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Exchange rate realignments and realignment expectations.

1. Introduction

From the mid 1980s to the exchange rate crises of 1992 and 1993 EC governments sought to use the ERM as an anti-inflation framework. Despite the rigour with which this policy was pursued it was evident that inflation was slow to adjust. Miller and Sutherland (1993) considered a number of theoretical explanations for the sluggishness of inflation. In that paper it was shown that overlapping nominal wage contracts could not generate a significant degree of inflation inertia. It was concluded that other explanations, such as lack of government credibility, must be more important in explaining the European experience.

Miller and Sutherland modelled the lack of credibility by assuming that the private sector expected exchange rate realignments to follow a Poisson arrival process with an exogenous arrival rate denoted [Pi]. When expectations of this form were built into a model of forward looking wage contracts it was shown that inflation could take a long time to adjust to a change in exchange rate regime. After a switch to a fixed rate the continued inflation would erode competitiveness and unemployment would increase. If the private sector continued to believe that realignments were possible the economy would become stuck in a recession where the high level of unemployment offset the inflationary effect of the lack of credibility.

The model was refined slightly by suggesting that the private sector would modify its beliefs about realignments in the light of the observed behaviour of the government. If there were no realignments it was suggested that the private sector would revise down its estimate of the arrival rate, [Pi]. Eventually the economy would return to full employment. In another paper Driffill and Miller (1993) derived the precise process that [Pi] would follow if the private sector used Bayesian updating. They showed that in the absence of realignments [Pi] would decline approximately exponentially.

However, Driffill and Miller demonstrated another implication of Bayesian learning in this context. They showed that if a realignment did actually occur the private sector would revise [Pi] sharply upwards. In other words a realignment would cause the private sector to expect more frequent realignments in the future. It can therefore be said that a realignment reduces the government's credibility. The obvious policy conclusion from the analysis is that realignments are bad in that they prolong the process of adjustment to equilibrium. The best policy the government can adopt is to fix the exchange rate permanently and to suffer the recession while expectations adjust. This was in effect the approach adopted by all ERM members in the period leading up to the 1992 crisis. It was frequently argued that realignments should be ruled out because they would only undermine the anti-inflation credibility of the system.

A major criticism of the Driffill and Miller analysis is that it models private sector expectations in a very mechanistic way. The private sector in the Driffill and Miller model believes that governments only realign out of habit. Governments are simply endowed with some exogenous tendency to realign which is unknown to the private sector. The private sector's problem is to form some view about the realignment rate of the government actually in power. It is obvious that in such a world the private sector will respond to a realignment by increasing their estimate of [Pi].

It is clearly not the case however, that governments simply realign out of habit. It is much more realistic to suppose that private sector agents believe that a realignment is motivated by a desire to boost aggregate demand. It would therefore be logical for private sector agents to believe that a realignment is more likely at times of recession and less likely when output is high. This is the main point of departure for this paper. Here we assume that private sector expectations of a realignment are linked to the state of the economy. Thus as an economy moves into recession expectations of a realignment increase and vice versa as output recovers.

We formalise this by assuming that [Pi], the private sector's estimate of the arrival rate of realignments, is given by the following expression

[Mathematical Expression Omitted]

Implicit in this equation is some assumption about what the private sector believes to be the objective function of the government. We assume that the private sector believes that the government cares about deviations of output, y, from some target level, [Mathematical Expression Omitted]. The decision to realign is, however, assumed to depend on other factors which are unobservable to the private sector (such as information on economic prospects known only to the government). Thus, from the point of view of the private sector, a realignment is not perfectly forecastable. The observation of a recession leads the private sector to believe that the balance of costs and benefits in the government's objective function has shifted in favour of a realignment. The private sector therefore revises the subjective probability attached to a realignment. The parameter [Lambda] measures the sensitivity of [Pi] to y.(1, 2, 3) (Output is measured as the deviation from the natural rate so the formulation in eq. (1) allows for the possibility that the private sector believes that the government is over-optimistic about the natural level of output, i.e. [Mathematical Expression Omitted]. Initially we will assume that [Mathematical Expression Omitted]. The implications of [Mathematical Expression Omitted] will be considered in the final section of the paper.)

We imbed eq. (1) in the same model of forward looking wages as used by Miller and Sutherland (1993) and Driffill and Miller (1993) and investigate the implications for inflation and the stability of the model. We demonstrate that, in contrast to Driffill and Miller, realignments can have favourable effects on the economy in that output and interest rates will be moved towards their equilibrium levels. We also demonstrate a number of other important implications of eq. (1). in particular we show that if realignment expectations are too sensitive to output there may be complete expectational instability. In addition, even if expectations are stable, the adjustment of the economy between realignments is slowed by expectations of the next realignment.

Section 2 of the paper presents the basic structure of the model we use. In Section 3 we derive the expected trajectory for the exchange rate, prices and wages when expectations of realignment are formed according to eq. (1). In that section we investigate the expectational stability of the model. In Section 4 we consider the actual (rather than the expected) behaviour of the model between realignments and show that realignment expectations slow down the speed of convergence to equilibrium. In Section 5 we analyse the effect of a realignment. In Section 6 we briefly consider the implications of the model for the effects of a switch in exchange rate regime from a downwards float to a fixed rate. We show there that a completely satisfactory account of the effects of realignment expectations may require some combination of the Driffill and Miller formulation and the relatmonship embodied in eq. (1).

2. The model

The overlapping contracts framework we use is based on Calvo (1983a, b). It allows expectations to affect current behaviour in both labour and financial markets and is thus suitable for analysing credibility effects. The model is described by the following four equations:

p(t) = [Delta] [integral of] x([Tau]) [e.sup.-[Delta](t - [Tau])] d[Tau] between limits t and -[infinity] or Dp = [Delta](x - p) (2)

x(t) = [Delta] [integral of] [[p.sub.e]([Tau]) + [Beta][y.sub.e]([Tau])][e.sup.-[Delta]([Tau] - t)] d[Tau] between limits [infinity] and t or Dx = [Delta](x - p - [Beta]y) (3)

y = -[Gamma](i - D[p.sub.e]) + [Eta](s - p + [p.sup.*]) (4)

Ds = i - [i.sup.*] (5)

where: p = the log of the price level; x = the log of the current wage contract; y = the log of output; s = the log of the nominal exchange rate; D = time differential operator; * indicates a foreign variable; and subscript e indicates the expectation conditional on time t information.

Equation (2) defines the price level at time t as the average of all outstanding wage contracts.(4) Contracts are assumed to be of a variable length, with the actual length being stochastic. The parameter [Delta] determines the rate at which contracts expire. The expected duration of a contract is therefore 1/[Delta]. The assumption of variable length contracts means that at any point of time contracts of different vintages co-exist. When a contract is renewed, allowance is made for future expected movements in prices and demand over all possible durations, as expressed in eq. (3).(5)

Equation (4) is a simple IS relationship which defines the process by which output is determined in the model. Here, output is simply assumed to be a negative function of expected real interest rates and a positive function of competitiveness, measured by the real exchange rate. Finally, to close the model, a simple international arbitrage equation (eq. (5)) is chosen. This assumes perfect capital mobility and implies that any lack of credibility in the exchange rate peg leads to an interest rate differential.

In order to form a reference case, it is worth quickly demonstrating the properties and equilibrium of the model when there is full credibility, that is when the expected movement in the fixed exchange rate is zero. By assuming rational expectations, we can use (2) to substitute for expected prices in (4) and then use the resulting expression to substitute for output in (3). This allows us to reduce the system to two dynamic equations in prices and contracts which can be written in matrix form as

[Mathematical Expression Omitted]


[Mathematical Expression Omitted]

and where [Mathematical Expression Omitted] is the parity at which the exchange rate is fixed. For convenience we have set [p.sup.*] = [i.sup.*] = 0. Equilibrium in this model will be given when Dp = Dx = 0. Imposing this condition on (6) yields the long run equilibrium [Mathematical Expression Omitted]. Thus equilibrium will lie on the 45 [degrees] line in Fig. 1, with the exact position being determined by the value of [Mathematical Expression Omitted]. With [Mathematical Expression Omitted] equilibrium is at E while a devaluation of size J shifts the equilibrium to E[prime].

For the model to be stable the necessary condition is that the eigenvalues of the coefficient matrix A should be of opposite sign. If this is the case then we can say that the model is saddle-point stable and it is possible to solve for the stable path. This represents the path along which adjustment to equilibrium will occur. It takes the form

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the equilibrium level of p and [Theta], the slope of the stable path, is derived from the stable eigenvector corresponding to the stable eigenvalue of the system. In Fig. 1 the stable path corresponding to equilibrium at E is represented by the line EC. A devaluation by amount J shifts the equilibrium to E[prime] with the corresponding stable path E[prime]C[prime]. If, for example, the starting point is C the impact effect of a devaluation is to cause the contract level, x, to jump to point C[prime]. Adjustment then takes place along E[prime]C[prime] towards the new equilibrium at E[prime].(6)

3. The expected path

We can now extend the model to examine the effects of imperfect credibility. Credibility (or a lack of it) feeds into the model as follows. Suppose government policy is not fully credible and thus the private sector does not believe official inflation forecasts - specifically they view the forecasts as underestimates. In labour markets, such a lack of credibility leads to upwards pressure on new wage settlements to compensate for this extra expected inflation. Similarly, in financial markets, this expected extra inflation will mean that agents are only willing to hold sterling if they are compensated by higher nominal interest rates.

We follow the approach adopted by Miller and Sutherland (1993) and Driffill and Miller (1993) who assume that the private sector believes that realignments follow a Poisson arrival process with arrival rate [Pi] and that realignments are of a fixed size J. In our case, however, [Pi] is given by eq. (1). Given this structure it is necessary to solve the model in two stages. In the first stage we solve for the expected evolution of the economy conditional on initial conditions for the exchange rate and outstanding wage contracts. The expected path of the economy will be characterised by smooth and continuous evolution of all variables. In particular the exchange rate will be expected to evolve according to the relationship Ds = [Pi]J. In the second stage of the solution process we solve for the behaviour of the economy as it moves along a realised path of adjustment. This path will contrast with the expected behaviour in that the exchange rate will be fixed for periods of time and will move discontinuously when realignments occur. Other variables will also be discontinuous at the time of a realignment.

The forecast path for the exchange rate is given by

[Mathematical Expression Omitted]

where the subscript e indicates an expected value conditional on information available at time t. Substituting eqs (8) and (2) into eq. (4) gives the following expression for expected output

[Mathematical Expression Omitted]


[Mathematical Expression Omitted]

After, substituting for expected output in eqs (3) and (8) we can derive equations for expected change in contracts and the expected change in the exchange rate. Taken together with (2), these equations define the expected evolution of the system as follows

[Mathematical Expression Omitted]


[Mathematical Expression Omitted]

Here [Mathematical Expression Omitted] has been set to zero so the private sector believes that the government's target level of output is equal to the natural level. We consider the implications of [Mathematical Expression Omitted] in Section 6.

Having derived the expectations system, we can now examine the conditions for its stability. Of the three dynamic variables in the system two are predetermined and one is a jump variable. The price level is based on all outstanding contracts and is therefore predetermined. The exchange rate is fixed by assumption so is therefore also predetermined at time t. The one jump variable is the new wage contract, x(t). Saddle-point stability therefore requires that one eigenvalue of the matrix B be positive and the other two be non-positive.

It is simple to show that the characteristic equation of B reduces to

-(1 - [Gamma][Lambda]J)[[Phi].sup.3] - ([Lambda]J[Eta] + [[Delta].sup.2][Beta][Gamma])[[Phi].sup.2] + [[Delta].sup.2][Beta][Eta][Phi] = 0 (11)

where the eigenvalues of B are denoted [Phi]. It therefore follows that

[Phi] = 0, -([Lambda]J[Eta] + [[Delta].sup.2][Beta][Gamma]) [+ or -] [square root of [([Lambda]J[Eta] + [[Delta].sup.2][Beta][Gamma]).sup.2] + 4[[Delta].sup.2][Beta][Eta](1 - [Gamma][Lambda]J)]/2(1 - [Gamma][Lambda]J) (12)

By inspection of the above, it can be seen that the signs of the eigenvalues depend critically on (1 - [Gamma][Lambda]J). If (1 - [Gamma][Lambda]J) [greater than] 0, then [Phi] is 0, + ve, -ve and the system is saddle-point semi-stable. The zero root implies that the model does not have a unique final equilibrium. In fact any point for which [s.sub.e] = [p.sub.e] = [x.sub.e] is an equilibrium of the model (with the final resting place depending on the initial conditions for s and p).

If the model is found to be (semi-)stable, then we can complete the solution to the forecasting system by solving for the stable path. The stable path is given by

[x.sub.e] = [[Theta].sub.1][s.sub.e] + [[Theta].sub.2][p.sub.e] (13)

where [[Theta].sub.1] and [[Theta].sub.2] are derived from the eigenvectors of B corresponding to the two non-positive roots. The following expressions for [[Theta].sub.1] and [[Theta].sub.2] are derived in Appendix 1

[[Theta].sub.1] = ([[Delta].sup.2][Beta][Gamma] - [Lambda]J[Eta]) + [square root of [([[Delta].sup.2][Beta][Gamma] - [Lambda]J[Eta]).sup.2] + 4[[Delta].sup.2][Beta][Eta]]/2[Delta], [[Theta].sub.2] = 1 - [[Theta].sub.1] (14)

Equation (13) is a crucial relationship in the model since it ties down the value of the wage contract in terms of the current price level and the current exchange rate parity and it incorporates the private sector's expectations about the evolution of the exchange rate.

If, however, (1 - [Gamma][Lambda]J) [less than] 0, the three eigenvalues are 0, + ve, + ve and the expectations system is unstable. In this case there is no stable path along which the expected system will converge to equilibrium. It is therefore not possible to derive a relationship such as (13) above. There is thus no basis on which to tie down the current value of the contract. We describe this as expectational instability. The explanation for this instability rests on the interpretation of the term [Gamma][Lambda]J. In this model there is an interaction between output and interest rates. Low output increases the expected arrival rate of realignments and therefore the expected rate of depreciation. It can be seen from eq. (8) that [Lambda]J determines the impact of output on the expected rate of depreciation. Via currency arbitrage [Lambda]J therefore determines the impact of output on the level of interest rates. Interest rates in turn affect output via aggregate demand and the parameter [Gamma] determines the strength of this effect. Thus low output causes high interest rates which further depress output. The overall strength of this feedback is measured by [Gamma][Lambda]J. If [Gamma][Lambda]J [greater than] 1 the feedback effect is so strong that the model becomes unstable. The implication of this result is that if the private sector believes that there is a strong link between the level of output and the government's propensity to realign the economy may become totally unstable.

4. Between realignments

Since the model is set in continuous time, the expectations system defined in eq. (10) predicts continuous realignments of the exchange rate and a smooth path of adjustment. On a particular realised path, of course, devaluations occur intermittently, with the system adjusting to each new parity between realignments. Thus, in order to define the actual behaviour of the system between realignments we need to derive the path of adjustment when there is no realignment but agents allow for the possibility of a realignment. To do this we make use of the stable path of the expectations system in eq. (13).

Recall that the subscript e indicates an expectation conditional on time t information. It must therefore be true by definition that [p.sub.e](t) = p(t) and [Mathematical Expression Omitted]. Thus the actual value of x at time t can be written as follows

[Mathematical Expression Omitted]

where the relationship [[Theta].sub.2] = 1 - [[Theta].sub.1] has been used to eliminate [[Theta].sub.2]. This can be substituted into eq. (2) to yield the following differential equation for p, conditional on a particular [Mathematical Expression Omitted]

[Mathematical Expression Omitted]

Equations (15) and (16) define the behaviour of the price level and the contract in the periods between realignments. In what follows we refer to eqs (15) and (16) as the 'partial credibility system' to indicate that it represents the behaviour of the economy during a period when the exchange rate is fixed but when there is some expectation of a realignment.

It can be seen from these two equations that, if there are never any realignments, the economy heads towards a point where [Mathematical Expression Omitted]. In what follows we shall refer to this point as the equilibrium of the partial credibility system. It should be noted, however, that this equilibrium is conditional on a particular value of [Mathematical Expression Omitted]. It should also be noted that this equilibrium is identical to the equilibrium of the full credibility system given in eq. (6) (i.e. the case where [Lambda] = 0). This can be explained as follows. At the point [Mathematical Expression Omitted] output is at its target level (i.e. [Mathematical Expression Omitted]). This implies [Pi] = 0 for any value of [Lambda]. In other words, there is full credibility regardless of the value of [Lambda]. The full credibility and partial credibility systems must therefore coincide at [Mathematical Expression Omitted].

Further implications of eqs (15) and (16) for the behaviour of the economy are illustrated in Fig. 1. For a given exchange rate parity the equilibrium point is at E. Equation (15) defines a linear relationship between the contract, x, and the price level, p, which is shown as the schedule ES. We shall refer to this relationship as the stable path of the partial credibility system. For comparison the full credibility stable path is illustrated as schedule EC. (As explained above, the two cases share the same equilibrium point). The slope of the partial credibility path is given by the expression 1 - [[Theta].sub.1]. It can be deduced from eq. (14) that this is increasing in [Lambda] so the slope of the partial credibility path is more positive (or less negative) than the slope of the full credibility path. The explanation for this is that for points to the right of E output is depressed below its natural (and target) level. This generates an expectation of devaluation. Wage bargainers therefore push wage settlements up in order to compensate for the possibility of inflation caused by a devaluation. Hence to the right of E, x is higher on ES than on EC.

Equation (16) can also be used to show that the introduction of realignment expectations in the form of eq. (1) slows the actual process of adjustment to the (conditional) equilibrium point. The root of eq. (16), denoted [Mathematical Expression Omitted], can be written as follows

[Mathematical Expression Omitted]

The speed of adjustment of the model between realignments is measured by the absolute value of [Mathematical Expression Omitted]. Inspection of eq. (17) shows that absolute value of [Mathematical Expression Omitted] falls as [Lambda] rises. Thus the speed of adjustment is lower in the partial credibility system. This can be illustrated by again referring to price level [p.sub.1] in Fig. 1. From this starting point there is a recession and prices must fall towards equilibrium in order to restore competitiveness. From eq. (2) it can be seen that the speed of price adjustment is proportional to (x - p). At price level [p.sub.1], x is lower on EC than on ES, hence the rate of deflation must be faster in the full credibility case than in the partial credibility case. (Remember that p [greater than] x for all points to the right of the 45 [degrees] line.) Realignment expectations therefore prolong the recession. It is also the case, for any given deviation of prices from equilibrium, realignment expectations increase the deviation of output from its natural rate (because the real exchange rate is the same but interest rates are higher).

These results imply that any shock that causes prices to deviate from equilibrium (for example a demand shock) causes a larger and more persistent deviation of output when there are realignment expectations of the form of eq. (1). In the previous section we demonstrated how realignment expectations could give rise to 'expectational' instability in the sense that there would be no stable expected trajectory for the economy. The results illustrated in Fig. 1 indicate a second sense in which the economy is destabilised by realignment expectations of the type embodied in eq. (1), namely, the economy becomes more sluggish in its response to shocks.

5. The effects of a realignment

We are now in a position to use the above framework to consider the effects of a realignment. We shall contrast the effects that arise in our framework with the effects that arise in the full credibility case and the effects that arise in the Driffill and Miller case. Each case can be explained with reference to Fig. 1. The full credibility stable path is marked EC with associated equilibrium point E (which corresponds to the current parity [Mathematical Expression Omitted]). The stable path for our case is marked ES. As explained above it is steeper than EC but passes through the same equilibrium point.

An example of a stable path in the Driffill-Miller case is shown as schedule DQ. This has an equilibrium displaced along the 45 [degrees] line at Q. It is shown in Appendix 2 that the price level at Q is given by [Mathematical Expression Omitted]. ([Psi] is defined in Appendix 2 and shown to be greater than zero). The size of the displacement is therefore proportional to the expected rate of realignment, [Pi]. In the Driffill-Miller case [Pi] is unaffected by the level of output so the expected rate of alignment is identical at all levels of p. The adjustment to wage contracts to take account of possible realignments is therefore also identical at all levels of p so the stable path DQ has the same slope as the full credibility case, EC. It is important to note that the equilibrium point Q is not a full equilibrium since it involves a permanent recession caused by the expectation of realignments. If there are no realignments the learning process assumed by Driffill and Miller results in [Pi] falling to zero. Thus, as time passes, the quasi-equilibrium point Q shifts towards the full equilibrium point E.

Assume that the price level is initially at [p.sub.1] which is higher than the price level at E but lower than at Q. This is a situation of recession for all three cases. Assume that a realignment takes place which raises the exchange rate to J. This raises the equilibrium point for our case and the full credibility case to E[prime]. The associated stable paths are E[prime]C[prime] and E[prime]S[prime]. The price level is backward looking so it cannot jump in response to the devaluation. There must therefore be an improvement in competitiveness. In the full credibility case the nominal interest rate is always fixed at the level of the foreign nominal rate so there is no impact from the devaluation. However, the real interest rate falls because the rate of inflation becomes less negative after the devaluation.(7) Thus, in the full credibility case, there is an unambiguous increase in output (because competitiveness has improved and the real interest rate has fallen).

It is immediately apparent that the arguments of the previous paragraph concerning inflation and competitiveness also apply to the case where realignment expectations are determined by eq. (1). This case differs, however, in the effect of a realignment on the nominal interest rate. The nominal interest rate is determined by the expected rate of devaluation as follows, i = Ds = [Pi]J = -[Lambda]yJ. The nominal interest rate is therefore proportional to the deviation of output from its equilibrium level. Thus, provided output moves closer to its equilibrium level after the devaluation, the nominal interest rate will fall. The conditions under which output moves towards its equilibrium level can be determined by inspecting eq. (9). This equation shows the dependence of output on inflation and competitiveness after the interdependence of the nominal interest rate and output has been solved out. It can be seen that output will be boosted by a devaluation provided [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are positive. Or equivalently provided (1 - [Gamma][Lambda]J) [greater than] 0. This is exactly the same as the condition required to ensure saddle-point stability of the expectations system (analysed in Section 3), i.e. that the feedback effect of output on interest should not be too strong. If this condition is satisfied, expectations of a further devaluation are reduced by a realignment and hence the nominal interest rate is also reduced.

The combined effect of less negative inflation and a lower nominal interest rate is a lower real interest rate. It is not possible to say however whether the fall in the real interest rate is more or less than in the full credibility case. In the full credibility case the nominal interest rate does not fall but the rate of inflation rises by more. It is not possible to tell which effect dominates.

The effects described above can now be compared to the effects in the Driffill and Miller case. Prior to the realignment the economy is on DQ heading towards the quasi-equilibrium at Q. As stated above, at Q the price level is given by [Mathematical Expression Omitted]. The realignment raises [Mathematical Expression Omitted] by the size of the realignment. The above expression shows that the quasi-equilibrium must therefore also be shifted by the size of the realignment. However, this is not the only effect. The learning process assumed by Driffill and Miller implies that expectations of further realignments are increased by the devaluation. Thus, in addition to the rise in [Mathematical Expression Omitted], there is also a step increase in [Pi]. The distance between the quasi-equilibrium and the full equilibrium therefore increases following a realignment. The combination of these two effects is represented by the shift of quasi-equilibrium to Q[prime] in Fig. 1. The associated stable path is marked D[prime]Q[prime].

In common with the other two cases considered the effect of a devaluation is to improve competitiveness. This follows because the price level cannot jump. In the previous cases inflation was made less negative by the realignment, i.e. the rate of inflation increased. The rate of inflation also increases in this case, but by more than in the previous two cases. The change in the rate of inflation can be measured by the vertical jump in x after the realignment. The fact that [Pi] increases in the Driffill-Miller case, which raises the stable path by more than in the other two cases, ensures that x jumps further.

The Driffill-Miller case also differs in terms of the nominal interest rate. The realignment causes expectations of more frequent realignments in the future so the nominal interest rate rises to compensate. This rise in the nominal interest rate is more than offset by the rise in inflation so the real interest rate falls after the realignment. It can be shown that the real interest rate falls by exactly the same amount in the Driffill-Miller case as in the full credibility case. This, combined with the improvement in competitiveness means that output is boosted by the same amount in both cases.

So far we have discussed only the impact effects of a devaluation. The impact effects on consumption and output are similar in all three cases. The main differences between the impact effects are confined to movements in nominal interest rates and the rate of inflation. The differences in the long run effects of a devaluation are, however, more significant. In the case where realignment expectations are determined by eq. (1) a realignment moves the economy closer to its long run equilibrium (or rather it moves the equilibrium closer to the current position of the economy). After the realignment output will be permanently higher than if there had been no realignment.

In the Driffill-Miller case the realignment moves output and the price level closer to the full long run equilibrium, just as in the other case, but it also pushes the quasi-equilibrium further away (because of the increase in [Pi]). The latter effect means that the price level continues to rise after the realignment (at a faster rate) and competitiveness and output begin to fall away from the levels achieved by the devaluation. Eventually, if there are no more realignments, [Pi] will fall again and the quasi-equilibrium will move back towards the full equilibrium. In the meantime, however, the economy will have to suffer a prolonged recession. This prolonged recession does not occur in the case where realignment expectations are determined by eq. (1). It is in this respect that the effects of a realignment are much more favourable in our case than they are in the Driffill-Miller case.

6. Switching from a floating to a fixed exchange rate

Up to this point we have been considering the behaviour of the model in a fixed rate system. We have not considered the implications of the model for the degree of inflation inertia displayed after a switch from a floating to a fixed rate regime. We will now examine this question.

Suppose that under floating exchange rates the money supply grows at a constant rate [Mu]. In dynamic equilibrium output would be at its natural rate (y = 0) and Dp = Dx = Ds = [Mu]. From the equations of the model we can tell that

x = p + [Mu]/[Delta] s and s = p (18)

i.e. the exchange rate and all other nominal variables will rise at rate [Mu] and the contract will be displaced above the price level by an amount [Mu]/[Delta].(8) In Fig. 2 the floating rate dynamic equilibrium is represented by a steady crawl up the 45 [degrees] line marked FF. Suppose the switch to a fixed rate takes place at time t = 0 at which point the exchange rate is zero. The economy will therefore be at point A immediately before the regime switch takes place. Assume that the exchange rate is fixed at its market value, zero, on entering the fixed rate system. There are two possible outcomes, depending on the government's target level of output [Mathematical Expression Omitted].

If [Mathematical Expression Omitted], as assumed in the paper up to this point, the post regime switch equilibrium will be at point E. Thus the impact effect of the regime switch will be for the contract to jump directly to the long run equilibrium point. Inflation will stop immediately. Thus, modelling realignment expectations in the form of eq. (1) with [Mathematical Expression Omitted] produces the same result as in the full credibility model. This can be explained as follows. When the government's target level of output is equal to the natural rate there are no expectations of realignment when the economy is in equilibrium. It was explicitly assumed above that the regime switch takes place in a situation of equilibrium so there are no realignment expectations generated. There is therefore no inertia in the inflation rate.

This changes, however, if we assume that the government's target output level, [Mathematical Expression Omitted], is greater than the natural level of output, i.e. [Mathematical Expression Omitted]. In other words, the private sector believes that the government is over-optimistic in its choice of output target. In this case the post regime switch equilibrium is not at the origin. Instead there is a quasi-equilibrium, of the same form as in the Driffill-Miller case, which is displaced from the true equilibrium.

Define the quasi-equilibrium of the model, as the situation where Dx = Dp = 0, but where [Mathematical Expression Omitted]. Imposing these conditions yields the following expression for the price level at the quasi-equilibrium(9)

[Mathematical Expression Omitted]

where the subscript Q denotes the value at the quasi-equilibrium. Recalling that [Mathematical Expression Omitted] is negative, it can be seen that for [Mathematical Expression Omitted] the quasi-equilibrium price level is above the true equilibrium price level. This case is illustrated in Fig. 2 where the quasi-equilibrium is at point Q and the true equilibrium is at point E (with [Mathematical Expression Omitted]). BQ is the stable path associated with the quasi-equilibrium. With [Mathematical Expression Omitted] the impact effect of a switch to a fixed rate is a jump in x from point A to point B. At this point, wages are greater than prices so eq. (2) tells us that the economy will inflate along the path BQ until the quasi-equilibrium Q, is reached. It can therefore be seen that by allowing for an over-optimistic target for output it is possible to generate inflation inertia. In effect this is being achieved by combining the endogenous realignment expectations of eq. (1) with an exogenous element (represented by [Mathematical Expression Omitted]) which is similar to the exogenous [Pi] used in Driffill and Miller.

The question now arises, however, about the determination of [Mathematical Expression Omitted]. In particular it would seem reasonable to suppose that if no realignment actually occurs as the system converges on Q, then agents will revise their view about [Mathematical Expression Omitted] in exactly the same way as agents learn about the value of [Pi] in the Driffill and Miller model. If such adjustment to expectations is allowed the quasi-equilibrium will shift towards the true equilibrium at E. Thus the economy will move towards full equilibrium if no realignments actually take place.

However, allowing for a learning process for [Mathematical Expression Omitted] logically should also introduce a learning effect of a realignment on [Mathematical Expression Omitted]. The act of realigning the currency should result in an upward revision of [Mathematical Expression Omitted] just as a realignment leads to an upward jump in [Pi] in the Driffill and Miller model. The long run effect of a realignment on [Mathematical Expression Omitted] will therefore be of the same form as in the Driffill-Miller model. The upward revision to [Mathematical Expression Omitted] pushes the economy away from the full equilibrium and implies that a period of recession is required in the future. Thus realignment expectations of the form of eq. (1) involve endogenous elements which imply that a realignment pushes the economy closer to equilibrium, through the effect of y and [Pi], and also elements which imply that a realignment pushes the economy away from equilibrium, through the effect of [Mathematical Expression Omitted] on [Pi]. The overall impact of a realignment depends on the relative strength of these two effects.(10)

7. Conclusion

This paper presents a model where the probability that private agents attach to a realignment is related to the level of spare capacity. Thus, if output is below the government's target level, agents attach a high probability to the possibility of a devaluation. The behaviour of the model is shown to depend crucially on the sensitivity of the private sector's subjective probability to the level of output. If the level of sensitivity is relatively low the model is found to be (semi-)stable in expected terms. Starting from a position of excess capacity (with a consequent expectation of devaluation) the economy is expected to converge to a long run equilibrium with full employment and zero inflation. If, on the other hand, the subjective probability is very sensitive, it is found that the model is completely unstable in expected terms. There is consequently no anchor for forward looking expectations.

We further show that, between realignments, the actual speed at which the economy converges to equilibrium is slowed by the presence of realignment expectations. The more sensitive the subjective probability is to the level of excess capacity the slower the economy adjusts. Demand shocks therefore give rise to larger and more persistent deviations of output from equilibrium. Both these conclusions suggest that, in circumstances where private sector expectations of devaluations are related to the level of output, a fixed exchange rate regime may be highly unstable and destabilising. In this sense this model can be seen as providing support for a form of the 'Walters' critique of the EMS (Walters 1990).

The model can also be seen as providing support for the view that tensions within the EMS might have been reduced in recent years had there been a realignment. The model shows that, when a country is in recession, a realignment moves the economy closer to equilibrium and therefore reduces private sector expectations of further realignments. This contrasts with the view that a realignment would have undermined anti-inflation credibility and pushed the EMS away from equilibrium.

This last conclusion must be qualified, however, by noting that the model, in its simplest form, fails to display significant inflation inertia following a switch from a inflationary equilibrium. Modifying the model to allow for inflation inertia re-introduces an exogenous element into realignment expectations which may react unfavourably in the event of a realignment. The long run effects of a realignment are ambiguous in this case.

1 In principle, it is possible to test eq. (1) against the data by, for instance, regressing some measure of realignment expectations on macroeconomic variables such as output and the trade balance. Measuring realignment expectations is, however, not straightforward. In target zone systems such as the EMS the observed interest differential between two currencies is influenced both by expectations of realignment and by expected movement of the exchange rate within the band. Measuring the former requires some assumption to be made about the latter. Rose and Svensson (1993) use one technique for disaggregating expectations and find no clear relationship between realignment expectations and most macroeconomic variables. However, Chen and Giovannini (1993), using a different technique, do find some weak evidence of a link between realignment expectations and macroeconomic variables.

2 Gros (1992) analyses a model where the expected rate of depreciation of the exchange rate is related to the level of domestic interest rates. Thus the government is assumed to view a realignment as a way of reducing pressure on domestic interest rates. A realignment is therefore more likely when interest rates are high. Gros considers the dynamic interaction of interest rates and exchange rates in this context. Our model differs from the Gros model in that realignment expectations are related to output (not interest rates) and we consider the wider macroeconomic stability of the model (including the interaction of realignment expectations and forward-looking wage contracts).

3 An alternative approach to this problem would be to specify an explicit objective function for the government and to derive its optimal realignment strategy. Minford (1993) does consider a model of this type where the government's objective function takes account of the output of the traded goods sector. Minford shows that a 'political equilibrium' arises where the exchange rate is overvalued. Drazen and Masson (1993) also use a model with an explicitly optimising government to consider the impact of external circumstances on the credibility of an anti-inflation policy. However, the extra complications caused by the explicit treatment of policy optimisation implies that the dynamic aspects of the economy are highly restricted in these models.

Ozkan and Suthertand (1993) also consider a model where an explicit government objective function is specified. The framework is used to analyse the causes and effects of a complete collapse of a fixed rate system (rather than a devaluation within a surviving system). The model, however, does not take explicit account of the dynamics created by forward looking wages and overlapping contracts (as this paper does).

4 We are thus normalising the profit mark-up to zero so that the price level is equal to the average level of wages. The price level, p(t), can thus also be interpreted as the average wage level. The latter should be clearly distinguished from x(t), which is the wage level for those workers signing contracts at time t.

5 In the integral form of eq. (3) it can be seen that the level of the contract wage set at time t is positively related to the level of prices and output from time t into the infinite future. On the other hand the forward rate of change of the contract wage (Dx) is negatively related to the current level of prices and output. This is because the price or output level at time t affects x(t) but not x(t + dt) (where dt is a small increment in time). A high price or output level at time t therefore raises x(t) relative to x(t + dt).

6 it was pointed out in footnote 4 that p can be interpreted as both the price level and the average wage level. Figure 1 therefore shows that this model displays inertia in both these variables. On the other hand the current contract level x(t) is fully flexible and is allowed to jump in response to new information (such as devaluations). It can be seen from eq. (2) that x determines the rate of inflation. One interpretation of the model is, therefore, that it implies inertia in price and wage levels while allowing perfect flexibility in the inflation rate.

7 From eq. (2) it can be seen that the rate of inflation is proportional to (x - p). At [p.sub.1] (x - p) is less negative on E[prime]C[prime] than on EC.

8 The latter effect arises because wage negotiators incorporate an allowance for future price rises in the current contract.

9 When [Mathematical Expression Omitted] the stable path of the partial credibility system becomes [Mathematical Expression Omitted]. Quasi-equilibrium is defined by Dp = Dx = 0 which, when combined with eq. (2), implies x - p. Imposing x = p in the equation for the stable path and rearranging yields eq. (19).

10 These two opposing effects are also evident in the model of Drazen and Masson (1993). It is likely that the two effects operate to different degrees for different countries. Thus for instance in the case of the United Kingdom it may be that the learning effect is relatively less important than the excess capacity effect on realignment expectation. A realignment of sterling may therefore have had a favourable effect on credibility. On the other hand, in the case of the Netherlands it may be that the learning effect is relatively more important. A realignment of the guilder may therefore lead to a loss of credibility.


CALVO, G. (1983a). 'Staggered Prices in a Utility-Maximising Framework', Journal of Monetary Economics, 12, 383-98.

CALVO, G. (1983b). 'Staggered Contracts and Exchange Rate Policy', in J. Frenkel (ed.), Exchange Rates and International Macroeconomics, University of Chicago Press, Chicago, IL.

CHEN, Z. and GIOVANNINI A. (1993). 'The Determinants of Realignment Expectations Under the EMS - Some Empirical Regularities', Discussion Paper No. 790, CEPR, London.

DRAZEN, A. and MASSON, P. (1993). 'Credibility of Policies versus Credibility of Policy Makers', Working Paper No. 4448, NBER, Cambridge, MA.

DRIFFILL, J. and MILLER, M. (1993). 'Learning and Inflation Convergence in the ERM', Economic Journal, 103, 369-78.

GROS, D. (1992). 'Capital Controls and Foreign Exchange Market Crises in the EMS', European Economic Review, 36, 1533-44.

MILLER, M. and SUTHERLAND, A. (1993). 'Contracts, Credibility, Common Knowledge: Their Influence on Inflation Convergence', IMF Staff Papers, 40, 178-200.

MINFORD, P. (1993). 'The Political Economy of the Exchange Rate Mechanism', Discussion Paper No. 774, CEPR, London.

OZKAN, F. G. and SUTHERLAND, A. (1993). 'A Model of the ERM Crisis', Discussion Paper No. 879, CEPR, London.

ROSE, A. and SVENSSON, L. E. O. (1993). 'European Exchange Rate Credibility Before the Fall', European Economic Review, 38, 1185-216.

WALTERS, A. (1990). Sterling in Danger, Fontana, London.


The stable-path for the expectations system

Assume the expectations system defined by eq. (10) has a single negative root, denoted [[Phi].sub.s], and a zero root. The stable solutions can therefore be written as follows

[Mathematical Expression Omitted]

where the column vectors on the right-hand side are the eigenvectors corresponding to the two relevant roots and [F.sub.1] and [F.sub.2] are constants determined by initial conditions for s and p. The elements [v.sub.1] and [v.sub.2] in the first eigenvector are given by the following

[Mathematical Expression Omitted]

Substituting from (A1) into the relationship [x.sub.e] = [[Theta].sub.1][s.sub.e] + [[Theta].sub.2][p.sub.e] yields the following

[v.sub.2][F.sub.1] [e.sup.[[Phi].sub.s][Tau]] + [F.sub.2] = [[Theta].sub.1]([F.sub.1] [e.sup.[[Phi].sub.s][Tau]] + [F.sub.2] + [[Theta].sub.2]([v.sub.1][F.sub.1] [e.sup.[[Phi].sub.s][Tau]] + [F.sub.2]) (A3)

which must hold for all [Tau]. Rearranging yields

(1 - [[Theta].sub.1] - [[Theta].sub.2])[F.sub.2] = ([[Theta].sub.1] + [[Theta].sub.2][v.sub.1] - [v.sub.2])[F.sub.1] [e.sup.[[Phi].sub.s][Tau]] (A4)


[[Theta].sub.2] = 1 - [[Theta].sub.1] and [[Theta].sub.1] = [v.sub.2] - [v.sub.1]/1 - [v.sub.1] (A5)

Substituting for [v.sub.1] and [v.sub.2] from (A2) yields the expression for [[Theta].sub.1] given in eq. (14).


Quasi-equilibrium in the Driffill-Miller case

In the Driffill-Miller case the expected rate of realignments is autonomous (in that it is not affected by the other variables in the model). With realignment execrations of that form, Miller and Sutherland (1993) show that the model can be reduced to the following system of two dynamic equations

[Mathematical Expression Omitted]

where A is defined in eq. (6) and b is defined as follows

[Mathematical Expression Omitted]

where [[Theta].sub.A] is the slope of the stable eigenvector of the matrix A. Quasi-equilibrium (denoted with a subscript Q) is defined as the point where Dp = Dx = 0 for a given value of [Pi]. Setting Dp = Dx = 0 in eq. (A6) yields

[Mathematical Expression Omitted]

where [Psi] = (1 - [[Theta].sub.A] - [Delta][Beta][Gamma])J/[Delta][Beta][Eta]. The negative eigenvalue of the matrix A, denoted [[Phi].sub.A], satisfies the characteristic equation [Mathematical Expression Omitted], and [[Phi].sub.A] and [[Theta].sub.A] satisfy [[Phi].sub.A] = [Delta]([[Theta].sub.A] - 1). Substituting these two relationships into the definition of [Psi] yields [Psi] = -(J/[[Phi].sub.A]) [greater than] 0.
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Author:Stansfield, Ed; Sutherland, Alan
Publication:Oxford Economic Papers
Date:Apr 1, 1995
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