In recent years, countries of the European Monetary System (EMS) have pegged their exchange rates to the deutsche mark in an attempt to reduce their inflation to German levels. Inflation rates, however, have been slow to adjust. Indeed, from the evidence of European inflation indices over the 1987 period, the authors of a 1991 report on “Monitoring European Integration” conclude, “There is no doubt … that inflation convergence has not occurred and it is not occurring. The differential between higher-inflation countries and Germany has recently fallen; but this has been almost entirely due to a surge in the German inflation rate which is believed to be temporary.” (See Begg and others (1991), p. 45.) We examine three possible reasons for the sluggish inflation response observed in the EMS.
Some observers have stressed the role of staggered wage setting in perpetuating inflation, and they recommend synchronizing pay settlements as an institutional solution (see, for example, Layard (1990)). This paper begins, therefore, with an investigation of the link between inflation persistence and overlapping wage contracts in the rational expectations model of John Taylor (1979), in which wage settlements hold for two periods, but only one of the two groups of workers makes a settlement each period. We do find some inflation persistence, but only for one period. In the context of the European exchange rate mechanism (ERM), for example, upward inflation pressure from high wage settlements made in the past would soon be overwhelmed by the downward pressure from transitory unemployment and from expectations of a noninflationary future. The methods used to illustrate the transition readily suggest alternative strategies for setting the exchange rate on entry to the ERM. We consider, in particular, the options of devaluing to avoid the initial recession and of revaluing to stop inflation in its tracks. In a related model of staggered wage contracts developed by Guillermo Calvo (1983a, 1983b), in which the price level is an average of contracts of random length, neither of these options proves necessary, however, as there is no inflation persistence despite the contracts.
It should be stressed that two assumptions play a key role in the demonstration that overlapping contracts are consistent with low or no inflation persistence after the implementation of anti-inflationary monetary or exchange rate policy. The first is that a change in policy is fully credible: wage setters know about the new policy and do not anticipate any reversal. The second is the assumption of rational expectations: forecasts are made using a model of the inflation process that is assumed to be common knowledge, so that wage setters know how the policy will work and know that others do too. If implementing the anti-inflationary policy involves a major change of regime, as in joining the ERM, these assumptions may not be appropriate.
Thus, for instance, the government’s commitment to a fixed exchange rate pegged to the deutsche mark may not be fully credible, at least for a while. Until the final stage of European integration, realignments will always be possible. Moreover, European interest differentials prevailing during the first stage indicated that market makers gave some credence to the prospect of a revaluation of the mark against its more inflationary partners in the ERM (a belief that has been justified by recent events). So a period may occur—possibly quite prolonged—during which the members of the ERM gradually gain credibility for their policy of pegging to a hard currency.
To allow for the lack of full credibility in the peg, we assume that the private sector expects random realignments of the exchange rate. For convenience, we use a model with Calvo contracts, which implies that the inflation adjustment depends only on the speed with which people come to believe the exchange rate peg. If it takes time for the peg to become credible, the inflation generated in that time will cause the price level to “overshoot.” Thus, when inflation does finally fall to German levels, the economy will be uncompetitive, a problem that will only be corrected by price increases that are lower than the German rate of inflation. If the price level is sticky, due to overlapping of wage contracts, this disinflation will be associated with a prolonged contraction of output, triggered by the lack of competitiveness itself. By combining overlapping contracts and a lack of policy credibility, we can thus explain the emergence of stagflation after a regime change.
It may also be the case that the second assumption, that of common knowledge of beliefs, is also inappropriate to the modeling of ERM membership. Even if individual wage setters are persuaded that the authorities will pursue a hard-currency policy, they may be unsure of what others believe and be tempted to keep wage claims up as a consequence. Frydman and Phelps (1983) argue that this lack of “common knowledge” may explain the failure of Latin American stabilization policies. Could it also be relevant for the delay in stabilizing inflation in Europe?
To investigate the effects of a lack of common knowledge on inflation, we weaken the usual rational-expectations assumption by supposing that each agent fully believes in the peg but assumes that other agents do not and will take time to learn. We find that, although slow convergence of inflation may arise from this lack of common knowledge, it is less pronounced than that implied by the lack of policy credibility discussed above.
We conclude that the rapid convergence of inflation in the presence of overlapping wage contracts can be significantly undermined when the regime shift is not seen as fully credible. Our analysis is conducted with reference to two popular macroeconomic models of wage contracts. An alternative approach to explaining inflation inertia might be to explore the role of near-rational behavior by imperfectly competitive firms along the lines suggested by Akerlof and Yellen (1985).
I. Wage Contracts and Inflation Persistence
In this section, we analyze the extent to which wage contract structures induce persistence in the inflation rate. We consider the model proposed by Taylor (1979), in which wage contracts have an exogenous fixed term, and that proposed by Calvo (1983a, 1983b), in which wage contracts have random lengths.
A model with Taylor-type two-period overlapping contracts takes the following form:
p = log of the price level,
x = log of the current new contract,
y = log of output,
s = log of the exchange rate (price of foreign currency), and an asterisk indicates a foreign variable.
There are two equal-sized groups of workers that negotiate their nominal wage rates in alternate periods. So, for instance, the group that settles in period t – 1 receives xt – 1 in period t – 1 and period t, while the group that settles in period t receives xt in period t and period t + 1. Thus, as shown in equation (1), the price level is a mark-up over the average of xt–1 and xt. The mark-up is set to zero for simplicity. The current contract, defined in equation (2), is the average of prices and demand pressure over the two periods of the contract (a hat over a variable indicates expectations at the time the contract is settled). Equation (3) is the IS relationship (describing goods market equilibrium), where, for simplicity, we assume that aggregate demand depends only on the level of competitiveness. For convenience, we set p* = 0.
Appendix 1 shows that when the exchange rate is fixed at
Figure 1.Taylor Contracts Before and After Entry
We assume that prior to joining a fixed-rate regime the money supply is growing and the exchange rate is depreciating at rate μ. The resulting (inflationary) equilibrium is as follows:
Thus, the price level tracks the exchange rate, and each new contract is set above the current price level (expected and actual) to allow for future anticipated inflation. Note that this allowance for inflation between the two periods of the contract only needs to be μ/2 because the current contract is an average of the price level in the two periods. In Figure 1, the inflationary equilibrium is illustrated by a series of steps along the line LAL with the position in each period indicated.
We now consider the transitional effects of a switch from the inflationary steady-state just described to a fixed exchange rate regime. We assume that the announcement of the peg is made at the start of period 1 and that the exchange rate is pegged at the level prevailing in the market in period 0. Initially in our analysis, we assume that the group setting its wage in period 1 is aware of the regime change before x1 is settled. (We consider the alternative assumption below.) For convenience, we set
This relationship is illustrated in Figure 1 by the schedule DN, which intersects the vertical axis at –xt–1 (the distance OD in Figure 1 is –x0 = – μ/2). In period 1, the contract wage and price level are given by point B, where TT and DN intersect, that is, by the point on the stable path at which equation (1) is satisfied. In subsequent periods, the contract wage and price level converge toward equilibrium by a series of steps down TT.
The implications of this solution path for inflation inertia are immediately apparent. The jump from point A in period 0 to point B in period 1 implies a rise in the price level of
The period of positive inflation following the change in regime is somewhat understated in this analysis because of the assumption that wage negotiators knew about the new regime before wages were set in period 1. It is simple to show the effects of the alternative assumption (that wages are set before the announcement) with reference to Figure 1. If wage negotiators in period 1 are unaware of the new regime, they will continue to forecast depreciation and inflation at rate μ and will set the contract in period 1 above the contract in period 0 by the amount μ. The contract is therefore given by the distance OA′ in Figure 1 where the distance AA′ is μ. The price level will consequently rise by μ between period 0 and period 1. Thus, the switch in regime does not immediately affect the inflation rate.
The negotiators who settle in period 2 do, however, know about the new regime and will therefore set a contract wage on the stable path TT. But because of the higher value of the contract wage set in the previous period, the schedule DN moves out to D′N′ (where the distance OD′ is –x1 – 3μ/2) and the contract wage and price level in period 2 are set at point B′. In subsequent periods, the contract wage and price level move toward equilibrium along TT. In sum, the effect of changing the timing of the announcement of the peg is to increase by one the number of periods of positive inflation.
It is worth noting that even though there is little persistence of inflation in this model, regardless of the timing of the announcement, there is a long recession. Output has to fall below capacity in order to drive prices down to their equilibrium level.
This analysis has concentrated on two-period contracts. More generally, Taylor-type contracts that exceed two periods will cause more persistence of inflation. However, the period of positive inflation following the switch to a fixed exchange rate is never longer than the length of the contract. Thus, for instance, n-period contracts will generate (depending on the timing of the announcement) either n – 1 or n periods of positive inflation after the regime change.
In the context of a two-period model, we show how the entry rate can be set so as to eliminate inflation immediately, though the revaluation needed to do so deepens the recession. Conversely, the entry rate can be set so as to avoid the recession but at a cost of higher initial inflation. Choosing an entry rate to maintain output is what Taylor (1984) describes as an “efficient” disinflationary policy because it leads (one period later) to price stability without any loss of output. This strategy is shown in Figure 2, in which the outcome shown in Figure 1 for an entry rate of
Figure 2.Entry Rates to Avoid Recession (N) or to End Inflation (D)
The act of devaluing before fixing the exchange rate requires that the chosen rate be underpinned by “precommitment,” as Backus and Driffill (1986) point out. Joining the ERM is just such a precommitment. It may not, however, be fully convincing, since a country can still “realign” its exchange rate. Therefore consider the alternative policy of revaluing the entry rate to achieve an immediate cessation of inflation. The outturn in period 1 for such a “cold turkey” policy toward inflation is also shown in Figure 2. Revaluing the currency sufficiently to shift the Taylor contract locus to T′T′ with a first-period equilibrium of D will stop inflation in its tracks in period 1. Prices would thereafter fall, as the long-run equilibrium price level with this entry rate would necessarily be on the 45° line at a lower price level, see point R in Figure 2. Setting
When we consider other sources of inflation inertia (as we do later in the paper), it proves more convenient to use the continuous-time model of contracts proposed by Calvo (1983a, 1983b). In this model, contracts do not have a fixed duration; instead their duration is random, with an exponential distribution through time. A model with Calvo contracts is described by the following equations:
where D is the time differential operator.
The fact that each individual contract has a lifetime that is exponentially distributed implies that the numbers of outstanding contracts are also exponentially distributed by vintage. In equation (8), the current price level is given as an average of all outstanding contracts. The current new contract is a forward-looking integral of expected future prices and demand pressure over the possible lifetime of the contract, as shown in equation (9). The aggregate demand relationship (equation (10)) is unchanged from the previous model.
The model can be rewritten as
It is easily shown that when the exchange rate is fixed at
Figure 3.Lack of Credibility and Learning
Again assume that before joining the fixed rate system the economy was in equilibrium with constant money growth at rate μ. This inflationary equilibrium implies a constant depreciation of the exchange rate at rate μ and constant growth in the price level at rate μ. It can be seen from equation (8) that the current contract must therefore be given by x = p + μ/δ. Clearly, the currently negotiated contract wage must be set higher than the current price level to compensate for anticipated inflation during the expected term of the contract. From equation (9), it can be seen that, in inflationary equilibrium, output is at its natural rate and that p = s (with p* = 0).
Again assume that the government chooses to peg the exchange rate at its market level on the date it joins the fixed rate regime. Thus, if the peg is put in place when p = 0, then
Why do Calvo contracts differ from Taylor contracts in this way? One might initially suspect that Calvo contracts do not give rise to inflation persistence because they are modeled in continuous time—the argument being that contracts signed at any given point have zero weight in the price level, so there is no constraint on the current contract wage falling directly to its new equilibrium. But it is possible to construct a discrete time analogue to the Calvo model of equations (8), (9), and (10), and such a model would display exactly the same lack of inflation inertia. The explanation for the difference between the Calvo model and the Taylor model lies in the implications of the two types of contracts for the structure of the price level. In the Taylor model, the price level is a moving average of past contract wages. In the example above, it is an average of the contract wage in the current period and the contract wage in the previous period. The price level in the Calvo model, on the other hand, is a weighted average of all past contract wages, with weights declining exponentially into the (infinite) past. In both models, immediately after a switch to a fixed rate recent contracts will tend to be too high relative to the new equilibrium, given that they were set at a time when agents were forecasting high inflation. The current price level in the Taylor model gives relatively high weight to recent contracts and therefore tends to overshoot the new equilibrium price level. The price level in the Calvo model, on the other hand, gives less weight (relative to the Taylor model) to recent contracts and more weight to older contracts (which are low relative to the current price level). The effect of the high valued recent contracts is cancelled out by the lower older contracts, so that the current price level in the Calvo model does not overshoot the new equilibrium level.
II. Calvo Contracts and Lack of Credibility
Predictions of prompt and painless adjustment do not conform to the experience of countries in Western Europe and elsewhere, which have tried to check inflation by pegging against hard currencies. Could this be because their pegs have not been fully credible? To capture key aspects of Latin American stabilizations, Calvo and Vegh (1991) analyze the case where the fixed peg is regarded as temporary, soon to be replaced by a crawling peg. They find that the temporary peg leads to an initial expansion of output, which is followed by a recession, and to an inflation of domestic costs and prices, which causes a sustained appreciation of the real exchange rate.
For EMS countries, however, it is shifting the peg by discrete realignments rather than switching to a crawling peg that is more relevant. To model the lack of credibility, we consider the impact of anticipated random realignments. Specifically, it is assumed that financial and labor markets perceive a constant probability of a devaluation of size J: the exchange rate peg suffers from a “peso problem.” The perceived probability of devaluation (per unit of time) is denoted π. (See Miller and Sutherland (1991) for a previous application of the Calvo model with repeated parity realignments of this sort.)
In the labor market, the expectation of realignments affects the forward-looking contract (defined in equation (9)) by modifying expected future prices and demand levels. Denote the rate of change of x conditional on the current value of
where θ < 1 is the slope of the stable path. At first sight, this expression (the derivation of which is explained in Appendix 2) appears paradoxical in that it seems to suggest that expected devaluations, in the form of a positive value for πJ, slow the rate of change of contract wages. However, it must be remembered that the contract wage is an “unstable” variable, which is free to jump onto the stable path. Any exogenous factor that causes a reduction in the rate of change of contract wages will bring about an upward jump in the level of contract wages onto a new stable path. This upward jump reflects the expectations of higher future price and demand levels, which are generated by the expectation of realignments.
When equation (12) is combined with the equation for price adjustment (and the variables are measured from the parity before any realignment), we find that the system has an additional constant term. Specifically,
where A remains as before and b = (θ – 1)πJ. Thus, the effect of the expected realignments is to shift the “equilibrium” of the model to the northeast, along the 45° line. In Figure 3, the new equilibrium is marked E on C′C′, which has the same slope as CC and a vertical intercept at πJ/δ. Because the equilibrium point E is associated with expectations of realignments that never occur, it is referred to as a “quasi-equilibrium.”
At this quasi-equilibrium, Dp = Dx = 0, and
measures the difference between quasi-equilibrium and true equilibrium. Hence q > 0 so long as πJ > 0.
Now consider a switch to a fixed exchange rate from a floating rate depreciating at rate μ that takes place when point A is reached. The contract wage jumps from point A to point B on the path C′C′. At point B, the rate of inflation is πJ. Thus, in contrast to the fully credible case, inflation does not fall directly to zero. As time passes, prices rise and the contract wage moves along C′C′ toward point E. The rise in prices erodes competitiveness (because the nominal exchange rate is fixed), causing inflation to slow, but at the cost of a recession that continues as long as expectations of a realignment persist.
The assumption that the private sector permanently expects realignments at rate πJ per unit time, despite the fact that no realignments actually occur, is obviously unsatisfactory. Some form of learning is likely, leading to a convergence of private sector expectations toward the true value of πJ (which is zero under a completely fixed rate regime). Thus, as time passes, the quasi-equilibrium point also moves toward the long-run equilibrium because q is proportional to π.
In Appendix 3, we indicate how a process of Bayesian updating leads to this sort of result, where the unknown value of π (a high value being denoted by πH or a low value by πL) is approximated by an estimate,
Since q is proportional to
In addition, x – q = θ(p – q) on the stable path, and Dp = δ(x – p), as in equation (8). This implies that
where, for convenience, we continue to assume that
where Φ = πH. So the adjustment of prices and contracts follows a curved path, such as that shown in Figure 3. The inflation that persists after the switch of regimes causes a recession, which fades as the private sector learns there will be no realignments.
The effect of rational learning in bringing down inflation is not foreseeable, and the authorities may be tempted by the “cold turkey” policy of revaluing to achieve prompt inflation reduction. If the entry rate is chosen so that the current price level is the quasi-equilibrium, the inflation rate will be zero on entry and will later fall as learning takes place. Formally,
The idea of using the exchange rate as a signal of a new policy stance is analyzed by Winckler (1991) using a game-theoretic framework.
III. Lack of Common Knowledge
The previous section demonstrated that if a switch in policy regime is not fully credible at the time it is implemented, the intended effects on private sector inflationary expectations are delayed until credibility is acquired, essentially by the government “sticking to its guns.” The assumption is that agents do not believe in their government’s commitment and learn from observing government policy, and that each individual agent regards his or her own beliefs as being held by all other agents. In this section, we examine the “lack of common knowledge,” along the lines suggested by Frydman and Phelps (1983, chap. 1). They emphasize that even when agents know the true parameters of the model and when the intended change of policy is announced,
The agent’s forecasting problem is further complicated when government policy alters the parameters of the process governing the behaviour of exogenous variables [… by] the problem that although an individual agent may know and believe the government’s announcement he does not know if other agents also know and believe in the change in policy…. Where the rational expectations approach predicts instantaneous movement to a new equilibrium, the difficulties faced by agents in their expectations formation lead to a protracted period of disequilibrium. (Frydman and Phelps (1983), p. 6, emphasis added.)
In the Calvo model, it was found that rational expectations and policy credibility ensured instantaneous adjustment to a new noninflationary equilibrium, despite the presence of overlapping wage contracts. To capture the issues discussed in Frydman and Phelps (1983), we continue to assume that the model parameters are known to all agents and also that the announced policy shift is believed by every agent. But because agents do not know about the beliefs of others, they are assumed to incorporate a model of learning into their forecasts of wage settlements, and the model they use for this purpose is the one suggested in equation (11).
This implies that, during the transition from one policy regime to another, agents misperceive the expectations of other agents: everyone knows the policy, so rather than learning about policy they are eliminating their errors in forecasting the behavior of others. This view of the world is rather asymmetric (agents are sure of their own beliefs but not sure of others’), yet it may be appropriate when a credible policy commitment by the government, which involves a considerable change of regime, has consequences for collective behavior that have not been fully spelt out. That this may be true of monetary disinflations has been argued by Edmund Phelps and Juan Carlos Di Tata in Frydman and Phelps (1983, chap. 2 and chap. 3), and it may also be applied to a change of exchange rate regime.
To see what happens under our stylized depiction of the lack of common knowledge, we first examine how forecasting the learning path of others affects current contracts. The result is that over time the price level mimics that which would have occurred with genuine learning, but because agents anticipate that which cannot be forecast if everyone is really learning, the price level is always closer to equilibrium.
As in the earlier section, we continue to assume that prices are a (infinite) moving average of past contracts (see equation (8)) and that contract wages are weighted forecasts of future prices and output levels (see equation (9)). We reject, however, the rational expectations assumption that the forecasts will immediately reflect the credibly fixed exchange rate and assume instead that the forecast for prices makes some allowance for learning.
Thus, it is assumed that each individual agent knows the exchange rate will not be devalued but believes that all other agents expect realignments at rate πJ. It is further assumed that each agent believes that all other agents will slowly learn that no realignments are to occur. The fact that each agent is infinitesimally small effectively means that he or she believes (wrongly) that the economy will evolve according to the learning model outlined earlier. Thus, the expected values of p and y, denoted
Consequently, following the change of exchange rate regime, each agent forecasts that inflation will remain high for a period while other agents learn that devaluations have halted permanently. In order to see the implications for the actual evolution of prices and contract wages, it is necessary to combine equations (19) and (20) with equations (8) and (9) to yield the following matrix system:
The roots of this system are conveniently displayed on the diagonal: three stable roots and one unstable root (δ). We assume that the contract wage will make the necessary jump to remain on the stable path. This allows
In this case, we find that the parameters entering the determination of the current contract wage are
The term θ could have either sign, but in these equations it is assumed to be positive. Specifically, we assume β = η = 1/2, so θ = 1/2. Using these parameter values and setting δ = Φ = 1/2 generates the results shown in Figure 4. In deriving these results, it is assumed that the price level begins at equilibrium. This assumption corresponds to the case where there is no revaluation on the date of the regime switch. As such, it must be the case that the expected path of prices will start from the current price level, so
Figure 4.Time Path of Inflation When Φ = 0.5
Note that the initial conditions are those in which a rational expectations solution of the model would lead to instant price stabilization; no individual actually expects any realignments and the price is at equilibrium. But we assume that each agent thinks that others expect devaluation
If the speed of learning that is postulated for others is much slower, then the period of rising prices will be that much more prolonged, as would the period of slack demand. For the second simulation, shown in Figure 5, Φ has been much reduced (Φ = 0.1), meaning that inflation lasts twice as long, with prices peaking at t = 6. In this case, the actual price level moves more closely in line with the forecast price level, since
Figure 5.Time Path of Inflation When Φ = 0.1
The example we have worked through is one in which agents can correctly forecast government policy but do not correctly forecast other agents’ expectations. For an open economy with a pegged exchange rate, it illustrates the point made by Phelps that “in order to reduce the price level (in relation to the accustomed trend), it is not sufficient that the central bank persuades each agent to reduce his private expectations of the money supply (in relation to past trend) by the warranted amount. The prevalence of this knowledge must be public knowledge—an accepted fact.” (See Frydman and Phelps (1983), p. 35.) For the open economy one needs to replace the money supply by the exchange rate in the quotation.
This example is subject to the criticism that people could discover that they are wrong about the forecasts of others or that the central bank would surely try to publicize the private credibility that its policies enjoy (by conducting a poll, for example). Nonetheless, the learning process may take time—so much time as to be judged a failure—and the task of persuading the general public that the policy is widely believed may well include “convincing them that groups that they regard as powerful have already bought it. Appealing over the heads of these groups to the people will not work unless you can simultaneously persuade them that traditionally powerful groups are no longer so strong as to be able to deflect you from your policy” (Bull’s comment on Di Tata in Frydman and Phelps (1983), p. 66–67).
This discussion is not intended to imply the impossibility of securing convergent beliefs; it is intended simply to emphasize the need for beliefs to be “re-equilibriated” after a change in policy, for reasons that are ignored in macroeconomic models with rational expectations. This process of re-equilibriation may take precious time and, as Frydman and Phelps (1981, p. 27) emphasize, “it may depend for its success on the power of institutions and norms external to the market (conventional theories, traditions, business practices, consensus over government policy, etc.).” What this suggests is that policymakers need to work actively to secure consensus rather than simply leaving it to rational expectations.
In this paper, we consider various explanations for a protracted process of inflation convergence after a switch to a hard currency peg. Assuming a fully credible peg and rational expectations, it appears that nominal stickiness in the form of Taylor contracts does not account for much inflation sluggishness. Giovannini (1990, p. 257) concludes that “this is not to say that nominal inertia is irrelevant, but only that additional explanations may be useful.” To focus clearly on these additional explanations, we proceed to work with Calvo contracts, for which nominal inertia implies no sluggishness for inflation.
First, we assume that agents anticipate random realignments, leaving the hard peg less than fully credible. The prospect of devaluation does impart an inflationary bias to the economy, but the inflation gets converted into economic slack if no realignments take place. It seems implausible that agents would permanently expect realignments in these circumstances, so we allow for Bayesian learning about the policy shift. This combination of nominal inertia and credibility that is slow to build generates sluggish inflation and a protracted recession after the peg is adopted. The roots of the process depend essentially on the expected life of contracts and on the speed of learning.
Then we follow the lead of Frydman and Phelps by relaxing the common knowledge assumption implicit in the usual rational expectations hypothesis. Each agent postulates that others are learning about the policy switch, whereas he or she knows it already happened. Since everyone knows about the true policy stance and is only learning about the expectations of others, the speed of adjustment is faster than in the case where everyone is learning about the policy itself. But if the intensity of realignments is not expected to be very high (πH is not high), then this process can be protracted.
If indeed the problem is simply one of learning, either about the policy shift or about the views of others, then presumably there is a good case for the authorities trying to signal the change of policy in a way that speeds up the learning process. Alternatively, the inertia might be caused by interactions between imperfectly competitive firms and unions, a finding that calls for a more strategic analysis than the staggered contract models used here.
From this equation, it is simple to check that the equilibrium level of x is
where K is a constant determined by an initial condition for xt – 1 and ρ is the stable root (the root that has an absolute value less than unity) of the characteristic equation
It is simple (but tedious) to check that p takes the following value:
To obtain an expression for the stable path observe that while on the stable path, equation A2 implies that the contract wage in period t – 1 is related to the contract wage in period t by the following expression:
This relation can be used to eliminate xt – 1 from equation (1), and after some rearrangement the following expression is found relating the current contract wage to the current price level:
where θ is the slope of the stable path. Using equation (A4) it follows that
which is the value of θ used in the text.
The assumed process for realignments implies that agents’ best forecast of the exchange rate at time τ based on information available at time t is
These equations can be solved to yield the following expressions for the expected price and output levels:
The realignment expectations discussed in the text can be derived from a model of Bayesian learning, as shown in Driffill and Miller (1991). The argument may be summarized as follows.
After the exchange rate is pegged, the public still believes that realignments of size J may occur as a Poisson process, but the public is not sure of the intensity, π. Assume specifically that the uncertainty is simply whether the intensity is high (πH) or low (πL), and suppose the public starts out immediately after the peg with initial probability PH(0) and PL(0) = 1 – PH(0) attached to each intensity, so
where time t is measured from the date at which the rate was first pegged.
If a realignment occurs at time t, it causes a discrete jump in PH(t) so
Thereafter, PH will decline exponentially from its new high value just as in equation (A13), measuring time from the date of the realignment.
Note that for large values of t
thus, if πL = 0, then for large t
the result we use in the text.
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