Self-Fulfilling Currency Crises: the Role of Interest Rates

Self-Fulﬁlling Currency Crises: The Role of Interest Rates

By CHRISTIAN HELLWIG,ARIJIT MUKHERJI, AND ALEH TSYVINSKI*

We develop a model of currency crises, in which traders are heterogeneously informed, and interest rates are endogenously determined in a noisy rational expectations equilibrium. In our model, multiple equilibria result from distinct roles an interest rate plays in determining domestic asset market allocations and the devaluation outcome. Except for special cases, this ﬁnding is not affected by the introduction of noisy private signals. We conclude that the global games results on equilibrium uniqueness do not apply to market-based models of currency crises. (JEL D84, E43, F32)

It is a commonly held view that financial parity equates the interest premium on domestic crises, such as currency crises, bank runs, debt assets to the expected currency depreciation. crises, or asset price crashes, may be the result Multiple equilibria arise when there are multi- of self-fulfilling expectations and multiple equi- ple values for the domestic interest rate pre- libria. For currency crises, this view is formu- mium that are consistent with uncovered interest lated in two broad classes of models. The first, parity. In one equilibrium, the interest rate tends pioneered by Maurice Obstfeld (1986), views a to be low, reserve losses are small, and there is devaluation as the outcome of a run on the a low likelihood of a devaluation. In another central bank’s stock of foreign reserves; in the equilibrium, interest rates are high, reserve second class of models, starting with Obstfeld losses are large, and there is a high chance of a (1996), devaluations are the result of the central devaluation. The sudden shifts in financial mar- bank’s inability or unwillingness to sustain the kets that characterize currency crises are then political or economic costs associated with high interpreted as a shift from one equilibrium to interest rates. In both types of environments, another.1 domestic asset markets play a central role. In Building on game-theoretic selection results the market equilibrium, an uncovered interest of Hans Carlsson and Eric E. van Damme (1993), this multiplicity view of crises has re- cently been challenged by Stephen Morris and Hyun Song Shin (1998), who argue that multi- * Hellwig: Department of Economics, University of Cali- fornia, Los Angeles, CA 90095 (e-mail: [email protected]); plicity is the consequence of assuming that fun- Mukherji: University of Minnesota; Tsyvinski: Department of damentals are common knowledge among Economics, Harvard University, Cambridge, MA 02138 market participants. Instead, they consider a (e-mail: [email protected]). This paper originated from stylized currency crises model, in which traders an earlier paper written by the late Arijit Mukherji and Aleh Tsyvinski. We thank two anonymous referees for excellent observe the relevant fundamentals with small comments and suggestions. We further thank Marios Angele- idiosyncratic noise, and show that this leads to tos, V. V. Chari, Pierre-Olivier Gourinchas, Patrick Kehoe, the selection of a unique equilibrium. Stephen Morris, Ivań Werning, and especially Andy Atkeson While their analysis highlights the critical role for useful discussions, and seminar audiences at UC-Berkeley, of the information structure for coordination Boston College, Boston University, Cambridge University, Harvard University, London School of Economics, Northwest- ern University Kellogg School of Management, Penn State, UC-Santa Cruz, Stanford University, UCLA, the Wharton 1 For related analyses of bank runs, debt crises, or asset School of the University of Pennsylvania, the Federal Reserve price crashes, see Douglas W. Diamond and Philip H. Bank of Minneapolis, the International Monetary Fund, the 1st Dybvig (1983), Guillermo A. Calvo (1988), Gerard Gen- Christmas Meetings of German Economists Abroad, the 2005 notte and Hayne E. Leland (1990), Harold L. Cole and SED meetings, and the 2005 World Congress of the Econo- Timothy J. Kehoe (2000), Gadi Barlevy and Pietro Veronesi metric Society for their comments. (2003), and many others.

1769 1770 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006 and multiplicity of equilibria, they also need to smaller supply of domestic bonds leads to a assume that the domestic interest rate premium higher equilibrium interest rate, and a larger is not market-determined, but exogenously loss of foreign reserves by the central bank.2 fixed. A currency crisis is then viewed as a Second, the domestic interest rate may influence coordinated run on foreign reserves, when the the devaluation outcome: a devaluation of the value of the domestic currency is out of line domestic currency may occur either because of with fundamentals. Their model, thus, not only large foreign reserve losses, or because of in- introduces a lack of common knowledge, but creases in the domestic interest rate (or a combi- also abstracts from an explicit model of domes- nation of both); our model thus embeds both tic asset markets. Moreover, it abstracts from Obstfeld (1986) and Obstfeld (1996) as special the interest parity condition that appears to be so cases. Finally, with heterogeneous beliefs, the do- central to many of the original multiple equilib- mestic interest rate serves as an endogenous pub- rium models. lic signal which aggregates private information. In this paper, we propose a new model of Our equilibrium characterization augments currency crises that allows for informational the optimality conditions for trading strategies differences among traders, while at the same and the devaluation outcome by a market- time accounting for the market forces of the clearing condition that determines interest rates original models. We use this model to reex- as a function of the underlying fundamentals amine the respective roles of the information and supply shocks. Combining these conditions, structure and the market characteristics in de- we arrive at a private information version of the termining uniqueness versus multiplicity; to do uncovered interest parity condition that deter- so, we compare the solution of our model mined equilibrium outcomes in the original when fundamentals are common knowledge multiple equilibrium models with common with the solution when traders have idiosyn- knowledge. cratic, noisy signals. As our main result, we In our model, optimal trading strategies are show that when domestic asset markets and always uniquely determined. Unlike in Morris interest rates are modeled explicitly, argu- and Shin (1998), multiplicity therefore does not ments for multiplicity remain valid even in originate from a coordination problem; instead, the presence of incomplete, heterogeneous in- it arises if there are multiple market-clearing formation. We conclude from this that the interest rates. If traders are sufficiently well simple global coordination game studied by informed, an increase in the interest rate may Morris and Shin does not fully capture the raise the expected devaluation premium by complex market interactions by which cur- more than the domestic bond return. This im- rency crises are characterized. Specifically, we consider a stylized model of a country’s domestic bond market with 2 This reflects the idea that an interest rate increase heterogeneously informed traders, who may reduces the amount of borrowing in domestic currency and either invest domestically or withdraw their leads to a capital outflow. Although we do not attempt to funds and invest in dollars. The interest rate on formally model this positive correlation between domestic interest rates and capital outflows, different motivations domestic bonds is endogenously determined in may be provided in the context of currency crises models. In a noisy rational expectations equilibrium along Obstfeld (1986), it results from inflationary expectations the lines of Sanford J. Grossman (1977), Gross- following a devaluation. In the presence of nominal rigidi- man and Joseph E. Stiglitz (1980), and Martin ties, our assumption may also be motivated by real invest- F. Hellwig (1980), with a shock to the domes- ment behavior and financial constraints, implying that domestic firms are less willing or able to borrow at higher tic bond supply preventing the interest rate interest rates; this view is put forth, for example, by the from perfectly revealing the state. Our model literature on “sudden stops” (cf. Calvo, 1998), or in many captures three key features of domestic inter- business cycle models of emerging market economies (see est rates that are absent from Morris and Pablo Andre´s Neumeyer and Fabrizio Perri, 2005). Neu- meyer and Perri further document a positive correlation Shin’s analysis. First, the domestic interest between net capital flows and domestic interest rates as a rate responds to the conditions in the domes- pervasive feature of business cycles in emerging market tic bond market, so that, in equilibrium, a economies. VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1771 plies that optimal trading strategies are non- est rate, and supply shocks are not too big, this monotone and lead to a backward-bending correlation is high enough so that the inferred demand for domestic bonds. For some realiza- increase in reserve losses and the devaluation tions of fundamentals and shocks, this generates probability exceeds the corresponding increase multiple market-clearing interest rates, and thus in domestic returns. As a result, the demand for multiple equilibria. In one equilibrium, a high domestic bonds is backward bending and there interest rate is associated with large reserve are multiple market-clearing interest rates. In losses and a high likelihood of devaluation. contrast, when the domestic bond supply is in- Another equilibrium leads to a low interest rate elastic and/or shocks are large, this correlation and small reserve losses, and a devaluation be- is low, and the inference drawn from the interest comes unlikely. To complete the description of rate is so weak that trading strategies are mono- our results, we sketch the forces behind such tone and there is a unique equilibrium. nonmonotone asset demands in our two leading Why does the introduction of noisy private cases, when devaluations are triggered, respec- signals have such dramatic effects on the equi- tively, by high interest rates or by large reserve librium set in Morris and Shin (1998), but little losses. to no effect in our model? Private information When devaluations are triggered solely by does not affect uniqueness versus multiplicity in high interest rates as in Obstfeld (1996), this our model, because multiplicity does not origi- nonmonotonicity is immediate and applies iden- nate from a coordination problem. The relevant tically to the common knowledge and private source of multiplicity is the nonmonotonicity of information versions of our model. Conditional optimal trading strategies, which is not affected on the interest rate, traders have no reason to by the introduction of noisy private signals. coordinate trading strategies and form a conjec- When the devaluation outcome depends only on ture of how the others are likely to trade—their the interest rate, there is no coordination motive private signal about fundamentals, together with in strategies, and it is perhaps not surprising that the interest rate, already tell them all they need introducing noisy private signals does not affect to know to infer the devaluation outcome. By the equilibrium set. When, instead, the devalu- increasing the cost of sustaining a fixed ex- ation outcome depends on reserve losses, there change rate, an increase in the interest rate is a coordination motive in trading strategies, expands the set of fundamentals for which a but it is not directly relevant for multiplicity, devaluation occurs. If the resulting increase in since it is resolved by the interest rate, which the expected devaluation probability exceeds endogenously gives the traders a signal about the corresponding increase in the domestic bond reserve losses. return, the traders will at some point switch In contrast, Morris and Shin (1998) repre- their asset holdings from domestic bonds to sents a special case of our reserve loss model, in dollars in response to a small interest rate in- which the domestic bond supply is infinitely crease, because they now anticipate that a de- elastic at a fixed, exogenous interest rate, so valuation is more likely to occur. that, by design, the interest rate remains com- When, instead, a devaluation is triggered by pletely uninformative. This is the only special reserve losses, as in Obstfeld (1986), traders do case, where observing the interest rate cannot have a coordination motive, since they need to resolve the coordination motive among traders, forecast the reserve losses (and hence the other in which case the change in the information traders’ strategies) to forecast the devaluation structure from common knowledge to private outcome. Remarkably, the domestic interest information has dramatic effects for the equi- rate resolves this coordination problem so that librium set. Moreover, the limiting case with equilibrium trading strategies are uniquely de- infinitely elastic supply at a fixed interest rate is termined. In equilibrium, the domestic interest quite different from the case where supply is rate is positively correlated with reserve losses, highly, but not infinitely, elastic, in which case which allows traders to infer the likely devalu- small movements in the interest rate remain ation outcome. If the domestic bond supply is informative of the equilibrium loss of foreign sufficiently responsive to variations in the inter- reserves. 1772 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006

Related Literature.—Following the original with endogenous interest rate determination in papers of Carlsson and van Damme (1993) and a noisy rational expectations equilibrium, in Morris and Shin (1998), several papers have which he concludes that there exists a unique studied the effects of exogenous public infor- equilibrium. His result appears as a special case mation in global coordination games (Christina of our model, in which the domestic bond sup- E. Metz, 2002; Morris and Shin, 2003, 2004) ply is inelastic, the impact of the interest rate for and have shown that this may restore multiplic- the devaluation outcome is small, and supply ity under fairly general conditions (Christian shocks are large enough to rule out multiple Hellwig, 2002). We build on these insights, but equilibria. depart from their general setup by considering Finally, the possibility of multiple equilibria more explicit market structures and modeling due to nonmonotone asset demand and supply the public information endogenously as an in- schedules has also been noted in traditional terest rate signal. Our paper is related also to REE asset pricing models in which coordina- George-Marios Angeletos et al. (forthcoming, tion problems are absent, such as Gennotte and 2006), who consider the informational effects of Leland’s (1990) analysis of market crashes. In costly policy measures (Angeletos et al., 2006) Barlevy and Veronesi (2003), multiplicity arises or equilibrium dynamics (Angeletos et al., from the interaction between informed and un- forthcoming). informed traders. Andrew G. Atkeson (2001) discusses the po- tential problems that the lack of a theory of prices poses for global coordination games.3 I. Model Description Closely related to our paper, Angeletos and Ivań Werning (2006) consider a version of Morris A. Players, Actions, and Payoffs and Shin’s currency crises game with endogenous information aggregation through the price We consider an economy populated by a of a “derivative” asset, or through noisy public measure one of risk-neutral traders, indexed signals of aggregate activity; in their model, by i ʦ [0, 1], and a central bank (CB). prices affect the coordination outcome only Initially, each agent is endowed with one unit through the information they provide. They of domestic currency. Traders can invest their show that equilibrium multiplicity may be re- endowment either in a domestic bond, or they stored by the endogenous public signal, pro- can go to the central bank and exchange the vided that private information is sufficiently domestic currency one for one for a dollar. precise. In this environment, they are the first to The investment in the domestic bond yields a show that multiplicity emerges from the equi- safe market-determined net interest rate r. librium price function, not from individual ac- The return to exchanging the domestic cur- tions which are uniquely determined. In our rency for a dollar is determined by whether a model, the multiplicity occurs within a primary devaluation occurs. If there is no devaluation, market, in which the interest rate not only ag- and the dollar is converted back into domestic gregates private information, but also has direct currency at the same level, its net return is effects on the traders’ payoffs. This highlights zero. If, however, the CB decides to abandon the role of interest rates in determining the the fixed exchange rate, the exchange rate ultimate devaluation outcome either directly or drops to two units of domestic currency for indirectly, as it does in the original multiple the dollar, and the net return on the dollar is equilibrium models of currency crises. one. These investment returns are summa- Nikola A. Tarashev (2003) analyzes a ver- rized in the following table: sion of Morris and Shin’s currency crises game No Devaluation devaluation 3 V. V. Chari and Patrick J. Kehoe (2004) use a noisy Dollar 1 0 rational expectations equilibrium approach to introduce Domestic bond rr prices in herding models. VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1773

B. Devaluation Decision mal distribution. Then, the cdf of the private signal distribution is given by ⌽(͌␤(x Ϫ ␪)). The central bank’s decision to devalue the We further assume that the Law of Large Num- domestic currency depends on the market- bers applies to the cross-sectional distribution determined domestic interest rate r; its loss of of private signals, so that with probability 1, foreign reserves A ʦ [0, 1], which measures the ⌽(͌␤(x Ϫ ␪)) also equals the fraction of trad- Յ total of dollars withdrawn by traders; and an ers who observe a signal xi x, when the unobserved fundamental ␪, which measures the realized fundamental is ␪. strength of the CB’s commitment to maintain a In stage 2, the domestic bond market and the fixed exchange rate. The net value of maintain- central bank open. Traders submit contingent ␪ Ϫ ʦ ing the fixed exchange rate is given by C(r, bids ai(r) [0, 1] to the central bank and ϭ Ϫ A), and the central bank will devalue, if and di(r) 1 ai(r) to the domestic bond market. only if These bid functions ai(r) and di(r) indicate the amount of dollars (ai(r)) and domestic bonds ␪ Յ ͑ ͒ (1) C r, A . (di(r)) that a trader wishes to acquire if the market-determined interest rate on domestic The fundamental ␪ may be interpreted as the bonds is r. The supply of dollars is guaranteed value of the peg in the absence of any reserve by the central bank. The supply of domestic losses or interest rate increases, and C(r, A) bonds is exogenously given by S(s, r), a con- measures the cost of having to defend the ex- tinuous function of the realized interest rate r, change rate in the event of high interest rates, or and an exogenous supply shock s. S(s, r)is losses of foreign reserves. For our analysis, we strictly increasing in s and nondecreasing in r, will focus on the two polar cases that represent so that an increase in interest rates reduces the the canonical models of currency crises: C(r, available quantity of domestic bonds. The sup- A) ϭ r allows for a scenario, in which the CB is ply shock s ʦ ޒ is independent of ␪ and the concerned exclusively with high domestic inter- private signals, and is normally distributed with est rates, such as in Obstfeld (1996). On the mean zero and precision ␦, s ϳ N(0, ␦Ϫ1). other hand, C(r, A) ϭ A represents the case in Once all bids are submitted and the supply which a devaluation is purely determined by the shock is realized, an interest rate r is selected to CB’s loss of foreign reserves. This corresponds clear the domestic bond market. to the modeling assumptions in Paul R. Krug- In stage 3, the CB decides whether to main- man (1979), Robert P. Flood and Peter M. Gar- tain the fixed exchange rate after observing ␪, r, ber (1984), or Obstfeld (1986).4 and the total amount of dollar withdrawals A. This decision follows mechanically from the C. Information Structure and Timing devaluation rule (1).

The currency crisis game has three stages. In D. Strategies and Equilibrium stage 1, nature selects ␪ ʦ ޒ from an improper uniform distribution over the entire real line.5 In stage 2, each trader observes a private Then, each trader observes an idiosyncratic, pri- signal xi and submits a bid function ai(r). We let ␪ vate signal about , denoted xi. Conditional on a(xi, r) denote the traders’ bidding strategy, ␪ , private signals are independent, and identi- which, conditional on a private signal xi and cally distributed according to a normal distribu- interest rate r, indicates the trader’s demand for ϳ N ␪ ␤Ϫ1 ⌽٪ ␾٪ 6 ϭ tion, xi ( , ). Let and denote, dollars. Respectively, we denote by d(xi, r) Ϫ respectively, the cdf and pdf of a standard nor- 1 a(xi, r) the bidding strategy for domestic bonds.

4 It is possible to extend our analysis to general cost functions C(r, A) that are nondecreasing in r and A. 6 Implicitly we are restricting attention to symmetric 5 This improper prior assumption is not essential for our bidding strategies, in which traders submit identical bidding results. We could extend our analysis to allow for normal or functions. It is straightforward to rule out equilibria with other proper prior distributions. asymmetric bidding strategies. 1774 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006

Integrating individual bidding strategies over have different beliefs about the likelihood of a x, we ﬁnd the total demand for dollars, or equiv- devaluation, and hence make different portfolio alently, the CB’s reserve losses, as a function of decisions in equilibrium. ␪ and r, denoted A(␪, r): As can be seen from (4), r affects optimal bidding strategies through two channels. On the (2) one hand, r determines the payoff of holding the domestic bond. This is captured by the right- hand side of the optimality condition p(x , r) Վ ͑␪ ͒ ϵ ͑ ͒ ͱ␤␾͑ͱ␤͑ Ϫ ␪͒͒ i A , r ͵ a x, r x dx. r. On the other hand, r may affect the trader’s expectations about the likelihood of a devalua- The demand for domestic bonds is then given tion. This is captured by the left-hand side of by D(␪, r) ϵ 1 Ϫ A(␪, r). Market clearing on the this optimality condition. The ability to submit domestic bond market requires that bids contingent on r enables the traders to take this effect into account in determining optimal (3) D͑␪, r͒ ϭ S͑s, r͒. bidding strategies. To complete the description of optimal strat- Therefore, the market-clearing condition egies, we determine the beliefs p(xi, r). When- characterizes a correspondence Rˆ (␪, s), such ever there exists (␪, s), s.t. r ϭ R(␪, s), so that r ␪ ʦ ˆ ␪ that for all and s, r R( , s) if and only if r is observed along the equilibrium path, p(xi, r) satisﬁes (3). We will impose assumptions on is pinned down by Bayes’s Law. These beliefs S(s, r) which guarantee that the set of market- are consistent with the devaluation outcome, clearing interest rates Rˆ (␪, s) is always non- which in equilibrium is determined from the empty. An equilibrium interest rate function market-clearing interest rate function R(␪, s) R(␪, s) is then a selection from Rˆ (␪, s), that is, a and the aggregate reserve losses by the central function that assigns a market-clearing interest bank, A(␪, r). On the other hand, if {(␪, s):r ϭ rate r to each possible realization of ␪ and s. R(␪, s)} is empty for some r, then r is never Now, consider a trader who observes signal realized as a market-clearing interest rate, xi. Let p(xi, r) denote this trader’s belief that a Bayes’s Law no longer determines p(xi, r), and devaluation occurs when the market-clearing beliefs are indeterminate. For our leading exam- interest rate is r. In other words, p(xi, r)isthe ples, we will provide a characterization of p(xi, posterior probability of a devaluation, condi- r) using Bayes’s Law on the equilibrium path. tional on observing a signal x and the market- Combining the conditions for optimal bid- clearing interest rate being r. A bidding strategy ding strategies, market-clearing and posterior a(xi, r) is optimal, if and only if beliefs, we have the following equilibrium definition:

(4) a͑xi , r͒ ϭ 1, if p͑xi , r͒ Ͼ r; DEFINITION 1: A Perfect Bayesian Equilib- a͑xi , r͒ ʦ ͓0, 1͔,ifp͑xi , r͒ ϭ r; rium consists of a bidding strategy a(xi, r), an interest rate function R ␪ s a reserve loss ͑ ͒ ϭ ͑ ͒ Ͻ ( , ), a xi , r 0, if p xi , r r. ␪ function A( , r), and posterior beliefs p(xi, r), ␪ ␪ such that (i) a(xi, r), A( , r), and R( , s) satisfy, Condition (4) states that a trader’s optimal respectively, (2), (3), and (4), given beliefs ␪ ϭ trading strategy compares the excess return on p(xi, r); and (ii) for all r such that {( , s):r ␪ domestic bonds r to the probability of devalua- R( , s)} is nonempty, p(xi, r) satisfies Bayes’s tion p(xi, r), which here corresponds to the Law. expected depreciation of the domestic currency. For a trader who is exactly indifferent between Below, we focus on a particular class of the two assets, condition (4) can thus be inter- equilibria in which bidding strategies a(xi, r) preted as an uncovered interest parity condition. are nonincreasing in the private signals xi, the With heterogeneous information, traders may information that is conveyed by r can be char- VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1775 acterized by a sufficient statistic z(␪, s), and the creasing in ␪, and there exists a unique ␪*(r)at equilibrium interest rate function R(␪, s) is con- which ditioned only on z(␪, s), not separately on ␪ and s. For this, we will need to make an additional (6) ␪*͑r͒ ϭ C͑r, A͑␪*͑r͒, r͒͒, functional form assumption about the domestic bond supply. where A(␪*(r), r) ϭ⌽(͌␤(x*(r) Ϫ ␪*(r))), and a devaluation occurs if and only if ␪ Յ ␪*(r). To II. Solving the Model constitute an equilibrium in monotone strategies, the thresholds ␪*(r) and x*(r) must thus In this section, we explain the main steps that jointly solve conditions (5) and (6), given pos- are required to solve our model. First, we re- terior beliefs p(x, r). strict attention to equilibria in monotone strat- To complete the equilibrium characteriza- egies. These are characterized by threshold tion, we discuss how the market-clearing in- rules x*(r) and ␪*(r) for r ʦ (0, 1), such that terest rate function R(␪, s) is determined, traders demand a dollar, whenever their private which in turn enables us to derive the belief Յ signal satisfies xi x*(r), and they buy a do- function p( x, r), which incorporates the in- Ͼ mestic bond, whenever xi x*(r). A devalua- formation conveyed in equilibrium by r.If tion occurs, if and only if ␪ Յ ␪*(r). These agents use a threshold rule characterized by thresholds may be conditioned on the interest x*(r), the demand for dollars is equal to the rate r. measure of agents who receive a signal below We first show that threshold strategies are x*(r), while the demand for domestic bonds mutually consistent: if the devaluation outcome equals the measure of agents whose signal is characterized by a threshold rule, the traders’ exceeds x*(r): D(␪, r) ϭ 1 Ϫ A(␪, r) ϭ 1 Ϫ optimal strategies are characterized by a thresh- ⌽(͌␤( x*(r) Ϫ ␪ )). In equilibrium, the mar- old rule as well, and vice versa. Suppose, first, ket-clearing condition requires that 1 Ϫ that the devaluation outcome is described by an ⌽(͌␤( x*(r) Ϫ ␪ )) ϭ S(s, r), for all (␪, s). arbitrary threshold rule ␪*(r) ʦ [0, 1], so that a Due to the endogeneity of equilibrium be- devaluation occurs if and only if ␪ Յ ␪*(r). liefs, complete equilibrium characterizations in Then, the traders’ belief function p(xi, r) can be noisy REE models of asset markets are often ϭ ␪ Յ ␪ ͉ rewritten as p(xi, r) Pr( *(r) xi, r). We difficult to obtain, unless specific functional conjecture (and will verify below) that p(xi, r)is form assumptions are made. We now make such strictly decreasing in x , with lim 3ϱ p(x , r) ϭ 0 an assumption about the functional form of S(s, i xi i and lim 3Ϫϱ p(x , r) ϭ 1. Then, for r ʦ (0, 1), r) that enables us to solve our model in closed xi i there exists a unique threshold x*(r), such that form.

(5) p͑x*͑r͒, r͒ ϭ r. Ϫ ASSUMPTION 1: S(s, r) ϭ⌽(s Ϫ ␥⌽ 1(r)), and s ϳ N(0, ␦Ϫ1). In words, a trader whose private signal is x*(r) is just indifferent between holding the domestic The parameter ␥ Ն 0 controls how sensitive bond and the dollar. If a trader’s signal is higher the domestic bond supply is to variations in the Ͼ ␥ ϭ than the threshold, xi x*(r), he strictly prefers interest rate. If 0, supply is inelastic, in to hold dollars, while a trader whose signal is which case the quantity of bonds traded in equi- Ͻ below the threshold, xi x*(r), strictly prefers librium does not respond to changes in the in- to hold the domestic bond. The reverse is also terest rate. If, instead, ␥ Ͼ 0, bond supply is true: given a threshold x*(r) below which trad- elastic. In this case, an increase in the domestic ers acquire the dollar, the total loss of dollar interest rate will lead to a smaller supply of reserves by the central bank is A(␪, r) ϭ bonds and, in equilibrium, a larger loss of for- ⌽(͌␤(x*(r) Ϫ ␪)), which is decreasing in ␪. eign reserves by the central bank, and this effect Therefore, for any r, ␪ Ϫ C(r, A(␪, r)) is in- becomes stronger the larger is ␥. In what 1776 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006 follows, we will refer to ␥ as the supply elas- (8) Rˆ ͑z͒ ticity parameter. With this assumption, the market-clearing condition can be rewritten as ϵ ͭr ʦ [0, 1] : z ϭ x*(r) ␥ 1 (7) x*͑r͒ Ϫ ⌽Ϫ1͑r͒ ϭ z ϭ ␪ Ϫ s. ␥ ͱ␤ ͱ␤ Ϫ ⌽Ϫ1(r) . ͱ␤ ͮ Equation (7) plays two important roles in our analysis. First, it guarantees market-clearing in (ii) If {z : r ϭ R(z)} is nonempty, the probabil- the domestic bond market. For a given threshold ity of devaluation p(xi, r) is given by x*(r), an interest rate R(␪, s) ϭ r clears the bond market, if and only if it satisfies (7), for all ␪ p͑x r͒ ϭ ⌽ͩ ͱ␤ ϩ ␤␦ ͩ␪ r and s. Equation (7) thus characterizes, for (9) i , *( ) given realization of z ϭ ␪ Ϫ s/͌␤ and interest rate r, the signal threshold x*(r)of xi ϩ ␦x*(r) ␥␦ the marginal trader whose indifference is re- Ϫ ϩ ⌽Ϫ1(r)ͪͪ. quired for market-clearing. 1 ϩ ␦ ͱ␤(1 ϩ ␦) Second, the market-clearing condition (7) enables us to characterize the information PROOF: provided by r: the left-hand side of (7) defines Part (i) is immediate from (7). For (ii), note ϭ ␪ Յ ␪ ͉ z as a function of the interest rate r and the that p(xi, r) Pr( *(r) xi, z(r)), where z(r) ␪͉ ϳ N ϩ threshold x*(r), while the right-hand side de- is determined from (7). Since xi, z ((xi fines z is in terms of the unobservable shocks ␦z)/(1 ϩ ␦); (␤ ϩ ␤␦)Ϫ1), Pr(␪ Յ ␪*(r)͉x, z) ϭ ␪ ⌽ ͌␤ ϩ ␤␦ ␪ Ϫ ϩ ␦ ϩ ␦ and s. The realization of z can therefore be ( ( *(r) (xi z)/(1 ))), from directly inferred from r and the knowledge of which (9) follows after substituting for z. x*(r). The variable z thus summarizes the information conveyed by r, providing a nor- Any monotone strategy equilibrium is thus mally distributed endogenous public signal of characterized by an interest rate function R(z), a ␪ ϳ N ␪ ␤␦ Ϫ1 , z ( ,( ) ). The precision of this belief function p(xi, r), and thresholds {x*(r), ␤␦ ␪ interest rate signal, , is increasing in the *(r)}, s.t. (5) and (6) are satisfied, p(xi, r)is precision of exogenous private signals, ␤; given by (9) on the equilibrium path, and R(z)is hence, the interest rate serves to aggregate selected from (8), for every realization of z.7 private information. At the same time, bigger This characterization also suggests a simple shocks in the domestic bond supply (a smaller strategy for solving the model in closed form: ␦) make r less informative. substituting (7) and (9) into (5), we determine At this point, we make a second restriction x*(r) and ␪*(r) from (5) and (6). From there, we by focusing on equilibria, in which r is con- construct the correspondence Rˆ (z) of interest ditioned only on the sufficient statistic z, but rates that are consistent with market-clearing. not on ␪ and s separately, so that the same As we will show below, in all our examples, R( z) is selected for any ␪, s, s.t. ␪ Ϫ s/͌␤ ϭ there exists a unique solution for the thresholds z. Lemma 1 characterizes the resulting equi- {x*(r), ␪*(r)}, which is continuous in r. Thus, librium beliefs. equilibrium trading strategies are always uniquely determined. If, however, there are realizations of z LEMMA 1 (Information Aggregation): Sup- pose that all other agents follow a threshold rule characterized by x*(r), and a devaluation 7 For interest rates that never occur in equilibrium, be- occurs, whenever ␪ Յ ␪*(r). Then, liefs and bidding strategies remain undetermined. This free- dom to choose out of equilibrium beliefs does not, however, affect optimal behavior on the equilibrium path. Our equi- (i) R(z) is selected from the correspondence librium characterization obtains for any selection of beliefs Rˆ (z) of market-clearing interest rates: out of equilibrium. VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1777 for which Rˆ(z) admits multiple values of r,it If the bond supply is not completely inelastic becomes possible to construct multiple market- (␥ Ͼ 0), a third effect arises from the restriction clearing interest rate functions from Rˆ(z), and our that the threshold x*(r) must be consistent with model admits multiple equilibria. market-clearing. For given state z, an increase in Now, for a given realization of z, market- r leads to a reduction in the supply of bonds. In clearing requires that the marginal trader who equilibrium, fewer traders buy bonds, and more clears the domestic bond market is exactly in- traders acquire dollars, as r increases. To be different between the domestic bond and the consistent with market-clearing, the fact that dollar. Formally, this requires that r ϭ p(x*(r), more traders demand dollars in equilibrium im- r), where x*(r) ϭ z ϩ (␥/͌␤)⌽Ϫ1(r). Substi- plies that the identity of the marginal trader tuting this into (9), an interest rate r therefore changes: his expectation about ␪ must increase, clears the market (r ʦ Rˆ (z)), if and only if and he must therefore become less optimistic about a devaluation. Formally, for given z, x*(r) is increasing in r, and so is the marginal trader’s (10) r ϭ ⌽ͩ ͱ␤ ϩ ␤␦ ͩ ␪*(r) Ϫ z posterior expectation of ␪, which equals z ϩ ␥/[(͌␤(1 ϩ ␦))⌽Ϫ1(r)]. This market-clearing ␥ effect of r suggests that an increase in r makes Ϫ ⌽Ϫ1(r)ͪͪ. the marginal trader less optimistic about a de- ͱ␤(1 ϩ ␦) valuation, which makes the domestic bond more attractive, reinforcing the direct payoff effect on Equation (10) is the central equilibrium con- the left-hand side of condition (10).8 dition of our model that determines the set of In summary, to determine whether our model market-clearing interest rates for a given real- admits multiple equilibria, we need to compare ization of z. Formally, it may be interpreted as the payoff, devaluation, and market-clearing ef- an uncovered interest parity condition which fects of r in the equilibrium condition (10). This must hold with equality for the marginal trader. is done in Lemma 2. The signal threshold x*(r) of the marginal trader is set so that the domestic bond market clears, LEMMA 2 (Multiplicity): Suppose the deval- which requires x*(r) ϭ z ϩ (␥/͌␤)⌽Ϫ1(r). uation threshold ␪*(r) is continuously differen- Clearly the market-clearing correspondence tiable. Then, there exist multiple market- Rˆ (z) is always nonempty. To understand under clearing interest rate functions, whenever for what conditions Rˆ (z) is single- or multi-valued, some r ʦ (0, 1), we need to discuss how the interest rate r affects (10), the condition which characterizes Rˆ (z). d␪* ␥ ϩ ͱ1 ϩ ␦ 1 (11) Ͼ Ϫ . First, r enters on the left-hand side as the direct dr ͱ␤͑1 ϩ ␦͒ ␾͑⌽ 1͑r͒͒ payoff from acquiring a domestic bond, measuring the direct payoff effect of the interest rate PROOF: for domestic bonds. All else equal, the payoff For any r ʦ (0, 1), there is at most one effect implies that an increase in r makes do- realization z ϭ zˆ(r), for which r is consistent mestic bonds more attractive and dollar assets with (10), and solving (10), we find zˆ(r) ϭ ␪*(r) Ϫ less attractive to the traders. Second, r enters on (␥ ϩ ͌1 ϩ ␦)/[(͌␤(1 ϩ ␦))⌽Ϫ1(r)]. the right-hand side of (10) through the equilib- Therefore, if there exists rЈ, such that rium devaluation threshold ␪*(r), measuring the impact of the interest rate on the devaluation d␪* ␥ ϩ ͱ1 ϩ ␦ 1 ͯ Ͼ Ϫ , outcome. The sign and magnitude of this deval- dr ͱ␤͑1 ϩ ␦͒ ␾͑⌽ 1͑rЈ͒͒ uation effect is ambiguous; as we will discuss r ϭ rЈ below, it actually depends on the specifics of the environment. As can be seen from conditions (9) and (5), these two effects are present for any 8 For this effect to exist, ␥ Ͼ 0 is essential. If ␥ ϭ 0, the trader with given signals x and z, not just for the domestic bond supply is inelastic and changes in r have no marginal trader. effect on the quantity of bonds traded in equilibrium. 1778 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006

͉ Ͼ Јϩ␧ Ͼ then dzˆ/dr r ϭ rЈ 0, and hence zˆ(r ) A. Equilibria with Common Knowledge Ј Ͼ ЈϪ␧ zˆ(r ) zˆ(r ). Moreover, since limr31 ϭϪϱ ϭϩϱ ␪ ʦ zˆ(r) and limr30 zˆ(r) , it follows Suppose that (0, 1] is common knowl- by continuity of zˆ(r) that zˆ(rЈ) ϭ zˆ(rЉ) ϭ edge among all traders. With a slight abuse of (zˆ(rٞ) for some rЉ ʦ (rЈ,1)andrٞ ʦ (0, rЈ), notation, we let p(␪, r), A(␪, r), and D(␪, r so that z ϭ zˆ(rЈ) is consistent with multiple denote, respectively, the traders’ beliefs about a market-clearing interest rates. Since ␪*(r) is con- devaluation, their optimal bidding strategies, tinuously differentiable, the same argument ap- and the resulting aggregate demand for dollars plies to any zЈ within a small open neighborhood and domestic bonds, for given ␪ ʦ (0, 1] and of zˆ(rЈ). We conclude that Rˆ (z) is multivalued realized interest rate r. over an open interval, and hence that there exist Clearly, p(␪, r) ϭ 1 whenever ␪ Յ r, and p(␪, multiple market-clearing interest rate functions. r) ϭ 0 otherwise. Therefore, optimal bidding strategies are characterized as follows: if r Ͼ 1, Lemma 2 provides a sufficient condition the domestic bond strictly dominates the dollar, under which, for given devaluation threshold and individual and aggregate demands for dol- ␪*(r), there are multiple market-clearing in- lars and bonds are A(␪, r) ϭ 0 and D(␪, r) ϭ 1. terest rates. This condition is also necessary If r Ͻ 0, the dollar strictly dominates the do- for multiplicity within the class of equilibria mestic bond, so that A(␪, r) ϭ 1 and D(␪, r) ϭ that we are constructing, that is, if condition 0. If r ʦ (0, 1), traders demand dollars (A(␪, (11) is not satisfied for any r, then there exists r) ϭ 1 ϭ 1 Ϫ D(␪, r)) if and only if ␪ Յ r, and a unique monotone strategy equilibrium in demand domestic bonds (A(␪, r) ϭ 0 ϭ 1 Ϫ which the interest rate is conditioned on z D(␪, r)) otherwise. Finally, when r ϭ 0, ␪ Ͼ r, only. Condition (11) compares the devalua- a devaluation does not occur, and traders are tion effect on the left-hand side against the indifferent between the dollar and the domestic payoff and market-clearing effects of r on the bond; any D(␪, r) ʦ [0, 1] is sustainable as part right-hand side. Multiplicity may result only of the demand schedule for bonds. Similarly, if the devaluation effect of r is positive and when r ϭ 1, a devaluation does occur, and again large enough to offset the payoff and market- traders are indifferent between bonds and dol- clearing effects of r,orif␪*(r) is locally lars, so that any D(␪, r) ʦ [0, 1] is part of the increasing in r with a slope sufficiently steep. demand schedule. Figure 1 illustrates the equi- In the next sections, we show how such pos- librium in the domestic bond market in a stan- itive devaluation effects lead to multiple equi- dard demand-and-supply graph, plotting the libria in the context of our leading examples, demand schedule D(␪, r), as characterized C(r, A) ϭ r and C(r, A) ϭ A. above, and an arbitrary supply function S(s, r). Unless the supply is perfectly elastic at some III. Devaluation Triggered by Interest Rates exogenously given interest rate level r, in which case the supply curve is horizontal, there are In this section, we discuss the version of our two market-clearing interest rates. Under our model in which devaluations are driven exclu- functional form assumption for S(s, r), both r ϭ sively by the cost of high interest rates, as in 0 and r ϭ 1 clear the domestic bond market, for Obstfeld (1996). Formally, we suppose that C(r, any s and ␪ ʦ (0, 1]. If r ϭ 0, then no deval- A) ϭ r, and a devaluation occurs if and only if uation will take place, the central bank will not ␪ Յ r. We proceed in two steps: first, we ex- lose any reserves, and S(s,0)ϭ D(␪,0)ϭ 1. On amine equilibrium outcomes with common the other hand, if r ϭ 1, there will be a deval- knowledge; then, we examine our model with uation, the reserve losses are equal to 1, and S(s, privately informed traders. By comparing the 1) ϭ D(␪,1)ϭ 0. two environments, we assess the role of the As can be seen from Figure 1, multiplicity of information structure and determine to what equilibria arises from the existence of multiple extent the insights of the common knowledge market-clearing prices, due to the nonmonotone environment carry over to the game with private demand schedule for domestic bonds. This non- information. monotonicity arises from the interaction between VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1779

tion problem among traders is critical for obtain- ing multiplicity (under common knowledge) or uniqueness (with private information).

B. Equilibria with Private Information

We next show how the same insights carry over into our market environment with private information. After substituting C(r, A) ϭ r into condition (6), we solve (5) and (6) for the thresholds ␪*(r) and x*(r)toﬁnd

(12) ␪*͑r͒ ϭ r

␥␦ Ϫ ͱ1 ϩ ␦ and x*͑r͒ ϭ r ϩ ⌽Ϫ1͑r͒. ͱ␤͑1 ϩ ␦͒

FIGURE 1. DEVALUATION TRIGGERED BY INTEREST RATES ␪ (Common knowledge) Substituting this solution for *(r) into the market-clearing correspondence (10), we find the following equilibrium characterization: the payoff effect of r for domestic bonds, and the PROPOSITION 1: Suppose that C(r, A) ϭ r. devaluation effect for the return on holding the Then, in any monotone strategy equilibrium, dollar. While an increase in r continuously raises the return to holding domestic bonds, it also leads (i) for all r s.t. {z : r ϭ R(z)} is nonempty, to a discrete increase in the payoff to holding the ␪*(r) and x*(r) are uniquely characterized dollar at r ϭ ␪, the level at which the interest rate by (12). becomes sufficiently high to trigger a devaluation. (ii) r ʦ Rˆ (z), if and only if Therefore, at r ϭ ␪, the traders shift their demand to dollar assets, and the demand for domestic bonds drops from 1 to 0. In other words, there are (13) r ϭ ⌽ͩ ͱ␤ ϩ ␤␦ ͩ r Ϫ z multiple market-clearing interest rates, because ␥ the devaluation effect of r locally dominates the Ϫ ⌽Ϫ1(r)ͪͪ. payoff effect.9 ͱ␤(1 ϩ ␦) Although this argument delivers multiple equilibria, the underlying logic is quite different from (iii) There are multiple equilibria, whenever the multiplicity argument in coordination games. ͌␤(1 ϩ ␦)/(͌1 ϩ ␦ ϩ ␥) Ͼ ͌2␲. When devaluations are driven by interest rates, once bids are conditioned on r, the actions taken PROOF: by other traders are irrelevant for any given trad- Parts (i) and (ii) are immediate from preced- er’s portfolio decision. Therefore, the traders do ing arguments. For (iii), note that d␪*/dr ϭ 1, not have an explicit motive to coordinate bids. In and therefore the condition of Lemma 2 is sat- contrast, in Morris and Shin (1998), a coordina- isfied, whenever for some r ʦ (0, 1),

␥ ϩ ͱ1 ϩ ␦ 1 Ͻ Ϫ1 1. 9 Notice that for this result, it is essential that the ͱ␤͑1 ϩ ␦͒ ␾͑⌽ ͑r͒͒ supply schedule is ﬁnitely elastic. If supply is inﬁnitely elastic at some given interest rate r, then the supply Since [␾(⌽Ϫ1(r))]Ϫ1 is maximized when r ϭ determines r, which in turn uniquely determines the Ϫ devaluation outcome. 1/ 2 and ⌽ 1(r) ϭ 0, the result follows from 1780 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006 noting that 1/␾(0) ϭ ͌2␲. This condition is also necessary for multiplicity within the class of equilibria under consideration.

Optimal bidding strategies for traders are thus determined for each interest rate r that is observed on the equilibrium path, but multiplicity may arise from the set of market-clearing interest rate functions. This should not appear to be surprising; after all, we already showed that optimal trading strategies under common knowledge were uniquely pinned down by comparing the fundamental ␪ to the realized interest rate r, but led to a nonmonotone trading strategy. The same logic applies here. The fundamental ␪ and the interest rate r uniquely determine the devaluation outcome, with ␪*(r) ϭ r being FIGURE 2. DEVALUATION TRIGGERED BY INTEREST RATES the exogenously given devaluation threshold. (Uniqueness versus multiplicity) Given their private signal xi, and the interest rate, each trader’s beliefs about a devaluation, and hence his optimal bidding strategies, are market-clearing effect becomes so strong that it uniquely pinned down. Multiple equilibria re- always outweighs the devaluation effect and sult because the devaluation effect locally dom- thus leads to a unique equilibrium. In this case, inates the payoff and market-clearing effects of the domestic bond supply becomes more and r, so that the demand for domestic bonds may more elastic at an exogenously given interest be locally decreasing in the interest rate r. rate, from which the devaluation outcome is When the supply elasticity is zero, ␥ ϭ 0, uniquely determined, for each fundamental ␪. the market-clearing effect is absent. In this This connects to our observation from the com- case, once traders are sufficiently well in- mon knowledge game, where there is always a formed, in that the precision of their overall unique equilibrium, when the domestic bond information, ␤(1 ϩ ␦), is sufficiently large, supply is infinitely elastic at an exogenously the devaluation effect locally dominates and given interest rate level. generates multiple equilibria. This parameter In summary, the argument for multiplicity condition is plotted in Figure 2. Uniqueness put forth by Obstfeld (1996) in models of versus multiplicity depends only on the over- devaluations triggered by high interest rates all precision of information, not on whether applies identically when traders are heteroge- this information results from the traders’ ex- neously informed. Given their private infor- ogenous private signals or the interest rate. In mation and the interest rate, traders form their particular, note that multiplicity may arise beliefs about the likely devaluation outcome, even if ␦ is close to 0, when the interest rate which uniquely determines optimal trading conveys little or no information about funda- strategies. If traders are sufficiently well in- mentals. formed, the devaluation effect of the interest When the supply elasticity is positive, ␥ Ͼ 0, rate leads to nonmonotone trading strategies: the market-clearing effect of r reinforces the in response to an increase in the domestic direct payoff effect. This expands the set of interest rate, traders shift their wealth into dol- parameters for which there is a unique equilib- lars if they anticipate that the interest rate in- rium and shifts the boundary of the uniqueness crease makes a devaluation more likely. As a range to the right, but does not otherwise affect result, there may be multiple market-clearing the main conclusions. However, holding the interest rates for some fundamental realizations, precision parameters ␤ and ␦ fixed, as the sup- a high interest rate associated with high likeli- ply elasticity becomes infinite (␥ 3 ϱ), the hood of a devaluation and large loss of foreign VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1781

FIGURE 3. DEVALUATION TRIGGERED BY RESERVE LOSSES (Common knowledge) reserves by the central bank, and a low interest prefer to invest in the dollar, so that A(␪, r) ϭ 1 rate associated with low likelihood of a deval- and D(␪, r) ϭ 0. If r ʦ [0, 1], however, agents uation and small reserve losses. must make an assessment of whether a devaluation is likely to occur to determine optimal IV. Devaluations Triggered by Reserve Losses bidding strategies. If traders take r as exogenously given without taking into account the In this section, we consider our second lead- fact that the observed r must clear domestic ing case, in which devaluations are triggered bond markets, then A(␪, r) ϭ 0, D(␪, r) ϭ 1 and exclusively by the loss of foreign reserves; a A(␪, r) ϭ 1, D(␪, r) ϭ 0 are both sustainable as prominent example of such a model is Obstfeld best responses and hence part of the demand (1986). Formally, we assume C(r, A) ϭ A,so schedule for dollar assets and domestic bonds, that a devaluation occurs if and only if ␪ Յ A. for any r ʦ [0, 1]: if traders expect no devalu- As before, we split our analysis in two parts. ation, they will all demand domestic bonds, in First, we discuss equilibria in the case where which case the central bank’s reserve losses are ␪ ʦ (0, 1] is common knowledge. Then, we 0, and a devaluation will indeed not occur. If, derive the equilibria of our private information instead, the traders do expect a devaluation, economy and compare them to the common they will demand dollars, and the resulting re- knowledge benchmark. We conclude this sec- serve losses of A(␪, r) ϭ 1 will force the deval- tion by discussing the relation between our re- uation that the traders are expecting. Finally, if sults and existing global games models of r ϭ 0, traders remain indifferent between the currency crises in Morris and Shin (1998, domestic bond and the dollar, as long as ␪ Ͼ 2004), Hellwig (2002), Tarashev (2003), and A(␪, r); hence, any D(␪, r) Ͼ 1 Ϫ ␪ can be Angeletos and Werning (2006). sustained as part of the demand correspondence. Likewise, if r ϭ 1, agents again remain indif- A. Equilibria with Common Knowledge ferent, as long as ␪ Յ A(␪, r), and hence any D(␪, r) Յ 1 Ϫ ␪ is sustainable. Suppose that ␪ ʦ (0, 1], s, and S(s, r) are In Figure 3, we plot the resulting demand cor- common knowledge. As before, if r Ͼ 1, agents respondence for domestic bonds, D(␪, r) ϭ 1 Ϫ strictly prefer the domestic bond, so that the A(␪, r), together with a supply function S(s, r), in aggregate demand for dollars and bonds is A(␪, a standard demand-and-supply graph. As long as r) ϭ 0 and D(␪, r) ϭ 1. If r Ͻ 0, agents strictly S(s, r) is not completely inelastic, there may be 1782 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006

FIGURE 4. DEVALUATION TRIGGERED BY RESERVE LOSSES (Off equilibrium inference from interest rate) multiple market-clearing interest rates (left panel). consistent conjecture of the equilibrium re- On the other hand, when the domestic bond sup- serve losses, thus resolving the coordination ply is perfectly inelastic, there exists a unique problem in the market, turns out to be central equilibrium (right panel). The inelastic supply ex- for understanding behavior on the equilibrium ogenously determines the CB’s loss of foreign path, when we go to the private information reserves, and thereby the devaluation outcome. version of our model. Interest rates adjust to clear the domestic bond To see how traders might use r to form a market: r ϭ 1, if ␪ Յ 1 Ϫ S, when there is a conjecture about the other traders’ strategies devaluation; and r ϭ 0, when ␪ Ͼ 1 Ϫ S, and there and aggregate reserve losses, notice that market- is no devaluation. The multiplicity result thus re- clearing implies S(s, r) ϭ D(␪, r) ϭ 1 Ϫ A(␪, r). lies on the bond supply and reserve losses varying If the domestic bond supply is positively, but with changes in the interest rate. finitely, elastic, the observation of r enables In the reserve-loss model, traders face an traders to infer the central bank’s equilibrium explicit coordination motive as they must loss of foreign reserves A(␪, r), which thus form a conjecture about each other’s trading enables them to predict the devaluation out- strategies to predict the resulting reserve come. Let r˜(␪, s) denote the interest rate level losses. The discussion above was based on the for which 1 Ϫ ␪ ϭ S(s, r˜). Whenever r Ն r˜(␪, s), premise that all traders form such conjectures A(␪, r) ϭ 1 Ϫ S(s, r) Ն ␪, so that traders expect by assuming that all the other traders behave a devaluation. Whenever r Ͻ r˜(␪, s), A(␪, r) ϭ optimally, and when the resulting strategies 1 Ϫ S(s, r) Ͻ ␪ and traders anticipate no deval- are inconsistent with market-clearing, the uation. In each case, the resulting demand for traders discard the market-clearing hypothe- domestic bonds is again uniquely pinned down sis. Instead, one might assume that traders by comparing r to the resulting devaluation form their conjectures about the other traders’ premium, 0 or 1; if r˜(␪, s) ʦ (0, 1), it is again strategies on the basis of market-clearing, backward bending. We sketch this case in Fig- without requiring necessarily that the implied ure 4 (left panel). strategies are optimal. The difference be- With this conjecture, multiplicity once again tween these two approaches matters only out arises because of a nonmonotonicity in optimal of equilibrium, when either optimality or trading strategies: the fact that r must be consistent market-clearing must be violated, and it might with market-clearing allows traders to infer the therefore appear purely semantic at this point. central bank’s reserve losses and hence the deval- However, the idea that the observation of the uation outcome. A higher r indicates a smaller interest rate r, together with the market- supply of domestic bonds, and larger loss of for- clearing condition, can be used to form a eign reserves. At r ϭ r˜(␪, s), a small increase in r VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1783 leads traders to the conclusion that the implied This leads to the following equilibrium charac- reserve losses become sufficiently large to trigger terization: a devaluation; in response, they shift their bidding strategies from domestic bonds back into dollars. PROPOSITION 2: Suppose that C(r, A) ϭ A. This nonmonotonicity in trading behavior gives Then, in any monotone strategy equilibrium: rise to multiple market-clearing interest rates. On the other hand, if the domestic bond sup- (i) For all r s.t. {z : r ϭ R(z)} is nonempty, ply is infinitely elastic at exogenously given r, ␪*(r) and x*(r) are uniquely characterized any quantity of domestic bonds is consistent by (16). with market-clearing, so r provides no informa- (ii) For given z, r ʦ Rˆ (z) if and only if tion about the reserve losses A (Figure 4, right panel). In that case, A(␪, r) ϭ 1 and A(␪, r) ϭ 0 (17) are both consistent with best-response behavior and market-clearing, implying multiple equilib- ␥␦ Ϫ ͱ1 ϩ ␦ r ϭ ⌽ͩ ͱ␤ ϩ ␤␦ͩ⌽ͩ ⌽Ϫ1(r)ͪ ria due to an explicit coordination problem, as 1 ϩ ␦ discussed above. The case with infinite supply ␥ elasticity differs from our model with finite Ϫ z Ϫ ⌽Ϫ1(r)ͪͪ. supply elasticity, because r remains completely ͱ␤(1 ϩ ␦) uninformative of the central bank’s reserve losses. This, however, is exactly the case on (iii) There are multiple equilibria whenever which Morris and Shin (1998) focus their analysis to show that the introduction of noise in the ͱ␤͑1 ϩ ␦͒ ␥␦ Ϫ ͱ1 ϩ ␦ observation of ␪ leads to the selection of a Ͼ ͱ ␲ ϩ ␦ 2 . unique equilibrium. ͱ1 ϩ ␦ ϩ ␥ 1

B. Equilibria with Private Information PROOF: Steps (i) and (ii) are again immediate. For We now return to our model with noisy pri- (iii), applying Lemma 2, and noting that vate signals. Using C(r, A) ϭ A, the conditions for the devaluation threshold ␪*(r) and the in- d␪* ␥␦ Ϫ ͱ1 ϩ ␦ difference condition for x*(r) can be written as ϭ ␾ dr 1 ϩ ␦

(14) ␪*͑r͒ ϭ ⌽͑ͱ␤͑x*͑r͒ Ϫ ␪*͑r͒͒͒; ␥␦ Ϫ ͱ1 ϩ ␦ 1 ϫ ͩ⌽ͩ ⌽Ϫ1 ͪͪ (r) Ϫ , 1 ϩ ␦ ␾͑⌽ 1͑r͒͒ (15) r ϭ ⌽͑ͱ␤ ϩ ␤␦(␪*(r) Ϫ x*(r)) there exist multiple equilibria whenever ␥␦ ϩ ⌽Ϫ1(r)ͪ. ␥␦ Ϫ ͱ1 ϩ ␦ ␥␦ Ϫ ͱ1 ϩ ␦ ͱ1 ϩ ␦ ␾ͩ⌽ͩ ⌽Ϫ1(r)ͪͪ 1 ϩ ␦ 1 ϩ ␦ Solving this pair of equations for {␪*(r), x*(r)}, ␥ ϩ ͱ1 ϩ ␦ Ͼ . we ﬁnd the following unique solution: ͱ␤͑1 ϩ ␦͒

␥␦ Ϫ ͱ1 ϩ ␦ Since the left-hand side reaches a maximum ␪ ͑ ͒ ϭ ⌽ͩ ⌽Ϫ1 ͪ at r ϭ 1/ 2, this condition is satisﬁed when- (16) * r ϩ ␦ (r) 1 ever 1 ͱ␤͑1 ϩ ␦͒ ␥␦ Ϫ ͱ1 ϩ ␦ and x*͑r͒ ϭ ␪*͑r͒ ϩ ⌽Ϫ1͑␪*͑r͒͒. Ͼ 1/␾͑0͒ ϭ ͱ2␲. ͱ␤ ͱ1 ϩ ␦ ϩ ␥ 1 ϩ ␦ 1784 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006

As before, this condition is also necessary for multiplicity within the class of equilibria under consideration.

Once again, optimal trading strategies and the devaluation outcome are uniquely determined for any interest rate r, but there may be multiple market-clearing interest rate functions. As we already discussed in the common knowledge context, this arises because, in equilibrium, the observation of r enables traders to make a noisy inference of the other traders’ actions through the market-clearing condition; r thus serves as an endogenous public signal of the foreign reserve losses A, which traders can use to forecast the devaluation outcome and hence uniquely determine optimal trading strategies. FIGURE 5. DEVALUATION TRIGGERED BY RESERVE LOSSES The possibility of multiple equilibria then (Uniqueness versus multiplicity) depends on whether the devaluation threshold ␪*(r) is increasing or decreasing in r. Both scenarios are possible, and which one material- dition relates the strength of the devaluation effect izes depends on the supply elasticity parameter to the informativeness of the interest rate r as a ␥ and the variance of the supply shocks ␦Ϫ1.If signal of equilibrium reserve losses: if ␥␦ Ͻ ␥␦ Յ ͌1 ϩ ␦, ␪*(r) is decreasing in r, the ͌1 ϩ ␦, the supply elasticity ␥ is low, and devaluation effect of r is negative and reinforces shocks are large (␦ is small). In this case, the the payoff and market-clearing effects, which interest rate has little impact on the realized implies that there is a unique market-clearing bond supply, carries little information about the interest rate, irrespective of ␤. On the other likely reserve losses, and hence has little influ- hand, if ␥␦ Ͼ ͌1 ϩ ␦, ␪*(r) is increasing in r, ence on the trader’s beliefs about a devaluation. the devaluation effect is positive and counter- On the other hand, if ␥␦ Ͼ ͌1 ϩ ␦, the bond acts the payoff and market-clearing effects, supply is sufficiently elastic and shocks are not which gives rise to multiple market-clearing too big, so that changes in r affect the realized interest rates once ␤ is sufficiently large. We bond supply, and become informative of the plot this uniqueness condition in Figure 5. central bank’s equilibrium reserve losses. The To understand when ␪*(r) is increasing in r, information carried by r then leads to nonmono- consider the conditions characterizing optimal tone trading behavior, whereby traders opti- trading strategies and the devaluation threshold, mally choose to withdraw dollars once the (14) and (15). For given ␪*(r) and x*(r), an in- interest rate (and therefore the implied reserve crease in r raises both the payoff to the domestic losses) are sufficiently high. This in turn gener- bond and the marginal trader’s expectation about a ates multiple market-clearing interest rates and devaluation, since for given x*(r), a higher r is multiple equilibria. associated with a lower value of the public signal To summarize, information transmitted by z. Intuitively, r serves as an endogenous signal not the interest rate about the equilibrium demand only of ␪ but also of the foreign reserve losses, for bonds and the central bank’s foreign reserve with a higher r being indicative of a smaller sup- losses plays a critical role here, because it en- ply of domestic bonds and a larger loss of foreign ables traders to resolve the coordination prob- reserves. The parameter condition on ␥ and ␦ lem that is present in the reserve loss model. determines whether the payoff effect or the effect Multiple equilibria then arise when the informa- on the likelihood of a devaluation dominates, tion about reserve losses is sufficiently precise and hence whether at the margin traders become to generate nonmonotone trading behavior and more or less aggressive as r increases. This con- multiple market-clearing interest rates. The in- VOL. 96 NO. 5 HELLWIG ET AL.: CURRENCY CRISES AND INTEREST RATES 1785 tuition for this result once again parallels our perfectly elastic supply. Taking the limit of discussion in the common knowledge case: ͌␤(␥␦ Ϫ ͌1 ϩ ␦)/(␥ ϩ ͌1 ϩ ␦)as␥ 3 ϱ, there through the market-clearing condition, a higher exists a unique equilibrium if and only if ͌␤␦ Յ interest rate leads traders to infer larger losses of ͌2␲; multiple equilibria necessarily arise once foreign reserves by the central bank, making a ␤ is sufficiently high. As ␥ becomes larger and devaluation more likely.10 If the domestic bond larger, fluctuations in r become smaller and supply is sufficiently elastic and supply shocks smaller, and r converges to an exogenously are not too large, increases in the interest rate given limit. This, however, does not imply that lead to a positive devaluation effect, which may r loses its value as a signal: as the supply counteract the direct payoff and market-clearing elasticity becomes large, even vanishingly small effects to generate multiple market-clearing in- fluctuations of r remain informative of changes terest rates for certain realizations of z. in the supply of domestic bonds and the amount of foreign reserve losses. C. Discussion This limit characterization as ␥ 3 ϱ has the following interpretation. Consider the global We conclude this section by discussing the games model with an exogenously fixed interest relation between our results and the results ob- rate (or infinitely elastic supply), but add an tained by Morris and Shin (1998), as well as exogenous public signal y ϳ N(␪, ␩Ϫ1) other global games models of currency crises in with mean ␪ and precision ␩. The main result of which devaluations are driven by reserve losses. the global games literature with exogenous pub- To apply the global games equilibrium selec- lic and private information (Morris and Shin, tion results, Morris and Shin (1998) consider an 2003, 2004; Hellwig, 2002) shows that there environment in which r is exogenously fixed. exists a unique equilibrium if and only if Their analysis is a special case of our reserve ␩/͌␤ Յ ͌2␲, that is, if the precision of pri- loss model, in which supply is infinitely elastic vate signals is sufficiently high relative to the at an exogenously given interest rate r; r then public signal precision. Once we replace the does not convey any information in equilibrium precision of the exogenous public signal ␩ with about fundamentals, and in the unique equilib- the endogenous precision of the interest rate ␤␦, rium, optimal strategies are characterized by a we find that our limit condition for multiplicity threshold xMS(r) and the devaluation threshold exactly replicates the condition with exogenous ␪MS(r), which must satisfy the following pair of information. This results because interest rate equations: r ϭ Pr(␪ Յ ␪MS͉xMS) ϭ⌽(͌␤ fluctuations vanish and r no longer has any (␪MS Ϫ xMS)) and ␪MS ϭ⌽(͌␤(xMS Ϫ ␪MS)). In the direct payoff implications in the limit as the unique equilibrium, ␪MS(r) ϭ 1 Ϫ r. supply elasticity becomes infinite. The interest As we already discussed above, for the case rate then affects trading strategies only through with common knowledge, when r is exog- the information it conveys in equilibrium. enously fixed, traders are unable to draw any This limiting case also mirrors the analysis in inference from r and therefore face an explicit Angeletos and Werning’s (2006) model of in- coordination problem. This leads to very differ- formation aggregation through a derivative as- ent equilibrium bidding strategies than the case set market. In their model, traders may trade a where supply is highly, but not infinitely, elastic derivative asset prior to the currency crises and inference from r remains possible using the game, which is modeled as in Morris and Shin. market-clearing condition. Since the derivative price serves only to aggre- An even stronger contrast arises when we gate information, but has no effects on payoffs compare the special case of Morris and Shin in the coordination game, uniqueness versus (1998) with our model with highly but not multiplicity is determined purely by the extent to which the price provides public information, as characterized by the uniqueness condition 10 Notice that in the model with devaluations driven by interest rates, traders did not need to make such indirect given above. Information aggregation then inference, since there was a direct link from the interest rate overturns the Morris-Shin limit uniqueness re- to the devaluation decision. sult and leads to multiplicity when exogenous 1786 THE AMERICAN ECONOMIC REVIEW DECEMBER 2006 private information and the endogenous public stylized global coordination game, which ab- signal are both sufficiently precise. stracts from the role of interest rates and Since ͌␤(␥␦ Ϫ ͌1 ϩ ␦)/(␥ ϩ ͌1 ϩ ␦) Յ markets. ͌␤␦, the condition for multiplicity becomes more Our paper provides a first step toward inte- stringent away from the limit. When supply is fi- grating insights from the global games literature nitely elastic, or ␥ Ͻϱ, the payoff effects associated into market-based models of currency crises. with fluctuations in r mitigate the informational ef- Although it provides a more explicit model of fects, and these payoff effects become larger as the financial markets, it remains stylized in many domestic bond supply becomes less and less elastic. respects, thus suggesting several avenues for When the supply elasticity parameter ␥ becomes future research. For example, we did not model sufficiently small, there is a unique equilibrium. If the supply of domestic bonds, the central bank ␥ ϭ 0, the bond supply is perfectly inelastic and objectives, or the devaluation premium from given by ⌽(s), the loss of foreign reserves is 1 Ϫ first principles. A first extension may therefore ⌽(s), and a devaluation occurs, if and only if ␪ Յ be to model these features explicitly in the con- 1 Ϫ⌽(s). Just as in the common knowledge game, text of a private information model. Second, our the devaluation outcome is then uniquely determined model may be used as a building block to un- by the exogenous fundamentals and the shocks to the derstand the effects of policy interventions in domestic bond supply, and the interest rate merely currency crises, when such interventions have adjusts to clear the domestic bond market. This case informational as well as allocative effects. An- underlies Tarashev’s (2003) argument for equilib- geletos et al. (2006) provide a step in this di- rium uniqueness.11 rection, but one would like to examine to what extent their analysis applies to more realistic V. Conclusion market settings. Third, one may consider how equilibrium outcomes are affected by richer in- In this paper, we have studied a stylized formational environments that allow, for exam- currency crises model with heterogeneously ple, for public information disclosures. 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