Structure of the Economy

We consider a two-period version of a standard model of monetary policy in a small, open economy of the type adopted, for example, by Obstfeld (1996, 1997), Jeanne (1997), and others. Our country is pegging its currency at a fixed rate against a foreign currency. We maintain that the purchasing power parity rule holds, so that, in the case of devaluation, the time t (for t=1, 2) depreciation rate is identical to inflation (π_t). We also assume that the unemployment rate (u_t) is described by

where u_n is the “natural” rate of unemployment⁴

The persistence in unemployment embedded in equation (1) will play an important role in our model. In fact, it implies that the information about the period t realization of the disturbance is relevant in forecasting time t+1 unemployment. Hence, any asymmetry concerning the information about the period t fundamental implies the presence of heterogeneous forecasts for time t+1 unemployment.

Many theoretical and empirical papers provide support for the inclusion of some unemployment persistence in a macroeconomic model. Blanchard and Wolfers (2000) suggest that the high and persistent level of unemployment that characterized many European countries in the eighties and in the nineties can be at least partly explained by the interplay of shocks with rigid market institutions. Lindbeck and Snower, in their 2001 survey, point at labor turnover costs as the main determinant of employment (and hence unemployment) persistence both at the firm and at the aggregate level. Whereas the above explanations mainly focus on labor demand factors, Ljungqvist and Sargent (1998) study the effects of the welfare state on the supply of labor. In their model, unemployment benefits depend on the workers' past earnings and induce jobless workers, whose skills depreciate during the unemployment spell, to significantly reduce their search effort for new jobs. In this framework, transient shocks can easily generate persistent unemployment because jobless workers—whose ability is shrinking over time—have difficulty in finding jobs that they prefer to their unemployment compensation. At the empirical level, the structural VAR analysis conducted by Balmaseda et al. (2000) supports, for almost all the OECD countries, an identification scheme based on unemployment being persistent but stationary. More recently, Fève et al. (2003) estimate, again for the OECD countries, a model including equation (1) and find strong evidence of a persistence parameter that assumes values close (but not equal) to unity.

In our model, when the peg is defended, and hence inflation is set to zero, the unemployment rate—denoted by the superscript p—is as follows:

Agents

There are three types of agents: wage-setters, speculators, and a policymaker.

There are an infinite number of wage-setters, who are indexed by the superscript z and are referred to as if they were all female. As we specify below, wage-setters need to decide their period 2 wages at time 1. Wage bargaining is completely decentralized: each worker determines her own wage on the basis of her private information about the state of the economic system. In particular, each wage-setter chooses the growth rate for her nominal wage (w₂^z) by targeting her expectations about period 2 inflation,

where E[·] denotes the conditional expectation operator. The relevant information set will be specified below. What we have in mind is an economic system in which the workers are concerned about their wage purchasing power, which they want to preserve. Moreover, they are aware that—if the increase in their wage is below (above) the inflation rate—they will be forced to supply an excessive (too low) amount of working hours, because they are substitutes for firms, which retain the “right to manage.”

With no loss of generality, we assume that the total mass of wage-setters is one.

Speculators, indexed by the superscript i, are risk-neutral, their measure is h, and they are assumed to be male. Each investor holds liquid wealth (inclusive of his borrowing potential) equal to unity, and he decides in both time periods whether to speculate against the domestic currency or not. If he speculates, he buys the foreign currency on the forward market,⁵

where W₂ⁱ is the expected profit yielded in period 2 by speculation, conditional on the time 2 information set; the payoff of not attacking is normalized to zero. The speculator's payoff in period 1 is

where W₁ⁱ is the expected profit yielded at time 1 by speculation, conditional on that time information, whereas p₁ⁱ is the subjective probability that the speculative attack is successful and the peg is abandoned in period 1. The discount factor, which plays no substantial role in our model, is unity.

Provided that the peg has not been abandoned in the past, the domestic policymaker decides, in each period, whether or not to abandon the peg in order to minimize her or his intertemporal loss function,

where L_t(·) is the period t loss. We assume that when the policymaker decides to abandon the peg, the devaluation is of the fixed size

.⁶

The first line is a standard quadratic loss function that applies when the peg is abandoned:

is the optimal unemployment level for the policymaker, and δ>0 is a parameter capturing the policymaker's “aversion” to inflation. This quadratic loss function applies also when the peg has been abandoned in a previous period. In this case, the time t inflation rate,

, is incorporated into the average wage and thus—via equation (1)—influences the period t unemployment.

The second line becomes relevant when the policymaker defends the peg. Its second addendum reflects the additional costs and benefits generated by keeping the peg. The benefit, v, is expressed as a share of GDP. This benefit captures a number of factors, such as increased investment and foreign trade generated by the reduction in exchange rate uncertainty or the political and economic advantages arising from the possibility of joining a monetary union (as might be the case for some central European countries).⁷

The net benefit of the defence is weighted by λ>1.

We now introduce the following:

Assumption 1. v<ch.

Assumption 1 implies that for sufficiently high values of α_t(·) the cost of the defense of the peg outweighs the per-period benefits (equation (3)). Hence, we shall consider both the case of a positive per-period benefit and the case of a negative one, making the discussion more general.

Timing of Events and Informational Structure

Every agent enters the first period having observed the past realizations of all the relevant variables. In line with the existing literature, we assume that he or she also knows the natural unemployment rate, the policymaker's preferences, the parameters characterizing eq. (1), and the structure of the stochastic disturbances. This knowledge is taken as understood: hence, we denote the time 1 common information set by

.

At the beginning of time 1, both speculators and wage-setters receive signals, denoted by s₁ⁱ and s₁^z, respectively, concerning the shock currently affecting the economic system. (More details about the structure of the signal will be provided shortly.) At this stage, speculators—whose private information set is

—decide whether to attack the peg, and the policymaker decides whether to resist or to devalue. Having observed whether the peg has been abandoned or not, the workers fix their next-period wage, contracting for it directly with their employers. Notice that the wage-setters' information set is

.

At this point, two comments are in order. First, we consider a “decentralized” bargaining environment because even the most casual observation suggests that agents' forecasts are different; thus, we found it of interest to analyze a framework in which agents' actions are driven by their diversified beliefs. Notice, moreover, that the polar case, based on the assumption of a single union determining wages, is equivalent to the hypothesis of common information at the wage-setting stage. Not surprisingly, such an assumption easily leads to multiple equilibria (as shown in the working paper version of this contribution) because it eradicates any coordination problem from the wage decision taken during period 1 and affecting period 2.⁹

Second, as already remarked, at the wage-setting stage any private information set incorporates the knowledge about the effectiveness of the defence of the peg. Hence, the one-period-ahead wage is decided upon by exploiting the informational content of the observation concerning the successfulness of the currency attack, a fact that is of remarkable importance in what follows. However, we assume that each agent must conclude her wage contract without knowing the wage agreed upon by other agents. If these data were available, each individual worker could extract information about the signals received by the others: this scenario would become more similar to a “centralized” wage-bargaining scenario (and the model would become analytically intractable).

In period 2, the realization of the time 1 disturbance and of the time 1 (average) contracted wage become common knowledge, so that

. Investors receive a signal about the time 2 disturbance (hence,

and decide again whether to attack the currency in the light of the new information; the policymaker decides whether to resist the attack or not. (Figure 1 summarizes the timing of events).

The timing of events.

The signal received by an agent on any particular date is private and potentially different from the signals received by others. The signal received by each agent—who knows only imperfectly the fundamental—concerns the realization of the stochastic disturbance η_t. To save on notation, we assume that the signals received by speculators and wage-setters are characterized by the same structure and by the same parameter values. Focusing, to save space, only on speculators, we specify the signal as

, where ε_tⁱ is a normal random variable with zero conditional mean and variance σ_ε. We assume that every ε_tⁱ is independent from η_t and over time (these hypotheses can be relaxed).

Notice that receiving a signal about the disturbance is equivalent to obtaining a signal about the level of unemployment that would prevail in the economy when the peg is defended. The normality of the disturbances ε_tⁱ, in conjunction with the normality of η_t, allows the exploitation of the Bayesian learning model to obtain

where

denotes the variance of

conditional on the information set

.

These expressions summarize the belief each agent i develops about the realization of the fundamental at the beginning of time t. When calculating

, the weight attributed to the private signal is positively linked to its precision and to the variance of the disturbances. Notice that the conditional expectation for the fundamental can be expressed as

We assume, following the existing literature, that the policymaker perfectly knows the state of the economy and hence the realization for η_t;¹⁰

which sum up the knowledge the policymaker has about individuals' signals.

Policy Analysis

When the policymaker opts to leave the peg, its loss (

is

while, if the fixed exchange rate is maintained, its loss becomes

For any given

(and hence for any given η₂), the policymaker's optimal decision is to defend the peg if

, an inequality that boils down to

Assume that

, where φ=(β²+δ)/2β. In this case, the left-hand side of the inequality above is negative. Thus, inequality (6) can be fulfilled only if Sign(v−chα₂(·)) is positive and therefore α₂(·)<v/ch. In this case, the above inequality is satisfied—and hence the peg is defended—when

where

. The obvious restriction α₂(·) ≥ 0 implies

. When

, the policymaker opts out of the fixed exchange regime even if nobody speculates against the currency.

Now consider the case

, so that the left-hand side of inequality (6) is nonnegative. In this case, (6) not only is satisfied when α₂(·)<v/ch but also can be fulfilled if α₂(·)≥v/ch. When α₂(·)≥v/ch, the defense of the peg requires the following further restriction:

Because α₂(·)≤1,

; when

the peg is defended even if every agent attacks.

Notice that equation (6) generates the tripartite classification for the fundamental that characterizes a large part of the literature.

The Share of Attackers

Morris and Shin analyzed in various contexts (e.g., 1998, 1999) the optimal behaviour of risk-neutral attackers in a global game framework. They propose a monotone-strategies equilibrium, where two key elements characterize the speculators' optimal behavior: all the agents who obtain a signal larger than a trigger k₂ attack the currency, whereas no speculator observing a signal lower than the trigger participates in the attack.

The policymaker is aware of this behavior, and so—given k₂—he or she exploits his knowledge about the individuals' signals [equation (5)] to compute the share of attackers for any given realization for η₂ (i.e., for any given level for the fundamental),

where ϕ(a, b, c) is the probability density function at a for a normal random variable with conditional mean b and conditional variance c; Φ(a) is the value of the cumulative density function for a standard normal computed at a. We will denote by ϕ(a) the probability density function at a for a standard normal distribution.

We now prove

PROPOSITION 1. For any given k₂, a unique threshold u(k₂) exists, such that the policymaker decides to opt out of the peg for u

and to defend the peg otherwise.

Proof. Refer to Appendix A.

Figure 2 shows how the share of attackers (equation 8) and the resistance curve (given by equations (7a) and (7b)) determine the resistance threshold u(k₂) as a function of the trigger signal k₂.

Determination of the resistance threshold u(k2).

The fact that u(k₂) can be uniquely determined as a function of k₂ helps to build some intuition for the equilibrium we are considering. In fact, because any speculator can figure out the threshold level u(k₂), he can compute his subjective probability that the peg is abandoned (which is, the probability that

). Because a speculator makes money only when the attack succeeds, his expected utility depends positively upon his assessment about the probability of a successful speculative episode, which is increasing in the speculator's private signal. Hence, the expected profit for any speculator is increasing in the signal he receives. Therefore, when an agent observing

finds it profitable to attack, he can be sure that all the speculators with larger signals will do the same. Similarly, when an agent receiving the signal

prefers not to attack, he can be sure that all the speculators observing a smaller signal will refrain from attacking.

Attackers' Payoffs

The expected profit for an attacker is given by the expected capital gain minus the cost that he must incur when speculating; in turn, the expected capital gain is given by inflation (

) multiplied by the subjective probability of a successful attack (which is the expression in the big square brackets is the equation below).

When speculators follow the equilibrium strategies we have described in the previous section, the expected profit for an agent i who decides to participate in the speculative attack—believing that the currency is attacked only by the agents who receive a signal larger than the threshold k₂—depends on his own signal and on the resistance threshold, which in its turn depends on k₂.

Hence, the period 2 expected profit for an attacker is

where I^k2 is a function indicating that—at time 2—the attack is run by all the agents observing a signal larger than or equal to k₂ (and only by them) and f is the cost that one must incur when speculating against a currency. Exploiting equation (4), we manipulate the above expression to write

Second-Period Currency Attack

We now analyze the equilibrium outcome of the speculative attack taking place in period 2. Because we need to determine simultaneously the resistance threshold u(k₂) and the trigger signal k₂, we need two equilibrium relations. The first one is equation (6), in which the share of attackers is given by (8),

where, for convenience, we have exploited the fact that

.

The second equation simply states that the “marginal speculator” (i.e., the one receiving the lowest signal allowing for participation in the attack (k₂)) makes zero expected profits:

Adapting the argument provided in Morris and Shin (1999) and Metz (2002), we now prove

PROPOSITION 2. If

, the period 2 equilibrium is unique.

Proof. Refer to Appendix A.

Thus, to guarantee the uniqueness of the “currency attack” equilibrium, we introduce the following:

So far, we have simply confirmed Morris and Shin's main result: provided that the variance of the private signals is sufficiently small in comparison to the variance of the true process, there is a unique period 2 equilibrium. It is the presence of agents receiving very high (larger than

or very low (smaller than

signals that induces the uniqueness of the equilibrium. In fact, for a speculator obtaining a very high (low) signal, it is a dominant strategy (not) to attack the peg, because he believes that the peg is (not) abandoned even if nobody (everybody) attacks. The knowledge that agents with “extreme” beliefs have a dominant strategy makes the same strategy dominant also for the speculators receiving “almost extreme” signals, because they are “almost sure” that a sufficient share of speculators will (not) attack the peg. Iterating this reasoning leads speculators to select the strategy prescribing to attack only upon receiving a signal larger than k₂.

The fact that equilibrium uniqueness is granted only if private signals are sufficiently precise can be understood by considering that the period 1 realization for the fundamental—being known by everybody at time 2—plays the role of a public signal. Therefore, it helps any speculator to make some inference about other speculators' beliefs, and therefore about their actions. If the public information is sufficiently precise relative to private signals, its coordinating effect becomes so large that it induces multiple equilibria.

However, the uniqueness result holds conditionally on the wage, which was determined during period 1 and—up to now—taken as given. Hence, to fully characterize the equilibrium, we need to connect the determination of the wage rate with the outcome of the currency attack.

Before we undertake this task, notice that the observation of the period 2 currency attack equilibrium implies the obtaining of some information about the realization of the stochastic disturbance η₂. When the attack fails, every agent understands that

, which means that

; moreover, notice that such information is common knowledge. As we shall argue below, the information revealed by the currency attack plays an important role.

Finally, notice that the two thresholds characterizing the equilibrium, k₂ and u(k₂), are computed without referring to any particular information set. Private signals, in addition to providing the rational basis for the adoption of monotone strategies, determine the identities of the attackers.

Wage Determination

As already mentioned, we assume that every wage-setter contracts her own wage on the basis of her private information about the state of the economy because this is a simple way to allow for an active role of time 1 informational differences. Notice that wage-setters know whether the peg has been abandoned or not; we focus our attention on the case where the peg has been defended and wage-setters are aware that, as suggested by our previous analysis,

, where

is determined during the currency attack phase and it is—for now—taken as given.

Because every wage-setter picks her wage by targeting her expectations about period 2 inflation (equation (2)), the wage she chooses in period 1 is

The determination of the time 2 individual wage for agent z is not trivial because, from the perspective of the first period, her time 2 expected inflation depends upon the average wage, which in turn is determined by the integral sum of individual wages. In addition, the average wage influences unemployment, via equation (1), and hence the likelihood of a successful period 2 defence, which in turn affects expected inflation.

We analyze the formation of individual inflation expectations in three steps. First, we investigate the inflation expectation, as it would be determined at the beginning of the second period on the basis of the common information set I₂ (which includes w^A₂). We then use this result to show how the wage-setters' subjective period 1 inflation expectations are formed, given the average wage. Finally, we determine the average wage.

At the beginning of the second period, wage-setters know the realization for η₁ and the average wage contracted during period 1, w₂^A. Hence, the expected inflation is

Solving the integral, we write

where we highlight the dependence of the fundamental

and of the resistance threshold

on the realized values for η₁ and

.

At time 2, any wage-setter can compute the threshold

as part of the unique solution for the system:

In period 1, each wage-setter—for any given average wage—can compute the one-period-ahead inflation expectation conditional upon her imperfect information about η₁. What any wage-setter does for a given average wage is to weight the inflation expectations conditional on η₁ by her subjective realization probabilities for η₁,

where

is the probability that agent z, receiving signal

and observing that

, assigns to the event that the true realization for the process affecting the fundamental is η₁.¹²

: what a wage-setter actually knows is her subjective estimate for the average wage. For a given

, the individual wage [equation (11)] becomes

At this stage, we must take into account the fact that any wage-setter z is aware that the other agents receive different signals and, hence, that using their information sets, they compute different wages. Any agent z can calculate, from the standpoint of her information set, the average wage,

where

is the probability that agent r receives signal

given that agent z has received signal

and that

.¹³

We define as a wage-setting equilibrium a situation in which (a) every wage-setter z computes her subjective inflation expectations, using equation (14), on the basis of the average wage w₂^A(z) obtained from equation (14), and (b) every private wage increase, being determined through equation (13), is consistent with the subjective inflation expectation and average wage.

Notice that the wage-setting equilibrium is conditional upon the true state of the economy (which affects the private signals) and upon the first-period resistance trigger,

, which, as we shall discuss in what follows, can be seen as the difference between the first-period resistance trigger, u(k₁), and the first-period expectation about the fundamental,

.

Finally, notice that with knowledge of the realization for η₁ the policymaker can compute the average wage as determined on the market:

First-Period Currency Attack

In the first period, speculators must decide whether to attack the peg, whereas the policymaker chooses whether to resist on the basis of his or her intertemporal loss function. As we did for the second period, we need to determine a trigger signal (which determines the speculators' behavior) and a resistance threshold for the policymaker. Notice that the policymaker's decision depends on his or her computations about

, because the second-period average wage affects his or her second-period loss.

Now, we encompass the possibility that there are multiple equilibrium distributions for wages. Following a relevant portion of the literature, we assume that, when equilibria are multiple, wage-setters coordinate using a “sunspot” variable. We denote by L the number of equilibria and by q_l the probability that wage-setters coordinate on a specific equilibrium (that is, that the sunspot variable takes the values related to a specific equilibrium). Obviously,

Now, notice that the speculators' first-period behavior can be characterized in the same way that we portrayed their time 2 conduct. In fact, because speculators are atomistic, they do not take into account the (actually negligible) impact of their first-period action on the probability of survival for the peg. Hence, their continuation value,

, is independent of their first-period decision with regard to the position they take. Because the continuation value is identical whatever action they take, speculators only care about their first-period expected payoff.

Thus, all those who receive a signal higher than the one obtained by the marginal speculator attack the peg:

where u(k₁) is the peg abandonment threshold for the policymaker.

When the policymaker decides to defend the peg against the first-period attack, he or she obtains in that period the following loss:

Defending the peg, the policymaker grants himself or herself the possibility of opting out of the fixed-exchange-rate regime during the second period. Because the policymaker can compute the average wage(s) that is (are) going to be established on the labour market given u^p₁ (which implies the knowledge of η₁), the expected time 2 loss is

where, to save on notation, the dependence of

and of u(k₂) on the average wage,

, and hence on the realization for the sunspot variable, is taken as understood.

Accordingly, the total loss involved by keeping the peg during the first period is

When the policymaker decides to float the currency at time 1 he or she obtains the following first-period loss:

The second-period expected loss is computed taking into account the fact that the second-period wages will be set equal to the second-period inflation,

:

As before, the total loss involved when the peg is abandoned during the first period is given by the sum of the losses in the two periods:

Because the peg is defended at time 1 whenever

, the resistance threshold is defined by the equality

which is the intertemporal counterpart of equation (9).

Numerical Results

The system composed of equations (9), (10), and (12) to (17) determines k₁, k₂, the resistance triggers u(k₁) and u(k₂), the individual wages

, the subjectively computed average wage

, the inflation expectations, and the true average wage

. This system cannot be solved analytically, and thus we need to rely on numerical techniques.¹⁴

We now provide numerical values for the parameters (β, δ, λ, ρ, f, σ_ε, and σ_η). In addition, we offer a quantitative assessment of the benefit obtained by maintaining the exchange rate (v) for the cost of the defence against speculation (determined by the product of c by h), for the natural rate, u_n, for the target level of unemployment,

, and for the inflation rate prevailing in case of free floating,

. A probability distribution for the sunspot variable must also be provided. Because in all of our calculations we have been able to identify at most two equilibria, what is actually needed is a value for the probability q of an equilibrium where wages are high (i.e., of a “pessimistic equilibrium”).

For some of our parameters a convincing value is easily provided. Some empirical studies suggest that, in the United States, the sensitivity of unemployment to wage growth and inflation is high; accordingly, the most frequently used value for β is one, when the calibration is based on annual data. We fix β to 0.5 to accommodate the fact that the relationship between inflation and unemployment seems to be looser in many countries—such as those in the Euro area—than in the United States. The standard deviation σ_η has been assigned the value 0.01, which implies that in two cases out of three, absent monetary policy surprises, the annual change in the unemployment rate is below 1%. The speculation cost f has been set to 0.03 to take account of transaction costs and of a small differential between the domestic interest rate and the “world” one.¹⁵

The parameter summarizing the policymaker aversion to inflation is clearly a more difficult choice. We opted to fix δ=0.5, which implies a policymaker rather averse to accommodation (compared with the one in the exercises proposed by Obstfeld, 1997).

We chose λ equal to 3, attributing a relatively high weight to the net benefits obtained by defending the peg, and ρ=0.9, which is in line with many empirical estimates. As we shall comment below, a high ρ tends to shrink the “multiple equilibria” area.

Further problems are posed by

, v, c, and h. Here we assume that

% because this value corresponds to the inflation rate that is optimal for a policymaker characterized by the proposed loss function, when unemployment is in the middle of the interval

. Then we take a shortcut: we assume that v=0.03 and ch =0.08 because this choice implies that

and

.¹⁶

, we obtain reasonable dimensions for the region in which the country is “ripe for attack”; we also set

.¹⁷

We now choose q=0.5, we add σ_ε=0.10% to the constellation of values described above, and we assume that the time 1 disturbance, η₁, takes its most likely value (zero) (hence u₁=E[u₁[mid ]I₁]). Having provided a baseline value for every parameter, we set to zero the first-period average wage (w₁^A) and we obtain model simulations conditioned on the initial unemployment value. The first column in Table 1 shows the initial condition, ranging from 7% to 9%; the second highlights the policy maker's strategy, providing the unemployment threshold,

. Notice that the peg is defended in all the cases we have considered except for u₁=9%, where u(k₁)<u₁. The third column describes the trigger signal characterizing the speculators' behavior (u′₁). This has been reported in terms of the fundamental—instead of the disturbance η₁—to help the reader's intuition (bear in mind that u′₁≡k₁(u₁)+u₁). The fourth column shows the first-period share of attackers implied by u′₁, which is α₁((u′₁−u₁)/σ_ε)≡α₁(k₁(u₁)/σ_ε), because η₁ = 0. As the fundamental increases, the attack trigger diminishes and the share of attackers increases, because the speculators—perceiving that the abandonment is more likely—are more inclined to attack. The fifth and sixth columns are crucial: they show the policymaker's computations of the two possible average wages, as determined on the market. The actual realization for

causes the economic system to move to the second or to the third section of Table 1.

The first column in the second section of Table 1 shows the second-period expected unemployment, determined by equation (1), when the average wage is low. The resistance threshold u(k₂) is presented in the second column, whereas the third shows the second-period trigger signal. These are both decreasing in the second-period unemployment, because a weaker fundamental reduces the resistance payoffs and makes the speculators more willing to attack, because they (subjectively) perceive that the abandonment probability increases. Column four provides the time 1 expected share of second-period attackers, which increases as the trigger signal decreases. The last column in the second section of Table 1 shows the expected probability of a second-period default, as computed by the government at time 1. The fact that the second-period resistance threshold and trigger signal are much higher than those characterizing the first period is due to the fact that an early abandonment of the peg is more appealing for the policy maker, because the reduction in unemployment it conveys exerts positive effects on the second period. The last line describes what happens when u₁=9% and hence the peg is abandoned at time 1. In this case, the surprise inflation reduces unemployment in period 1 (by

; because the first-period unemployment drops to 4%, which is the “natural level,” its expected value for the second period does not move (refer again to equation (1)).

The third section of Table 1 shows what happens when the wage-setters select—during the first period—the “pessimistic equilibrium.” In this case, when the peg survives the attack at the beginning of time 1, the expected second-period fundamental is higher due to the negative effect of higher wages on employment. Accordingly, the resistance threshold for the policymaker and the trigger signal for speculators are lower, whereas the expected share of attackers and the expected probability of default are higher.

Having established that multiple equilibria can emerge, we characterize the “indeterminacy area” for σ_ε∈{0.05%, 0.10%, 0.15%, 0.20%}, considering three alternatives for the probability of a pessimistic equilibrium at the wage-setting stage. In the first scenario, wage-setters are (and are known to be) “optimistic,” so that q=0.1; in the intermediate case q=0.5; in the third one pessimism prevails (q=0.9). As before, we consider the case η₁=0; we define as “multiple equilibria” a situation in which our routine—starting from different initial conditions—computes two equilibrium average wages the difference of which is larger than or equal to one percentage point.

The “multiple equilibria” range is summarized in Table 2, where we denote by

and by

, respectively, the lowest and the highest value for unemployment such that our routine computes two equilibria.

When

, the peg is maintained in period 1, and the equilibrium wage distribution is unique, being characterized by wages low enough to guarantee probabilities of a successful second-period defence close to unity.

For

, the policymaker keeps the peg, but there are two equilibrium wage distributions, implying respectively high and low probabilities of a time 2 defence of the peg.

Finally, when

, the peg is abandoned in the first period.

Table 2 provides also the “trigger signals” that correspond to

and

, that is,

and

, therefore characterizing the speculators' behavior. Not surprisingly, as the threshold decreases, the corresponding trigger signal gets larger, because the speculators—perceiving that the fundamental is sounder–are less inclined to attack.

Several remarks are in order. First, the range for the fundamental for which we have found multiple equilibria is relevant, going from 1.26% to 2.33%.

Second, the comparison of the results obtained for q=0.1 and for q=0.9 shows that, when the wage-setters coordinate with a low probability on the “high-wages” equilibrium, the threshold

is higher than in case of a high probability for the pessimistic outcome. The intuition for this result is clear: the policy maker is more willing to resist a first-period attack when she or he knows that the probability of a second-period low-wages equilibrium is high. In fact, when wages are low, the second-period currency attack is characterized by a lower share of attackers because—ceteris paribus—the fundamental is stronger; hence, the benefit of having kept the peg in the first period is higher. The policymaker's willingness to resist also influences the first-period behavior of the speculators, who become less inclined to attack. Therefore, in the case under scrutiny, the coordinating device used by the wage-setters influences the strategies adopted by the other players during the first-period attack, so that

, and hence the resistance areas are significantly affected.

It may be surprising to notice that the results for q = 0.1 and for q = 0.5 are identical. This is due to the fact that, for q = 0.5, the benefits obtained by the policy maker thanks to the first-period defence are already high enough to imply resistance for any u₁ such that the “low-wages” equilibrium exists. Hence, a reduction in q is ineffective in increasing the resistance area: for the policy maker it is pointless to enhance her or his resistance threshold. In fact, to defend the peg with a higher fundamental is tantamount to resisting in a situation where the low-wage equilibrium does not exist. This policy cannot be optimal because a high-wage growth implies an almost sure abandonment of the peg in the second period, undermining the advantages of a first-period resistance.¹⁸

Third, we notice that—in most cases—the upper bound for the interval (

is increasing with the standard deviation for the private signal. This contrasts with the effect emphasized by Metz (2002). In her paper, when—as is the case—the fundamental is perceived ex ante as unsound, an increase in the variance for private signals tends to make speculators more aggressive. In fact, if private signals get less precise, speculators tend to give more prominence to the public information (that is, to the ex ante expected value for the fundamental), which tells them that the fundamental situation is gloomy. Hence, for a given expected fundamental, the unemployment rate and the signal trigger get lower. In our model, when u₁ is close to

, information is relatively homogeneous among wage-setters; thus, they can coordinate on low wages. To appreciate this point, consider the case in which

. When those agents who have been induced by their signal to believe that

observe that the peg has survived, they must significantly revise downward their belief about u₁. Hence, wage-setters' information becomes more homogeneous, allowing coordination on the “low-wage” equilibrium, which enhances the benefits of keeping the peg and therefore raises the defense trigger. This effect is more important the larger the standard deviation for private signals: the higher the signals' precision, the lower is the effect of further information. In most cases, this positive relationship between the standard deviation and the resistance trigger outweighs the negative effect highlighted by Metz, hence the link between the upper bound of the multiple equilibria interval and the variance for private signals is positive.

Fourth, the lower bound for the multiple equilibria interval (

is mildly decreasing with σ_ε. An increase in the variance in private signals implies that a greater proportion of agents receive a high signal. This tends to make it easier to coordinate on a pessimistic equilibrium, which increases the “multiple equilibria” range, reducing

.¹⁹

Notice that the reduction in the pessimistic expectation area does not necessarily imply that the policymakers should try ex ante (i.e., before observing u₁) to let individual agents receive precise signals (i.e., that a transparent policy should be pursued). In fact, a decrease in σ_ε, although reducing the area where the economic system finds itself in the multiple equlibria region, increases the area where the peg is abandoned in period 1.

We have found that the error autocorrelation parameter, ρ, is relevant in determining the interval where multiple equilibria are possible, as documented in Table 3. More specifically, the lower is ρ, the wider that interval. The intuition for this result is simple: when ρ is low, the policymaker is aware that unemployment, ceteris paribus, is going to decrease more quickly in the future (equation (1)). Hence, he or she is more willing to resist the attack and keep the peg during the first period. This remarkably increases the upper bound for the “multiple equilibria” region,

.

To verify that the information released by the failure of the period 1 currency attack actually plays a crucial role, we have modified our routines to check what happens when wage-setters disregard this piece of information. We have found that, in this case, the multiplicity in the equilibrium wage distributions never arises. In this case, in fact, every wage-setter thinks that there are some agents obtaining an extremely high (low) signal on u₁; these wage-setters are thought to fix a percentage wage increase equal to

(to 0). The wage setters receiving an in-between signal determine their average wage, taking into account the existence of all the other agents and hence also of those asking for a wage increase as high as

(as low as 0). This ties both the ends of the wage distribution, which turns out to be unique. In contrast, when agents exploit the information that

, there are no agents that consider sure an inflation rate as high as

and one end of the wage distribution is “loose.” Intuitively, when the agents receiving a signal slightly lower than

are pessimistic, they coordinate around a high-wage equilibrium, which can prove consistent with the fundamental and with the other agents' expectations. If, on the contrary, these wage setters are optimistic, they choose low wages, which are as well part of an equilibrium.

The “narrowing down” of the subjective distributions for η₁ characterizing the agents who receive a signal close to

is portrayed in Figure 3a. There, the probability distribution

is drawn for three different signals, the difference between the lowest one (continuous line) and the intermediate one (dashed line)—0.0015—being the same as that between the intermediate and the highest one (dotted line). Figure 3b depicts the probability distributions

, given the values for

used to draw Figure 3a. Again, the presence of the threshold

induces the agents receiving high signals to tighten the distributions for the other agents' signals. These effects induce wage-setters receiving high signals to compute “relatively homogeneous”

.

(a) Probability distribution functions $P(\eta _1\,|\,{s_1^{z} \cap (\eta _1 \le \tilde {\eta} _1 = 0))}$ for $s_1^{z} = \{-0.0030, - 0.0015,0\}$. (b) Corresponding probability distribution functions $P(s_1^{r}\,|\,{s_1^{z}\; \cap}$ ${(\eta _1 \le\tilde {\eta} _1 = 0))}$.

Hence, the informational content of

makes agents, and especially agents receiving high signals, less uncertain about the fundamental and about the other agents' perceptions. Moreover, as already remarked, every agent knows that the fundamental is not in the “opt out” region. Therefore, they can coordinate their expectations.