Fighting Against Currency Depreciation, Macroeconomic Instability and Sudden Stops

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 848 December 2005 Fighting Against Currency Depreciation, Macroeconomic Instability and Sudden Stops 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publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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/.

Fighting Against Currency Depreciation, Macroeconomic Instability and Sudden Stops ∗ Luis-Felipe Zanna † First Draft: October 2002 This Draft: November 2005 Abstract Inthispaperweshowthatintheaftermathofacrisis,agovernmentthatchangesthenominalinterest rate in response to currency depreciation can induce aggregate instability in the economy by generating self-fulfilling endogenous cycles. In particular if a government raises the interest rate proportionally more than an increase in currency depreciation then it induces self-fulfilling cyclical equilibria that are able to replicate some of the empirical regularities of emerging market crises. We construct an equilibriumcharacterizedbytheself-validationofpeople’sexpectationsaboutcurrencydepreciationand by the following stylized facts of the “Sudden Stop” phenomenon: a decline in domestic production and aggregate demand, a significantly larger currency depreciation, a collapse in asset prices, a sharp correction in the price of traded goods relative to non-traded goods and an improvement in the current account deficit. Keywords: Small Open Economy, Interest Rate Rules, Currency Depreciation, Multiple Equilibria, Sudden Stops, Collateral Constraints. JEL Classifications: E32, E52, E58, F41. ThispaperisanupdatedversionofChapter3ofmydissertationattheUniversityofPennsylvania. IamgratefultoMartín ∗ Uribe, Stephanie Schmitt-Grohé and Frank Schorfheide for their guidance and teaching. I also received helpful suggestions and benefitted from conversations with Roc Armenter, Martin Bodenstein, David Bowman, Bora Durdu, Chris Gust, Dale Henderson, Sylvain Leduc, Gustavo Suarez and seminar participants at the International Economics Brown Bag Lunch at the University of Pennsylvania, the International Finance Workshop at the Federal Reserve Board and the fall 2005 SCIEA Meetings. Allerrorsremainmine. Previousversionsofthispapercirculatedunderthetitle: “InterestRateRulesandMultiple Equilibria in the Aftermath of BOP Crises”. The views expressed in this paper are solely the responsibility of the author and shouldnotbeinterpretedasreflectingtheviewoftheBoardofGovernorsoftheFederalReserveSystemorofanyotherperson associated to the FederalReserve System. Board of Governors of the Federal Reserve System, 20th Street and Constitution Avenue, NW, Washington, D.C., 20551. † Tel.: (202)452-2337. Fax: (202)736-5638. Email: Luis-Felipe.Zanna@frb.gov.

1 Introduction One of the most controversial issues that emerged with the Asian Crisis of 1997 was the appropriate interest rate policy to fight against currency depreciation. There was a debate between two opposite views. On one hand, the IMF advocated for higher interest rates to prevent excessive currency depreciation. They claimedthatthispolicycouldreducecapitaloutflowsbyraisingthecostofcurrencyspeculation,andinduce capital inflows by making domestic assets more attractive. This would restore the confidence in domestic currencies and stop their accelerated depreciation.1 On the other hand, some critics of the IMF policy prescription argued that this policy would exacerbate the depreciation process.2 They argued that raising interest rates could aggravate the recession that the Asian economies were sliding into and weaken significantly the balance sheets of the banking and corporate sectors. This in turn would generate expectations of future financial crises, external debt defaults, and currency depreciations. Hence the critics recommended to lower interest rates. Althoughtheseviewsprescribedoppositepolicyrecommendations,bothviewshadsomethingincommon: to some extent they conceived the interest rate policy as a reaction function. They advocated for changes in the nominal interest rate “in response to” some macroeconomic indicators such as currency depreciation. In fact some of the theoretical and empirical works motivated by the debate have modelled, implicitly or explicitly, the interest rate policy as a feedback rule responding to some measure of nominal depreciation.3 After all this oversimplified reaction function captures both the concern of the Asian economies about currencyundervaluationandtheuseoftheinterestrateasanexclusiveinstrumenttofightagainstcurrency depreciation. Littleisknownaboutthemacroeconomicconsequencesoftheseinterestratefeedbackpoliciesincountries that have been hit by a crisis. In this paper we study some of the possible consequences. Our main result is that in the aftermath of a crisis, an interest rate policy that calls for changing the nominal interest rate in responsetocurrencydepreciationcaninduceaggregateinstabilityintheeconomybygeneratingself-fulfilling endogenous cycles. In other words, this policy can cause cycles in the economy that are driven by people’s self-fulfilling expectations and not by fundamentals. Surprisinglythisresultholdsbothwhenagovernmentraisesorwhenitlowerstheinterestrateinresponse toanincreaseinthenominaldepreciationrate. Inbothcasesthispolicyinducesacontinuumofself-fulfilling cyclical equilibria. But if a government raises the interest rate proportionally more than the increase in currency depreciation then it induces self-fulfilling cyclical equilibria that are able to replicate some of the empirical regularities of emerging market crises. In particular we construct an equilibrium characterized by the self-validation of people’s expectations about currency depreciation and by the following stylized facts 1See Stanley Fischer(1998)among others. 2See forinstance Furman and Stiglitz (1998)and Radeletand Sachs(1998)among others. 3See forinstance Cho and West(2001),Goldfajn and Baig (1998),and Lahiriand Vegh (2003),among others. 1

labelledbyCalvo(1998)asthe“SuddenStop”phenomenon: adeclineindomesticproductionandaggregate demand,asignificantlylargercurrencydepreciation,acollapseinassetprices,asharpcorrectionintheprice of traded goods relative to non-traded goods and an improvement in the current account deficit.4 We derive these results in the context of a typical small open economy model with incomplete financial markets and traded and non-traded goods. We augment this model by adding some features that have been proved to be useful in explaining some of the stylized facts of the aftermath of a crisis. First, in accord with Burnstein, Eichenbaum and Rebelo (2005a,b) we introduce slow adjustment in the price of the non-traded goodandnon-tradeddistributionservicesforthetradedgood. Thesecharacteristicsarecrucialinexplaining the large movements in real exchange rates after large devaluations. Second, following Christiano, Gust and Roldos (2004) we assume that firms require working capital to hire labor and international working capital topurchaseanimportedintermediateinput. Thesefeaturesbecomeimportanttoobtainadeclineinoutput wheninterest rates rise inthe midst of a crisis.5 And thirdas inthe newliterature about currencycrises we introduceacollateralconstraint: internationalloansmustbeguaranteedbyphysicalassetssuchascapital.6 This provides us with the following definition of a crisis. A crisis is a time when the constraint is binding. It is possible to provide a basic explanation of why the previously mentioned policy rule can induce selffulfilling fluctuations, although non-cyclical, in a sticky-price economy that in “good times” does not face a collateralconstraint. Intheory, theUncoveredInterestParity(UIP)conditiontogetherwiththepolicyrule, that links the nominal interest rate to the current depreciation rate, determine the dynamics of the nominal depreciation rate and the nominal interest rate.7 Both are determined independently of the dynamics of other nominal and real variables of the economy. As a consequence of this we can construct the following self-fulfilling equilibrium. Suppose that in response to a sunspot, agents in the economy expect a higher non-traded goods inflation rate. Since the monetary authority does not react to these expectations then the realinterestratemeasuredintermsofthenon-tradedinflationcandecline. Inresponse,householdsincrease desired consumption of non-traded goods which leads firms to rise their prices. But by doing this, firms end validating the original non-traded inflation expectations. Although appealing this intuition is incomplete unless we show that all the equilibrium conditions of the economy in “good times” are satisfied on the entire equilibrium path. We accomplish this goal in Section 2 of the paper. We consider a simple model that abstracts from the collateral constraint, distribution services and loan requirements but considers price stickiness for non-traded goods. The importance of considering this simple set-up is that, under some extra assumptions about the household’s utility function, it allows us 4Rigorously Calvo (1998)refersto a “reversal” ofthe currentaccountdeficit. 5See also Lahiriand Vegh (2002). 6Thisideaofmodellingthecrisisasanunxepectedbindingcollateralconstraintcapturestheessenceofthe“SuddenStops.” For works that introduce collateral constraints see Braggion, Christiano and Roldos (2005), Caballero and Krishnamurthy (2001),Christiano,Gustand Roldos (2004),Krugman (1999),Mendoza and Smith (2002),and Paasche (2001)among others. 7As we will see in Section 3, for our results it is not necessary to assume that the UIP condition holds. Here we assume it to highlightthe basic intuition. 2

to derive analytical results. We can show that interest rate policies that respond to currency depreciation rates can induce real indeterminacy or multiple equilibria in the economy.8 These analytical results are useful for two reasons. First and in contrast to the explanation provided above, we can use them to construct self-fulfilling equilibria that are based on expectations of a different variablefromthenon-tradedinflationsuchascurrencydepreciation. Secondtheyareusefulinmakingclear the main features of the model that explain the presence of real indeterminacy. They are the following: the interestratepolicy,thepresenceofprice-stickinessfornon-tradedgoodsandtheexclusivedependenceofthe policyoncurrencydepreciation. InSection2 weelaborate ontheir interactionandtheirroles inourresults. Onceweintroduceabindingcollateralconstraint,distributionservicesandtheneedofworkingcapitalto hire productive factors, it is not longer possible to characterize analytically the equilibrium of the economy. We have to rely on numerical simulations for a sensible calibrated version of this augmented model. Neverthelesstheself-fulfillingequilibriummechanismexplainedbeforeisstillattheheartoftheresultsfromthese simulations. As we show in Section 3 of the paper the simulations confirm the results of the simple model: interestratepoliciesthatrespondtocurrencydepreciationratesarepronetoinducemultipleequilibria. But in this case, as mentioned before, these equilibria are cyclical because of the binding collateral constraint. This is not surprising since the seminal work by Kiyotaki and Moore (1997) shows that a binding collateral constraintinducescreditcyclesthatmayamplifybusinesscyclesinaneconomy. Inourworkthisconstraint in tandem with the previously mentioned policy rule and the presence of price stickiness is what leads to “endogenous cycles” or multiple cyclical equilibria. A reader that is familiarized with the interest rate rules literature in closed and open economies may find an interesting connection between our results and this literature.9 This connection poses the question of whether a rule that reacts aggressively and positively to past CPI-inflation and aggressively or timidly to currentdepreciationwillprecludethepreviousmultipleequilibriaresults. Afteralltheinterestrateliterature claims that aggressive rules to past inflation are more likely to guarantee a unique equilibrium. In Section 3 we show that even in the augmented model such a policy can still induce real indeterminacy as long as the rule responds positively or negatively to current depreciation. Therefore it is the response to depreciation what opens the possibility of real indeterminacy. Webelieveourresultsarenovelandimportantbecauseoftheirimplications. Firsttheyprovideapossible explanation of why the empirical literature has not been able to obtain conclusive evidence about whether higher interest rates can cause nominal exchange depreciation or instead appreciation in the aftermath of 8From now on we will use the terms “multiple equilibria” and “real indeterminacy” (a “unique equilibrium” and “real determinacy”) interchangeably. By real indeterminacy we mean a situation in which the behavior of one or more (real) variables of the economy is not pinned down by the model. This situation implies that there are multiple equilibria, which in turn opensthe possibility ofhaving fluctuationsin the economy generated by endogenous beliefs thatare ofthe sunspottype; i.e.,they are based on stochastic variablesthatare extrinsic in Cassand Shell’s (1983)terminology. 9See for instance Benhabib et al. (2001), Taylor (1999), and Woodford (2003) among others. See also Zanna (2003) for a determinacy ofequilibrium analysisforinterestrate rulesin smallopen economies. 3

a crisis.10 This literature has tried to control for the variables that influence the nominal exchange rate. But our results suggest that there can be potential influences that may depend on “sunspots” which in turn can induce self-fulfilling cycles in the nominal exchange rate (or the nominal depreciation rate) as well as in other variables. Clearly these influences do not depend on fundamentals and their effect is something that the empirical literature should take into account. Second, to the extent that these interest rate policies can induce multiple self-fulfilling equilibria in the economy then they can be costly in terms of macroeconomic instability and welfare. In other words these policies can lead to “sunspot” equilibria that are characterized by a large degree of volatility of some macroeconomic aggregates such as consumption; and provided that agents are risk averse, then these rules can induce equilibria where welfare can decrease. This has not been studied in the previous literature on interest rate policies during and in the aftermath of the Asian crisis. For instance, Lahiri and Vegh (2000, 2003) and Flood and Jeanne (2000) focus on the fiscal and output costs of higher interest rates before and after a crisis. In addition Lahiri and Vegh (2003) claim that there is a non-monotonic relationship between welfare and the increase in interest rates. Whereas Christiano, Gust and Roldos (2004) and Braggion, Christiano and Roldos (2005) explore conditions under which a cut (rise) in the interest rate in the midst of a crisis will stimulate output and employment and improve welfare. Third our results cannot be understood as arguments that favor a particular view of the previously mentioned debate. They represent a simple example of policy induced macroeconomic instability regardless of whether the government increases or decreases the interest rate in response to currency depreciation. Whatiscrucialinouranalysisisthefeedbackresponseofthenominalinterestratetonominaldepreciation. In this regard, we unveil a peril that may be present in previously mentioned policy recommendations that has not been discussed before. Fourthbyconstructingaself-fulfillingequilibriumthatreplicatessomeofthestylizedfactsofthe“Sudden Stops” we suggest that these interest rate policies may have contributed to generating the dynamic cycles experienced by the Asian economies in the aftermath of the crisis. The remainder of this paper is organized as follows. In Section 2 we consider the simple set-up of the economy in “good times” and pursue analytically the characterization of the equilibrium for the interest rate policy that responds to currency depreciation. In section 3 we add to the simple model a collateral constraint, non-traded distribution services and the loan requirements to hire factors of production. Since we assume that the collateral constraint is binding, then this augmented set-up represents the economy in the aftermath of a crisis or the economy in “bad times.” Through a calibrated simulation of the economy we pursue the determinacy of equilibrium analysis and confirm the results derived in Section 2. In section 4 we use this augmented set-up and construct a self-fulfilling equilibrium that captures the stylized facts of a 10See the review of this literature by Montiel (2003). Some of the papers in this literature are Basurto and Ghosh (2000), Caporaleetal. (2005),Choand West(2003),Dekleetal. (2001,2002),Furman and Stiglitz(1998),Goldfajn and Baig(1998), Goldfajn and Gupta (1999),and Gould and Kamin (2000)among others. 4

“Sudden Stop.” Finally in Section 5 we present some concluding remarks. 2 The Simple Model: The Economy in “Good Times” Inthissectionwedevelopasimpleinfinite-horizonsmallopeneconomymodel. Theeconomyispopulated by a continuum of identical household-firm units and a government who are blessed with perfect foresight. Before we describe in detail the behavior of these agents we state a fewgeneral assumptions and definitions. There are two consumption goods: a traded good and a composite non-traded good whose prices are denotedbyPT andPN respectively. Forthissimplemodelweassumethatthelawofonepriceholdsforthe t t traded good. Then PT = PT where is the nominal exchange rate and PT is the foreign price of the t E t t ∗ E t t ∗ traded good. Later we will relax this assumption. We also normalize the foreign price of the traded good to one implying that PT = . t E t The real exchange rate (e ) is defined as the ratio between the price of traded goods and the aggregate t price of non-traded goods, e = /PN. From this definition we deduce that t E t t (cid:18) e =e t (1) t t − 1 µ πN t ¶ where (cid:18) = / is the gross nominal depreciation and πN =PN/PN is the gross non-traded inflation. t E t E t − 1 t t t − 1 2.1 The Government The government issues two nominal liabilities: money, Mg, and a domestic bond, Bg, that pays a t t gross nominal interest rate R . It does not have access to foreign debt and makes lump-sum transfers to t the household-firm units, τ , pays interest on its domestic debt, (R 1)Bg, and receives revenues from t t − t s b e g ig = ni B or t g a . ge. Its budget constraint is described by mg t +bg t = m (cid:18) g t t − 1 +τ t + Rt − (cid:18) 1 t bg t − 1 where mg t = M Et t g and t Et We assume that the government follows a Ricardian fiscal policy. That is, the government picks the path of the lump-sum transfers, τ , in order to satisfy the intertemporal version of its budget constraint in t conjunction with the transversality condition lim bg t =0. t →∞ s t − = 1 0 (cid:18)s R + s 1 Ontheotherhandmonetarypolicyisdescribed(cid:84)as(cid:19)anin(cid:20)terest-ratefeedbackrulewherebythegovernment maneuvers the nominal interest rate of the domestic bond in response to currency depreciation. As we mentioned earlier, the motivation of this rule comes from the debate about the appropriate interest rate policytofightagainstcurrencydepreciationintheaftermathoftheAsiancrisis. Tosomeextentthediverse policy recommendations conceived the interest rate policy as a reaction function. In fact some of the works inspiredbythisdebatesuchasChoandWest(2001),GoldfajnandBaig(1998),andLahiriandVegh(2003) 5

among others, have already considered describing the interest rate policy, implicitly or explicitly, as a rule that reacts to some measure of nominal depreciation. Specifically we assume that the government can implement the following rule (cid:18) R =R¯ρ t (2) t ¯(cid:18) ³ ´ where ρ(.) is a continuous, differentiable and strictly positive function in its argument with ρ(1) = 1 and ρ ρ (1)=0; and R¯ and¯(cid:18) are the targets of the nominal interest rate and the nominal depreciation rate (cid:18) ≡ 0 6 that the government wants to achieve.11 In this sense the rule responds to the deviation of the current depreciation rate from the depreciation target.12 Wealsoassumethattherulecanrespondpositivelytothedeviationofthenominaldepreciationratefrom itstarget,ρ >0,capturingthepolicyrecommendationoftheIMFpolicymakers; oritcanreactnegatively, (cid:18) ρ <0,describing,tosomeextent,thepolicyrecommendationsoftheoppositeview. Neverthelessweexclude (cid:18) the cases ρ = 1,1.13 In other words the interest rate policy corresponds to (2) with ρ ρ (1) = 0 and (cid:18) − (cid:18) ≡ 0 6 either ρ >1 or ρ <1. | (cid:18)| | (cid:18)| 2.2 The Household-Firm Unit There is a large number of identical household-firm units. They have perfect foresight, live infinitely and derive utility from consuming, not working and liquidity services of money. The intertemporal utility function of the representative unit is described by ∞ βt U(cT)+V(cN)+H(hT)+L(hN)+J(m ) (3) t t t t t t=0 X £ ¤ where β (0,1) corresponds to the discount rate, cT and cN denote the consumption of traded and non- ∈ t t traded goods respectively, hT and hN are the labor allocated to the production of the traded good and the t t non-traded good, and m refers to real money holdings measured with respect to foreign currency. The t specification in (3) assumes separability in the single period utility function among consumption, labor and real money balances. By doing this we remove the distortionary effects of transactions money demand.14 Moreover we introduce separability in the utility derived from cT, cN, hT and hN which will allow us to t t t t 11Forsimplicity we also assume thatthese targetscorrespond to the steady-state levels ofthese variables. 12Belowwewillconsiderotherinterestratepoliciesthatdifferintermsofthetimingoftheruleandontheinclusionofother argumentssuch as the CPI-inflation rate. 13The reason of this is that our analysis relies on a loglinearized system of equations that describes the dynamics of the economy. Thecasesofρ =1orρ = 1introduceaunitrootinthissystem precludingthepossibilityofusingthissystem to (cid:18) (cid:18) − pursue a meaningfuldeterminacy ofequilibrium analysis. 14Becauseofthiswecanwritetherealmoneybalancesthatentertheutilityfunctionintermsofforeigncurrency,mt Mt, withoutconsequencesforourresults. ≡ Et 6

derive analytical results in the determinacy of equilibrium analysis. To complete the characterization of the utility function we also make the following assumption. Assumption 1. a) U(.), V(.), H(.), L(.) and J(.) are continuous and twice differentiable; and b) U(.), V(.), and J(.) are strictly increasing (U dU > 0, V > 0, J > 0) and strictly concave (U < 0, T ≡ dcT t N m TT V < 0, J < 0) whereas H(.) and L(.) are strictly decreasing (H dH < 0, L < 0) and concave NN mm T ≡ dhT t N (H 0, L 0). TT NN ≤ ≤ The representative household-firm unit is engaged in the production of a flexible-price traded good and a sticky-price non-traded good by employing labor from a perfectly competitive market. The technologies are described by yT =F h˘T and yN =G h˜N t t t t ³ ´ ³ ´ where h˘T and h˜N denote the labor hired by the household-firm unit for the production of the traded good t t and the non-traded good respectively. The technologies satisfy the following assumption. Assumption 2. F(.)and G(.)are continuous, twice differentiable, strictly increasing (F dF >0, T ≡ dhT t G >0 ), and strictly concave (F <0, G <0 ). N TT NN Consumption of the non-traded good, cN, is a composite good made of a continuum of intermediate t differentiated goods. The aggregator function is of the Dixit-Stiglitz type. Each household-firm unit is the monopolistic producer of one variety of non-traded intermediate goods. The demand for the intermediate good is of the form CNd P˜ t N satisfying d(1)=1 and d (1)= µ with µ>1 where CN denotes the level t P t N 0 − t of aggregate demand for t³he n´on-traded good, P˜N is the nominal price of the intermediate non-traded good t produced by the household-firm unit and PN is the price of the composite non-traded good. The unit that t behavesasamonopolistintheproductionofthenon-tradedgoodsetsthepriceofthegooditsupplies, P˜N, t taking the level of aggregate demand for the good as given. Specifically the monopolist is constrained to satisfy demand at that price. That is P˜N G h˜N CNd t (4) t ≥ t PN Ã t ! ³ ´ Following Rotemberg (1982) we introduce nominal price rigidities for the intermediate non-traded good. 2 The household-firm unit faces a resource cost of the type γ P˜ t N π¯N , that reflects that it is costly 2 P˜N − having the price of the good that it sets grow at a different rat µ e f t r − o 1 m π¯N, ¶ the steady-state level of the gross non-traded inflation rate. There are incomplete markets. The representative household-firm unit has access to two different risk freebonds: adomesticbondissuedbythegovernment,B , thatpaysagrossnominalinterestrate,R anda t t foreignbond,b ,thatpaysagrossforeigninterestrateR .Inaddition,theunitreceivesawageincomefrom ∗t t∗ working, W hT +hN , lump-sum transfers from the government, τ , and dividends from selling the traded t t t t ¡ ¢ 7

good and the non-traded composite good. Then its flow budget constraint in units of the traded good can be written as m R b cN m t +b t ≤ (cid:18) t − 1 + t − (cid:18) 1 t − 1 +w t hT t +hN t +τ t +Ω t − cT t − e t (5) t t t where b = Bt, w = Wt and15 ¡ ¢ t t Et Et 2 1 P˜N P˜N γ P˜N Ω = F hˇT w hˇT t CNd t e w h˜N t π¯ R b +b (6) t t − t t − e t P t N t à P t N ! − t t t − 2 ÃP˜ t N 1 − ! − t∗ − 1 ∗t − 1 ∗t £ ¡ ¢ ¤ −   Equation(5)saysthattheend-of-periodrealfinancialdomesticassets(moneyplusdomesticbond)canbe worthnomorethantherealvalueoffinancialdomesticwealthbroughtintotheperiodplusthesumofwage income, transfers and dividends (Ω ) net of consumption. The dividends described in (6) correspond to the t differencebetweensalerevenuesandcosts, takingintoaccountthatthroughthefirm-sidetherepresentative unit can hold foreign debt, b . For holdings of foreign debt the agent pays interests, (R 1)b . ∗t t∗ − 1− ∗t − 1 Besides the budget constraint the household-firm unit is subject to an Non-Ponzi game condition n lim t 0 (7) t t 1 ≥ →∞ − R s∗ s=0 Q where n =b +m b . t t t − ∗t The representative household-firm unit chooses the set of sequences cT, cN, hT, hN, hˇT, h˜N, P˜N, b , { t t t t t t t ∗t b , m in order to maximize (3) subject to (4), (5), (6) and (7), given the initial condition n and t t } ∞t=0 − 1 the set of sequences {R , R , (cid:18) , e , PN, w , τ , CN . Note that since the utility function specified in (3) t∗ t t t t t t t } implies that the preferences of the agent display non-sasiation then both constraints (5) and (7) hold with equality. TheAppendixcontainsadetailedderivationofthenecessaryconditionsforoptimization. Imposing these conditions along with the market clearing conditions in the labor market, the equilibrium symmetry (P˜N =PN and h˜N =hN), the market clearing condition for the non-traded good t t t t G hN =cN + γ πN π¯N 2 (8) t t 2 t − ¡ ¢ ¡ ¢ and the definitions πN =PN/PN , d(1)=1 and d(1)= µ we obtain t t t − 1 0 − R R = t (9) t∗ (cid:18) t+1 H (hT) T t =F hT (10) −U (cT) T t T t ¡ ¢ 15By having only one real wage wt for hT t and hN t we are implicitly assuming that there is perfect labor mobility between the production oftraded and non-traded goods. 8

U (cT)=βR U (cT ) (11) T t t∗ T t+1 βR V (cN)= tV (cN ) (12) N t πN N t+1 t+1 V (cN ) πN π¯N πN πN π¯N πN µcN µ 1 N t+1 t+1− t+1 = t − t + t − mc (13) V (cN) β βγ µ − t ¡N t ¢ ¡ ¢ µ ¶ wheremc = LN(hN t ) correspondstothemarginalcostofproducingthenon-tradedgood. Inaddition t −VN(cN t )GN (hN t ) equilibrium in the traded good market implies that b b = R 1 b +cT F hT (14) ∗t − ∗t − 1 t∗ − 1− ∗t − 1 t − t ¡ ¢ ¡ ¢ The interpretation of these equations is straightforward. Condition (9) corresponds to an Uncovered Interest Parity condition (UIP) that equalizes the returns of the foreign and domestic bonds. Equation (10) makes the marginal rate of substitution between labor (assigned to the production of the traded good) and consumption of the traded good equal to the marginal product of labor in the production of the traded good. Equations (11) and (12) are the standard Euler equations for consumption of the traded good and consumption of the non-traded good. Equation (13) corresponds to the augmented Phillips curve for the sticky-price non-traded goods inflation.16 And (14) corresponds to the current account equation. 2.3 Capital Markets We introduce imperfect capital markets using the following ad-hoc upward-sloping supply curve of funds on the world capital market b b R =R f ∗t with f ∗t >0, f(1)=1, f (1)=ψ >0, (15) t∗ ∗ ¯b 0 ¯b 0 µ ∗¶ µ ∗¶ where f b∗t corresponds to the country-specific risk premium and R is the risk free international interest ¯b ∗ ∗ rate. Th³is sp´ecification captures the idea that the small borrowing economy faces a world interest rate, R , t∗ that increases when the stock of the debt issued by the country, b , is above its long run level, ¯b . Then as ∗t ∗ the external debt grows so does the risk of default, and in order to compensate the lenders for this risk, the economy has to pay them a premium over the risk free international interest rate. Thereasonforintroducing(15)ismerelytechnical. Bydoingso,we“closethesmallopeneconomy”and avoid the unit root problem as discussed in Schmitt-Grohé and Uribe (2003). This will allow us to obtain meaningful results from the determinacy of equilibrium analysis once we log-linearize the equations of the 16We would have derived a similaraugmented Phillipscurve ifwe had follow Calvo’s(1983)approach. 9

model.17 Our results are invariant to other approaches to “close the small open economy” such as complete markets or convex portfolio adjustment costs. Finally throughout this paper we will also assume that the long-run level of foreign stock of debt is positive as stated in the following assumption. Assumption 3. The long-run level of the foreign stock of debt is positive: ¯b >0. ∗ 2.4 A Perfect Foresight Equilibrium We are ready to provide a definition of a perfect foresight equilibrium in this economy. Definition 1 Given the initial condition b , the steady-state level of foreign debt ¯b and the depreciation ∗1 ∗ − target ¯(cid:18), a symmetric perfect foresight equilibrium is defined as a set of sequences cT, cN, hT, hN, b , { t t t t ∗t (cid:18) , πN, R , R satisfying: a) the market clearing conditions for the non-traded and traded goods, (8) t t t t∗ } ∞t=0 and (14), b) the UIP condition (9), c) the intratemporal efficient condition (10), d) the Euler equations for consumption of traded and non-traded goods, (11) and (12), e) the augmented Phillips curve, (13), f) the monetary policy (2) and g) the ad-hoc upward-sloping supply curve of foreign funds (15). Notethatthisdefinitionignoresthebudgetconstraintofthegovernmentanditstransversalitycondition. The reason is that by following a Ricardian fiscal policy the government guarantees that the intertemporal version of its budget constraint in conjunction with its transversality condition will be always satisfied. In addition real money balances do not appear in the definition. This is because monetary policy is described as an interest rate rule and real balances enter in the utility function in a separable way. In fact once we solvefor cT,cN,hT,hN,b ,(cid:18) ,πN, R ,R itispossibletoretrievethesetofsequences λ ,e ,m ,b , { t t t t ∗t t t t t∗ } ∞t=0 { t t t t w , mc using (1), (5), and equations (40), (41), (43), (45) and (47) that are presented in the Appendix. t t } ∞t=0 2.5 The Determinacy of Equilibrium Analysis In order to pursue the determinacy of equilibrium analysis we will log-linearize the system of equations that describe the dynamics of this economy around a steady state c¯T, c¯N, h¯T, h¯N, ¯b , ¯(cid:18), π¯N, R¯, R¯ . In ∗ ∗ { } the Appendix we characterize this steady state. Log-linearizing the equations of Definition 1 around the steady-state yields Rˆ =ρ ˆ(cid:18) with ρ =0 and either ρ >1 or ρ <1 (16) t (cid:18) t (cid:18) 6 | (cid:18)| | (cid:18)| 17The“unit-rootproblem”thatiscommonlypresentinsmallopeneconomymodelsarisesbecauseofassumingthatR = 1. t∗ β Toseewhy,usethisassumptiontogetherwithcondition(48)todeducethatλt=λt+1.Thisisanequationthathasaunitroot and thatintroducesaunitrootin theentiredynamicalsystem ofthesimpleset-up. Then itisnotvalid to apply thecommon technique of linearizing the system around the steady state and studying the eigenvalues of the Jacobian matrix in order to characterize localdeterminacy ofthe dynamicalsystem. See Schmitt-Grohé and Uribe (2003). 10

Rˆ =Rˆ +ˆ(cid:18) (17) t t∗ t+1 cˆN=cˆN ξN Rˆ πˆN (18) t t+1− t − t+1 ³ ´ πˆN =βπˆN +βϕcˆN (19) t t+1 t 1+ψ ˆb = ˆb +κcˆT (20) ∗t β ∗t 1 t − µ ¶ cˆT=cˆT ξT Rˆ ξTˆ(cid:18) (21) t t+1− t − t+1 ³ ´ where xˆ =log xt and t x¯ ¡ ¢ U V H L ξT = T >0 ξN = N >0 σT = T >0 σN = N >0 −U c¯T −V c¯N H h¯T L h¯N TT NN TT NN F G ωT = T >0 ωN = N >0 (22) −F h¯T −G h¯N TT NN 1 F h¯TσTωT (µ 1)c¯N c¯N(σN +ωN) 1 κ= c¯T + T >0 and ϕ= − + >0 ¯b ∗ · (σT +ωT)ξT ¸ · βγ¯(cid:18)2 ¸· G N h¯NσNωN ξN ¸ whose signs are derived using Assumptions 1, 2 and 3. Equations (16)-(21) correspond to the reduced log-linear representations of the policy rule, the UIP condition, the Euler equation for consumption of the non-traded good, the augmented Phillips curve, the current account equation, and the Euler equation for consumption of the traded good, respectively. Our main goal in this Section is to show that the rule in (16) is prone to induce multiple equilibria in the economy described by equations (17)-(21). Proving this implies that this policy can cause fluctuations in the economy that are driven by people’s self-fulfilling beliefs and not by fundamentals. In fact, before we provide a formal proof of the existence of multiple equilibria, it is worth developing a simple intuition of why this policy rule can induce self-fulfilling equilibria. To do so it is sufficient to concentrate on equations (16)-(19) in order to construct the following argument. Note that given the international interest rate, Rˆ , then the policy rule (16) and the UIP condition (17) t∗ determinethedynamicsofthedepreciationrate,ˆ(cid:18) ,andthenominalinterestrate,Rˆ .Moreimportantlythe t t nominal interest rate, Rˆ , is not affected by either the non-traded good inflation, πˆN, or the consumption of t t the non-traded good, cN. Taking this into account we can construct the following self-fulfilling equilibrium. t Assume that agents in response to a sunspot expect a higher non-traded good inflation in the next period. Since the interest rate policy does not react to these expectations then the real interest rate measured with 11

respect to the expected non-traded good inflation, Rˆ πˆN , declines. This stimulates consumption of t − t+1 the non-traded good according to (18). And as a response to this, firms raise the price of the non-traded good inducing a higher non-traded inflation as can be seen in (19). Hence the original beliefs of a higher non-traded good inflation are validated. This simple intuition is appealing but incomplete unless we show that all the equilibrium conditions of the economy (16)-(21) are satisfied on the entire equilibrium path. In other words we need to characterize formally the equilibrium of this economy. To accomplish this goal we first manipulate equations (16)-(21) and write them in the matrix form ˆ(cid:18) ρ ψ(1+ψ) ψκ 0 0 ˆ(cid:18) t+1 (cid:18) − β − t  ˆb   0 1+ψ κ 0 0  ˆb  ∗t β ∗t 1  cˆT = 0 ψ(1+ψ)ξT 1+ψκξT 0 0  cˆ − T  (23)  t+1   β  t        πˆN   0 0 ³ 0 ´ 1 ϕ  πˆN    t+1     β −     t    cˆN   ρ ξN 0 0 ξN 1+ϕξN  cˆN    t+1     (cid:18) − β     t      ³ ´   Jc Thenweusethissystemtofindandtocomparethedimensionoftheunstablesubspaceofthesystemtothe | {z } number of non-predetermined variables.18 If the dimension of this subspace is smaller than the number of non-predetermined variables then, from the results by Blanchard and Kahn (1980), we can infer that there existmultipleperfectforesightequilibria. Thisformsthebasisfortheexistenceofself-fulfillingfluctuations. The following Proposition states the main result of the determinacy of equilibrium analysis: an interest rate policy that raises or lowers the nominal interest rate in response to current currency depreciation can lead to real indeterminacy, or, equivalently, to multiple equilibria. Proposition 1 If the government follows an interest rate policy rule such as Rˆ = ρ ˆ(cid:18) with ρ = 0 and t (cid:18) t (cid:18) 6 either ρ >1 or ρ <1, then there exists a continuum of perfect foresight equilibria in which the sequences | (cid:18)| | (cid:18)| ˆ(cid:18) , ˆb , cˆT, πˆN, cˆN converge asymptotically to the steady state. In addition { t ∗t t t t } ∞t=0 a) if ρ >1 then the degree of indeterminacy is of order 1.19 | (cid:18)| b) if ρ <1 then the degree of indeterminacy is of order 2. | (cid:18)| Proof. TheeigenvaluesofthematrixJc in(23)correspondtotherootsofthecharacteristicpolynomial c(v)= Jc vI =0. Using the definition of Jc in (23) this polynomial can be written as P | − | c(v)=(v ρ ) f(v)=0 (24) P − (cid:18) P 18Thedimension oftheunstablesubspaceisgiven bythenumberofrootsofthesystem thatareoutsidetheunitcircle. See Blanchard and Kahn (1980). 19The degree of indeterminacy is defined as the difference between the number of non-predetermined variables and the dimension ofthe unstable subspace ofthe log-linearized system. 12

where 1+ψ 1+ψ 1 1 f(v)= v2 1+ +ψκξT v+ v2 1+ +ϕξN v+ P − β β − β β · µ ¶ ¸· µ ¶ ¸ ByLemma4intheAppendixweknowthatthecharacteristicpolynomial f(v)=0hasrealrootssatisfying P v < 1, v > 1, v < 1 and v > 1. The fifth root of c(v) = 0 is v = ρ . Clearly if ρ > 1 then | 1 | | 2 | | 3 | | 4 | P 5 (cid:18) | (cid:18)| v >1whereasif ρ <1then v <1.Thereforeusingthis,thecharacterizationoftherootsof f(v)=0 | 5 | | (cid:18)| | 5 | P and(24)wecanconcludethefollowing. If ρ >1then c(v)=0hasthreeexplosiverootsnamely v >1, | (cid:18)| P | 2 | v >1 and v >1. While if ρ <1 then c(v)=0 has two explosive roots namely v >1 and v >1. | 4 | | 5 | | (cid:18)| P | 2 | | 4 | Thereforeregardlessofwhether ρ >1or ρ <1thereareatmostthreeexplosiveroots. Giventhatthere | (cid:18)| | (cid:18)| are four non-predetermined variables, ˆ(cid:18) , cˆT, πˆN and cˆN, then the number of non-predetermined variables t t t t is greater than the number of explosive roots. Applying the results of Blanchard and Kahn (1980) it follows that there exists an infinite number of perfect foresight equilibria converging to the steady state. Finally parts a) and b) follow from the difference between the number of non-predetermined variables and the number of explosive roots when ρ >1 and ρ <1 respectively. | (cid:18)| | (cid:18)| Table 1: Determinacy of Equilibrium Analysis The Simple Model The Augmented Model Degree of Responsiveness Degree of Responsiveness Interest Rate Policy ρ <1 ρ >1 ρ <1 ρ >1 | (cid:18)| | (cid:18)| | (cid:18)| | (cid:18)| Forward-Looking Rˆ =ρ ˆ(cid:18) with ρ =0 M M M M t (cid:18) t+1 (cid:18) 6 Contemporaneous Rˆ =ρ ˆ(cid:18) with ρ =0 M M M M t (cid:18) t (cid:18) 6 Backward-Looking Rˆ =ρ ˆ(cid:18) with ρ =0 M U M M or U t (cid:18) t − 1 (cid:18) 6 Note: M refers to multiple equilibria and U refers to a unique equilibrium Proposition 1 has two important implications. First provided that the fiscal policy is Ricardian, then the interest rate policy considered in this Proposition will not pin-down the level of the nominal exchange rate.20 Hencethesamepolicyalsoinducesnominalindeterminacyoftheexchangeratelevel. Second,thereis 20A Non-Ricardian fiscal policy combined with the monetary policy under study will determine the level of the nominal 13

nothing in the characterization of the equilibrium that prevents us from constructing self-fulfilling equilibria that are based on expectations of a different variable from the non-traded inflation. For instance we can construct a self-fulfilling equilibrium driven by people’s beliefs about currency depreciation. We will pursue this exercise in Section 4. The results of Proposition 1 also pose the question of whether policies that respond exclusively to either the future depreciation rate (a forward-looking policy, Rˆ = ρ ˆ(cid:18) ) or to the past depreciation rate (a t (cid:18) t+1 backward-looking policy, Rˆ = ρ ˆ(cid:18) ) can still induce multiple equilibria. The answer to this question is t (cid:18) t 1 − affirmative and the characterization of the equilibrium under these policies is provided in the Appendix. Inparticularwefindthatforward-lookingpolicies,Rˆ =ρ ˆ(cid:18) withρ =0,canleadtomultipleequilibria t (cid:18) t+1 (cid:18) 6 when the interest rate response coefficient to future depreciation satisfies either ρ >1 or ρ <1. On the | (cid:18)| | (cid:18)| contrary backward-looking policies, Rˆ = ρ ˆ(cid:18) with ρ = 0, that are very aggressive with respect to past t (cid:18) t − 1 (cid:18) 6 depreciation and satisfy ρ > 1 will guarantee a unique equilibrium whereas timid policies that satisfy | (cid:18)| ρ <1,canleadtorealindeterminacy. TheseresultsaswellastheresultsfromProposition1arepresented | (cid:18)| in Table 1 in the columns labeled as “The Simple Model.”21 Itisimportanttounderstandthefeaturesofthemodelthatallowfortheexistenceofmultipleequilibria. After all by unveiling them in the simple model of this section will also help us to understand the results in the richer set-up of the next section. The crucial features are the following: the description of monetary policy as an interest rate feedback rule, the introduction of price-stickiness in non-traded goods and the exclusive dependence of the rule on currency depreciation. By Sargent and Wallace (1975) we know that the first feature by itself leads to nominal indeterminacy of the exchange rate level (price level) in a flexible price model under a Ricardian fiscal policy. The second characteristic together with the rule elucidate why nominal indeterminacy turns into real indeterminacy. And finally the first two features in tandem with the exclusiveresponseoftheruletocurrencydepreciationiswhatexplainswhy,atleastforforward-lookingand contemporaneous rules, multiple equilibria arise regardless of the degree of responsiveness of the rule.22 Although the results of this section are interesting, it is clear that the model suffers from at least two drawbacks. On one hand there is no specific feature in the model that captures the fact that the economy is inacrisis. Ontheotherhandsomeofthedynamicsofconsumptionandinflation(ofnon-tradedgoods)that aresupportedasaself-fulfillingequilibriumarecompletelyatoddswiththestylizedfactsofa“SuddenStop.” exchange rate if ρ >1 butnotif ρ <1. | (cid:18)| | (cid:18)| 21Our results will not be affected if we model monetary policy as Rt = ρ((cid:18)t+s) with s = 1,0,1. The reason is that in − (2) we have assumed that the target depreciation rate coincides with the long run steady state depreciation rate and the determinacy of equilibrium analysis is pursued using a log-linearized version of the system of equations that describe the dynamics of the economy around the steady state. In addition our general results still hold if we describe monetary policy as ∆Rt=Rt − Rt − 1=ϕ( (cid:18)t ¯(cid:18) +s)with s= − 1,0,1and ϕ(1)=0.This resembles the implicit descriptions in some ofthe empirical workssuchasGouldandKamin(2000),Caporale,CipolliniandDemetriades(2005)andDekle,HsiaoandWang(2001,2002). 22In Zanna (2003) we show that in this simple set-up in order to guarantee a unique equilibrium a rule must respond aggressivelytothenon-tradedinflationbuttimidlytodepreciation. Arulethatrespondsaggressivelytocurrencydepreciation stillopensthe possibility ofmultiple equilibria regardlessofits response to non-traded inflation. 14

Inparticulartheintuitionthatweprovidedtoconstructaself-fulfillingequilibriumimpliesthatconsumption of non-traded goods and inflation are positively correlated. On the contrary, the typical stylized facts of a crisis suggest that they are negatively correlated: there is a strong decline in consumption accompanied by an increase in inflation. In order to correct these drawbacks we will enrich the current model with some features that have been proved to be useful in explaining some stylized facts of a crisis. This defines the objective of our next section. 3 The Augmented Model: The Economy in “Bad Times” In this Section we introduce some features that will enrich the simple model in several dimensions. First weintroducedistributioncosts(services)forthetradedgood. Thistogetherwithpricestickinessarecrucial to explain the large movements in real exchange rates after large devaluations. Second, we assume that the household-firm units require working capital (loans) to hire labor and international working capital (loans) to purchase an imported intermediate input. This characteristic is important to obtain a decline in output and demand in the midst of the crisis when interest rates rise. Third we impose a collateral constraint: international loans must be guaranteed by physical assets such as capital. This provides a definition of a crisis. Acrisisisatimewhentheconstraintisbindingandtheshadowpriceoftheconstraintisgreaterthan zero. Fourth we assume non-separabilityin the utility function between the two types of consumption. This willguaranteethatourpreviousresultsarenotdrivenbythespecificassumptionofseparability. Weproceed to explain how we introduce these features in the model and their influence in the previous equations. 3.1 The Additional Features AsinBurnsteinetal. (2003)weassumethatthetradedgoodneedstobecombinedwithsomenon-traded distribution services before it is consumed. In order to consume one unit of the traded good it is required η units of the basket of differentiated non-traded goods. Let P˘T, PT and PN be the price in the domestic t t t currency that the household-firm unit receives from producing and selling the traded good, the price that it pays to consume this good and the general price level of the basket of differentiated non-traded goods, respectively. Hence the consumer price of the traded good is simply PT = P˘T +ηPN. And since PPP t t t holds at the production level of the traded good (P˘T = P˘T ) and the foreign price of the traded good is t E t t ∗ normalized to one (P˘T =1), we have that PT = +ηPN. t ∗ t E t t The production of the non-traded good is still demand determined by P˜N P˜N G h˜N,KN CNd t +ηCTd t (25) t ≥ t PN t PN Ã t ! Ã t ! ³ ´ where d(1)=1, d (1)= µ, CN denotes the level of aggregate demand for the non-traded good, P˜N is the 0 − t t nominalpriceoftheintermediatenon-tradedgoodproducedbythehousehold-firmunitandCT corresponds t 15

to the level of aggregate consumption of the traded good. But now the demand requirements come from two sources.23 They come from consumption of non-traded goods CNd P˜ t N that provide utility and from t PN t non-traded distribution services ηCTd P˜ t N that are necessary to bri³ng o´ne unit of the traded good to t PN t the household-firm unit. Note that we ³assum´e that there is no difference between non-traded consumption goodsandnon-tradeddistributionservices. Asaconsequence,inequilibriumthebasketofnon-tradedgoods required to distribute traded goods will have the same composition as the non-traded basket consumed by the household-firm unit. TheintroductionoftheloanrequirementsandthecollateralconstraintinthemodelfollowsChristianoet al. (2004). Thehousehold-firmunitrequiresdomesticloanstohirelabor(hˇT andh˜N)andinternationalloans t t tobuyanimportedinput(I )thatwillbeusedintheproductionofthetradedgood. Theseloansareobtained t at the beginning of the period and repaid at the end of the period. In this sense they represent short-term debt and differ from long-term foreign debt b . We do not model, however, the financial institutions that ∗t 1 − providetheseloans. Insteadweassumethatthedomesticloansareprovidedbythegovernmentwhereasthe foreignloansaresuppliedbyforeigncreditors.24 Fortheseloanstheunitpaysinterests(R 1)W (hˇT+h˜N) t − t t t and (R 1)P˜TI that are accrued between periods, where R is the domestic nominal interest rate and R t∗ − t t t ∗ is the international interest rate. The latter is assumed to be constant and equal to 1. β In contrast to the simple model we assume that the production technology of the traded good needs labor (h˘T), an imported input (I ) and capital (KT). In addition, the technology for the non-traded goods t t requires labor (h˜N) and capital (KN). That is t yT =F h˘T,I ,KT and yN =G h˜N,KN t t t t t ³ ´ ³ ´ Furthermore,asinChristianoetal. (2004)andMendozaandSmith(2002),amongothers,capitalisassumed to be time-invariant, does not depreciate and there is no technology to making it bigger. Under these new features the dividends that the household-firm unit receives can be written as 2 1 P˜N γ P˜N Ω = F h˘T,I ,KT + t G h˜N,KN t π¯ (26) t t t e t P t N t − 2 ÃP˜ t N 1 − !  ³ ´ ³ ´ − w R hˇT w R h˜N R I R b +b  − t t t − t t t − ∗ t − ∗ ∗t − 1 ∗t where w t = W Et t and e t = PE t N t . To model the crisis we follow closely Christiano et al. (2004) by imposing a collateral constraint on the household-firm unit 23See Corsettietal. (2005). 24To formalize this point we could introduce financial institutions in the model that behave in a perfectly competitive way and supply the aforementioned loans. Thiswould notchange ourmain results. 16

R b +R I +w R hˇT +h˜N φ qNKN +qTKT (27) ∗ ∗t − 1 ∗ t t t t t ≤ t t ³ ´ ¡ ¢ whereqN andqT representtherealvalue(inunitsofforeigncurrency)ofoneunitofcapitalfortheproduction t t of the non-traded and traded goods respectively, and φ is the fraction of these stocks that foreign creditors accept as collateral. The constraint (27) says that the total value of foreign and domestic debt that the representative household-firm unit has to pay to completely eliminate the debt of the firm by the end of period t cannot exceed the value of the collateral. The crisis makes this constraint unexpectedly binding in every period henceforth without the possibility of being removed. Finally we will assume non separability in the utility function between the two types of consumption. Instead of having U(cT)+V(cN) as the specification in (3) we define U˜(cT,cN). But we will still assume t t t t separability among consumption, labor and real money balances. Withallthesefeaturestheproblemoftherepresentativehousehold-firmunitdoesnotchangesignificantly withrespecttotheoneinthesimplemodel. Butthecollateralconstraintrepresentsanextraconditionthat affects the decisions of the representative agent. We proceed to study how this constraint in tandem with the other features influence the optimal conditions of a symmetric equilibrium. 3.2 The New Equilibrium Conditions Theproblemthatthehouseholdfirmunithastosolveissimilartotheonepresentedinthesimplemodel. Theagentchoosesthesetofsequences cT,cN,hT,hN,hˇT,h˜N,I ,P˜N,b ,b ,m inordertomaximize { t t t t t t t t ∗t t t } ∞t=0 ∞ βt U˜(cT,cN)+H(hT)+L(hN)+J(m ) t t t t t X t=0 h i subject to the budget constraint m R b η cN m t +b t ≤ (cid:18) t − 1 + t − (cid:18) 1 t − 1 +w t (hT t +hN t )+τ t +Ω t − 1+ e cT t − e t t t µ t¶ t and the constraints (7), (25), (26), and (27), given the initial conditions b , b , and m and the set of ∗1 1 1 − − − sequences {R , R , (cid:18) , e , PN, w , τ , CN, CT . t∗ t t t t t t t t } From this problem we derive the optimization conditions which together with symmetry conditions and market clearing conditions can be used to find the laws of motion of the economy. These laws correspond to (1), (2), (27) with equality, G hN,KN =cN + γ πN π¯ 2 +ηcT (28) t t 2 t − t ¡ ¢ ¡ ¢ R =R (1+ζ )(cid:18) (29) t ∗ t+1 t+1 17

H (hT) w T t = t (30) −U˜ (cT,cN) 1+ η T t t et 1+ η U˜ (cT,cN)= βR tU˜ (cT ,cN ) where πT =(cid:18) et+1 (31) T t t πT t+1 T t+1 t+1 t+1 t+1³ 1+ η ´ et ³ ´ βR U˜ (cT,cN)= tU˜ (cT ,cN ) (32) N t t πN N t+1 t+1 t+1 U˜ (cT ,cN )(πN π¯N)πN (πN π¯N)πN µ cN +ηcT µ 1 N t+1 t+1 t+1− t+1 = t − t + t t − mc (33) U˜ (cT,cN) β βγ µ − t N t t ¡ ¢µ ¶ b b =(R 1)b +R I +cT F hT,I ,KT (34) ∗t − ∗t − 1 ∗ − ∗t − 1 ∗ t t − t t ¡ ¢ F hT,KT,I =w (1+ζ )R (35) T t t t t t ¡ ¢ w e (1+ζ )R mc = t t t t (36) t G hN,KN N t ¡ ¢ F hT,KT,I =(1+ζ )R (37) I t t t ∗ ¡ ¢ where λ ζ and λ are the Lagrange multipliers of the collateral constraint and the budget constraint. The t t t latter multiplier evolves according to the asset pricing equation λ =βR (1+ζ )λ (38) t ∗ t+1 t+1 Equations (28)-(34) are basically equivalent to equations (8)-(14) in the simple model.25 Therefore they have a similar interpretation. On the other hand equations (35)-(37) correspond to the optimal conditions that determine the household-firm unit demands for labor (for the production of the traded and non-traded good) and for the imported input. A comparison between the laws of motion of the simple model and the augmented model reveal that the introductionofdistributioncosts,thecollateralconstraint,andtherequirementofloanshassomeimportant consequences. Ononehanddistributionservicesaffecttherelativepriceofthetradedgoodattheconsumer level with respect to the nominal exchange rate. In the simple model this relative price was equal to one. In the augmented model this price is 1+ η which depends on the distribution costs parameter η. From (30) et and(31)itisclearthatthroughthisprice, distributioncostsaffectinequilibriumtheoptimalintratemporal decisions between labor and consumption of the traded good as well as the optimal intertemporal choices 25In the simple modelFT hT t =wt (cid:3) (cid:4) 18

for consumption of the traded good. On the other hand distribution services generate an extra demand of non-traded goods as is captured by the last term, ηcT, of the right hand side of (28). This extra demand t also influences the dynamics of non-traded goods inflation as can be seen in (33). Thebindingcollateralgeneratesanendogenousriskpremiumasreflectedbythe“modified”UIPcondition in (29). In fact because of the constraint, the “effective” international nominal interest rate that domestic agents pay becomes (1+ζ )R . Thus raising external debt b not only requires the payment of interests t+1 ∗ ∗t (R b ) but also tightens the binding constraint (ζ > 0) generating an additional interest cost. Note ∗ ∗t t+1 also that in contrast to the simple model, in the augmented model we have assumed that the international interest rate (R ) is constant and equal to 1. Nevertheless in this context this typical assumption of the ∗ β small open economy literature does not cause the unit-root problem.26 Therequirementofloanstohirelaborintandemwiththebindingcollateralconstrainthasanimportant effect on the labor demand decisions. By looking at the right hand sides of (35) and (36) it is possible to conclude the following. Ceteris paribus the necessity of short-term loans, the binding constraint, and the fact that in the short run ζ >0 imply that an increase in the “effective” interest rate (1+ζ )R , will push t t t the cost of hiring labor up. In response to this, the demand for labor for the production of the traded and non-traded goods will contract. As mentioned before equation (37) corresponds to the optimal condition that determines the demand for the imported input. This condition equalizes the marginal product of this input to the effective cost of foreign working capital (1+ζ )R necessary to import it. As the constraint tightens and ζ goes up, the t ∗ t effective cost raises and the demand for the imported input decreases. Furthermore the purchases of this input influence the market equilibrium condition for the traded good as represented by equation (34). Since this equation describes the behavior of the current account then a decrease in the imported input can cause animprovementinthecurrentaccountdeficit. Thisimprovementisalmostimmediategiventheassumption that external short term loans to finance the intermediate input have to be repaid at the end of the period and not at the beginning of next period. Finally although identical household-firm units will not trade capital in equilibrium, it is possible to derive the equilibrium value of the prices of capital. The equilibrium value of these asset prices correspond to 1 1 qT = F K + (cid:18) R t+ t 1 − q t T +1 and qN = (m e c t t)G K + (cid:18) R t+ t 1 − q t N +1 (39) t (³1 φ´ζ ) t (1 ³φζ )´ − t − t where F and G are to the marginal products of capital in the production of the traded good and non- K K traded good respectively. 26Under this assumption and with the binding constraint, condition (38) becomes λt = (1+ζ t+1 )λt+1 which clearly does notpresenta unitroot. 19

3.3 The Determinacy of Equilibrium Analysis The definition of equilibrium in this set-up is the following. Definition 2 Given b , R , KN, KT and the depreciation target ¯(cid:18), a symmetric perfect foresight equi- ∗1 ∗ − librium is defined as a set of sequences cT, cN, ζ , hT, hN, I , b , mc , e , qT, qN, w , (cid:18) , πN, R { t t t t t t ∗t t t t t t t t t } ∞t=0 satisfying equations (1), (2), (27) with equality, (28)-(37) and (39). The methodology to pursue the determinacy of equilibrium analysis for this augmented model is the sameastheoneforthesimplemodel. Welog-linearizethesystemofequationsthatdescribetheequilibrium dynamicsaroundtheperfect-foresightsteadystateandcharacterizethedimensionoftheunstablesubspaceof thesystemcomparingittothenumberofnon-predeterminedvariables. Bylog-linearizingwearefollowingthe same approach that Kiyotaki and Moore (1997) adopt to solve for an equilibrium of a model with a binding collateral constraint. By doing this we are precluding the possibility of exploring non-linear equilibrium dynamics.27 Nevertheless the (log)linear approximation is what allows us to pursue the determinacy of equilibrium analysis for such a complex system. In this set-up the steady state is calculated taking into account that the collateral constraint is binding. But by construction the shadow price of this constraint at the steady state is equal to zero. To see this use (38) in tandem with β = 1 to obtain ζ¯ = 0.28 Nevertheless, in the short run the shadow price of the R ∗ collateral constraint may vary as ζ changes. When ζ >0 is high, then the collateral constraint tightens. t t It is not possible to derive analytical results for the log-linearized augmented model. Then we pursue some numerical simulations. To do so we need to choose some specific functional forms. For consumption and labor preferences we use U˜(cT,cN)= (α)1 a cT t a −a 1 +(1 − α)1 a cN t a −a 1 ( a − a 1 )(1 − σ) − 1 t t h ¡ ¢ 1 ¡σ ¢ i − H(hT)+L(hN)= ς hT 1+δ + hN 1+δ t t −1+δ t t h¡ ¢ ¡ ¢ i where α (0,1), ς,σ,a>0 and δ 0 whereas for technologies we utilize ∈ ≥ χ F hT t ,KT,I t = (cid:39) ϑ 1 (hT t )θT (KT)1 − θT χ −χ 1 +(1 − (cid:39))[ϑ 2 I t ] χ −χ 1 χ − 1 ½ ¾ ¡ ¢ h i 27Topursueadeterminacyofequilibrium analysistotheaugmentednon-linearmodelisaverydifficulttask. Infactmostof theaforementionedworksthatincludeacollateralconstraintandthatsimulateequilibriumdynamicsforthenon-linearsystem, do not characterize the equilibrium. They assume that the equilibrium that is found is the unique and relevant equilibrium whose propertiesshould be studied. 28Sinceζ¯=0then to beable to log-linearize thesystem ofequationsofDefinition 2,wedefine the new variable ζn 1+ζ t ≡ t whose steady state value is ζ¯n=1. 20

G hN,KN =(hN) θN (KN)1 θN t t − where θN,θT,(cid:39) (0,1) and ϑ ,ϑ ,χ> ¡ 0. ¢ 1 2 ∈ InordertoassignvaluestotheparametersofthemodelweusemainlythecalibrationofChristianoetal. (2004). Theonlyvaluesthatarenottakenfromthisworkaretheintratemporalelasticityofsubstitution(a), the parameter related to distribution services (η), the parameter that governs the degree of price stickiness (γ), and the parameter associated with the degree of imperfect competition (µ).29 We do not pick any particular value for the interest rate response coefficient to currency depreciation (ρ ) since we will study (cid:18) how this parameter affects the determinacy of equilibrium. We choose values for “a” and η that are in line with similar values used in the distribution services literature.30 Since there are no robust estimates of a New-Keynesian Phillips curve for emerging economies we choose values for γ and µ that are consistent with the values used in the closed economy literature about nominal price rigidities.31 Table 2 summarizes the parametrization. Table 2 R β R¯ γ µ η φ α a σ ∗ 1.06 0.943 1.16 17.5 6 0.85 0.185 0.7 0.4 2 ς δ (cid:39) ϑ ϑ θT χ θN KT KN 1 2 4.59 5 0.6 1.4 3.5 0.5 0.7 0.64 1 2 Usingthisparametrizationwecanstudyhowthedeterminacyofequilibriumvarieswithrespecttotheresponsecoefficienttodepreciation(ρ )ofthepolicyrule(2)andotherstructuralparameters. Asanillustrative (cid:18) casewefocusontheexperimentofcharacterizingtheequilibriumwhilewevarythedegreeofresponsiveness to current currency depreciation (ρ ) and the intratemporal elasticity of substitution (a) keeping the rest (cid:18) constant. The results are presented in Figure 1. In this Figure a cross “x” denotes combinations of these parameters under which the policy induces multiple cyclical equilibria whose degree of indeterminacy is of order one. On the other hand, a dot “.” represents parameter combinations under which the policy induces multiple cyclical equilibria whose degree of indeterminacy is of order two.32 FromFigure1wecanderivethefollowingconclusions. Intheaugmentedset-upapolicythatrespondsto currentcurrencydepreciationbyraising(ρ >0withρ =1)orlowering(ρ <0withρ = 1)thenominal (cid:18) (cid:18) 6 (cid:18) (cid:18) 6 − 29Notethattargetnominaldepreciationrate,¯(cid:18) canbefoundevaluating(29)atthesteadystate. Thatis,¯(cid:18)= R¯ .Thevalue ofR¯ thatwe take isclose to the one in Christiano etal. (2004). R∗ 30See Burnstein etal. (2005a,b). 31See Schmitt-Grohé and Uribe (2004)among others. 32The presence of cyclical equilibria is associated with the existence of non-explosive complex roots in the multidimensional system. 21

3 2 1 0 -1 -2 -3 0.5 1 1.5 2 2.5 3 3.5 Intratemporal Elasticity of substitution (a) ) ρ ( noitaicerpeD tnerruC ot ssenevisnopseR fo eergeD ε Contemporaneous Policies Multiple Cyclical Equilibria of Order 1 Multiple Cyclical Equilibria of Order 2 Multiple Cyclical Equilibria of Order 2 Multiple Cyclical Equilibria of Order 1 Figure 1: This Figure shows the characterization of the equilibrium for interest rate policies Rˆ = ρ ˆ(cid:18) varying t (cid:18) t the degree of responsiveness to currency depreciation (ρ ) and the intratemporal elasticity of substitution (a). It is (cid:18) assumed that ρ = 1,0,1. A cross “x” denotes parameter combinations under which the policy induces multiple (cid:18) 6 − cyclicalequilibriawhosedegreeofindeterminacyisoforderone. Adot“.”representsparametercombinationsunder which the policy induces multiple cyclical equilibria whose degree of indeterminacy is of order two. interestratecaninducemultipleequilibriaregardlessoftheintratemporalelasticityofsubstitution“a”. This suggests that the results in the augmented set-up are similar to the ones in the simple set-up. Nevertheless there is an important distinction. In the augmented model because of the binding collateral constraint, the previously mentioned policy leads to self-fulfilling “cycles” or, equivalently, to multiple “cyclical” equilibria. This should not be a surprise. It is just a consequence of two mechanisms working together. On one hand fromtheresultsinthesimplemodelwehavethatthispolicycaninduceself-fulfillingnon-cyclicalfluctuations. On the other hand from Kiyotaki and Moore (1997) we know that the introduction of a binding collateral constraintcancause“creditcycles”. Hencethecombinationofthetwomechanismscanleadto“self-fulfilling cyclical equilibria.” Experiments with respect to other structural parameters different from the intratemporal elasticity of substitution (a) lead to similar results.33 Based on this we state the following proposition. Proposition 2 Under a crisis when the collateral constraint is binding if the government follows a contemporaneous interest rate policy that responds to currency depreciation (Rˆ = ρ ˆ(cid:18) with ρ = 0 and either t (cid:18) t (cid:18) 6 ρ > 1 or ρ < 1), then there exists a continuum of “cyclical” perfect foresight equilibria in which the | (cid:18)| | (cid:18)| 33The resultsare available from the authorupon request. 22

sequences cˆT, cˆN, ζˆ , hˆT, hˆN, Iˆ, ˆb , mˆc , eˆ, qˆT, qˆN, wˆ , ˆ(cid:18) , πˆN, Rˆ converge asymptotically to the { t t t t t t ∗t t t t t t t t t } ∞t=0 steady state. Are these results specific to the type of interest rate policy we are considering? To answer this question wecanpursueasensitivityanalysisconsideringotherrules. Inparticularwecancharacterizetheequilibrium intheeconomyunderpoliciesthatrespondexclusivelytoeitherfuturedepreciation(forward-lookingpolicies Rˆ =ρ ˆ(cid:18) ) or past depreciation (backward-looking policies Rˆ =ρ ˆ(cid:18) ). The numerical results from this t (cid:18) t+1 t (cid:18) t 1 − analysis are presented in the Appendix. They imply that the answer to the previous question is no: as long as the interest rate policy responds to the depreciation rate, then the policy can induce multiple equilibria. Forward-looking policies Rˆ = ρ ˆ(cid:18) with ρ = 0 always induce multiple cyclical equilibria when either t (cid:18) t+1 (cid:18) 6 ρ >1 or ρ <1. Except for the presence of cycles these results still agree with the ones from the simple | (cid:18)| | (cid:18)| model. On the other hand, for backward-looking policies Rˆ =ρ ˆ(cid:18) with ρ =0, the coefficient of response to t (cid:18) t − 1 (cid:18) 6 pastdepreciationρ playsanimportantroleinthecharacterizationoftheequilibrium: timidrulessatisfying (cid:18) ρ <1 always induce multiple equilibria; aggressive rules with ρ >1 canguarantee aunique equilibrium. | (cid:18)| | (cid:18)| Nevertheless being aggressive with respect to past depreciation (ρ > 1) is not a sufficient condition to | (cid:18)| guarantee a unique equilibrium. It is only a necessary condition. In other words, backward-looking rules can still induce aggregate instability by generating self-fulfilling cyclical fluctuations. These results as well as the results for a contemporaneous policy Rˆ =ρ ˆ(cid:18) from Proposition 2 are summarized in Table 1 in the t (cid:18) t columns labeled as “The Augmented Model.” FromthisTablewecanobservethat,tosomeextent,thepossibilityofmultipleequilibriaarisesregardless ofwhetherthenominalinterestrateisraisedorloweredinresponsetoeithercurrent,futureorpastcurrency depreciation. Inthissenseourresultsdonotprovideanysupporttoanyofthepolicyrecommendationsthat were part of the debate about the right interest rate policy in the aftermath of the Asian crisis. They just pointoutsomeof thenegativeconsequencesof usingthenominal interestrateasanexclusiveinstrumentto respond to past, current or future currency depreciation. It is possible to argue that in the aftermath of a crisis governments may maneuver the nominal interest rate in response not only to currency depreciation but also to inflation. This poses the question of whether an interest rate policy that reacts to both the CPI-inflation and the depreciation rate will induce aggregate instability in the economy by generating multiple equilibria. The answer to this question is affirmative making our previous results stronger. It is the reaction to currency depreciation what opens the possibility ofmultipleequilibria. Toseethiswecanstudyarulethat,besidesreactingtocurrentcurrencydepreciation, responds aggressively and positively to past CPI-inflation. That is, in log-linearized terms Rˆ =ρ πˆcpi +ρ ˆ(cid:18) with ρ >1, ρ =0 and either ρ >1 or ρ <1 t π t − 1 (cid:18) t π (cid:18) 6 | (cid:18)| | (cid:18)| 23

Responding to Past CPI-Inflation and Current Depreciation 3 2 Multiple Equilibria of Order 1 (Cyclical or Non-Cyclical) 1 Multiple Equilibria of Order 1 (Cyclical or Non-Cyclical) 0 MCE (2) -1 -2 Multiple Equilibria of Order 1 (Cyclical or Non-Cyclical) -3 1 1.5 2 2.5 3 3.5 4 4.5 5 Degree of Responsiveness to Past CPI-Inflation (ρ ) π ) ρ ( noitaicerpeD tnerruC ot ssenevisnopseR fo eergeD ε Unique Equilibrium Unique Equilibrium Figure 2: This Figure shows the characterization of the equilibrium for the rule Rˆ = ρ πˆcpi + ρ ˆ(cid:18) varying t π t 1 (cid:18) t the degrees of responsiveness to past CPI-inflation (ρ ) and to current currency depreciation (ρ−). It is assumed π (cid:18) that ρ = 1,0,1. A cross “x” denotes parameter combinations associated with multiple equilibria whose degree (cid:18) 6 − of indeterminacy is of order one. These equilibria can be cyclical or non-cyclical. A dot “.” represents parameter combinations associated with multiple equilibria whose degree of indeterminacy is of order two. These equilibria are cyclical and we name these combinations as “MCE(2)”. The white regions represent parameter combinations under which there exists a unique equilibrium. The reason for analyzing this policy is that the literature of interest rate rules claims that an aggressive backward-looking rule with respect to inflation is more prone to guarantee a unique local equilibrium than forward-looking and contemporaneous rules.34 Hence by analyzing such a rule we can isolate and unveil the mechanism that opens the possibility of indeterminacy. As argued before this mechanism is associated with the response to currency depreciation. TheresultsoftheanalysisareshowninFigure2. Westudyhowvariationsofthedegreesofresponsiveness topastCPI-inflationandcurrentdepreciation,ρ andρ ,affectthecharacterizationoftheequilibrium. From π (cid:18) this figure it is clear that even backward-looking rules that respond aggressively with respect to the CPIinflation can induce multiple equilibria. In fact under the celebrated “Taylor coefficient” ρ = 1.5, any π rule will lead to real indeterminacy as long as the response to currency depreciation is positive or negative. Interestingly if the rule is positively aggressive with respect to current depreciation (ρ > 1) then it would (cid:18) be necessary to have an excessively aggressive rule with respect to inflation (ρ >5) to avoid the possibility π of self-fulfilling equilibria. 34See Woodford (2003). 24

Inthissectionwehavepointedoutsomepossibleperilsofrespondingtocurrencydepreciationbyraising or lowering the nominal interest rate in the aftermath of a crisis. But there is still a relevant question that has not been answered: if we believe that the governments of the Asian economies followed these interest rate policies then is it possible to support the stylized facts of the aftermath of the crisis as one of these self-fulfilling equilibria? An affirmative answer to this question will make our previous theoretical results more credible. The next section provides an answer to this question. 4 Constructing a Self-fulfilling Cyclical Equilibrium In this Section we use the augmented model in tandem with an interest rate policy that responds to currency depreciation in order to construct a self-fulfilling cyclical equilibrium that replicates some of the stylized facts of the aftermath of emerging crises. In particular we construct an equilibrium characterized by the self-validation of people’s expectations about currency depreciation and by some of the stylized facts of the “Sudden Stop” phenomenon. WeassumethegovernmentfollowsaninterestratepolicyRˆ =ρ ˆ(cid:18) ,thatrespondsaggressively(ρ >1) t (cid:18) t | (cid:18)| to current depreciation, ˆ(cid:18) .35 This policy captures the immediate reaction of the government to present t conditions about currency depreciation. Unfortunately in the empirical literature that emerged after the Asian crisis there are no robust estimates for the parameter ρ . As we discussed before this is one of (cid:18) the problems and drawbacks of the literature. For illustrative purposes we set ρ = 1.5 which implicitly (cid:18) reflectsthatgovernmenttendstoincreaseproportionallythenominalinterestratemorethantheincreasein currency depreciation. From Figure 1 we know that varying ρ will not preclude the possibility of multiple (cid:18) equilibriaalthoughitchangesthedegreeofindeterminacy. Sincewewanttoconstructaself-fulfillingcyclical equilibrium in which a sunspot affects exclusively people’s expectations about one variable of the economy such as currency depreciation, then we need a value of ρ that satisfies ρ > 1. This implies a degree of (cid:18) | (cid:18)| indeterminacy of order one.36 By choosing ρ = 1.5 and keeping the rest constant, we achieve this goal.37 (cid:18) As long as ρ >1, increasing or reducing ρ will not change the qualitative results that we will present and (cid:18) (cid:18) that capture some of the stylized facts of the “Sudden Stops.” Itisimportanttoclarifythatintheconstructionofourself-fulfillingequilibriumweareassumingexogenously the occurrence of a crisis. That is, the binding collateral constraint is exogenously imposed at time t = 0 as in Christiano et al. (2004). In this sense we are only interested in studying what happens in the economy at and in the aftermath of the crisis. In what follows, therefore, we concentrate exclusively in the 35Itisalsopossibletoconstructself-fulfillingequilibriawiththeforward-lookingandbackward-lookingrulesthatinprinciple replicate mostofthe stylized facts. 36Ifthe degree ofindeterminacy were 2,then we would have an extra degree offreedom. We could assume that the sunspot affectsthe expectationsofan extra variable differentfrom the depreciation rate. In this sense we are being conservative. 37Thatis,inthiscasethedynamiclog-linearizedsystemthatdescribestheeconomyhascomplexandnon-explosiveeigenvalues and the numberofnon-predetermined variables exceeds the numberofexplosive eigenvalues by one. 25

Nominal Depreciation Rate (ε) 20 15 10 5 0 -5 0 2 4 6 8 10 SS morf noitaiveD % Nominal Interest Rate (R) 30 20 10 0 -10 0 2 4 6 8 10 SS morf noitaiveD % Non-Traded Inflation (π N ) 25 20 15 10 5 0 -5 0 2 4 6 8 10 SS morf noitaiveD % Traded Inflation (π T ) 25 20 15 10 5 0 -5 0 2 4 6 8 10 SS morf noitaiveD % CPI-Inflation (π cpi ) 20 15 10 5 0 -5 0 2 4 6 8 10 SS morf noitaiveD % Traded Consumption (cT) 2 0 -2 -4 -6 -8 0 2 4 6 8 10 SS morf noitaiveD % Non-Traded Consumption (cN) 3 2 1 0 -1 -2 -3 0 2 4 6 8 10 SS morf noitaiveD % Aggregate Consumption (c) 2 0 -2 -4 -6 0 2 4 6 8 10 time (years) SS morf noitaiveD % Labor Costs (w.hT + w.hN ) 10 0 -10 -20 -30 0 2 4 6 8 10 time (years) SS morf noitaiveD % time (years) Figure 3: Impulse Responses of a self-fulfilling equilibrium when at time t = 0 people expect a higher nominal depreciationrateˆ(cid:18) =20%.Thefigureshowsthatthisequilibriumreplicatesadeclineofconsumptionoftradedand 0 non-tradedgoodsthatispresentina“SuddenStop”. ItalsoreplicatesarelativelymoderateincreaseinCPI-inflation, ahigherdepreciationrateandahighernominalinterestrate. Allthevariablesaremeasuredaspercentagedeviations from the steady state. equilibrium dynamics of the economy at and after t=0. Imagine that when the crisis hits the economy and the collateral constraint binds at time t = 0, people develop expectations, in response to a sunspot, of a 20% higher nominal depreciation. By the determinacy of equilibrium analysis we know that these expectations will be self-validated. More importantly these expectationswillcauseachainofeventsthataredescribedbytheimpulseresponsefunctionsofsomeofthe macroeconomic variables presented in Figures 3 and 4. In these figures all the variables but the multiplier of the collateral constraint are measured as percentage deviations from the steady state. A quick inspection reveals that at time t = 0 and for some subsequent periods, the self-fulfilling equilibrium captures the following stylized facts of the “Sudden Stops”: a decline in the aggregate demand (consumptions of traded andnon-tradedgoodsandaggregateconsumption),acollapseinthedomesticproduction(tradedoutputand 26

Multiplier of the Collateral Constraint 0.15 0.1 0.05 0 -0.05 0 2 4 6 8 10 level Imported Intermediate Input (I) 5 0 -5 -10 -15 -20 -25 0 2 4 6 8 10 SS morf noitaiveD % Marginal Cost (mc) 25 20 15 10 5 0 -5 0 2 4 6 8 10 SS morf noitaiveD % Traded Output (yT) 5 0 -5 -10 -15 0 2 4 6 8 10 SS morf noitaiveD % Non-Traded Output (yN) 2 0 -2 -4 -6 0 2 4 6 8 10 SS morf noitaiveD % Current Account Deficit 2 1 0 -1 -2 -3 -4 0 2 4 6 8 10 SS morf noitaiveD % Price of Traded Goods Relative to Non-traded Goods 40 30 20 10 0 -10 0 2 4 6 8 10 SS morf noitaiveD % Price of Traded Capital (qT) 10 0 -10 -20 -30 0 2 4 6 8 10 time (years) SS morf noitaiveD % Price of Non-Traded Capital (qN) 10 0 -10 -20 -30 0 2 4 6 8 10 time (years) SS morf noitaiveD % time (years) Figure 4: Impulse Responses of a self-fulfilling equilibrium when at time t = 0 people expect a higher nominal depreciation rateˆ(cid:18) = 20%. The figure shows that this equilibrium replicates some of the “Sudden Stops” stylized 0 facts: a collapse in the domestic production (of the traded and non-traded good), a collapse in asset prices (prices of capital), a sharp correction in the price of traded goods relative to non-traded goods and an improvement in the current account deficit. All the variables are measured as percentage deviations from the steady state. non-traded output), a collapse in asset prices (prices of traded and non-traded capital), a sharp correction in the price of traded goods relative to non-traded goods, an improvement in the current account deficit, a moderatehigherCPI-inflation,ahighercurrencydepreciationrateandahighernominalinterestrate. After some periods the cycles are quickly dampened and the economy converges to the steady state. To provide an explanation of these results is not an easy task given the interaction and interdependence of all the variables. Nevertheless we can give the following logical arguments. The self-validated increase in the nominal depreciation rate implies an increase in the nominal exchange rate and, by the rule, leads to an increase in the nominal interest rate at t=0. Since the rule is aggressive, the nominal interest rate rises by more than both the expected non-traded good inflation rate and the expected traded good inflation rate at time t=1.38 Hence the real interest rate at t=0 measured in terms of either the expected traded inflation 38Thisisprobablythecasebecausethereissluggishpriceadjustmentforthepriceofthenon-tradedgoodthatinturnaffects 27

or the non-traded inflation at t=1 goes up. Provided that this induces an intertemporal substitution effect inconsumptionthatmorethanoffsetsanyintratemporalsubstitutioneffect,thenconsumptionofthetraded good and consumption of the non-traded good decline at t=0. This can be inferred from (31) and (32). As a result of this, aggregate consumption also decreases implying, to some extent, that the model is able to capture the initial decline in aggregate demand present in the “Sudden Stops.” Sincetherealvalueofcapitalasacollateralisexpressedintermsofforeigncurrencythenthepreviously mentioned increase of the nominal exchange rate at t = 0 reduces this value. Hence, by the collateral constraint, less international loans will be available to buy the intermediate imported input. As long as the costs of hiring labor rise, due to a higher interest rate, then a shortage in international loans will be translated into a reduction in the demand for the imported input at t =0. This can be deduced from (27) taking into account that b is a predetermined variable. ∗t 1 − At the same time, a higher interest rate in response to currency depreciation will also push up the costs of loans to hire labor utilized in the production of traded goods. This implies that the household-firm unit willcutbacklaborintheproductionofthetradedgoodwhichintandemwiththereductionintheimported input leads to a decrease in output at t = 0.39 On the other hand the supply of the non-traded good is demand determined. This supply satisfies consumption of non-traded goods and distribution services for traded goods. Consequently the previously mentioned decrease in demand of both goods causes a decrease in non-traded output (labor) at t=0. Thus the model is able to capture the decline in output of both nontraded and traded goods. And provided that the decrease in traded output is smaller than the contractions in consumption of the traded good and the imports of the intermediate input then an improvement in the current account is also possible as can be inferred from (34). As the collateral constraint tightens and the nominal interest rate rises then the “effective” nominal interest rate (1+ζ )R increases. This pushes marginal costs of producing the non-traded good up forcing t t the household-firm unit to raise the price of this good; which in turn leads to a higher non-traded goods inflation rate as can be deduced from (36) and (33). To some extent price-stickiness will guarantee that the increase in non-traded inflation is smaller than the increase the depreciation rate. This together with distribution services has two important consequences. First as a consequence of large depreciations there will be a sharp correction in the price of the traded good relative to the price of non-traded good at the consumer level P t T . Second in the short run the consumer-price-index (CPI) inflation will be below the PN t depreciation rat³e. Bo´th facts agree with some empirical regularities present in the aftermath of a crisis. The real value of a unit of capital for traded output (respectively for non-traded output) in terms of foreign currency is determined by the net present value of the flows of the marginal product of capital in the price ofthe traded good through the existence ofdistribution services. 39Thiscanbededucedfrom(35)andtheproductiontechnologyofthetradedgood. Laborcostsincreasenotonlybecausethe nominalinterestrateincreasesbutalsobecausethecollateralconstrainttightens. Thatisthereisanincreaseinthe“effective” nominalinterestrate (1+ζ t )Rt. 28

the production of the traded good (respectively non-traded good). This can be seen by iterating forward equations (39). Then provided that capital is constant in the analysis, the reduction of labor and the intermediateinputwillcauseadeclineinthemarginalproductofcapital. Thisinturnwillaffectnegatively the real value of capital for traded output. A similar mechanism will lead to a decrease in the real value of capital for non-traded output. Thus asset prices fall capturing another stylized fact of the “Sudden Stops.” 5 Concluding Remarks In this paper we show that in the aftermath of a crisis, a government that changes the nominal interest rateinresponsetocurrencydepreciationcaninduceaggregateinstabilityintheeconomybygeneratingselffulfillingendogenouscycles. Inthissenseapolicythatoriginallyattemptstostabilizethenominalexchange rate and the whole economy leads to complete opposite effects. Wealsoshowthatifagovernmentraisestheinterestrateproportionallymorethananincreaseincurrency depreciation then it induces self-fulfilling cyclical equilibria that are able to replicate some of the empirical regularities of emerging market crises. In fact we construct an equilibrium based on the self-validation of people’s expectations about currency depreciation that is able to replicate the following “Sudden Stop” stylized facts: a decline in domestic production and aggregate demand, a moderate higher CPI-inflation, a significantly larger currency depreciation, a higher nominal interest rate, a collapse in asset prices, a sharp correction in the price of traded goods relative to non-traded goods and an improvement in the current account deficit. The implications of our results are interesting. Previous works have emphasized that this interest rate policy can cause fiscal and output costs. Our results suggest that this policy can be also costly to the extent that it can induce macroeconomic instability in the economy by opening the possibility of “sunspot” equilibria. These equilibria that are not driven by fundamentals can be associated with a large degree of volatility for some macroeconomic aggregates such as consumption. Provided that agents are risk averse then volatile consumption will cause a decrease in welfare. Our results also provide a possible explanation of why the empirical literature has not been able to disentangle the relationship between interest rates and the nominal exchange (depreciation) rate in the aftermathofacrisis. Thisliteraturehastriedtocontrolforthevariablesthatinfluencethenominalexchange rate. Butourresultssuggestthattherecanbepotentialinfluencesthatmaydependon“sunspots”whichin turn can induce self-fulfilling cycles in the nominal exchange rate (or the nominal depreciation rate) as well as in other variables. Clearly these influences do not depend on fundamentals and their effect is something that the empirical literature should take into account. Anissuethatwehavenotaddressediswhetherthe(self-fulfilling)sunspotequilibriaarelearnablebythe agentsoftheeconomy. Itwasimplicitlyassumedthatagentscouldcoordinatetheiractionsonanyparticular 29

equilibrium. WecouldrelaxthisassumptionandusetheExpectationalStabilityconceptdevelopedinEvans and Honkapojha (2001) to pursue a learnability analysis. We leave this for future research. A Appendix This Appendix has two parts. The first part includes material that supports the analysis for the simple model of Section 2. The second part includes the simulations of the determinacy of equilibrium analysis for forward-looking and backward-looking rules for the augmented model in Section 3. A.1 The Simple Model A.1.1 The First Order Conditions of the Household-Firm Unit Problem in the Simple Model The representative household-firm unit chooses the set of sequences cT, cN, hT, hN, hˇT, h˜N, P˜N, b , { t t t t t t t ∗t b , m in order to maximize (3) subject to (4), (5), (6) and (7), given the initial condition n and the t t } ∞t=0 − 1 set of sequences {R , R , (cid:18) , e , PN, w , τ , CN . The first order conditions correspond to (5) and (7) with t∗ t t t t t t t } equality and U (cT)=λ (40) T t t U (cT) T t =e (41) V (cN) t N t H (hT) T t =w (42) −U (cT) t T t L (hN) N t =w e (43) −V (cN) t t N t w 1= t (44) F hˇT T t ¡ ¢ w e mc = t t (45) t G h˜N N t ³ ´ λ CN P˜N λ P˜N P˜N P˜N λ CN P˜N 0 = t t d t γ t t π¯N t + t mc t t d t (46) e t à P t N ! − e t ÃP˜ t N 1 − !P˜ t N 1 à P t N − t ! e t 0 à P t N ! − − λ P˜N P˜N +βγ t+1 t+1 π¯N t+1 e t+1 à P˜ t N − ! P˜ t N 30

R 1 J (m )=U (cT) t − (47) m t T t R µ t ¶ λ =βR λ (48) t t∗ t+1 βR λ = tλ (49) t (cid:18) t+1 t+1 where mc e t t λt and λ t correspond to the Lagrange multipliers of (4) and (5) respectively. We will focus on a symmetric equilibrium in which all the monopolistic producers of sticky-price nontraded goods pick the same price. Hence P˜N =PN. Since all the monopolists face the same wage rate, W , t t t and the same production function, G hN , then they will demand the same amount of labor h˜N =hN. In t t t equilibrium the money market, the do¡mes¢tic bond market, the labor markets, the non-traded goods market and the traded good market clear. Therefore m =mg (50) t t b =bg (51) t t hT =hˇT (52) t t hN =h˜N (53) t t G hN =cN + γ πN π¯N 2 (54) t t 2 t − ¡ ¢ ¡ ¢ and b =R b +cT F hT (55) ∗t t∗ − 1 ∗t − 1 t − t ¡ ¢ Combining (48) and (49) yields the uncovered interest parity condition (9). From (42) and (44) we can derive equation (10). Using conditions (40) and (48) we obtain the Euler equation for consumption of the traded good that corresponds to (11). Utilizing (1), (40), (41), and (49) we derive the Euler equation (12) forconsumptionofthenon-tradedgood. Andfinallyusingthenotionofasymmetricequilibrium,conditions (4), (41), (43), (45), (53), (46) and the definitions πN = PN/PN , d(1) = 1 and d(1) = µ we can derive t t t − 1 0 − the augmented Phillips curve described by equation (13). A.1.2 Characterization of the Steady State in the Simple Model We use βR =1, (1), (8)-(15), and the condition that at the steady state x =x¯ for all the variables to ∗ t derive π¯N =¯(cid:18) βR¯ =¯(cid:18) 1=βR¯ R¯ =R ∗ ∗ ∗ µ 1 L (h¯N) − V (G(h¯N))= N µ N −G h¯N µ ¶ N U (1 1/β)¯b +F h¯T = H¡T (h¯ ¢T) T − ∗ −F h¯T T ¡ ¡ ¢¢ ¡ ¢ 31

c¯N =G h¯N c¯T =(1 1/β)¯b +F h¯T ∗ − ¡ ¢ ¡ ¢ ThenitissimpletoprovethatundersomeassumptionsthatincludeAssumptions1,2,and3,andgivenµ>1, β (0,1)and¯(cid:18)>1,thereexistsasteadystate c¯N,c¯T,h¯T,h¯N,π¯N, R¯,R¯ fortheeconomythatprovideda ∗ ∈ { } particular¯b satisfies these equations with c¯T,c¯N,h¯T,h¯N >0 and π¯N,R¯,R¯ >1. In particular to guarantee ∗ ∗ that there exist c¯T,c¯N,h¯T,h¯N > 0 we need some Inada-type assumptions such as U (0) = V (0) = , T N ∞ U ( )=V ( )=0, F (0)=G (0)= , and F ( )=G ( )=0.40 T N T N T N ∞ ∞ ∞ ∞ ∞ A.1.3 Forward-Looking and Backward-Looking Rules in the Simple Model Inordertopursue thedeterminacyof equilibriumanalysisforforward-lookingrules, Rˆ =ρ ˆ(cid:18) , weuse t (cid:18) t+1 this rule and equations (17)-(21) to obtain ˆb 1+ψ κ 0 0 ˆb ∗t β ∗t 1  cˆT   ψ(1+ψ)ξT 1+ψκξT 0 0  cˆ − T  t+1 β t = (56)  πˆN   0 ³ 0 ´ 1 ϕ  πˆN    t+1     β −     t      cˆN t+1       ρ (cid:18) ψ β( ( ρ 1 (cid:18) + − ψ 1 ) ) ξN ρ ( (cid:18) ρ ψ (cid:18)− κξ 1 N ) − ξ β N 1+ϕξN       cˆN t       ³ ´   Jf Then the determinacy of e|quilibrium analysis deliv{ezrs the results stated in th}e following Proposition. Proposition 3 If the government follows a forward-looking interest rate rule such as Rˆ = ρ ˆ(cid:18) with t (cid:18) t+1 ρ = 0 and either ρ > 1 or ρ < 1, then there exists a continuum of perfect foresight equilibria in which (cid:18) 6 | (cid:18)| | (cid:18)| the sequences ˆb , cˆT, πˆN, cˆN converge asymptotically to the steady state. The degree of indeterminacy { ∗t t t t } ∞t=0 is of order 1.41 Proof. TheeigenvaluesofthematrixJf in(56)correspondtotherootsofthecharacteristicpolynomial f(v) = Jf vI = 0 whose definition is provided in (59). By Lemma 4 we know that the characteristic P − polynomia¯l f(v)¯= 0 has real roots satisfying v 1 < 1, v 2 > 1, v 3 < 1 and v 4 > 1. Therefore f(v) = ¯ P ¯ | | | | | | | | P 0 has only two explosive roots, which means that the matrix Jf in (56) has two explosive eigenvalues. Given that there are three non-predetermined variables namely cˆT, πˆN and cˆN, then the number of nont t t predeterminedvariablesisgreaterthanthenumberofexplosiveroots. ApplyingtheresultsofBlanchardand Kahn (1980) it follows that there exists an infinite number of perfect foresight equilibria converging to the steady state. In addition the difference between the number of non-predetermined variables and explosive roots implies that the degree of indeterminacy is of order 1. Toderivetheresultsforbackward-lookingpoliciesweusethelog-linearizedversionoftherule,Rˆ =ρ ˆ(cid:18) t (cid:18) t 1 − together with equations (20)-(17) to obtain the log-linearized system 40Detailsare available from the authorupon request. 41The degree of indeterminacy is defined as the difference between the number of non-predetermined variables and the dimension ofthe unstable subspace ofthe log-linearized system. 32

Rˆ 0 ρ 0 0 0 0 Rˆ t+1 (cid:18) t  ˆ(cid:18)   1 0 ψ(1+ψ) ψκ 0 0  ˆ(cid:18)  t+1 − β − t  ˆb   0 0 1+ψ κ 0 0  ˆb   ∗t   β  ∗t 1    cˆT   =  0 0 ψ(1+ψ)ξT 1+ψκξT 0 0     cˆ − T   (57)  t+1   β  t        πˆN   0 0 0 ³ 0 ´ 1 ϕ  πˆN    t+1     β −     t    cˆN   ξN 0 0 0 ξN 1+ϕξN  cˆN    t+1     − β     t      ³ ´   Jb We use this system to pursue the determinacy of equilibrium analysis and derive the results in the | {z } following proposition. Proposition 4 Assume the government follows a backward-looking rule such as Rˆ =ρ ˆ(cid:18) with ρ =0. t (cid:18) t − 1 (cid:18) 6 a) if ρ > 1 then there exists a unique perfect foresight equilibria in which the sequences Rˆ , ˆ(cid:18) , | (cid:18)| { t t ˆb , cˆT, πˆN, cˆN converge to the steady state. ∗t t t t } ∞t=0 b) if ρ < 1 then there exists a continuum of perfect foresight equilibria in which the sequences | (cid:18)| Rˆ , ˆ(cid:18) , ˆb , cˆT, πˆN, cˆN converge asymptotically to the steady state. In addition the degree of { t t ∗t t t t } ∞t=0 indeterminacy is of order 2. Proof. TheeigenvaluesofthematrixJb in(57)correspondtotherootsofthecharacteristicpolynomial b(v)= Jb vI =0. Using the definition of Jb in (57) this polynomial can be written as P − ¯ ¯ ¯ ¯ b(v)= v2 ρ f(v)=0 (58) P − (cid:18) P ¡ ¢ where f(v) is defined in (59). By Lemma 4 we know that the characteristic polynomial f(v) = 0 has P P real roots satisfying v < 1, v > 1, v < 1 and v > 1. The fifth and the sixth roots of b(v) = 0 are 1 2 3 4 | | | | | | | | P v 5 =√ρ (cid:18) and v 6 = − √ρ (cid:18) . Clearly if | ρ (cid:18)| >1 then | v 5 | >1 and | v 6 | >1 whereas if | ρ (cid:18)| <1 then | v 5 | <1 and v < 1. Therefore using this, the characterization of the roots of f(v) = 0 and (58) we can conclude the 6 | | P following. If ρ >1then b(v)=0hasfour explosiveroots namely v >1, v >1, v >1 and v >1. | (cid:18)| P | 2 | | 4 | | 5 | | 6 | While if ρ <1 then b(v)=0 has two explosive roots namely v >1 and v >1. Therefore if ρ >1 | (cid:18)| P | 2 | | 4 | | (cid:18)| the number of explosive roots is equal to the number of non-predetermined variables (ˆ(cid:18) , cˆT, πˆN and cˆN). t t t t Hence applying the results of Blanchard and Kahn (1980) it follows that there exists a unique equilibrium. This completes the proof for a). On the contrary by the previous analysis if ρ < 1 the number of explosive roots, 2, is less than the | (cid:18)| number of non-predetermined variables (ˆ(cid:18) , cˆT, πˆN and cˆN),4. Applying the results of Blanchard and Kahn t t t t (1980) it follows that there exists an infinite number of perfect foresight equilibria converging to the steady state. Thedegreeofindeterminacyisthedifferencebetweenthenumberofnon-predeterminedvariablesand the number of explosive roots. This completes the proof for b). 33

A.1.4 Lemmata and Proofs for the Results in the Simple Model Lemma 1 Consider the characteristic polynomial (v)=v2 +Tv +D = 0. If either a) P(1) < 0 or b) P P( 1)<0 then the roots are real. − Proof. FirstrecallfromAzariadis(1993)thatasufficientconditiontohaverealrootsisthatT2 4D 0. − ≥ To prove a) note that (1) < 0 means that (1) = 1 T +D < 0. But this implies that 4T 4 > 4D P P − − which in turn leads to T2 4D>T2 4T +4=(T 2)2 0. Hence the roots are real. Next we prove b). − − − ≥ ( 1) < 0 means that (1) = 1+T +D < 0. But this implies that 4T 4 > 4D that in turn leads to P − P − − T2 4D>T2+4T +4=(T +2)2 0. Hence the roots are real. − ≥ Lemma 2 The roots of the characteristic polynomial 1+ψ 1+ψ (v)=v2 1+ +ψκξT v+ =0 P bcT − β β µ ¶ are real and satisfy v <1 and v >1. 1 2 | | | | Proof. First using the definition of (v), Assumptions 1, 2 and 3 and definitions in (22) we obtain bcT P that (1)= ψκξT <0 bcT P − 1+ψ ( 1)=2 1+ +ψκξT >0 P bcT − β µ ¶ Since (1) < 0 then by Lemma 1 we know that the two roots are real. In addition from Azariadis bcT P (1993), having (1)<0 and ( 1)>0 imply that one root lies inside of the unit circle and the other bcT bcT P P − one lies outside the unit circle. Without loss of generality we can conclude that υ <1 and υ >1. 1 2 | | | | Lemma 3 The roots of the characteristic polynomial 1 1 (v)=v2 1+ +ϕξN v+ =0 P πNcN − β β µ ¶ are real and satisfy v <1 and v >1. 3 4 | | | | Proof. First using the definition of (v), Assumptions 1, 2 and 3 and definitions in (22) we obtain πNcN P that: ( 1) = ϕξN < 0 and (1) = 2 1+ 1 +ϕξN > 0. Since ( 1) < 0 then by P πNcN − − P πNcN β P πNcN − Lemma 1 we know that the two roots of (v) =³0 are r´eal. In addition from Azariadis (1993), having πNcN P ( 1) < 0 and (1) > 0 imply that one root lies inside of the unit circle and the other one lies πNcN πNcN P − P outside the unit circle. Without loss of generality we can conclude that that υ <1 and υ >1. 3 4 | | | | Lemma 4 The roots of the characteristic polynomial f(v)= (v) (v)=0 (59) bcT πNcN P P P 34

where 1+ψ 1+ψ (v)=v2 1+ +ψκξT v+ P bcT − β β µ ¶ 1 1 (v)=v2 1+ +ϕξN v+ P πNcN − β β µ ¶ are real and satisfy v <1, v >1, v <1 and v >1. 1 2 3 4 | | | | | | | | Proof. ByLemma2weknowthatthecharacteristicpolynomial (v)=0hastworealrootssatisfying bcT P v <1and v >1.OntheotherhandbyLemma3weknowthatthecharacteristicpolynomial (v)= 1 2 πNcN | | | | P 0 has two real roots satisfying v < 1 and v > 1. Therefore from these Lemmata and the definition of 3 4 | | | | f(v)=0 the result of the Lemma follows. P A.2 The Augmented Model A.2.1 Forward-Looking and Backward-Looking Policies in the Augmented Model We pursue the determinacy of equilibrium analysis for forward-looking policies (Rˆ =ρ ˆ(cid:18) ) and backt (cid:18) t+1 ward looking policies (Rˆ = ρ ˆ(cid:18) ). We use the parametrization of Table 2, and as an illustrative case t (cid:18) t 1 − we focus on the experiment of characterizing the equilibrium while we vary the degree of responsiveness to future (past) currency depreciation (ρ ) and the intratemporal elasticity of substitution (a) keeping the rest (cid:18) constant. TheresultsarepresentedinFigure5. Thetoppanelshowstheresultsforaforward-lookingpolicy whereas the bottom panel presents the results for backward-looking policies. Fromthetop-panelwecaninferthatforward-lookingpoliciesalwaysinducemultiplecyclicalequilibriaas long as ρ =0 and either ρ >1 or ρ <1. On the other hand, for backward-looking rules, the coefficient (cid:18) 6 | (cid:18)| | (cid:18)| of response to past depreciation, ρ , plays an important role in the characterization of the equilibrium. In (cid:18) particulartimidruleswithrespecttopastdepreciation(ρ <1) alwaysinducemultipleequilibriaregardless | (cid:18)| of the intratemporal elasticity of substitution (a). While aggressive rules (ρ >1) can guarantee a unique | (cid:18)| equilibrium. Nevertheless being aggressive with respect to past depreciation (ρ > 1) is not a sufficient | (cid:18)| condition to guarantee a unique equilibrium. It is only a necessary condition. As before varying other structural parameters different from the intratemporal elasticity of substitution (a) in tandem with ρ lead to similar results.42 The following proposition summarizes these results. (cid:18) Proposition 5 Under a crisis when the collateral constraint is binding, 1. If the government follows a forward-looking policy (Rˆ = ρ ˆ(cid:18) with ρ = 0 and either ρ > 1 or t (cid:18) t+1 (cid:18) 6 | (cid:18)| ρ <1), then there exists a continuum of “cyclical” perfect foresight equilibria in which the sequences | (cid:18)| cˆT, cˆN, ζˆ , hˆT, hˆN, Iˆ, ˆb , mˆc , eˆ, qˆT, qˆN, wˆ , ˆ(cid:18) , πˆN, Rˆ converge asymptotically to the steady { t t t t t t ∗t t t t t t t t t } ∞t=0 state. 42The resultsare available from the authorupon request. 35

3 2 1 0 -1 -2 -3 0.5 1 1.5 2 2.5 Intratemporal Elasticity of substitution (a) ) ρ ( noitaicerpeD erutuF ot ssenevisnopseR fo eergeD ε Forward-Looking Policies Multiple Cyclical Equilibria of Order 1 Backward-Looking Policies 3 Unique Equilibrium 2 Multiple Equilibria of Order 1 1 0 Multiple Equilibria of Order 2 (Cyclical or Non-Cyclical) -1 Multiple Cyclical Equilibria of Order 2 -2 Unique Equilibrium -3 0.5 1 1.5 2 2.5 Intratemporal Elasticity of substitution (a) ) ρ ( noitaicerpeD tsaP ot ssenevisnopseR fo eergeD ε Multiple Cyclical Equilibria of Order 1 Multiple Cyclical Equilibria of Order 1 Figure 5: This Figure shows the characterization of the equilibrium for forward-looking (top-panel) and backwardlooking (bottom-panel) policies varying the degree of responsiveness to currency depreciation (ρ ) and the intratem- (cid:18) poral elasticity of substitution (a). It is assumed that ρ = 1,0,1. A cross “x” denotes parameter combinations (cid:18) 6 − underwhichthepolicyinducesmultipleequilibriawhosedegreeofindeterminacyisoforderone. Adot“.”represents parametercombinationsunderwhichthepolicyinducesmultipleequilibriawhosedegreeofindeterminacyisoforder two. IntheFigureitisalsospecifiedwhetherthesemultipleequilibriaarecyclicalornon-cyclical. Thewhiteregions represent parameter combinations under which there exists a unique equilibrium. 2. If the government follows a backward-looking policy (Rˆ t =ρ (cid:18) ˆ(cid:18) t 1 with ρ (cid:18) ≶0), − a) if ρ <1 then there exists a continuum of perfect foresight equilibria, possibly “cyclical”, in which | (cid:18)| the sequences cˆT, cˆN, ζˆ , hˆT, hˆN, Iˆ, ˆb , mˆc , eˆ, qˆT, qˆN, wˆ ,ˆ(cid:18) , πˆN, Rˆ converge asymptotically { t t t t t t ∗t t t t t t t t t } ∞t=0 to the steady state. b) ρ > 1 is a necessary but not sufficient condition for the existence of a unique perfect foresight | (cid:18)| equilibriumwherethesequences cˆT,cˆN,ζˆ ,hˆT,hˆN,Iˆ,ˆb ,mˆc ,eˆ,qˆT,qˆN,wˆ ,ˆ(cid:18) ,πˆN, Rˆ converge { t t t t t t ∗t t t t t t t t t } ∞t=0 asymptotically to the steady state. 36

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