Dynamic Speculative Attacks

DYNAMIC SPECULATIVE ATTACKS Abstract The paper presents a model of rational Bayesian agents with speculative attacks in a regime of exchange rate which is pegged within a band. Speculators learn from the observation of the exchange rate within the band whether their mass is sufficiently large for a successful attack. Multiple periods are necessary for the existence of speculative attacks. Various defense policies are analyzed. A trading policy by the central bank may defend the peg if it is unobserved and diminishes the market’s information for the coordination of speculators. Introduction A regime of fixed exchange rates is for speculators an open invitation to a stag hunt. If they act in coordination, their combined reserves dwarf those of the central bank, thus forcing a devaluation and a distribution of good profits to the hunters. The game has been formalized by Obstfeld (1996) who also provides some empirical justification: under perfect information on the strategies and the payoffs of individuals, there are two equilibria in pure strategies, as in any standard stag hunt game: either speculators cooperate and gain, or they don’t in which case there is neither gain nor loss. Coordination games with multiple equilibria raise the issue of expectations on the choice of the equilibrium and imperfect information. Carlsson and Van Damme (1993a) and (1993b) “resolve” the issue of multiplicity of a one-period coordination game by assuming that agents have different expectations. Each agent makes his decision not knowing the actions of the others, but educing what others will do in a mental process which takes place before all agents make their unique decision1 . The problem of multiple equilibria disappears because a contagion process from the agents with extreme beliefs leads all other agents either to action or to inaction. This “global game” method has been applied to the model of Obstfeld by Morris and Shin (1998) who assume that agents have different beliefs on a fundamental parameter of the economy. A speculative attack succeeds only if the mass of hunters exceeds some level which is an increasing function of the funda1 See Binmore (1987), Guesnerie (1992) who use the method in the case of strategic substi- tutability. For an exposition, see Chamley (2002). 1 mental. Such an assumption is justified by the consideration of a central bank which defends the currency only if the cost of defense (which is increasing in the mass of hunters and decreasing in the fundamental parameter), does not exceed an exogenous ceiling. The model “resolves” the multiplicity of equilibria and generates conclusions on the feasibility of fixed exchange rates. However, it does not embody some important features of currency markets. (i) An essential property of the one-period model is that all agents have to make a decision once and simultaneously2 . There is no interaction between the learning from others’ actions and strategic decisions (i.e., to speculate now or delay after some observation of others’ actions). Actual currency attacks may be short but during an attack agents intensively observe the actions taken by others and react quickly to their perceptions of these actions. (ii) The global game method requires the distribution to have a tail of agents on each side for whom the dominant strategy is to attack or not to attack the currency. In a multi-period context, one tail may disappear (e.g., the tail of agents with high expectations who exercise their option to attack the currency), and the method is not applicable in the following period unless a new shock “regenerates” the tail (as in Morris and Shin, 1998). If the period is short (as it 2 Morris and Shin (1998b) consider a sequence of one-period models with no strategic de- cisions. The state of nature θt evolves randomly from period to period. The assumption that θt is learned exogenously in period t + 1 rules out social learning. In any case, the exogenous learning seems a bit slow compared to the time frame of a speculative attack. Corsetti et al (2000) consider the strategic behavior of a large player and infinitesimal players. Given the assumption of the model, the outcome is trivial with the large player moving first. 2 should be in exchange markets), such a new shock is asymptotically equivalent to a discontinuous jump of the parameters and may not be plausible3 . (iii) A fixed cost of transaction plays a critical role in the model. When that cost is vanishingly small, the difference between the sustainable exchange rate and the fundamental exchange rate is also vanishingly small. A realistic value of this cost as a fraction of the transaction is very small for positions which last just a few days. (iv) The model assumes that the exchange rate is fixed with absolutely no room for variations. Unless the currency belongs to a monetary zone such as the Euro between January 1, 1999 and January 1, 2002, some fluctuations as in the regime before 1999 is allowed. These fluctuations have two opposite effects. The first is to provide a channel of communication. A speculative attack typically induces a depreciation of the exchange rate within the allowed band before any devaluation. This communication increases the risk of an attack. The second effect reduces this risk by introducing a penalty if the attack fails: after a failed attack, the price of the foreign currency falls thus generating a loss for the holders of the foreign currency. This penalty is incurred without any ad hoc transaction cost. (v) The payoff after the abandonment of the fixed rate is independent of the actions of the players if the attack is successful. All the bids are carried at the “old” exchange rate and the speculators gain from the devaluation. But actual 3 For some distributions of agents however, there is a unique equilibrium even when one of the tails of dominance is erased by past learning (Chamley, 1999). 3 players in a dynamic game face the risk of arriving at the window too late. This is obviously the reason why they may want to rush, like in a bank run. The model should incorporate the trade-off between “going early” at a smaller cost (with a favorable exchange rate) and little information, and delaying for more information with the risk of coming too late. (vi) In most previous models of speculative attacks, the central bank has perfect information on the state of the world. However, a critical factor for the success of a speculative attack is the distribution of characteristics of the speculators about which the central bank may not have superior information. The one-period approach leaves little room for the role of the central bank during a speculative attack. These issues are addressed here with a new model where agents act strategically in a multi-period context. The equilibrium exchange rate is allowed to fluctuate within a band and a devaluation takes place only if the ceiling of the band is reached. As emphasized previously, any speculator in a currency attack wonders how many other agents believe that the fundamentals are such that an attack will succeed, how others think about others’ beliefs, and so on. This feature is modeled here by assuming that there is a mass of speculators with relatively high beliefs that there is a large mass of agents with high beliefs and so on. The mass of these speculators is the uncertain parameter in the economy. For simplicity, there are two states of nature. In one of the two states, the “high” state, the mass of speculators is sufficiently large to induce a devaluation if all speculators buy the foreign currency while that mass is “sub-critical” in the other “low” state. 4 The emphasis is not on the resolution of multiple equilibria as in the global game approach, but on the opportunities offered by the dynamic setting for speculative attacks when agents have heterogeneous expectations on the mass of potential speculators. In all the cases considered here, there will be an equilibrium with no speculative attack. Under some conditions, there will also be an equilibrium with a speculative attack. The role of policy, if any, will be to abolish this second equilibrium. The strategic aspect is critical: the interesting cases occur when the parameters of the model are such that if there is one-period (and thus no opportunity to learn from others), there is a unique equilibrium with no attack4 . Payoff externalities have a particular property here. When speculators face a coordination problem (e.g. attack a currency), there is some incentive to delay in order to get more information on others5 . But there is also a premium for the agents who invest earlier in the coordination, (e.g. when the asset price is still relatively low in a financial market, before the bank run really starts or at the beginning of a political revolution). If the agent delays too much, he comes too late for the gain of the attack and does not get anything. The model presented here builds on a standard structure of a financial market 4 Gale (1995) has shown how a finite number of agents may achieve coordination in multiple periods when each agent can precipitate a subgame with fewer agents in which coordination is dominant. Rodionova and Surti (2000) analyze the case of speculative attacks with a finite number of players and Corsetti et al. (1999) assume one large agent who can induce small and competitive agents to act. 5 The incentive for delay generates strategic substitutability in Chamley and Gale (1994), and strategic complementarity when agents are heterogeneous in Chamley (2000). 5 with noise traders. The main feature of the model is that speculators observe the exchange rate at the end of each period and place market orders for the next period. Following Hellwig (1982) and Blume, Easley and O’Hara (1994), these orders depend on all the information available at the time they are placed and the rational expectations about the price in the next period. Speculators may delay the timing of their attack. An equilibrium is constructed analytically for any finite number of periods by backward induction. In all periods, the subgame has an equilibrium in which there is no speculation for all remaining periods. Under some conditions on the beliefs of the agents, there are other equilibria with speculative attacks. The purpose of the analysis is not to construct a model with a unique equilibrium in which there may be a speculative attack but to analyze how a speculative attack may be facilitated by the learning from prices in markets, and how such equilibria with a speculative attack can be prevented by policy. Proposition 1 shows that a higher number of periods extends the set of beliefs for which a speculative attack is an equilibrium. The equilibrium strategy which is analyzed here tends to a stationary solution when the number of periods tends to infinity (Proposition 2). A numerical example with Gaussian noise illustrates some properties of the model. Policies are analyzed in the last section. A speculative attack can be prevented by widening the band of fluctuations or through the trading by the central bank. If the central bank intervenes by stabilizing the exchange rate (i.e., selling when the exchange rate increases), and this policy is anticipated by rational speculators, a speculative attack is more likely. Speculative attacks may be prevented either by a rationally anticipated intervention which amplifies the fluctuations of the exchange rate, or by a random intervention which cannot be anticipated. 6 1 The model There is a finite number of periods, T + 1, and a continuum of speculators, called agents, of mass θ ∈ {θ0 , θ1 }. The value of θ defines the state which is selected by nature before the first period. In each period, an agent can at most one unit of foreign currency, also called asset. This constraint embodies a credit constraint. At the beginning of the first period, all agents hold only the domestic currency, also called currency. In period T + 1, all agents undo their position: by assumption, all agents must hold only the currency at the end of period T + 1; if an agent holds the asset at the beginning of period T + 1, he sells it in period T + 1. The game is effectively played during T periods. Market orders At the beginning of any period, an agent who holds the currency can place a buy order for the asset, and an agent who holds the asset can place a sell order. The orders are market orders, i.e., they specify a quantity to be traded (conditional on no devaluation) at whatever the market price in the period. An order depends on the information available at the beginning of the period and on the rational expectations, given that information, on the transaction price6 . Market orders embody the sequential nature of trades. Another assumption which is used in the micro-structure of financial markets is that of limit orders. When agents can place limit orders, they submit their entire demand schedule contingent on the information revealed by the equilibrium price. Orders are executed only at the equilibrium price and agents would not change their orders after the closing of 6 The informational properties of a financial market with market orders have been analyzed by Hellwig (1982), Blume, Easly and O’Hara (1994) and Vives (1995). 7 the market if they had an opportunity to do so. No rationing can occur. Such a setting is not appropriate for a situation which is similar to that of a bank run and where agents would change their trade after the equilibrium price is known. We will consider only symmetric equilibria. Because agents are risk-neutral, the payoff of placing an order of amount a < 1 for the asset will be equivalent to that of an order of 1 placed with probability a. We will assume the second formulation to facilitate the presentation. In this way, an agent either holds one unit of the asset (and is an asset holder), or no amount of the asset (and is a currency holder). The market for the asset There is a game in period t if no devaluation (a process described below) has taken place before period t. Let λt−1 be the fraction of agents who hold the asset at the beginning of period t. The mass of agents who hold the asset at the beginning of period t is therefore λt−1 θ. By assumption, no agent holds the asset at the beginning of the first period: λ0 = 0. We will consider only symmetric strategies: there is one strategy (possibly random) for the asset holders and one for the currency holders. Let ζt be the fraction of agents who place a buy order for the asset in period t. By an abuse of notation, ζt will define the strategy of currency holders who place a buy order with probability ζt /(1 − λt−1 ). The strategy of the asset holders will be simpler as we will see later: if some agents buy in an equilibrium, no asset holder sells, and if some asset holders sell, they all sell. Since there will be no ambiguity, an order will be a buy order. Suppose ζt > 0. Given the quantity of orders ζt θ, the demand for the asset by the agents in period t (a stock) is (λt−1 + ζt )θ. The total 8 demand for the asset in period t is the sum of the agents’ endogenous demands, and of an exogenous noise ηt . The introduction of noise traders is standard in financial markets and facilitates trading between agents with asymmetric information (Grossman, 1981). The noise traders in period t undo their position at the beginning of the next period. The distribution of ηt is stationary. The terms ηt are serially independent and the mean of ηt is equal to zero. The supply of the asset is a linear function of the price p of the asset: S(p) = (p − 1)/a, where a is a fixed parameter. The value of S represents the net supply when the price departs from the middle of the band of exchange rate fluctuations which can be interpreted as a long-run value as determined by real trade and policy. This long-run price is an equilibrium value when there is no speculative attack, and it is normalized to 1. The supply schedule S(p) can be defined as minus the net demand of risk-averse “market-makers” who place price contingent orders which may take into account the information revealed by the transaction price. Their net demand is of the form κE[pt+1 |pt − pt ]/V ar(pt+1 |pt ), where pt and pt+1 are the prices of the asset in period t and t+1, and κ > 0 is a parameter. Because market makers have a lower ex ante subjective probability on state θ1 than the speculators, they interpret the market data differently. (The speculators are more optimistic about their high mass). For simplicity, we assume that the market makers have ex ante a vanishingly low probability that there is a high mass of speculators and that a devaluation will take place. Hence, contingent on the observation of the equilibrium price, the revised probability of a devaluation is still vanishingly low: E[pt+1 |pt ] = 1 and the variance of pt+1 in the subjective distribution is constant. In this case, their net demand is of the form (1 − p)/a with a = V ar(pt+1 |pt )/κ. If the current price pt has a significant impact on the 9 expected value E[pt+1 |pt ], a higher price pt shifts the demand curve up, and the effect is equivalent to a less elastic demand curve. We will consider below the impact of such a lower elasticity on the properties of the model. The schedule S(p) may also incorporate the strategy of other agents whose trades depend only on pt . The central bank is assumed to perform the function of a clearing house by matching the trade orders. The central bank may also use its reserves for trading. In that case, its net supply is incorporated in the schedule S(p). Policies of the central bank will be discussed in Section 5, and no trading by the central bank will be considered until then. The regime of the exchange rate within a band of fluctuations stipulates that the price of the asset is allowed to fluctuate in a band below a threshold value 1 + γ, with γ > 0. If the equilibrium price (to be defined later) is above 1 + γ, a devaluation takes place according to a specification which will be given later. The event of a price below 1 − γ will have a negligible probability and will be ignored. Assuming no devaluation prior to period t, let pt be the price determined by the equation   pt = 1 + a (λt−1 + ζt )θ + ηt . (1) If pt ≤ 1 + γ, the price which clears the demand and the supply is within the band and is equal to pt in (refeq:equiprice). There is no devaluation in period t. All buy orders are satisfied. The fraction of agents who hold the asset at the beginning of the next period is λt = λt−1 + ζt . If pt > 1 + γ, the price at which supply and demand are equal is greater than that allowed by the band of fluctuations. Let X̄ be the critical mass of the 10 demand (speculators and noise traders), i.e., the highest value of the demand which can be accomodated by an equilibrium in the band. From equation (1), X̄ is defined such that γ = aX̄. A devaluation takes place when the demand is higher than the critical mass. All orders cannot by executed. To simplify the process, it is assumed that: first, noise traders of period t − 1 undo their positions; second, noise traders of period t place their orders. All these orders are executed (even if the total amount exceeds the critical mass)7 . The amount of the asset which is available for new orders without devaluation is therefore M ax(X̄ − λt−1 θ − ηt , 0). (We will see later that if new orders come in, no asset holder sells). Suppose that the mass of new orders in period t is strictly positive: ζt > 0. (The case ζt = 0 will be described below). By assumption, all the agents’ new orders are executed with the same probability and with the highest probability. The transaction price is the highest possible in the band, 1+γ, and the probability of execution of a buy order is π= M ax(X̄ − λt−1 θ − ηt , 0) . ζt θ By construction, 0 ≤ π ≤ 1. If a devaluation takes place, the price of the foreign asset is set at 1 + A, where A > γ is a fixed parameter8 . 7 This assumption is made to simplify the process. The probability of a demand greater than the critical mass at that stage is very small. 8 One could consider the case where the amount of the devaluation is determined by the intensity of the attack, or by an equibrium mechanism. Such an effect would enhance the strategic complementarity but would not alter the properties of the model. The issue is discussed 11 Information The true state θ is not observable. At the beginning of the first period, all agents have a subjective probability µ0 of the high state θ1 . At the end of each period t, if a devaluation takes place, the game ends. If no devaluation takes place, a subgame begins in period t + 1. Agents observe the price pt in period t. They use this observation to update in a Bayesian fashion their belief from µt−1 to µt . Since the strategies are common knowledge, agents know the fraction of agents who place orders in period t. Hence, the fraction of agents who hold the asset at the beginning of the next period, λt , is known. We will show that the subgame which begins in period t depends only on the period t and on (λt−1 , µt−1 ). Payoffs The payoff of an agent is the sum of the discounted values of the trades in all periods, valued in the (domestic) currency. The discount factor δ, 0 < δ < 1, embodies a positive difference between the rate of return in the domestic currency and that of the foreign asset, for a fixed exchange rate. Such a positive difference ensures that in the context of the model, the band of the asset price is “sustainable”: if there is no speculative attack, speculators prefer to hold the currency (or to sell the foreign asset which they may own), in an equilibrium. If the difference between interest rates were not strictly positive (or δ = 1), the band might not be sustainable9 . In the next section, we will introduce a mild briefly at the end of the paper. 9 For example, if the support of the distribution of noise traders extends beyond γ/a, holders of the foreign asset never sell, and if the number of periods is sufficiently large, all agents with an option to buy the asset exercise it with no delay. 12 sufficient condition on the discount rate for the sustainability of the exchange rate band. 2 Zero-equilibria Suppose that in some period t, all asset holders sell and no currency holder buys. This situation occurs at least in the last period. The value of holding an asset at the beginning of such a period t is denoted by V̄ and is given by Lemma 1. Lemma 1 If in some period (e.g., in period T + 1), all asset holders sell and no currency holder buys, the value of selling the asset is V̄ = 1 − β,  with β=a η>X̄ (η − X̄)dF (η). All formal results are proven in the appendix. We will assume that 0 ≤ β < 1. The main purpose of the model is to analyze how the endogenous behavior of speculators can trigger a devaluation with no “real shock” to the economy. We assume that the probability of a devaluation generated solely by the noise traders α is small:  α= η>X̄ dF (η) ≥ 0. (2) The next result shows that under some conditions on (α, β), a speculator does not have an incentive to buy or to hold the asset if he is the only one to do so. Lemma 2 For any δ ∈ (0, 1) and any β ∗ ∈ (0, 1), there exists α∗ > 0 such that if α < α∗ and β < β ∗ , the following property holds: for any (λt−1 , µt−1 ), if there has been no devaluation before period t, the subgame which begins in period t has an equilibrium in which all asset holders sell in period t and no agent places an order in period t or after. This equilibrium is called the zero-equilibrium. The value of β ∗ does not need to be small. 13 In the last period T + 1, the zero-equilibrium is the only equilibrium, by assumption. The value of holding the asset at the beginning of the last period is therefore V̄ , while the value of the option to buy the asset is zero. We now proceed by backward induction to determine any equilibrium in the subgame which begins in period t ≤ T . 3 Equilibria with speculation 3.1 Value functions In any period t, three value functions will be important. They will be defined by backward induction. 1. The payoff of an order in period t depends on (i) the fraction of agents who place an order in the same period, ζt , (ii) the fraction of agents who hold the asset at the beginning of the period, λt−1 , (iii) the speculator’s probability µt−1 of the state θ1 , which is determined by the history of past prices. The payoff is thus a function u(ζt ; λt−1 , µt−1 ) where (λt−1 , µt−1 ) are “initial conditions” in period t. We will show that in an equilibrium, the strategy ζt depends only on (λt−1 , µt−1 ). The value of an order in equilibrium will be a function of (λt−1 , µt−1 ) and will be denoted by Ut (λt−1 , µt−1 ). 2. The value of holding the asset at the beginning of period t and keeping the asset until the end of the period (not selling), in an equilibrium, is denoted by Vt (λt−1 , µt−1 ). Since an asset holder does not have to buy the asset, Vt ≥ Ut . 3. The agents who don’t hold the asset have the option to buy. The value 14 of delay (i.e., keeping the option until at least the next period), will be denoted by wt (ζt , λt−1 , µt−1 ). In period T + 1, all agents sell by assumption. The payoff of a buy order is set at zero by convention and the payoff of selling the asset if it is held at the beginnin of the period is given in Lemma 1. For all (λT , µT ) ∈ [0, 1) × (0, 1), VT +1 (λT , µT ) = V̄ = 1 − β, and UT +1 (λT , µT ) = 0. We now proceed by backward induction for all periods t ≤ T . 3.2 Equilibria The evolution of beliefs If there is no devaluation in period t, the equilibrium price is pt = 1 + ayt . Using this equation, the observation of pt is equivalent to the observation of the total demand yt = (λt−1 + ζt )θ + ηt , (3) which conveys a signal on the state θ. Since agents know the strategies and λt−1 + ζt , the observation of yt is equivalent to the observation of the variable zt = θ + ηt . λt−1 + ζt (4) The variance of the noise term is reduced when more agents place an order. The information conveyed by the market (conditional on no devaluation) increases with the fraction of agents who place orders. Recall that the belief at the beginning of period t (probability that θ = θ1 ) is denoted by µt−1 . Let f (η) be the density of η (which is independent t). If there is no devaluation in period t, the belief on θt in the next period is determined by the Bayesian updating formula µt (yt ; λt−1 + ζt , µt−1 ) µt−1 f (yt − (λt−1 + ζt )θ1 ) = . 1 − µt (yt ; λt−1 + ζt , µt−1 ) (1 − µt−1 )f (yt − (λt−1 + ζt )θ0 ) 15 (5) Payoff of an order From the description of the trades, a devaluation takes place in period t if yt > X̄, in which case the ex post payoff of an order is (A − γ) M ax(X̄ − λt−1 θ − η, 0) . ζt θ If no devaluation takes place in period t, the ex post payoff is the value of holding the asset in the continuation of the game minus the purchase price, 1+ayt , i.e, u(θ, η) = δ Vt+1 (λt−1 + ζt , µt ) − (1 + ayt ). In this expression, the continuation value Vt+1 depends on the fraction of speculators holding the asset λt = λt−1 + ζt , and on the belief µt at the end of period t which has been expressed in (5). The payoff of an order is the expected value of all ex post payoffs for all possible values of θ and η. Denoting by F (θ, η; µt−1 ) the cumulative distribution10 of (θ, η), this payoff is  ut (ζt ; λt−1 , µt−1 ) =  + yt X̄ (A − γ) M ax(X̄ − λt−1 θ − η, 0) dF (θ, η; µt−1 ) ζt θ  (6) δ Vt+1 (λt−1 + ζt , µt ) − (1 + ayt ) dF (θ, η; µt−1 ). The method of backward induction which is used here can characterize all equilibria of all subgames but such a complete characterization is beyond the scope of the paper. We will assume that if in period t there is no equilibrium strategy with ζt > 0 (no new order comes in), then the speculative attacks stops completely11 . 10 θ and η are independent; the distribution of θ depends on the belief µt−1 , and the distri- bution of ηt is independent of the period. 11 Since the incentive to hold is weaker than that to buy, there may be a level of belief such that no speculator buys but asset holders don’t sell. The price in period t may convey sufficient information to induce a resumption of new orders in period t + 1 and eventually a successful attack. While such an equilibrium is theoretically possible, we assume that agents don’t coordinate on it. 16 Such an assumption is possible because the zero-equilibrium is an equilibrium in any period (Lemma 2). In the subgame which begins in period t, there may be multiple equilibrium values for ζt > 0. (An example will be given below). When there are such multiple equilibrium strategies, we assume that agents coordinate on the highest equilibrium value of ζt . Assumption 1 In any period t, if there is no strictly positive equilibrium value of ζt for buy orders, agents coordinate on the zero-equilibrium and the game ends at the end of the period. If there are multiple equilibrium strategies with strictly positive ζt , agents coordinate on the highest such value. This coordination rule is common knowledge. Arbitrage and the payoff of delay By Assumption 1, the game ends in period t + 1 if the payoff of an order in that period is negative. Hence, in equilibrium, the payoff of delay is equal to that of making a final decision in the following period, either to place a buy order in period t + 1 or to never place a buy order. This one-step property is the same as in Chamley and Gale (1994) and is a consequence of Assumption 1. The payoff of delay is therefore equal to  wt (ζt ; λt−1 , µt−1 ) = δ yt < X̄   M ax Ut+1 (λt−1 + ζt , µt ), 0 dF (θ, η; µt−1 ), (7) Since the mass of asset holders is not greater than one, ζ ∈ [0, 1 − λt−1 ]. A necessary condition for ζ to be an equilibrium value in period t is that the payoff of a buy order is at least equal to that of delay: ut (ζ; λt−1 , µt−1 ) ≥ wt (ζ; λt−1 , µt−1 ). 17 (8) Case a: If there is no ζ > 0 which satisfies (8), then the equilibrium value is ζt = 0 and by Assumption 1, the game ends in period t with the zero-equilibrium. All speculators hold only the currency for all remaining periods. Case b: If there is a value ζ > 0 such that (8) is satisfied, by Assumption 1, the equilibrium value ζt is defined as   ζt = M ax ζ ∈ (0, 1 − λt−1 ] | ut (ζ; λt−1 , µt−1 ) ≥ wt (ζ; λt−1 , µt−1 ) . In general, the strategy in period t is a function ζt = φt (λt−1 , µt−1 ) with   φt (λt−1 , µt−1 ) = M ax 0, ζ ∈ (0, 1−λt−1 ]|ut (ζ; λt−1 , µt−1 ) ≥ wt (ζ; λt−1 , µt−1 ) (9) When ζt = φt > 0, a speculative attack takes place. It is of one of the following two types: (i) If ζt = 1 − λt−1 , all agents who have an option place an order. Since there cannot be new orders in period t + 1, the attack either succeeds in period t or it fails with all agents selling the foreign asset in period t + 1. (ii) If 0 < ζt < 1 − λt−1 , by continuity of ut and wt , (8) must be an equality, ut (ζt ; λt−1 , µt−1 ) = wt (ζt ; λt−1 , µt−1 ). In such an equilibrium, there is an arbitrage between buying in period t at a relatively low price with less information, and delaying until period t+1 to get more information while facing the risk of missing the benefit from a devaluation in period t. The strategy φt (λt−1 , µt−1 ) determines the payoff of an order, in equilibrium: Ut (λt−1 , µt−1 ) = ut (φt (λt−1 , µt−1 ), λt−1 , µt−1 ). (10) Value of holding the asset Suppose that the belief µ is such that some agents place a buy order. These agents buy after the agents who already own the asset and face a higher price. 18 . In this situation, if some agents find it profitable to place a new order, then the agents who already own the asset and have the same information strictly prefer to hold rather than to sell. This property is formalized in Lemma 3. Lemma 3 In an equilibrium, if some agents place a buy order in period t (and φt (λt , µt ) > 0), then no asset holder sells in period t. The value of holding the asset at the beginning of period t depends on whether the speculative attack continues in the period. Using Lemma 3, this value satisfies the following recursive equations. • If φt (λt−1 , µt−1 ) > 0, Vt (λt−1 , µt−1 ) =  yt > X̄ (1 + A)dF (θ, η; µt−1 )  + δ (11) yt < X̄ Vt+1 (λt−1 + φt , µt )dF (θ, η; µt−1 ). • If φt (λt−1 , µt−1 ) = 0, by Assumption 1 the equilibrium is the zero-equilibrium and Vt (λt−1 , µt−1 ) = V̄ . Backward determination of the equilibrium In the last period T + 1, we have UT +1 ≡ 0 and VT +1 ≡ V̄ = 1 − β. Assume that for t ≤ T , Ut+1 and Vt+1 are given. The payoffs of a buy order in period t is determined by (6) and that of delay by (7). The policy function φt (λt , µt ) is then determined by (9). This function determines Ut by (10) and Vt by (11). The equilibrium is determined for all periods by backward induction. The previous definition of the policy function generates an equilibrium which is stable in the sense that a small deviation of all currency holders from the equilibrium strategy induces a stabilizing reaction. Indeed, consider first the 19 corner solution and assume that ζt = 1 − λt > 0. Ruling out an event with probability zero, we can assume that ut (ζt ; λt−1 , µt−1 ) > 0 = wt (ζt ; λt−1 , µt−1 ). Assume that a perturbation occurs in the form of a small reduction of ζt . Using the expression of u in (6) with no future period, one can see that the optimal response is still to place an order with probability 1. Suppose now that there is arbitrage with 0 < φt (λt−1 , µt−1 ) < 1 − λt , and consider the difference between the payoff of an order and that of delay: D(ζ) = ut (ζ; λt−1 , µt−1 ) − wt (ζ; λt−1 , µt−1 ). By definition of φt , D(ζ) < 0 for all ζ ∈ (φt , 1 − λt−1 ] and its derivative is not equal to 0 at the point ζ = φt . Hence the graph of ut (ζ; λt−1 , µt−1 ) cuts that of wt (ζ; λt−1 , µt−1 ) from above at ζ = ζt : if all currency holders but one reduce (increase) their probability of placing an order, the optimal response is to place an order with probability one (zero). The reaction is stabilizing. 4 Equilibrium properties In the model, holding the currency at the beginning of period t is an option to buy. The expected gain from placing an order is Ut . The value of the option, which is the same as the value of the currency for a speculator, is therefore 1+M ax(Ut , 0). The next result establishes a relation between the value of holding the asset and the value of holding the currency. Lemma 4 For any (λ, µ) ∈ (0, 1) × (0, 1), the difference between the values of holding the asset and holding the currency is equal to    Vt (λ, µ) − 1 + M ax Ut (λ, µ), 0  where G(ζ, λ, µ) is an increasing function of ζ ∈ (0, 1 − λ]. 20  = G φt (λ, µ), λ, µ , The monotonicity of G is intuitive. A higher value of ζ raises the probability of a devaluation, and the capital gain on the asset increases. The effect on the asset holders is greater than on the option holders because the gain of the option holders is dampened by the higher purchase price if there is no devaluation, and a lower probability of execution if the devaluation takes place. Let us now compare the policy functions in two consecutive periods φt and φt+1 . Suppose that for some value (λ, µ), 0 < φt+1 (λ, µ) < 1 − λ, and that there is an arbitrage between delay and no delay in period t + 1 if (λt , µt ) = (λ, µ). Assume that φt+1 ≥ φt+2 for all values of (λ, µ) ∈ (0, 1) × (0, 1). An intuitive argument indicates that if in equilibrium the payoff of an order is at least as high as that of delay in period t + 1 for the initial condition (λt , µt ) = (λ, µ), then in period t for the same initial condition, (λt−1 , µt−1 ), the payoff of an order is at least as high as that of delay when ζt = φt+1 (λ, µ). The difference between the payoff of an order and that of delay is the sum of two terms. The first is the capital gain from a devaluation in period t, net of the purchase price if no devaluation takes place. This term depends only on the strategy ζt and the initial conditions which by assumption are the same in periods t and t + 1. The second term is the expected premium of holding the asset over the payoff of an order in the next period, δE[Vt+1 −M ax(Ut+1 , 0)], conditional on no devaluation in period t. By the induction hypothesis, φt+1 ≥ φt+2 and using Lemma 4, this term is higher in period t + 1 than in period t + 2 (for the same initial conditions). Since there is arbitrage in period t + 1, the difference between the payoff of an order and that of delay must be non negative in period t. By definition of the policy function φt in (9), φt (λ, µ) ≥ φt+1 (λ, µ). The argument 21 which is detailed in the appendix leads to the following result. Proposition 1 Under Assumption 1, for any (λ, µ) ∈ (0, 1)×(0, 1), φt−1 (λ, µ) ≥ φt (λ, µ). For any t and (λ, µ) ∈ (0, 1) × (0, 1), φt (λ, µ) is non decreasing in the number T of periods of the game. The mass of speculators who attack the currency is an increasing function of the number of periods, ceteris paribus. Another description of the result is that the set of values (λ, µ) of the initial conditions at the beginning of a period t over which a speculative attack takes place is larger when the number of periods which follow period t is higher. More periods provide more opportunities for speculators to “communicate” through the market prices during an attack and thus a higher incentive to participate in a speculative attack. Let φTt (λ, µ) be the policy function in period t when the game has T periods (not counting the last one). The subgame which begins in period t with conditions (λ, µ) and number of periods T is the same as that which begins in period t − 1 with the same condition (λ, µ) and number of periods T − 1: −1 (λ, µ). φTt (λ, µ) = φTt−1 Hence, φTt +1 = φTt−1 ≥ φTt , by Proposition 1. If the number of periods in the game T increases (T > t), the sequence of policy functions φTt (λ, µ) is monotone increasing in T . Since φTt is bounded by one, it tends to a limit. The equilibrium solution tends to a stationary solution when T tends to infinity. Proposition 2 For any fixed t and (λ, µ), if the number of periods T increases, the policy function φTt (λ, µ) does not decrease. If T → ∞, φTt (λ, µ) tends to a limit φ(λ, µ). 22