding α = 0.31. In contrast, obtaining an estimate of σ is more difficult. As explained before, σ is related to the CRRA coefficient σ through σ =1 γ(1 σ) which spans over a wide − − range. The RBC literature usually sets σ = 2. This value and 1 γ = 0.03 imply σ = 2.03. Since the e e − value of σ determines whether consumption and real money balances are either Edgeworth substitutes or e complements we will use different values for the CRRA σ. For instance, we let σ 0.8,1.5,2,2.5 which in ∈{ } tandem with 1 γ =0.03 leads to σ 0.79,1.51,2.03,2.55 respectively. − ∈{ e } e Givenσ 0.79,2.03 whichcorrespondstoCRRAofσ˜ 0.8,2 weconstructFigure2. Itpresentsthe ∈{ } ∈{ } bifurcation (or orbit) diagrams for the degree of openness α. The left panel considers the case when money and consumption are complements by setting σ˜ = 0.8. The right panel corresponds to the case when they are substitutes as σ˜ = 2. With α (0,1) on the horizontal axis and R > 1 on the vertical axis, the solid t ∈ lines in the diagram correspond to stable solutions of period n. The left and right panels of the figure show how by increasing α 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 α = 0, both panels show that for low degrees of openness the economy, that is described by the mapping R =f(R ) in (27), always settles t+1 t on a stable steady state equilibrium after a long enough series of iterations. It settles on the passive steady state, RL, for σ˜ = 0.8 and on the active steady state, R , for σ˜ = 2. Once α reaches some threshold a ∗ 25See Holman (1998)among others. 17

Figure 2: Orbit-bifurcation diagrams for the degree of openness, α. R denotes the nominal interest rate. t The diagrams show the set of limit points as a function of α, under two different coefficients of risk aversion (CRRA) σ˜ =0.8 and σ˜ =2, and under an active forward-looking rule. Depending on α, an active forwardlooking rule may drive the economy into period-2 cycles, period-4 cycles,...period-n cycles and even chaotic dynamics. stableperiod-2cycleappears,asindicatedbythefirstsplitintotwobranchesinbothpanels. Asweincrease α 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 α, yielding cycles of period-8, period-16 and so on. Finally for sufficiently high α 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. FromFigure2wealsoseethatwhenconsumptionandmoneyarecomplementsthencyclicalandchaotic dynamicsoccuraroundthepassivesteadystate; whereasiftheyaresubstitutesthenthesedynamicsappear 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. Inordertosummarizeandcomparetheresultsofthelocalandglobaldeterminacyofequilibriumanalyses weconstructFigure3. Itshowsthecombinationsofthedegreeofopennessandtheriskaversionparameter, α and σ, for which there is local and/or global (in)determinacy. For σ 0 and α [0,1] we plot two ≥ ∈ threshold frontiers: the flip bifurcation frontier for period-2 cycles around the passive steady state, αf(σ) andthefrontierαd(σ)forbothlocaldeterminacyandperiod-2cyclesaroundtheactivesteadystate. Regions featuringalocallyuniqueequilibriumarelabeledwitha“U”,whilethosefeaturinglocallymultipleequilibria are labeled with an “M”. Clearly, “U” appears everywhere but below the curve αd(σ) implying that local 18

1 C ycles and Chaos Around the Cycles and Chaos Passive SS Around the Active SS Flip Flip Bifurcation and Bifurcation Local α Frontier αf(σ) U Indetermi nacy Frontier αd(σ) U M U 0 σf 1 σd σ Figure3: Equilibriumanalysisforanactiveforward-lookinginterestraterule. Thisfigureshowsacomparison 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. determinacy occurs for a wide range of (α,σ) combinations. In fact note how local determinacy coexists with global indeterminacy. 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 σ, while our results show that they can basically appear for any σ >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 19

the non-policy structural parameters, in particular, given the degree of openness α and the risk aversion parameter σ, what elasticity to inflation will eliminate cyclical and chaotic dynamics? To answer this question we can do the following simple exercise.26 Thebifurcationthresholdsthatdeterminetheexistenceofcyclicaldynamicscanbeimplicitlyrepresented by χ = Υ where χ depends on α, σ, γ, and θ and Υ depends on R and A . Then we can keep R fixed N ∗ R ∗ ∗ as well as α, σ, γ, and θ (that determine χ) and use χ=Υ to solve for the bifurcation thresholds in terms N of the elasticity ξ A , subject to ξ >1. This will help us to find values of ξ that preclude the existence of ≡ R ∗ cycles. It is simple to show that, for the case of χ<0, there cannot exist cycles when27 R 1 χ RL 1 ∗ − 1 (30) ≤ − A − µ ¶ and for the case of χ>0 when R χ 1 ∗ (R 1) (31) ∗ ≥ − A − µ ¶ Let’s consider the case of χ > 0. We notice that if the interest rate target is set such that R < 1+χ, ∗ then inequality (31) always holds for any active interest rate rule. If instead, R > 1+χ, then inequality ∗ (31) is equivalent to ξ 1 , which means that cycles are ruled out if the interest rate rule is not too ≤ 1 −R∗ χ − 1 active. To illustrate this we use the calibration in Table 2 and set α=0.4 and CRRA =2 (or equivalently σ =2.03).Underthisparametrizationχ>0andtherightpanelofFigure2suggeststhatthereareperiod-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 χ < 0, inequality (30) can be written as RL ≥ 1 1 − + Υ χ, for Υ ≡ R∗ A− 1 = R R ∗ ∗ −ξ 1. First of all we notice that, since in equilibrium RL > 1, then the last inequality always holds when 1+χ 1, i.e. 1 Υ ≤ − χ+Υ ≤ 0. Given the definition of elasticity, the latter can also be written as ξ ≥ ξH where ξH ≡ R R ∗ ∗ ( − −χ 1 ) . Hence a sufficiently active rule satisfying ξ > ξH does not allow for equilibrium cycles. Nevertheless, even if we had ξ < R R ∗ ∗ ( − −χ 1 ) , still RL ≥ 1 1 − + Υ χ could hold. From the implicit definition of RL in eq Υ uation (23) and the related proof in Proposition 1, the last inequality is equivalent to 1 1 Υ R 1 ∗−Υ 1(χ+Υ) ≥ 1 R + ∗ χ , which − − dependssolelyonχ,R andξ.ThisinequalitydefinesimplicitlyathresholdhξL belowwhicihcyclesareruled ∗ out. Summarizing when χ < 0 then rules with elasticities ξ > 1 that satisfy either ξ > ξH or ξ < ξL will precludetheexistenceofcyclicaldynamics. Forinstance,usingthecalibrationinTable2andsettingα=0.4 and CRRA= 0.8 (or equivalently σ = 0.79), we know that the left panel of Figure 2 suggests that there are chaotic dynamics around the passive steady state. But for this parametrization we obtain ξH = 8.75 and ξL =1.22. Thus these complex dynamics can be ruled out as long as the elasticity to inflation is either 26Assumptions0-3 willnotbe violated by the following analysis. 27The formalderivation is available upon request. 20

ξ <1.22 or ξ >8.75. According to these results, the Taylor principle ξ A >1 can be still a viable policy recommendation ≡ R ∗ forequilibriumdeterminacy,evenfromaglobalequilibriumdynamicsperspective. Butthereisanadditional requirement: an upper bound on the rule’s elasticity to inflation.28 5 Distribution Costs and Imperfect Exchange Rate Pass-Through The previous results were derived assuming that the Law of One Price held at the consumption level for traded goods and normalizing their foreign price to one. This in turn implied that there was a perfect exchange rate pass-through into import prices. In this Section we relax this assumption by introducing non-tradeddistributioncostswhichinturnimpliesimperfectexchangeratepass-throughintoimportprices. Our goal is to understand how this can affect the previous global results of Section 3. We follow Burnstein, Neves and Rebelo (2003) by assuming that the traded good needs to be combined withsomenon-tradeddistributionservicesbeforeitisconsumed. Inordertoconsumeoneunitofthetraded gooditisrequiredη unitsofthenon-tradedgood. Let P˜T andPT bethepricesinthedomesticcurrencyof t t thesmallopeneconomythatproducersoftradedgoodsreceiveandthatconsumerspay, respectively. Hence the consumer price of the traded good can be written as PT =P˜T +ηPN. (32) t t t TosimplifytheanalysisweassumethattheLawofOnePriceholdsfortradedgoodsattheproductionlevel and normalize the foreign price of the traded good to one (P˜Tw =1). Thus P˜T = P˜Tw = . t t E t t E t The presence of distribution services leads to imperfect exchange rate pass-through into import prices. To see this we combine (32), P˜T = and e = /PN to obtain t E t t E t t e η πT t = e t − + 1 η (cid:18) t + e +η πN t µ t − 1 ¶ µ t − 1 ¶ where πT t ≡ P P t T t − T 1 , (cid:18) t ≡ Et E − t 1 and πN t ≡ P P t N t − N 1 correspond to the (gross) inflation of import prices, the (gross) nominal depreciation rate and the (gross) non-traded inflation respectively. Clearly if η = 0 then there is perfect pass-through of the nominal depreciation rate into the inflation of import prices and it is measured by dπT t =1. This is the case that we already studied. But if η >0 then we obtain imperfect exchange rate d(cid:18)t pass-throughmeasuredby d d π (cid:18) T t t = et et − 1+ 1 η ∈ (0,1).Astheparameterofdistributioncosts,η,increasesthen − the degree of exchange rate pass-t³hrough´decreases. Inordertopursueadeterminacyofequilibriumanalysisweproceedasbefore. Weobtainareducednon- 28Note that although it is possible to rule out cyclical and chaotic dynamics, liquidity traps are still feasible. To eliminate them itisnecessary to implementsome ofthe fiscal-monetary regimes proposed by Benhabib etal. (2002a). 21

Figure 4: Orbit-bifurcation diagrams for the distribution costs parameter (η). R denotes the nominal t interest rate. The diagrams show the set of limit points as a function of η, under two different sets of values of the degree of openness and the CRRA, (σ˜,α) = (0.8,0.4) and (σ˜,α) = (2,0.4), and under an active forward-looking rule. Depending on η, which is related to the degree of exchange rate pass-through, an active forward-looking rule may lead the economy into cyclical and chaotic dynamics. linear form of the model that describes the dynamics of this economy. Nevertheless in contrast to the case of perfect exchange rate pass-through it is not possible to derive explicitly a difference equation similar to (22) that depends exclusively on the nominal interest rate R . Under imperfect exchange rate pass-through t the dynamics of the economy are determined by the system e t+1 +η α e t = R ∗ R t − 1 R∗ A− 1 e +η e R R 1 µ t ¶ µ t+1¶ µ t¶µ ∗− ¶ e t +η [α(1 − σ)+σ] (1 − α)e t+1 +η σ e t+1 (cid:19) 1+ 1 σ − θ θ N N (cid:20) = R t+1 − 1 R t (1 − γ)(1 − σ) . e +η (1 α)e +η e R R 1 µ t+1 ¶ µ − t ¶ µ t ¶ ·µ t+1 ¶µ t − ¶¸ Sinceitisnotpossibletoderiveanalyticalresultswerelyonnumericalsimulations. Theobjectiveofthe simulationsistoassesstheimpactofvaryingthedistributioncostsparameterη (andthereforethedegreeof exchange rate pass-through) on the previous results about cyclical and chaotic dynamics. To do so we use the parametrization of Table 2 and construct Figure 4. We use two pairs of values for the degree of openness and the CRRA (that depends on σ). The first pair sets α=0.4 and CRRA=0.8. According to the left panel of Figure 2 for theses values there are chaotic dynamics aroundthepassivesteadystate. Thesecondpairsets α=0.4andCRRA=2. Forthesevaluesthe 22

rightpanelof Figure2suggeststhatthereareperiod-2cyclesaroundtheactivesteadystate. Inthecontext of the present discussion these results associated with the selected pairs correspond to perfect exchange rate pass-through or η =0. Increasing η, or in other words decreasing the degree of exchange rate pass-through, has a non-trivial impact on these results as can be confirmed by Figure 4. Consider the left panel first, where α = 0.4 and CRRA=0.8. Starting from η = 0 as η increases (as the degree of exchange rate passthroughdeclines)thentheeconomymovesfromdisplayingchaoticdynamicsintodisplayingperiodiccyclical dynamics. As η continues increasing the period of cycles decrease. Beyond η 0.6 cycles disappear and the ≈ only fixed point that subsists corresponds to the passive steady state. On the other hand, the right panel of Figure 4 shows the results for α = 0.4 and CRRA=2. For η = 0 the economy presents period-2 cycles around the active steady state as mentioned before. However as η increases (as the degree of exchange rate pass-through declines) cycles also disappear and the only attractor that subsists is the active steady state. The results of this analysis can be summarized in the following Proposition. Proposition 4 Forward-lookingrules are moreprone to inducecyclical and chaoticdynamics thehigherthe degree of exchange rate pass-through. 6 Sensitivity Analysis Throughoutthewholepaper,theanalysiswaspursuedunderthefollowingassumptions: 1)international completefinancialmarkets,2)themeasureofinflationintherulewastheCPI-inflation,3)aforward-looking ruleand4)realmoneybalancesenteredintoutilityviawhatCarlstromandFuerst(2001)calla“cash-when- I’m-done” timing. In this section, we study the consequences of relaxing these assumptions. To simplify the analysis we still assume perfect exchange rate pass-through. We will show that our previous global findings of Section 3 hold even if we consider incomplete financial markets or if we introduce a forward-looking rule that responds to a different measure of inflation such as the non-traded inflation. That is, the degree of openness still plays an important role for the existence of cyclical and chaotic dynamics. We also argue that cyclical and chaotic dynamics are less likely to occur under backward-looking rules whiletheycanstillappearundercontemporaneousrulesdependingonthedegreesoftradeopennessandrisk aversion. And finally we prove that the alternative timing for real money balances, known as the “Cash-in- Advance” timing, affects the previously derived bifurcation thresholds, but does not preclude the existence of cyclical and chaotic dynamics. 6.1 Incomplete Markets TheassumptionaboutcompletemarketsisnotessentialfortheresultsderivedinSection3aboutcyclical and chaotic dynamics. Assuming incomplete markets leads to the same global dynamics results. To see this assume that the agent is blessed with perfect foresight and has access to an international bond bw and a t 23

domesticbondB issuedbythegovernment.Theformerpaysaconstantinternationalinterestrate,Rw,and t the latter pays an interest rate, R . Using this and the assumptions in the Subsection 2.1 we can rewrite the t agent’s budget constraint as B + bw+M =Rw bw +M +R B + yT +PNyN τ cT PNcN t E t t t E t t − 1 t − 1 t − 1 t − 1 E t t t t −E t t −E t t − t t The agent chooses the sequences cT, cN, hT, hN, Md, bw, B in order to maximize (1) subject to { t t t t t t t } ∞t=0 (2) and (3), the previously mentioned budget constraint and corresponding transversality conditions, given Md , bw , and B and the time paths of R , , PN, Rw and τ . The first order conditions of this problem − 1 − 1 − 1 t E t t t for bw and B correspond to t t λ =βRwλ (33) t t+1 βR λ λ = t t+1 (34) t (cid:18) t whereas the conditions for cT,cN,hT,hN,Md can be written as (11)-(14). { t t t t t} As is common in the small open economy literature, we assume βRw = 1, which implies by (33) that λ =λ .Thisinturnmeansthatthereisaunitrootinthesystemofequationsthatdescribethedynamics t t+1 of the economy. This prevents us from using (log)linearization techniques in order to pursue a meaningful local determinacy of equilibrium analysis. Hence we cannot derive local results similar to the ones in the Subsection 3.1. But this does not change the previous results from the global determinacy of equilibrium analysis, since theglobaldynamicsarestillgovernedby(22). Toseethisuseλ =λ and(34)toobtain(cid:18) =βR whichis t t+1 t t identical to (19). This condition together with λ =λ , (5), (7), (11)-(14), and market clearing conditions t t+1 canbeusedtoderiveanidentical equationto(20)whichintandemwithaforward-lookingruleallows usto find an identical difference equation to (22). Then the cyclical and chaotic dynamics results that we derived before still hold under incomplete markets. Nevertheless, introducing incomplete markets has an interesting consequence for the behavior of the currentaccount. Toexplore this, recall thatindeterminacyof thenominal interestrateinourmodel implies real indeterminacy because of the non-separability in the utility function between consumption and money. That is, one can obtain all the remaining real endogenous variables as a functions of R . In particular t hN =hN(R ), cT = α ψ hN and hT =hT. Using these and the definition of the current account t t t 1 − αλθN t t bw bw =(Rw 1)bw + hT θT cT t − t − 1 − t − 1 t − t ¡ ¢ wecandeducethatcyclesofthenominalinterestratemaycausecyclesoftheaccumulationofforeignbonds and therefore cycles of the current account. 24

6.2 Choosing A Different Measure of Inflation: The Core Inflation In the specification of the rule, the government can react to a measure of inflation different from the CPI-inflation. For instance it can respond to the core-inflation which can be defined as either the inflation of the traded goods or the inflation of the non-traded goods. These two cases can be considered as extreme cases of cases of a rule that reacts to the full-inflation πf (cid:18)ω πN 1 − ω when the government picks the t ≡ t t weight ω [0,1].29 In the former case it picks ω =1, whereas in t¡he la¢tter case it chooses ω =0. ∈ To analyze these cases we derive a difference equation similar to (22) where the only difference is that the parameter χ is substituted by (σ 1)(1 ω)(1 γ)(1 θ ) χf − − − − N . ≡ σ[θ +α(1 θ )]+(1 α)(1 θ ) N N N − − − Using this difference equation and setting sequentially ω = 1 and ω = 0 in χf we obtain the following. Cyclical and chaotic dynamics are not present under a rule that reacts to the future traded inflation, (cid:18) t+1 (ω = 1); only liquidity traps are possible in this case. Nevertheless by reacting to the future non-traded inflation πN , forward-looking rules can still induce cycles and their existence can depend on the degree of t+1 openness of the economy. To illustrate this point we rely on simulations and using the parametrization in Table 2 we construct Figure 5. It presents the orbit-diagram for a forward-looking rule that reacts exclusively to the non-traded inflation where the relative risk aversion coefficient CRRA is σ˜ = 1.5 (or equivalently σ = 1.51). From this figure it is clear that for low degrees of openness (α), the previously mentioned rule leads the economy to the active steady state. But as the degree of openness increases the same rule can induce period-2 cycles in the economy. Hence the appearance of these cycles depend on the degree of openness. 6.3 Contemporaneous and Backward-Looking Rules In this subsection we consider varying the timing of the rule. We will study contemporaneous and backward-looking rules that still respond to the CPI inflation. 6.3.1 Contemporaneous Rules Motivated by the recent estimations by Lubik and Schorfheide (2003) for the United kingdom, Canada, Australia and New Zealand, we study the determinacy of equilibrium for rules that respond to current CPI A inflation: R t = 1+(R ∗ − 1) π π ∗ t R∗− 1 with R ∗ = π ∗ /β and R t > 1. Under these rules the equilibrium 29CPIinflation is clearly a specific¡cas¢e offullinflation forω=α. 25

Figure 5: Orbit-bifurcation diagram for the degree of openness, α. R denotes the nominal interest rate. t The diagram shows the set of limit points as a function of α, when the coefficient of relative risk aversion (CRRA) σ˜ =1.5 and when the active forward-looking rule responds exclusively to the non-traded inflation. Depending on α this rule can induce cyclical dynamics in the economy. dynamics are described by R t+1 − 1 χ R ∗ − 1 R∗ A− 1 = R t − 1 χ R ∗ (35) R R 1 R R · t+1 ¸ · t+1 − ¸ · t ¸ t whereχdefinedin(21)dependsonαandσ.Anexplicitrepresentationforeithertheforwardorthebackward dynamics of (35) is not available in this case. Although it is feasible to derive some analytical results as before, for reasons of space, we only present some numerical simulations. These are sufficient to make the point that the degree of openness still affects the appearance of complex dynamics.30 In addition in the subsequent analysis we will focus on characterizing the existence of cycles of period 2 and 3 because of the following reasons. By Sarkovskii (1964)’s Theorem the existence of period-2 cycles is a necessary (but not a sufficient) condition for the existence of cycles of any higher order, while the existence of period-3 cycles implies the existence of cycles of every possible period. Furthermore, by Li and Yorke (1975), if a map possesses a period-3 cycle then it also features topologically chaotic trajectories.31 Using the parametrization of Table 2 we construct Figure 6. This Figure presents some qualitative properties of the global dynamics of the model for contemporaneous rules for different degrees of openness while setting σ = 2.55 (when consumption and money are Edgeworth substitutes). Each row of panels in Figure6containsplotsofthemapR =f(R )implicitlydefinedinthedifferenceequation(35),itssecond t+1 t iterate R =f2(R ) and its third iterate R =f3(R ), respectively, for a given degree of openness α. t+2 t t+3 t Going from the bottom to the top row we increase openness by considering α = 0.01,0.38,0.90 . In { } 30The analyticalresultsofthispartofthe paperare available upon request. 31See Lorenz (1993)fora precise statementofthese two theorems. 26

1.04 1.03 1.02 1.01 1 1 1.01 1.02 1.03 10.0 = α 1.04 1.03 1.02 1.01 1 1 1.01 1.02 1.04 1.04 1.03 1.03 1.02 1.02 1.01 1.01 1 1 1 1.01 1.02 1.03 1 1.01 1.02 1.03 83.0 = α R = f (R) t+1 t 1.025 1.02 1.015 1.01 1.005 1 1 1.01 1.02 1.03 1.08 1.06 1.02 1.04 1.01 1.02 1 1 1 1.01 1.02 1 1.005 1.01 1.015 9.0 = α Active Contemporaneous Rules R t+2 = f 2(R t ) R t+3 = f 3(R t ) 1.025 1.025 1.02 1.02 1.015 1.015 1.01 1.01 1.005 1.005 1 1 1 1.01 1.02 1 1.005 1.01 1.015 Figure6: Thisgraphshowsthefirst,thesecondandthethirditeratesoftheimplicitmappingR =f(R ) t+1 t defined in (35) for different degrees of openness of the economy (α). From the figure it is possible to infer that depending on α, two-period cycles and three-period cycles around the active and the passive steady states may arise. 27

all the panels the straight line corresponds to the 45o degree line. The first result to notice is that for α= 0.01,0.38 , namely an almost closed and moderately open economy respectively, the second and third { } iterates, R =f2(R ) and R =f3(R ), have fixed points different from the steady state values R and t+2 t t+3 t ∗ RL, implying the existence of cycles of period-2 and 3. Then by the Sarkovskii’s (1964) Theorem and the Li and Yorke’s (1975) Theorem, f(R ) features cycles of any order, as well as aperiodic cyclical dynamics t (chaos). But there is an important difference between the dynamics for α = 0.01 and those for α = 0.38. In the former case, cycles and chaos appear around the active steady state, while in the latter they occur aroundthepassivesteadystate. Finally,forveryopeneconomies(α=0.90),nocyclesandchaoticdynamics appear at all. The monotonicity of the map f(R ) implies that liquidity traps are the only type of global t equilibrium multiplicity in this case. Given σ = 2.55 (or equivalently σ = 2.5) it is possible to find the exact numerical values of the α thresholds triggering a qualitative switch in dynamics. We find that period-2 cycles appear around the e active steady state when α (0.001,0.22); and period-3 cycles occur around the active steady state when ∈ α (0.001,0.16). Pushing α up, period-2 cycles appear around the passive steady state for α (0.25,0.43), ∈ ∈ whereas period-3 cycles (and therefore chaos) exist for α (0.33,0.38). Finally for α > 0.39 only liquidity ∈ traps exist. Overall the existence of cyclical and chaotic dynamics is robust to changing the timing of the rule from forward-looking to contemporaneous. The key difference is that now endogenous global fluctuations can only occur for the case when consumption and money are Edgeworth substitutes. Nevertheless for this case conclusions are quite different in the following sense: very open economies seem to be the less prone to endogenous cycles. Furthermore the qualitative changes induced by a variation in the degree of openness, for a fixed σ >1, can be even more dramatic under a contemporaneous rule since they might even trigger a switch from one steady state to the other as the focus of fluctuations. The following proposition summarizes these results. Proposition 5 Suppose the government follows an active contemporaneous rule. Then: 1. if consumption and money are Edgeworth complements, i.e. σ (0,1), there cannot be equilibrium ∈ cycles of any periodicity for any degree of openness α (0,1); ∈ 2. if consumption and money are Edgeworth substitutes, i.e. σ >1, (a) there cannot be equilibrium cycles of any periodicity if the economy is sufficiently open; (b) cyclical and chaotic dynamics occur around the passive steady state for intermediate degrees of openness; (c) cyclical and chaotic dynamics appear around the active steady state occur if the economy is sufficiently closed. 28

6.3.2 Backward-Looking Rules We conclude the analysis of different timings for the rule by studying backward-looking rules defined as R t ≡ 1+(R ∗ − 1) π π t − ∗ 1 R∗ A − 1 with ξ ≡ R A ∗ >1. This specification in tandem with (20) conform a system of twofirst-orderdiffe¡rence¢equationsthatcanbeusedtopursuetheglobaldeterminacyofequilibriumanalysis. Since it is very difficult to derive analytical results, we rely on simulations in order to assess whether for different values of α and σ, the system presents cycles or chaos. The simulation results show that these types of dynamics are not present regardless of the degree of openness α when σ 0.79,1,51,2.03,2.55 ∈ { } (or equivalently when the CRRA coefficient σ 0.8,1.5,2,2.5 ). The interest rate converges to either the ∈{ } active or the passive steady-state. e Nevertheless this is not sufficient evidence to conclude that backward-looking rules will preclude the existence of cyclical dynamics. In fact in Airaudo and Zanna (2005) we show that these rules may still lead to these dynamics in open economies that face nominal price rigidities. 6.4 Real money balances timing and cycles Inourset-uptherealmoneybalancesthatprovidetransactionservices,andthereforeutility,arethoseleft after leaving the goods market. This is the traditional timing adopted in money-in-utility-function models. Carlstrom and Fuerst (2001) call this timing the “Cash-When-I’m-Done” (CWID) timing and argue that is counterintuitive.32 They suggest an alternative timing: the standard “Cash-In-Advance” (CIA) timing, where the real money balances entering the agent’s utility are those left after leaving the bond market but before entering the goods market. In this Subsection we construct a simple example showing that the results presented in Section 3 for active forward-looking rules are not driven by the CWID timing. Following Woodford (2003), we introduce a CIA timing by altering the budget constraint of the representative agent as follows R 1 1 E Q W W τ t − Md+ yT +PNyN cT PNcN t t,t+1 t+1 ≤ t −E t t − R t R E t t t t −E t t − t t t t ¡ ¢ whileleavingunchangedalltheremainingelementsofthemodelpresentedinSection2. Bythesameanalysis we pursued in that section we obtain the non-linear reduced form R t+1 − 1 χ = R ∗ (R t − 1)χ+R∗ A− 1 (36) · R t Ψ +1 ¸ (R ∗− 1) R∗ A− 1 R t 1+Ψχ thatdescribestheequilibriumdynamicsoftheeconomyunderforward-lookingrules. Thedifferencebetween 32A discussion on money-in-utility timing and cycles is already present in the papers of Fukuda (1993, 1995), pointing out that,undermoneygrowthrules,non-separabilitybetweenconsumptionandrealbalancesisnotanecessaryconditionforcycles toexist(asonemighthaveconjecturedfromthepreviousworksofMatsuyama(1991)andObstfeldandRogoff(1983)). Under CIA timing cyclesoccureven underfullseparability. 29

(36) and its CWID counterpart (22) is the coefficient Ψ 1 − θN(1 − σ) , which is positive for σ (0,1) ≡ (1 − σ)(1 − γ)(1 − θN) ∈ and negative for σ >1.33 The structural parameter χ is unchanged and still defined by (21). The following Proposition suggests that under the CIA timing, cycles are still possible and their existence depends on the degree of openness α. Nevertheless this different timing implies different bifurcation thresholds for α from the ones derived in Section 3. Proposition 6 Consider the definition of χ in (21) and define Υcia 1 − R∗ A− 1 R∗− 1 . Suppose Υcia > 0 ≡ (cid:19) 2R ∗(cid:20) and that the government follows an active forward-looking rule. If consumption and money are Edgeworth substitutes, i.e. σ > 1, then there exist period-2 cycles when χ < Υcia, a condition that is satisfied for sufficiently open economies, i.e α>αˆ. Proof. See the Appendix. 7 Conclusions In this paper we show that active interest rate rules can have perverse effects in a small open economy by inducing endogenous cyclical and chaotic dynamics. Our main contribution is to show that the existence of these dynamics depends on some particular features of open economies such as the degree of openness (measured by the share of tradable goods in consumers’ preferences) and the degree of exchange rate passthrough (implied by the presence of non-traded distribution services). In our model a forward-looking rule that responds to CPI-inflation is more prone to lead to cyclical and chaotic dynamics the more open the economyandthehigherthedegreeofexchangeratepass-through. IfconsumptionandmoneyareEdgeworth complements 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 interest rate target set by the monetary authority. We also discuss how the government could design a forward-looking rule in order to rule out complex dynamics, while still preserving the Taylor principle at the target steady state. The analysis implies a “modified Taylor principle”: for a given interest rate target, the interest rate rule responsiveness to CPI inflation should satisfy some lower (upper) bound depending on whether consumption and real balances are Edgeworth complements or substitutes. The existence of cyclical and chaotic dynamics and their dependence on the degree of openness are in general robust to different inflation timings in the rule (forward-looking versus contemporaneous rules), to theuseofalternativemeasuresofinflation(CPIversusCoreinflation),aswellastochangesinthetimingof realmoneybalancesinliquidityservices(“Cash-When-I’m-Done”timingversus“Cash-In-Advance”timing). Clearly the analytical tractability of our model comes from one of its limitations: we are considering a flexible price and perfectly competitive economy. Without any sort of nominal rigidity, monetary policy 33One could think ofthe CWID setup as one where Ψ=1. 30

does not have any role for active stabilization. Nevertheless as we show in Airaudo and Zanna (2005), the existence of cycles induced by interest rate rules and the important role played by some open economy features in the characterization of the equilibrium are not precluded by introducing sticky-prices. Thispaperleavesopensomeinterestingresearchquestions. Firstasstandardintheliterature, wemodel the government as an automaton that always follows the same monetary rule. That is, it implements an active interest rate rule with a given parametrization, independently from the evolution of inflation itself and other monetary variables. One might consider the possibility of an endogenous monetary policy regime switching featuring a change in the monetary instrument (from an interest rate rule to money growth and viceversa) or a change in the responsiveness to inflation triggered by an endogenous variable monitored by the central bank. Finallywehaveimplicitlyassumedthatagentsintheeconomycancoordinatetheiractionsandlearnthe particular equilibria that we studied. Relaxing this assumption and introducing learning as in Evans and Honkapojha (2001) can be another avenue to explore in future research. A Appendix This first part of the appendix presents the statements of some Lemmata and their proofs as well as some of the proofs of the Lemmata stated in the paper. The second part presents the proofs of the main Propositions of the paper. A.1 Lemmata and Proofs A.1.1 Proof of Lemma 1 P a r n o o o n f - . pre L d e e t te φ r f m ≡ ine 1 d + var R R A i ∗ ∗ a χ − − b 1 l 1 e, b t e he th e e qu sl i o li p b e riu o m f (2 is 6) lo a c n a d lly d u efi n n iq e u Υ e d if ≡ and 1 2 ( o R n ∗ ly − if 1) φ³f 1 − > R A 1 ∗ ,´i. > e. 0 t . he Si l n in ce ea R ri t ze i d s mapping (26) is explosive and therefore the target steady state is the unique bo¯und¯ed PFE. ¯ ¯ First suppose that χ<0. This together with Assumption 0 ( A >1) and th¯e z¯ero lower bound R >1, R t ∗ implies that φf >1. Hence the equilibrium is locally unique. Now suppose that χ>0. This in tandem with Assumption0andthezerolowerboundleadstoφf <1,meaningthatinordertohaveauniqueequilibrium we need φf < 1. Simple algebra shows that χ < Υd implies φf < 1. Then the map is explosive and the − − equilibrium is unique. Ontheotherhand,itissimpletoshowthatifχ>Υdthen 1<φf <1. Hencethemapisnon-explosive, − i.e. from any initial condition R off the target steady state R , R will eventually converge to R . This 0 ∗ t ∗ continuity of PFE paths is the source of local multiple equilibria. A.1.2 Proof of Lemma 2 Proof. From point 2 of Lemma 6, the assumption χ (Υw,0) implies that the mapping f(.) satisfies ∈ the zero-lower-bound condition only for R Rl,Ru (1,+ ). Moreover within Rl,Ru the mapping t ∈ ⊂ ∞ ¡ ¢ ¡ ¢ 31

f(.) looks like an inverted logistic mapping with a minimum at RJ and f (RL) < 0 (see the right panel of 0 Figure 1). With these properties of f(.), the proof of Point 1 follows from the fact that f (R )<0 for any 0 t R Rl,RJ . t ∈ To¡prove¢Point 2, take any R t [R,R ∗ ] whose first iterate is R t+1 = 1 J(R t )χ(α 1 ,σ) − 1 . If R t ∈ − ∈ R,RJ , then R = f(R ) > f RJ R since f (R ) < 0 for any R <hRJ; moreover Ri < R if t+1 t 0 t t t+1 ∗ ≡ f R.Similarly,ifR (RJ,R ]thenR =f(R )<f(R )=R sincef (R )>0forR >RJ,aswell £min ≥¤ t ∈ ∗¡ ¢ t+1 t ∗ ∗ 0 t t as R =f(R )>f RJ =R. Hence we have shown that for any R [R,R ], R =f(R ) [R,R ]. t+1 t t ∗ t+1 t ∗ e ∈ ∈ Then f : [R,R ] [R,R ]. The attractive property of the set is straightforward to show and therefore ∗ → ¡ ¢∗ omitted. Point3isbasedonthefactthatasufficientconditionfortheexistenceofperiod-2cyclesisf (RL)< 1 0 − which implies an unstable passive steady state. To show this we define an auxiliary function g(R ) t ≡ R f2(R ) such that period-2 cycles are the zeros of g(R ). The assumption f f RJ R requires a t t t min − ≡ ≥ distinction between the two cases: a) f RJ >R and b) f RJ =R. ¡ ¢ e Incasea), themap-invariantsetis[R,R ].Clearlyg RL =g(R )=0andg(R) 0.Also,bythechain rule,g (Rss)=1 [f (Rss)]2,withRss b ¡ ein ∗¢ geith e eroneofth ¡ etw ¢ oste ∗ e adystates. But ≤ thenf (R )= A >1 0 − 0 ¡ ¢ 0 ∗ R ∗ implies that g (R )<0. What is ambiguous is the sign of g RJ . If g RJ =0 then the period-2 cycle is 0 ∗ obviously RJ,R .Ifg RJ <0,bycontinuityoff(R )on[R,R ],thenthereexistsapointRc RJ,R t ¡ ¢∗ ¡ ¢ ∈ ∗ such that g(Rc) = 0. Since the set [R,R ] is map invariant the second point of the period-2 cycle belongs © ª ¡ ¢ ∗ ¡ ¢ to the same set.34 If instead g RJ >0, nothing guarantees that the function g(.) has zeros other than the two steady states. A sufficient condition for having another zero is that g RL = 1 f RL 2 < 0, i.e. ¡ ¢ 0 − 0 f RL < 1. If this holds, then g(R)=0 at some point between RL and RJ. From the definition of f(.) it 0 − ¡ ¢ £ ¡ ¢¤ is s ¡ imp ¢ le to show that f 0 (RL)< − 1 is equivalent to χ> 1 2 Υw where Υw ≡ RL 1 − R∗ A− 1 − 1. Since we are considering χ > Υw, in case a) period-2 cycles around the passive steady stat³e occur wh´en χ 1Υw,0 . ∈ 2 In case b) all conditions of case a) hold, although in this case g RJ < 0 is always true. Therefore by a ¡ ¢ similar argument there are always period-2 cycles. ¡ ¢ A.1.3 Proof of Lemma 3 Proof. Let Υk 1 R∗ (R 1). The restriction χ 0,Υk implies, by Point 1 of Lemma 6, that the ≡ − A ∗ − ∈ mapping f(.) look³s like a´logistic map with a maximum ¡ at RJ, ¢ f 0 (RL) > 1 and f 0 (R ∗ ) < 0 (see the left panel of Figure 1). The proofs of Points 1 and 2 are similar to their counterparts in Lemma 2, so both are omitted. Point 3 involves searching for a flip bifurcation at the active steady state. Let’s define an auxiliary functiong(R ) R f2(R ).Period2cyclesarethenzerosofg(R ).Theassumptionf f RJ R t t t t max ≡ − ≡ ≤ requires a distinction between the two cases: f RJ =R and f RJ <R. ¡ ¢ e If f RJ < R, the set invariant under mapping f(.) is RL,R . Clearly g RL = g(R ) = 0,and ¡ ¢ ¡ ¢ ∗ e e g 0 RL ¡ <0, ¢ but g RJ R0. If g RJ =0, a period-2 cycle is, £ by con ¤ struction, R ¡ J,R ¢ . If g RJ >0, by e continuity of the function g(.) there must exist a point Rc RL,RJ such that g(Rc)=0. As Rc belongs ¡ ¢ ¡ ¢ ¡ ¢ ∈ © ª ¡ ¢ to the map invariant set RL,R , f2(Rc) RL,R as well: a period-2 cycle exists. If instead g RJ <0, ∈ ¡ ¢ by continuity, a sufficient condition for period-2 cycles is that g (R )=1 [f (R )]2 <0, i.e. f (R )< 1 £ ¤ £ ¤ 0 ∗ − 0 ∗ 0 ¡ ∗ ¢ − 34We could actually show that this second point belong to the intervak R,RL ,meaning that the equilibrium path jumps deterministically from the leftto the rightneighborhood ofthe passive steady state. (cid:3) (cid:4) 32

given that over these parametric ranges f (R ) < 0 always. Using the definition of f(.) in (27) and by 0 ∗ simple differentiation, this is in fact equivalent to χ< Υk. 2 The case of f RJ = R is trivial. All the properties of the previous case hold, but now g RJ = RJ RL >0 always. So by similar arguments, a period-2 cycle always exists. − ¡ ¢ e ¡ ¢ A.1.4 Lemma 4 Lemma 4 Keep γ and θ constant and let χ(α,σ), χ , χ and µ(σ) be defined as in (21), (24) and N max min 1 Υi (25). Consider any real number Υi (0,χ ) and define a function αi(σ) −µ(σ) for σ >1. ∈ max ≡ (cid:107)1 Υi (cid:108) −χmin (cid:107) (cid:108) 1. Over the domain (1,+ ), the function αi(σ) satisfies the following properties: ∞ (a) it is continuously differentiable, strictly increasing and strictly concave; 1 (b) lim αi(σ)= and lim αi(σ)= 1 Υi − 1 Υi (0,1) σ 1+ −∞ σ + − χ min − χ max ∈ → → ∞ h i h i 1 Υi (c) αi(σ)T0 for σ Tσi where σi −χmin >1; ≡ (cid:107)1 Υi (cid:108) −χmax (cid:107) (cid:108) 2. For any given σ >1, χ(α,σ)TΥi if and only if αSαi(σ). 3. For α (0,1) and σ >1 we have that χ(α,σ)<Υi if and only if α>max 0,αi(σ) . ∈ © ª Proof. Firstofallrecallthedefinitionsofχ(α,σ),χ ,χ andµ(σ)in(21),(24)and(25). Thentake max min any real number Υi (0,χ ) and any σ >1. If we solve explicitly the equation χ(α,σ)=Υi with respect ∈ max 1 to α, for a given σ, after some simple algebra, the solution is a function αi(σ) 1 Υi − 1 Υi . ≡ − χ min − µ(σ) We first prove point 1 for σ (1,+ ). h i h i ∈ ∞ The statement that αi(σ) is continuously differentiable, strictly increasing and strictly concave over the domain (1,+ ), follows from its definition, χ (0,+ ), χ ( 1,0), Υi (0,χ ), and the fact ∞ max ∈ ∞ min ∈ − ∈ max that for σ > 1, the function µ(σ), defined in (25) satisfies the following. µ : (1, ) R, it is continuously ∞ → differentiable, strictly increasing and strictly concave with respect to σ and satisfies lim µ(σ) = 0 and σ 1+ lim µ(σ)=χ (0,+ ). This proves Point 1(a). → σ + max ∈ ∞ → G ∞ iven that lim µ(σ) = 0 and lim µ(σ) = χ then for Υi (0,χ ) and χ ( 1,0) we have σ 1+ σ + max ∈ max min ∈ − → → ∞ 1 that lim αi(σ) = and lim αi(σ) = 1 Υi − 1 Υi . The assumption of Υi (0,χ ) σ 1+ −∞ σ + − χ min − χ max ∈ max togethe → r with χ ( 1,0) gu → ara ∞ ntees that h 1 Υi i − 1h 1 Υi i (0,1). This proves Point 1(b). min ∈ − − χ min − χ max ∈ h i h i 1 Υi Point 1(c) follows from solving µ(σ) = Υi with respect to σ and obtaining σi = −χmin . As Υi (cid:107)1 Υi (cid:108) ∈ −χmax (0,χ ) and χ ( 1,0), it must be that σi >1. Using the definition of σi and the p(cid:107)revious(cid:108)two points max min ∈ − 1(a) and 1(b) then the rest of 1(c) follows. Sinceforσ >1andα (0,1),thefunctionχ:(0,1) (1, ) Riscontinuouslydifferentiable, strictly ∈ × ∞ → decreasing with respect to α and satisfies χ(α,σ) > 0 then it follows that χ(α,σ) T Υi if and only if αSαi(σ). This proves Point 2. Toprove Point 3 we proceedas follows. Frompoint2 of thepresent Lemma, we knowthat χ(α,σ)<Υi if and only if α>αi(σ). From point 1(c) of the same Lemma we have also that αi(σ)T0 for σ Tσi. Since 33

α (0,1) then χ(α,σ) < Υi for any α (0,1) when σ σi, as in this case αi(σ) 0; while χ(α,σ) < Υi ∈ ∈ ≤ ≤ only for α > αi(σ) > 0 if σ > σi. If we write this compactly, we have that χ(α,σ) < Υi if and only if α>max 0,αi(σ) . © ª A.1.5 Lemma 5 Lemma 5 Keep γ and θ constant and let χ(α,σ), χ , χ and µ(σ) be defined as in (21), (24) and N max min 1 Υi (25). Consider any real number Υi (χ ,0) and define a function αi(σ) −µ(σ) for σ (0,1). ∈ min ≡ (cid:107)1 Υi (cid:108) ∈ −χmin (cid:107) (cid:108) 1. Over the domain (0,1), the function αi(σ) satisfies the following properties: (a) it is continuously differentiable, strictly decreasing and strictly concave; (b) limαi(σ)=1 and lim αi(σ)= ; σ 0 σ 1 −∞ → → − 1 Υi (c) αi(σ)T0 for σ Sσi where σi −χmin (0,1). ≡ (cid:107)1 Υi (cid:108) ∈ −χmax (cid:107) (cid:108) 2. For any given σ (0,1), χ(α,σ)RΥi if and only if αTαi(σ). ∈ 3. For α (0,1) and σ (0,1) we have that χ(α,σ)>Υi if and only if α>max 0,αi(σ) . ∈ ∈ © ª Proof. The proof is omitted since it is very similar to the proof of Lemma 4. A.1.6 Lemma 6 Lemma 6 Define the scalars Υp ≡ R∗ A− 1 > 0, Υk ≡ 1 − R A ∗ (R ∗ − 1) > 0 and RJ ≡ 1 1+ Υ χ p . Recall the − definition of f(R t ) in (27)-(28) for R t (1,+ ). ³ ´ ∈ ∞ 1. Suppose that χ > 0. Then a) f : (1,+ ) (1,+ ) and it is continuously differentiable; b) f(.) has ∞ → ∞ a global maximum at RJ with RJ > RL; c) lim f(R ) = lim f(R ) = 0; d) f RL > 1 always, t t 0 Rt→ 1+ Rt→∞ while f 0 (R ∗ )T0 depending on χTΥk. ¡ ¢ 2. Suppose that χ ( Υp,0). Let Rl and Ru be the solutions to J(R ) = 1 with Rl 1,RL and t ∈ − ∈ Ru > R . Then a) f : Rl,Ru (1,+ ) and it is continuously differentiable ; b) within Rl,Ru , ∗ → ∞ ¡ ¢ f(.) has a unique minimum at RJ; c) lim f(R ) = lim f(R ) = + ; d) f (R ) > 1 always, ¡ ¢ Rt→ Rl+ t Rt→ Ru − t ∞ 0 ∗ ¡ ¢ while f 0 RL T0 depending on χSΥw with Υw RL(1 Υp) 1 ( Υp,0). ≡ − − ∈ − ¡ ¢ Proof. Inordertosimplifythenotationintheproofandthestatements,wedefinethefollowingscalars: Υp ≡ R∗ A− 1, Υw ≡ RL(1 − Υp) − 1 and Υk ≡ 1 − R A ∗ (R ∗ − 1). From the zero-lower-bound restriction and the assumption of an active rule, it is clear th³at Υp >´0 and Υk >0. Recallthedefinitionoff(R )in(27). From(28),J(R )isalwayspositiveandcontinuouslydifferentiable t t t fo h r en any R t ∈ (1,+ ∞ ) and any χ ∈ R. Furthermore, since [R ∗− R 1] ∗ R∗A− 1 > 1 and Υp ∈ (0,1) (as A > R ∗ > 1) lim J(R t )=0 for any χ R (37) Rt→ + ∞ ∈ 34

and 0 if χ> Υp lim J(R )= − (38) Rt→ 1+ t ( + ∞ if χ< − Υp From (29) we also observe that J 0 (R t ) T 0 if and only if R t S 1 1+ Υ χ p , as well as that 1 1+ Υ χ p T 1 if and only if χ+Υp T 0. As a consequence of these inequalities, when χ+ − Υp < 0 we have 1 1+ Υ χ p − < 1, and therefore − J (R )<0 for any R >1. 0 t t Next we prove point 1. Suppose that χ > 0. As χ+Υp > 0 then from (29) and (37) we infer that J(R ) is single-peaked at RJ 1+χ > 1, with lim J(R ) = lim J(R ) = 0. An inspection of (27) show t s that for χ>0, f(R )> ≡ 1 a 1 n − d Υp it is continuo R u t s→o 1 v + er the t dom R a t in → + (1 ∞ ,+ ) t if and only if J(R ) (0,1). t t ∞ ∈ But J(R ) > 0 and with a maximum at RJ when χ > 0. Then since J RJ < 1 holds, we have that t J(R ) (0,1) for any R (1,+ ).35 This in turn implies that f : (1,+ ) (1,+ ). Furthermore, t ∈ t ∈ ∞ ¡∞¢→ ∞ f(.) is continuously differentiable since J(.) is continuously differentiable as well. This proves Point 1(a). Point 1(b) follows from (29) together with χ > 0; while Point 1(c) is obtained from the definition of f(.) in (27) and (37)-(38). The complete characterization of f(.) for χ > 0 is achieved by noticing that being continuousandcrossingthe45degreelinetwicewithinitsdomain(seeProposition1)itmustnecessarilybe that RJ >RL. But then f(.) must cross the 45 degree line at the low steady state RL with a slope bigger than1. Thatisf RL >1.36 Neverthelessthesignof f (R )remainsambiguous becausesign f (R ) = 0 0 ∗ 0 ∗ { } sign { J 0 (R ∗ ) } =si ¡ gn ¢ RJ − R ∗ and RJ TR ∗ . Simple algebra shows that this last inequality is equivalent to χTΥk. This prove © s Point 1( ª d). Supposethatχ ( Υp,0)thenthefunctionJ(.)hasthesamepropertieswederivedatthebeginningof ∈ − theproofforthecaseofχ>0.ThusJ(.)issingle-peakedatRJ >1,with lim J(R )= lim J(R )=0. t t HoweverthefactthatRL <R (seeProposition1)combinedwithχ<0im R p t l → y 1 t + hatalthou R g t h → R + J ∞ RL,R ∗ ∗ ∈ we now have that J RL >J(R )>1. As RJ is a global maximum, J RJ >1 as well. These properties ∗ ¡ ¢ together with lim J(R ) = lim J(R ) = 0 imply that there exist two values, Rl and Ru, such that ¡ ¢t t ¡ ¢ J Rl = J(R R u) t→ = 1+ 1 where R R l t→ + ∞ 1,RL and Ru > R 37 Given the definition (27) of f (.) we can see ∗ ∈ that it is not well-defined at both Rl and Ru. Moreover, as lim f(R ) = lim f(R ) = and f( ¡ R ) ¢ < 0 for any R > Ru and an ¡ y R ¢ 1,Rl , then any in R it t i → al R c l o − nditi t on R Rt / → R R u+ l,Ru t would −∞ violate t t t 0 ∈ ∈ the zero-lower-bound condition and therefore cannot be part of a PFE of our model. The range R ¡ ¢ ¡ ¢ t ∈ Rl,Ru is therefore necessary for an iterated sequence to be supported as an equilibrium. Within the i ¡ nterval ¢ Rl,Ru , J(.) satisfies J(R t ) > 1 and is continuously differentiable, implying J(R t )χ 1 ∈ (0,1) for any χ < 0. It follows that f(R ) > 1 and f(.) is continuously differentiable for any R Rl,Ru . ¡ ¢ t t ∈ This proves Point 2(a). Point 2(b) follows from (29) together with χ < 0, while Point 2(c) is obtained ¡ ¢ from the definition of f(.) in (27) and (37)-(38). Since f(.) crosses the 45 degree line at both RL and R (see Proposition 1) it must be that f (R ) > 1. Nevertheless the sign of f RL remains ambiguous ∗ 0 ∗ 0 as sign f 0 RL = − sign J 0 RL = sign RL − RJ and RL T RJ. Simple ¡ alg ¢ ebra shows that this last ineq © ual ¡ ity i ¢ s ª equivalent © to χ ¡ S¢ R ª L(1 − Υp © ) − 1. Fin ª ally Observe that an active rule implies − Υp < RL(1 Υp) 1=Υw and that, from Assumption 3, RL 1 <RL 1<Υp. Then RL(1 Υp) 1<0 such − − RL− − − − 35Proving that J RJ <1involessome algebra. Itisavailable from us upon request. 3 3 6 7 S T i h m is pl s e ho a u lg l e d b b ra e (cid:3) a cl l e so ar (cid:4) p f r r o o v m es d t r h w a a t in R l g t im →th 1 e f 0 fu (R nc t t ) io ∈ n (0 J , ( 1 . ) ) . ,which is continous,hump-shaped,with maximum biggerthan 1 and limitsequalto zero. 35

that Υw < Υf < 0. Using this and Υp < RL(1 Υp) 1 = Υw we conclude that Υw ( Υp,0). This − − − ∈ − completes the proof for Point 2(d). For χ Υp < 0, it can be shown that the mapping f(.) features no critical point and it is strictly ≤ − increasing. A.2 Proofs of the Propositions A.2.1 Proof of Proposition 1 Proof. Let the left hand side and the right hand side of equation (23) be denoted as LHS(Rss) and RHS(Rss) respectively. Since R >1 we have that ∗ lim LHS(Rss)=(R ∗ 1) R∗ A− 1 >0 and lim RHS(Rss)=0. Rss 1 − Rss 1 → → ItissimpletoverifythatgivenR ∗ >1andRss >1thenLHS(Rss)islinearinRsswithslope(R ∗ 1) R∗ A− 1 > − 0 and RHS(Rss) slopes upwards as well, ∂RH ∂R S s ( s Rss) =R ∗ R ∗ A − 1 (Rss − 1) R∗ A− 1 − 1 >0. Moreover A >1, R >1 and Rss >1 are sufficient conditions for RHS(RSS) to be strictly concave. That R ∗ ∗ is ∂2R ∂( H R S ss (R )2 ss) =R ∗ R∗ A− 1(Rss − 1) R∗ A− 1 − 2 R∗ A− 1 − 1 <0. Given that LHS(Rss) is continuous an³d linear an´d RHS(Rss) is continuous and strictly concave for any Rss > 1, a sufficient condition for a second solution Rss = RL (1,R ) to exist is that the slope of the ∗ ∈ RHS(Rss) at R ∗ is smaller than the slope of LHS(Rss) at R ∗ . In other words, R ∗ R∗ A− 1(R ∗ − 1) R∗ A− 1 − 1 < (R ∗ − 1) R∗ A− 1 , or equivalently R A ∗ >1. Moreover since RHS(Rss) is strictly concave then no other solution to (23) different from R and RL exists. ∗ A.2.2 Proof of Proposition 2 Proof. Recall the definition of χ(α,σ) in (21). For σ (0,1) and α (0,1) we have that χ : (0,1) ∈ ∈ × (0,1) R. Then Point 1(a) follows from point 1 of Lemma 1 and the fact that χ(α,σ)<0 if σ (0,1) and → ∈ α (0,1). ∈ Next we prove point 1(b). For σ (1, ) and α (0,1) we have that χ : (0,1) (1, ) R. Define ∈ ∞ ∈ × ∞ → Υd 1(R 1) 1 R∗ .Theproofofpoint1(a)ofLemma1makesclearthatχ=Υdisthethresholdvalue ≡ 2 ∗ − − A ofχthatdifferen³tiatesb´etweenactiverulesleadingtoeitherauniqueequilibriumormultipleequilibria. Since we are interested in how the pair (α,σ) affects local equilibrium determinacy, we look for values of α that, 1 forgivenσ,solveχ(α,σ)=Υd.Simplealgebrashowsthatthissolutionisαd(σ)= 1 Υd − 1 Υd , − χ min − µ(σ) a function relating α to σ. Since Assumption 1 holds, that is Υd (0,χ ), thenhwe can uise Lhemma 4 ito ∈ max deduce that the function αd(σ) is equivalent to the function αi(σ), defined in that Lemma, for Υi = Υd. Therefore, αd(σ) inherits all the properties of αi(σ). FromPoint3ofLemma4,wecanconcludethatforanyσ >1thenα>αd withαd max 0,αd(σ) . min min ≡ This is a necessary and sufficient condition for χ(α,σ) < Υd to hold. Which, by Lemma 1, implies local © ª equilibrium determinacy. 36

Toprovepoint2wefollowthesamestepsofproving1(b)takingintoaccountthatfrompoint2ofLemma 4 we can deduce that σ > σd and α (0,αd(σ)) leads to χ(α,σ) > Υd. Which by point 2 of Lemma 1 ∈ implies multiple equilibria. A.2.3 Proof of Proposition 3 Proof. The proof of Point 1 combines the results of Point 3 in Lemma 2 with Lemma 5. DefineΥp ≡ R∗ A− 1,Υw ≡ RL(1 − Υp) − 1andΥf ≡ 1 2 Υw.FromPoint3ofLemma2, theflipbifurcation thresholdisχ=Υf.Sinceχisafunctionofαandσ thenthisthresholdcanbeexpressedinthe(α,σ)-space as the solution to χ(α,σ) = Υf with respect to α, for any arbitrary σ (0,1). Simple algebra shows that 1 ∈ the solution is a function αf(σ) 1 Υf − 1 Υf . Observe that an active rule implies Υp < ≡ − χ min − µ(σ) − RL(1 Υp) 1=Υw andthat,fromhAssumptiionh3, RL 1 <i RL 1<Υp.ThenRL(1 Υp) 1<0which − − RL− − − − implies that Υw <Υf <0. Using this and Υp <RL(1 Υp) 1=Υw we conclude that Υw ( Υp,0). − − − ∈ − From this and Assumption 2 we can see that Υf (χ ,0) satisfying the assumptions in Lemma 5. Hence ∈ min αf(σ) shares all the properties of the function αi(σ) for Υi = Υf stated in that Lemma. By Point 3 of Lemma 5, for an arbitrary σ (0,1), the flip bifurcation condition χ(α,σ) > Υf in Lemma 2 is therefore ∈ equivalent to α > αf where αf max 0,αf(σ) . From an application of Point 1 of Lemma 5 we can min min ≡ also see that such minimum degree of openness, αf , is positive and decreasing in σ, for σ 0,σf , but © miªn ∈ constant and equal to zero for σ σf,1 . ∈ ¡ ¢ The proof of Point 2 is omitted since it is very similar to the proof of Point 1. But it uses the fact that £ ¢ σ >1 and Point 3 of Lemma 3 together with Lemma 4 instead. A.2.4 Proof of Proposition 6 Proof. We provide a sketch of the proof. Let K(R ) and J(R ) be the left and the right hand sides of t+1 t (36) respectively. First of all notice that for σ > 1, we have Ψ<0 and χ> 0. To simplify the notation let Γ (1 Ψ)χ. ≡ − Then consider the function K(.). Since Γ > 0, it is simple to show that lim K(R ) = 0 and t+1 Rt+1→ 1+ lim K(R )=+ . Furthermore simple algebra shows that K(.) is strictly increasing over the entire t+1 Rt+1→ + ∞ ∞ domain (1,+ ). But then K(.) is also globally invertible such that there exist a well defined function ∞ f(R )=K 1(J(R )) describing the forward equilibrium dynamics. t − t Then consider the function J(R t ). Since χ+ R∗ A− 1 >0 for any α ∈ (0,1), then Rt li → m 1+ J(R t )=0, while: lim J(R )= + ∞ if Γ> 1 − R∗ A− 1 t Rt→ + ∞   0 if Γ< ³1 − R∗ A− 1´ ³ ´  Using the definition of Γ, let’s consider the case of χ< 1 − R∗ A− 1 such that lim J(R )=0. It is possible (cid:19) (1 − Ψ) (cid:20) Rt→ + ∞ t to show that J(.) has a critical point at RJ = 1+Ψχ >1. Therefore if χ< 1 − R∗ A− 1 the function J(.) (1 − R∗ A− 1) (cid:19) (1 − Ψ) (cid:20) issingle-peaked. ThisinturnintandemwiththefactthatK(.)isstrictlyincreasingovertheentiredomain (1,+ )impliesthattheequilibriummappingf(.)lookslikealogisticmap. Thenifcyclesexisttheyhaveto ∞ becenteredaroundtheactivesteadystate. Toshowthiswecanuseasimilarargumenttotheonedeveloped 37

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