A currency crisis model with an optimising policymaker

Journal of International Economics 44 (1998) 339–364

A currency crisis model with an optimising policymaker a b, F. Gulcin Ozkan , Alan Sutherland * a

Department of Economics and Finance, Brunel University, Uxbridge, Middlesex, UB8 3 PH, UK b Department of Economics, University of York, Heslington, York, YO1 5 DD, UK

Received 12 January 1995; received in revised form 19 August 1995; accepted 27 February 1997

Abstract A model is presented in which the abandonment of a fixed exchange rate regime is triggered by an optimising policymaker who wants to loosen monetary policy and boost aggregate demand. Agents in the foreign exchange market know the policymaker’s objective function and build expectations of a regime switch into interest differentials. The resulting rise in interest rates affects the policymaker’s decision to switch regime. It is shown that a rational expectations equilibrium exists where the fixed rate is abandoned in response to adverse demand shocks. In some circumstances multiple equilibria arise which may lead to self-fulfilling crises.  1998 Elsevier Science B.V. Keywords: Speculative attacks; Balance of payments crises JEL classification: F31; F32

1. Introduction Prior to the ERM Crises of 1992 and 1993 the literature 1 on speculative attacks and balance of payments crises emphasised the role that limited reserves play in precipitating the collapse of a fixed rate. In this paper we present an alternative model where a switch from a fixed to a floating exchange rate is triggered by an *Corresponding author. Tel.: 11 44 1904 433780; fax: 11 44 1904 433759; e-mail: [email protected] 1 See Agenor et al. (1992), Blackburn and Sola (1993) and Willman (1992) for recent surveys of that literature. 0022-1996 / 98 / $19.00  1998 Elsevier Science B.V. All rights reserved. PII S0022-1996( 97 )00025-1

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optimising policymaker. In this model the important state variable is a demand shock variable. The policymaker is assumed to be setting policy so as to maximise a utility function which depends on demand pressure. Shocks cause deviations of demand pressure from the policymaker’s preferred level and therefore reduce the policymaker’s utility relative to that which could be obtained if the exchange rate were floating. The policymaker’s optimising problem is to choose the trigger level of the demand shock at which to leave the fixed rate system 2 . The private sector knows the policymaker’s preferences and its policy options and is therefore able to forecast the collapse of the fixed rate. This is built into domestic interest rates so that the interest differential between domestic and foreign rates widens as the demand shock rises 3 . This hastens the onset of the collapse. Unlike in existing models, foreign exchange reserves play no explicit part in determining the timing of the collapse. The switch of exchange rate regime is the result of an optimising decision by the policymaker. Existing models of speculative attacks and balance of payments crises fall into two broad categories, those where a collapse is the inevitable consequence of some fundamental imbalance and those where a collapse results from self-fulfilling expectations. The first category originates from the Krugman (1979) model (which is an application to the foreign exchange market of the Salant and Henderson (1978) model of the price of gold) where a balance of payments crisis is generated by a monetary authority which operates a policy of domestic credit expansion while simultaneously fixing the exchange rate. Foreign exchange reserves inevitably run out and the fixed rate has to be abandoned. Krugman shows that, with forward looking exchange markets, the final stage of the crisis involves a sudden discrete loss of reserves in a ‘‘speculative attack’’. The second category of models arises from the Obstfeld (1986) model which differs from the Krugman model in that it does not depend on a policymaker adopting an unsustainable policy stance 4 . Instead the possibility of a speculative

2 Flood et al. (1989) also model exchange rate regime switches as being generated by an optimal state-contingent policy strategy. Their model differs from ours in two important respects. Firstly in their model a switch of regime is triggered by changes in the variability of shocks, whereas in our model it is the level of the shock which is important. And secondly, the Flood et al. model does not incorporate forward looking behaviour on the part of agents in financial markets. Another example of an optimal state-contingent regime switch is analysed in De Kock and Grilli (1993). In that model a regime switch follows adverse shocks to the fiscal deficit. The policymaker abandons the fixed rate so that it may exploit the inflation tax. 3 Gros (1992) analyses a model where private sector expectations of devaluation increase as interest rates rise. Gros, however, does not explicitly model the linkage in terms of the policymaker’s optimising problem. Stansfield and Sutherland (1995) consider a similar linkage between realignment expectations and economic activity in a model of overlapping wage contracts. Again, in that paper, the policymaker’s optimising problem is not modelled explicitly. 4 Flood and Garber (1984) provide an earlier example of a model of self-fulfilling attacks in the context of a collapse of a gold standard.

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attack is generated by private sector expectations of a loosening of monetary policy after a collapse of the fixed rate regime. The expectation of a devaluation makes it rational for private sector agents to join an attack when one occurs. An attack exhausts reserves and forces the authorities to abandon the fixed rate. If the authorities do in fact then loosen monetary policy the exchange rate depreciates and the expectations of speculators are fulfilled. Thus a rational expectations equilibrium can exist with a speculative attack even when the initial policy stance is sustainable. The crises in the European Exchange Rate Mechanism of 1992 and 1993 provide a very obvious testing ground for these models and our proposed alternative. The crisis which preceded the exit of the pound and the lira from the ERM in September 1992 certainly involved a large and sudden loss of reserves for the UK and Italian authorities. This is consistent with the prediction of the Krugman model, but it is not clear that the other features of that model are relevant to this period. The central cause of a regime collapse in the Krugman model is the unsustainability of a fixed exchange rate when the authorities are allowing domestic credit to expand. Policy inconsistencies of this type do not appear to have been a major factor in generating the ERM crisis. Eichengreen and Wyplosz (1993) argue that the Obstfeld model does provide a good explanation for the ERM crisis in Autumn 1992. They suggest that up to that point ERM members were against the principle of realignments because of the requirement in the Maastricht Treaty that participants in EMU should not have devalued in the two years prior to Stage 3. Eichengreen and Wyplosz point out, however, that if a speculative attack exhausted a country’s reserves it would be forced to accept a realignment. Having been forced to accept the principle of a realignment it would no longer be eligible for Stage 3 of EMU and would therefore have no incentive to resist a large devaluation. Thus Eichengreen and Wyplosz argue that the Maastricht Treaty created the very conditions needed for self-fulfilling speculative attacks 5 . We believe, however, that there are a number of important features of the ERM crisis which the Obstfeld (1986) model does not capture but which are incorporated in our model. In particular the Obstfeld model gives no role to the pressures put on the ERM by German unification. In both the September 1992 and the July 1993 crises there was a widespread belief in financial markets that high German interest rates (which acted as a negative demand shock in the rest of Europe) were placing an intolerable burden on ERM members such as Italy, the UK and France. Speculators were clearly concerned that these governments would at some point

5

Eichengreen et al. (1994) carry out an extensive analysis of macro data surrounding the crisis period in the ERM and find that there was no statistically significant differences between the behaviour of fundamentals during this period as compared to non-crisis periods. They conclude that this is evidence against the ‘‘fundamentals’’ based models of speculative attacks.

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find it acceptable to abandon the ERM in order to loosen monetary policy. This lack of credibility resulted in even higher interest rates in the countries concerned, which, in turn, increased the pressure. Interaction of this type between credibility and interest rates appeared to be an important element in the crises. It is also not clear what role the loss of foreign exchange reserves played in the decision of each government to abandon its fixed rate. In theory the governments concerned could have defended their currencies by raising interest rates to whatever level necessary. Their decision not to do so reveals something about their preferences. It appears that the decision to leave the ERM was to some extent an optimising decision rather than a policy action forced upon the authorities by a loss of reserves (as is true in both the Krugman and Obstfeld models). We argue that the model presented in this paper is more consistent with the basic features of the ERM crisis as we have just described them than either of the other two models 6 . In Section 2 of the paper we describe our basic framework. In Section 3 we discuss the behaviour of the term structure of interest rates and show the important linkage between expectations of regime collapse and interest differentials. Section 4 describes the policymaker’s optimisation problem and illustrates a rational expectations equilibrium. In Section 5 we show how a small modification to the model can generate multiple equilibria and we discuss the implications for a self-fulfilling crisis.

2. The model We assume that exchange rate policy is determined by a policymaker who is attempting to maximise the following utility function

6

Obstfeld (1994) also argues that a model of an optimising policymaker is necessary to understand the collapse of the ERM. He presents two such models. In the first the policymaker must choose between distortionary taxes and inflation (through devaluation) as a means of raising revenue. In the second model the policymaker must choose between price stabilisation within the fixed rate system and output stabilisation in the floating rate system. The second of these models has some similarities with ours but it also differs in a number of important respects. Firstly, the source of forward-looking behaviour in our model is the foreign exchange market. In the Obstfeld (1994) model it is the labour market which is forward-looking. Secondly, the Obstfeld model is set in a static framework whereas our model is dynamic. Andersen (1994) proposes a model which is similar to the second of Obstfeld’s models. Isard (1994) presents a model where realignments within a fixed exchange rate system are triggered by an optimising policymaker. Again this model differs from ours in that the optimising problem facing the policymaker is essentially static in nature. Bensaid and Jeanne (1994) present a deterministic dynamic model of an optimising policymaker choosing to switch regimes. The dynamics in their model arise because of learning on the part of the private sector.

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`

W(t) 5 Et

E [Z 2 y (t)]e 2

2d (t 2t )

dt

(1)

t

where d is the policymaker’s discount rate and Et is the expectations operator conditional on information available at time t. The variable Z represents the (flow) benefits of membership of a fixed rate system. We do not model these benefits explicitly but suggest that they may arise from such factors as anti-inflation credibility and the elimination of excess volatility 7 . Alternatively they may take the form of increased influence for the domestic country in other aspects of international relations. Z is assumed to be positive, at a fixed exogenous level, while the domestic country is a member of the fixed rate regime. In a floating rate regime Z is zero. In order to keep the analysis as simple as possible we assume that once a country has switched to a floating rate regime it cannot regain positive Z by switching back into the fixed rate regime. In a sense this can be regarded as a form of punishment for a policymaker who has reneged on an initial commitment to a fixed rate system. It will become apparent below that this assumption implies that it is never optimal to re-enter the fixed rate system 8 . The variable y is an index of demand pressure in the goods market and is given by the following expression y 5 2 g i T 1 hs 2 b´

(2)

where i T is the domestic nominal interest rate with maturity T 9 , s is the log of the nominal exchange rate (defined as the domestic currency price of foreign currency)

7

Flood and Rose (1995) provide evidence that fixed rates eliminate excess volatility. We are therefore in effect modelling the switch of regimes as irreversible. Krugman and Rotemberg (1992) on the other hand model the collapse of a fixed rate as a reversible state-contingent regime switch. In their model the authorities re-enter the fixed rate system immediately stochastic shocks take the economy back to the state at which the regime collapsed. Flood et al. (1989) also model exchange rate regime switches as reversible. But in their case lump-sum costs of adjustment imply that entry and exit take place at different trigger points. In this paper we rule out the possibility of a reversible regime switch in order to keep the analysis simple. But the techniques we use can be extended to the reversible case. 9 For reasons that will become apparent later, it is necessary to set 0,T ,`. Ozkan and Sutherland (1994), (1995) use a version of this model where demand is assumed to depend on a composite interest rate which averages over all maturities. In the present version of the model we assume that demand (and therefore the policymaker’s welfare) depends on one particular maturity. This makes the solution of the model more complicated but avoids the ad hoc nature of the composite interest rate structure previously adopted. Despite being less ad hoc the present structure is still restrictive in the sense that real-world policymakers clearly do care about the whole term structure of interest rates. For present purposes, however, Eq. (2) captures the essential features that we require. The effects on equilibrium of varying T will be considered. 8

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and ´ is a random shock variable. Implicit in this construction is the idea that there are some nominal rigidities in the economy which mean that changes in the nominal interest rate and the nominal exchange rate have real effects 10 . In Eq. (2) we are assuming that the effect of a high nominal interest rate can be offset by a depreciation of the nominal exchange rate (and vice versa). Eqs. (1) and (2) incorporate the notion that the policymaker wishes to stabilise demand around some target level of y (which is normalised to zero). The stochastic shock variable, ´, is assumed to be driftless brownian motion d´ 5 s dz

(3)

where z is a standard brownian motion process with unit variance. This is the only source of noise in the model. We do not make any specific assumptions about the identity of the shock. But an obvious source of shocks in the case of the collapse of the ERM was the level of German interest rates. This can be modelled in the present framework by setting b equal to g and reinterpreting i as the differential between domestic and foreign interest rates. We restrict the policymaker’s choice of policy regime to either a completely fixed rate (where realignments are not possible) or a floating rate. We assume that initially a fixed rate regime is in force (with a parity of s50 for convenience). In order to fix the exchange rate the policymaker varies the money supply and hence the interest rate. The interest rate is then effectively determined by the private sector’s expectation of devaluation. Uncovered interest parity is assumed to hold so domestic interest rates can be written as the sum of the corresponding foreign interest rate and the expected rate of change of the exchange rate. To see how interest rates are determined consider a pure-discount bond of maturity t. For convenience to eliminate unnecessary terms we normalise foreign interest rates to be zero at all maturities. Thus uncovered interest parity implies the following [Et [s(t 1 t )] 2 s(t)] it 5 ]]]]]] t

(4)

where it indicates an interest rate over the period t and Et is the expectations

10 It is not in fact necessary to make any strong assumptions about the true effects of monetary policy on the real economy. It is sufficient for the purposes of our model to assume that the policymaker believes monetary policy to have real effects in the way represented by Eq. (2) and for the private sector to be aware that the policymaker holds these beliefs. We acknowledge that our structure is unsatisfactory in that Eq. (2) implies that nominal variables have real effects even in the very long run. Ideally we should allow some dynamic convergence to monetary neutrality. However, the additional dynamics implied would make the model intractable.

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operator conditional on information available at time t. Eq. (4) determines the interest rate for each and every individual maturity 11 . Before proceeding to solve the model it is useful to summarise the optimising problem facing the policymaker. While the currency is in the fixed rate system the policymaker has to use interest rates to keep the exchange rate at its parity. Shocks cause deviations of y from its target level. The policymaker therefore faces a trade-off between the benefits of the fixed rate system represented by Z and the costs represented by the y 2 term in the objective function. This contrasts with the floating rate regime where the policymaker is free to use monetary policy to stabilise y around its target level but where the flow benefits of the fixed rate are lost forever. The optimal strategy for a policymaker who is currently operating a fixed rate is to choose upper and lower trigger values of ´ (denoted ´U , ´L ) at which to switch to a floating rate 12 .

3. Interest rate behaviour in the fixed rate regime Obviously the prospect of the shock variable hitting one of the trigger points within the lifetime of a T period bond, and thus causing a step change in the exchange rate, must affect i T . This link between the behaviour of the domestic interest rate and expectations of the breakdown of the fixed rate is one of the main relationships in the model. In the fixed rate regime domestic interest rates are determined by two factors. The first is the probability of the shock variable hitting one of the trigger points, and the second is the size of the step-change in the exchange rate that is implemented by the policymaker on leaving the fixed rate system 13 . The second of these two factors is easily derived. In the floating rate regime the policymaker can use the exchange rate to stabilise y against shocks. It follows immediately from the form of the objective function that the optimal

11 Eq. (4) is in fact a log-linear approximation to uncovered interest parity which omits some terms arising from Jensen’s inequality. The omitted terms have no important implications for the analysis which follows. Note that Eq. (4) is only appropriate in the case of pure-discount bonds, i.e. bonds which pay no coupon. 12 For an intuitive discussion of the optimality of trigger strategies in stochastic problems see Dixit (1993) pp. 32–34, which provides references to formal proofs. 13 Notice that with brownian motion shocks the probability of hitting a trigger point in the next instant of time is effectively zero (unless the shock variable is very close to a trigger point). This implies that the instantaneous interest rate will be completely unaffected by the prospect of a regime switch until the moment it happens. Similarly the interest rate on a pure-discount bond with infinite maturity will be unaffected by the prospect of a regime switch because the step change in the exchange rate is averaged over an infinite horizon. For this reason we consider finite values of T which are strictly greater than zero.

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policy is to stabilise y completely at its target level (which is zero). Once in the floating rate regime it is not optimal to re-enter the fixed rate system. This fact, combined with the assumption of brownian motion shocks, ensures that the expected change in the exchange rate, and therefore the interest rate, is zero in the floating rate regime. The stabilisation of y at zero therefore requires the following relationship between the spot rate and the shock variable to hold

b s5] ´ h

(5)

The interpretation of this relationship is simple. In the floating rate system, when there is a rise in ´ the policymaker loosens monetary policy by just enough to allow the exchange rate to depreciate and offset the depressing effect of ´ 14 . Fig. 1 plots this relationship as the linear schedule SS and shows clearly the step-change in the exchange rate that is implemented when a regime switch occurs at one of the trigger points. It is now possible to derive a differential equation which defines the behaviour of domestic interest rates in the fixed rate system. At this stage it is necessary to distinguish between the private sector’s expectations of the two trigger points and the policymaker’s optimal choice. The expected trigger points will be denoted ´ eU and ´ eL while the policymaker’s optimal choice of trigger points will be denoted ´U and ´L . Obviously, in due course we will show that a rational expectations equilibrium will be where ´U 5 ´ eU and ´L 5 ´ Le . The equilibrium trigger points will be denoted ´¯ U and ´¯ L . As explained earlier, it is assumed that the policymaker cares about a particular maturity of interest rate, i T . To understand the behaviour of the interest differential at any given maturity, t, it is useful to start by considering the expected value of the spot exchange rate at that maturity. Consider Et [s(t1t )] where the current time is t and t is the time horizon. It is intuitively obvious that the expected spot rate will be a function of ´. As ´ approaches the upper trigger point, for instance, the probability of hitting that trigger point within a given period of time rises so the expected future spot rate must rise. It is also obvious that the expected future spot rate must be a function of t since the longer the time horizon the higher the probability of hitting a trigger point within that horizon. Thus we can write Et [s(t 1 t )] 5 h(´, t )

(6)

Svensson (1991) shows that h(.,.) satisfies the following partial differential equation 14

Implicitly we are assuming that the policymaker has sufficient control over monetary policy to allow complete stabilisation of y. In particular monetary policy can be adjusted instantaneously to offset shocks as they occur. The policymaker therefore does not face a problem of choosing the best intermediate target for monetary policy.

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Fig. 1. Demand shocks, the exchange rate and interest rates.

s2 ht (´, t ) 5 ] h´´ (´, t ) 2

(7)

The derivation of this equation is outlined in Appendix A. The appropriate boundary conditions are the following h(´, 0) 5 0 ´ eU # ´ # ´ eL

(8)

b h(´ eL , t ) 5 ] ´ eL h

(9)

and 0#t #`

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b h(´ eU , t ) 5 ] ´ eu h

0#t #`

(10)

Condition (8) states that the expected exchange rate over zero time is the current exchange rate (which is zero in the fixed rate system). Condition (9) states that when the shock variable hits the lower trigger point the expected value of the exchange rate is simply the size of the revaluation imposed by the government at that point (given by Eq. (5)). Condition (10) states the same for the upper trigger point. Solutions to partial differential equations of this type are given in many text books. The relevant solution, which is derived in Appendix B, is the following

S

D S

` np (´ 2 ´ eL ) b s 2 n 2 p 2t ]]]] h(´, t ) 5 ] ´ 1 cn exp 2 ]]]] sin h (´ eu 2 ´ eL ) 2(´ Ue 2 ´ Le )2 n 51

O

D

(11)

where

S

D

e e 2b (´ U 2 ´ L ) cn 5 2 ]] ]]] sin(np ) 2 ´ eu cos(np ) 1 ´ eL hnp np

(12)

and where p is the mathematical constant. Having derived h(.,.) the solution for the interest rate is [h(´, t ) 2 s¯ ] it 5 ]]]] t

(13)

where s¯ is the parity in the fixed rate system (which we set at zero). The solution for the interest rate that enters the policymaker’s welfare function (i.e. the interest rate at maturity T ) is simply given by replacing t with T in Eq. (13)15 . For the purpose of solving for the policymaker’s optimising problem we only require i T . It is, however, interesting to consider the implications of an approaching crisis for the entire term structure. Numerical examples of the relationship between interest rates for two different maturities and the shock variable are illustrated in Fig. 1 using the following parameter values: b 50.5, h 50.5, s 51.0, ´ eU 52.0 and ´ eL 5 22.0. Intuitively it is clear that the effect of the anticipated regime switch will be to drive a wedge between the domestic and foreign interest rates. At values of ´ close to the upper trigger point there is a high probability that a regime switch will be triggered and a devaluation implemented. This is reflected in a rise in interest rates as the upper trigger point is approached. Likewise, as ´ 15 Note that this derivation of the term structure ensures that all arbitrage opportunities, both across currencies and across different maturities in the same currency, are eliminated. This is true at all times up to and including the moment a crisis occurs. Note also that by applying the appropriate weighting function (i.e. the Gamma distribution) to Eq. (13) and integrating over all maturities yields the composite interest rate used in Ozkan and Sutherland (1994), (1995).

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approaches the lower trigger point there is an increasing probability that a revaluation of the currency will be triggered, so interest rates fall 16 . Fig. 1 plots the effect of ´ on two maturities, t 51.0 and t 50.5. It is apparent

Fig. 2. Demand shocks and the term structure of interest rates.

16

In principle, it is possible to test this aspect of the model 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 (1994) 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.

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that the short maturity interest rate reacts relatively little when the shock variable is far away from a trigger point as compared to the long maturity interest rates. This is because the long maturity rate picks up the possibility of a regime switch far in the future while the short rate reacts only to imminent events. On the other hand, at values of the shock variable close to a trigger point the short rate reacts much more than the long rate. This is because a step change in the exchange rate of a given size is much more important to a short maturity bond than a long maturity bond. Fig. 2 plots the term structure of interest rates for two given values of the shock variable (using the same parameter values as given above). This illustrates three facts. Firstly, when the shock variable moves closer to the upper trigger point the entire term structure rises. Secondly, the prospect of a regime switch gives the

Fig. 3. Expected trigger points and interest rates.

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term structure a hump shape. Very short term interest rates are low because the regime switch is relatively improbable over a short horizon. Very long term rates are low because the step change in the exchange rate is averaged over a long maturity. Thirdly, as the shock variable moves closer to a trigger point the hump in the term structure moves towards the lower end of the maturity range. An important corollary of this last feature is that the interest differential for any given level of the shock variable and for any given maturity is a function of the expected regime switch points. This effect is illustrated in Fig. 3 for t 51.0. Changing from ´ eU 5 2 ´ eL 52 to ´ eU 5 2 ´ eL 51.5 has the effect of pivoting the interest rate schedule. This linkage between expected trigger points and interest rates plays a vital role in the model.

4. The policymaker’s behaviour and equilibrium The objective of the policymaker is to choose the values of ´U and ´L so as to maximise W. The structure of the model however suggests that the equilibrium values of ´U and ´L are not obtained by simply maximising W with respect to ´U and ´L . The dependence of the policymaker’s objective function on interest rates and thus on private sector expectations immediately raises the issue of credibility. The policymaker’s choice of ´U and ´L will depend on the extent to which the policymaker can influence private sector expectations. In turn private sector expectations will only be influenced by the policymaker’s choice of ´U and ´L if private sector agents have good reason to believe that the policymaker will implement a regime switch at the announced values of ´U and ´L . In fact it turns out to be the case that optimal policy in this model is ‘‘inconsistent’’ in the sense discussed by Kydland and Prescott (1977). For a given state of the world (summarised by the value of ´) a policymaker who believes that he can influence expectations will choose a pair of trigger points. But when the state of the world changes he will find that the trigger points chosen are no longer optimal given the new value of ´. He will therefore be induced to choose a new pair. Clearly the initial choice of trigger points would only be credible if the policymaker was able to precommit to that choice. An attempt by a policymaker to precommit to such a policy is, in practice, almost never observed. We therefore choose not to consider precommitment in the context of this model. Instead we concentrate on the equilibrium that arises when policymakers make their choices while taking private sector expectations as given 17 . 17 Ozkan and Sutherland (1994), in the context of a simpler version of this model, use the term ‘‘state-inconsistency’’ to describe the policymaker’s incentive to change trigger points as the state variable changes. In turn, in that paper we used the term ‘‘state-consistent equilibrium’’ to describe the equilibrium without commitment. The difference between the precommitment solution and the state-consistent solution are extensively analysed in that paper.

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The analysis proceeds in two stages. In the first stage we take the private sector’s expectations, ´ eU and ´ eL , as given and derive the policymaker’s optimal choice of ´U and ´L . Obviously, in a rational expectations equilibrium private sector expectations should correspond to the policymaker’s choice. Therefore, in the second stage we will look for equilibria where ´ eU 5 ´U and ´ eL 5 ´L 18 . In each regime the value of the policymaker’s objective function can be expressed as a function of the current value of the shock variable. Thus we can write W5V(´). The solution for the value function in the floating rate regime is easily obtained given that in the floating rate regime it is optimal to set y50 and it is not optimal to re-enter the fixed rate system. The policymaker’s flow payoff in the floating rate system is zero forever so V(´)50 for all ´. Turn now to the policymaker’s value function in the fixed rate regime. Here the policymaker’s welfare is a function of the interest rate, which in turn is a function of the expected trigger points. We could show this explicitly by including ´ eU and ´ Le as arguments in the value function V(.). However in what follows, to avoid complicating the notation, we suppress these arguments. Nevertheless we emphasise again that the linkage between private sector expectations and the policymaker’s optimising problem is a vital feature of the model. Ito’s lemma implies that V(.) satisfies the following equation while in the fixed rate regime

s2 ] V 0(´) 5 d V(´) 1 [g i T (´) 1 b´ ] 2 2 Z 2

(14)

See Appendix C for an explicit derivation of Eq. (14). Eq. (14) is a secondorder differential equation so two boundary conditions are required in order to tie down a particular solution. There are two value matching conditions which ensure that at a trigger point V has the same value in both the fixed and flexible rate regimes (i.e. there is continuity of V across trigger points). These conditions are V(´U ) 5 0

(15)

V(´L ) 5 0

(16)

since welfare is zero in the floating rate regime. Two further boundary conditions are required to tie down the optimal values of ´U and ´L . These are provided by the following ‘‘smooth pasting’’ conditions

18 The main focus of interest in this paper is on rational expectations equilibria. We believe that the solution strategy we adopt provides a relatively simple and intuitive method for tying down a rational expectations equilibrium. It is important to stress, however, that our solution procedure is not intended to be a description of the out-of-equilibrium behaviour that allows the economy to move towards equilibrium.

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V 9(´U ) 5 0

(17)

V 9(´L ) 5 0

(18)

which can loosely be thought of as first order conditions. (See Dixit and Pindyck (1994) for a derivation of ‘‘smooth pasting’’ as an optimality condition in a regime switching problem.) In general it is not possible to obtain explicit solutions to the policymaker’s optimisation problem (because of the nonlinearity introduced by the dependence of i T on ´). It is, however, possible to use standard techniques to obtain numerical solutions 19 . We will confine attention to symmetric solutions where ´U 5 2 ´L . Henceforth we therefore only refer to ´U and the private sector expectation ´ eU . Fig. 4 plots values for the policymaker’s optimal choice of ´U against values of private sector expectations, ´ Ue , using the following parameter values: b 50.5, h 50.5, s 51.0, g 50.5, d 50.05, Z51.0, T51.0 20 . Schedule RR in Fig. 4 shows clearly the linkage between the policymaker’s choice of trigger points and private sector expectations. This plot can be thought of as a ‘‘best response function’’ for the policymaker 21 . (Notice that Fig. 4 only plots values of ´U which are below the 458 line, i.e. values of ´U which are less than or equal to ´ Ue . This is because the value of the interest rate is undefined outside the range of the expected trigger points. It is thus not possible to obtain an optimal trigger point outside that range.) A rational expectations equilibrium for this model is clearly given by a point where the best response function intersects the 458 line. In Fig. 4 this is given by ´¯ U . It is now possible to offer an interpretation of the ERM crisis in relation to this model. The demand shock, ´, can be interpreted as the level of the German interest rate. In 1992 rising German interest rates were translated into high interest rates in other ERM countries through uncovered interest parity. In turn this caused depressed demand. In terms of the model this is reflected in a deviation of y from its target level. In 1992 it was clear that financial markets anticipated that depressed output in the UK and other ERM countries could result in devaluations. This raised interest differentials and pushed non-German interest rates even higher. This effect is captured in the present model, which demonstrates that a rational 19 Technically Eqs. (14)–(18) define a ‘‘free-boundary value problem’’, i.e. the position of the trigger points is unknown and has to be determined as part of the solution process. We use a shooting technique to obtain solutions to Eq. (14) which satisfy the value matching conditions and a search algorithm to locate the trigger points which satisfy the smooth pasting conditions. 20 The structure of the model is intentionally made symmetric so considering symmetric solutions is natural. It is possible that asymmetric solutions exist, but we do not consider them in this paper. It is important to note that in generating Fig. 4 it is always the case that ´ eL 5 2 ´ eU . 21 We emphasise that the use of the term ‘‘best response function’’ is not intended to indicate anything about the out-of-equilibrium behaviour that allows the economy to move towards equilibrium.

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Fig. 4. The equilibrium trigger point.

expectations equilibrium exists where financial market expectations are fulfilled by an optimising policymaker 22 . The fact that a rational expectations equilibrium exists, where the fixed rate does not collapse immediately, is not obvious a priori. At first sight one might predict that expectations of a devaluation should result in an immediate reallocation of private sector portfolios and a catastrophic loss of reserves for the domestic monetary authority. The reason this does not happen is that, while the shock 22

Ozkan and Sutherland (1995) used a simpler version of this model to consider the impact of various policy measures designed to delay a currency crisis of the type represented by a breakdown at ´¯ U . The model used in that paper is simpler in that the policymaker is assumed to care about a composite interest rate which averages over all maturities. This allows a closed-form solution to be obtained for the interest differential and the policymaker’s value function. The model of this paper is more general in that it allows a solution to be obtained for the entire term structure of interest rates.

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variable is between the trigger points, the probability of a collapse is less than unity. Speculative flows can therefore be restrained by an appropriate interest differential. When the shock variable approaches a trigger point the probability of the collapse does approach unity but this again need not cause a problem for bonds with a maturity greater than zero because a finite interest differential can still be found to compensate for the expected devaluation. The only interest rate that becomes infinite at the time of collapse is the instantaneous interest rate. The key question addressed by this model is whether the policymaker is prepared to accept the interest rate behaviour implied by the expectations of collapse 23 . This is reflected in the weight placed on the interest rate and the maturity of interest rate that is included in the welfare function. Table 1 shows the effect on the equilibrium trigger point of varying the parameter g, which is effectively the weight placed on interest differentials in the welfare function. When g is zero the trigger point is 3.26. The higher the weight the lower the trigger point. Clearly, if there is a very high weight placed on the interest rate the trigger point will be very low. This implies that the expected survival time of the fixed rate system will be very low. Table 2 illustrates the effect on the equilibrium trigger point of varying T, the maturity of interest rate that enters the welfare function. The longer the maturity the higher is the trigger point. This reflects the fact that long maturity interest rates are much less sensitive to the expectation of a devaluation. On the other hand as T approaches zero the equilibrium trigger point approaches zero. In other words, if the policymaker cares about very short term interest rates the expected survival time of the fixed rate will be very short. A further important feature of the model,which is illustrated in Fig. 4, is that the Table 1 Equilibrium and g

g

´¯ U

0.00 0.25 0.50 1.00 2.00

3.260 2.428 1.813 1.156 0.690

23 In the description of the welfare function we chose to place emphasis on the link between interest rates, aggregate demand and welfare. There may be other reasons, unconnected to demand effects, for the policymaker to care about interest rates. For instance, in order to achieve the level of interest rates necessary to defend the currency, the policymaker must reduce the money supply. This can be achieved through reductions in domestic credit or through reductions in foreign exchange reserves. The former route may be impossible for the policymaker to implement while the latter route may be unpalatable. It can therefore be argued that a dislike for high interest rates is, to some extent, equivalent to a dislike for low levels of reserves. In this way it may be possible to recast this model in a form which gives some role to the level of reserves in generating a crisis.

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Table 2 Equilibrium and T T

´¯ U

0.10 0.25 0.50 1.00 2.00

0.316 0.723 1.240 1.813 2.304

optimal choice of ´U is increasing in ´ Ue . This is easily explained. As shown in Fig. 3, increasing ´ eU causes the interest rate to fall for every positive level of ´. This reduces the costs to the policymaker of staying in the fixed rate regime. The policymaker therefore delays a switch of regime. An upward sloping ‘‘best response function’’ creates the possibility of multiple crossing points and therefore multiple equilibria. However, for the particular set-up of the model under discussion here experimentation with a wide range of parameter values suggests that the equilibrium is unique. The next section discusses a modified version of the model where multiple equilibria do arise. This allows an alternative interpretation of the ERM crisis in terms of self-fulfilling attacks.

5. Multiple solutions Obstfeld (1994) emphasises the role of multiple equilibria and self-fulfilling attacks in the collapse of fixed exchange rate regimes. A very simple modification of the present model generates behaviour of this type. In this section 24 we introduce an explicit cost of switching regimes which is related to the size of the step-change in the exchange rate which is implemented at the time of a regime switch. The step-change in the exchange rate is linearly related to ´ so we assume that the regime switch causes a one-off welfare penalty for the policymaker of the form C 5 a´ 2

(19)

Thus the cost of switching regime rises as ´ deviates from zero. Implicit in this set-up is the notion that policymakers find large changes in the exchange rate (of either sign) much more unpleasant than small changes. This could be because large changes indicate major crises and generate a severe loss of public confidence. Or it 24 With an upward sloping best response function, all that is required for multiple equilibria is that the model be sufficiently nonlinear. In principle extra nonlinearities can be introduced in a number of different ways. In this section we concentrate on just one example.

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may be because large changes in the nominal exchange rate lead to large changes in nominal goods prices in the medium run. In terms of the formal solution to the model this modification is very easy to deal with. The form of the solution for the interest differential is unaffected. The differential equation that defines the value function is also unaffected. The only changes are in terms of the boundary conditions which must be imposed on the value function. The value matching condition given in Eq. (15) now becomes V(´U ) 5 2 a´ U2

(20)

i.e. the value of the welfare function at the time of switching must be equal to the cost of switching. The smooth pasting condition given in Eq. (16) becomes V 9(´U ) 5 2 2a´U

(21)

Corresponding value matching and smooth pasting conditions hold at the lower trigger point. Again solutions can only be obtained numerically. Fig. 5 plots an example of ‘‘the best response’’ function for the policymaker with the new structure, using the parameter values: b 50.5, h 50.5, s 51.0, g 50.5, d 50.05, Z50.1, T51.0, a 50.6. (Again notice that, for reasons explained previously, the best response function is only defined for values of ´U less than or equal to ´ Ue .) It is apparent that there are now two crossing points—one at 0.863 and a second at 4.223. The economic interpretation of these two equilibria naturally introduces the notion of self-fulfilling expectations. If the expected trigger point is the higher of the two equilibria, interest rates are relatively low. This reduces the costs of staying in the fixed rate regime and encourages the policymaker to switch at the higher trigger point. In contrast, if the expected trigger point is the lower of the two equilibria then interest rates are higher and the policymaker is induced to switch sooner. Obstfeld (1994) argues (in the context of a static model) that multiple equilibria of this type are potentially important in interpreting the ERM crisis. A selffulfilling crisis can be thought of as a switch in financial market expectations from the higher equilibrium to the lower. Suppose, in the context of the current model, initially financial markets expect a switch at the higher of the two equilibria and the current value of the shock variable is between the two equilibria. A switch in expectations to the lower equilibrium induces a jump in interest rates which causes an immediate exit from the fixed rate regime. Note that even if there is no switch in expectations both of the equilibria in Fig. 5 will generate a collapse of the fixed rate sooner or later. There is therefore a sense in which both fundamentals and expectations are still important. But in addition to these two equilibria it is possible to deduce that there is a further equilibrium in this model where fundamentals do not cause a collapse. Suppose the private sector believe that a collapse will never take place. In other words suppose the expected trigger point is infinite. In this case the interest differential

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Fig. 5. Multiple equilibria.

will be zero for all levels of ´, and y will simply be given by 2 b´. Now assume the policymaker does in fact choose to set the trigger point at infinity. It is simple to show that this will be optimal. The policymaker’s value function in this case will be given by the following expression 25 (Z 2 b 2 ´ 2 ) V(´) 5 ]]] d

(22)

If a . b 2 /d (as is the case for the parameters used to generate Fig. 5) it is clear 25

Flow welfare is given by Z2 b 2 ´ 2 . Eq. (22) is simply the discounted value of flow welfare over the infinite future.

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that V(´). 2C for all values of ´. It is therefore never optimal to leave the fixed rate. Thus a rational expectations equilibrium exists in which a regime switch never takes place. If the initial state is one where expectations are consistent with this third type of equilibrium, and a change in expectations takes place to one of the other equilibria, then the resulting crash can be viewed as having arisen purely from self-fulfilling expectations.

6. Conclusion The currency crisis model presented in this paper emphasises the policy optimising problem facing the policymaker in choosing an exchange rate regime. The model also allows explicit attention to be paid to the role of high interest rates in the centre country in generating (or prolonging) a negative demand shock in the domestic economy. The utility loss to the policymaker generated by this shock gives rise to expectations in financial markets of an abandonment of the fixed exchange rate, which in turn further increase domestic nominal interest rates and therefore hasten the collapse. The preferences of the policymaker, and private sector perceptions of those preferences, play a crucial role in determining the timing of the regime switch. In the introduction we suggested that the existing literature was based around two models: the unsustainable-policy model of Krugman (1979) and the selffulfilling attack model of Obstfeld (1986). Our model differs from both these models in a number of important respects, particularly in the explicit treatment of the policymaker’s optimising problem, but it also shares a number of features of both approaches. One form of our model shares the property of the Krugman model that the breakdown of the fixed rate regime is triggered by a fundamental variable (in Krugman’s case it is reserves, in our case it is the demand shock variable). However, it would be wrong to suggest that in our model the collapse is the inevitable consequence of an unsustainable policy stance. There is no fundamental reason in our model why the policymaker could not sustain the fixed rate indefinitely. In another version of our model presented in Section 5 we show that it is possible to generate multiple equilibria and a self-fulfilling attacks in the spirit of Obstfeld (1986), (1994). In this case, if financial markets believe that the demand shock must be very high before a collapse takes place, then interest rates will be low and the policymaker will be happy to stay in the fixed rate system. But if financial markets believe that a regime switch will take place at a low level of the demand shock then interest rates will be high and the policymaker will be induced to switch. In this version of the model an equilibrium exists where indefinite membership of the fixed rate regime is optimal for all levels of the shock variable. A shift away from this equilibrium can therefore be interpreted as a purely expectations induced crisis.

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In the introduction we argued that the basic structure of our model was very relevant to the crises that hit the European Exchange Rate Mechanism in 1992 and 1993. Those crises were characterised by high interest rates in the centre country (which acted as a negative demand shock) and a perception amongst private sector speculators that, sooner or later, various members of the ERM would find it advantageous to leave the ERM and devalue. These expectations added to the pressure on interest rates and appeared to hasten the eventual near collapse of the system. Our model provides a framework for considering these issues. Furthermore, our model can handle both fundamentals driven crises (as may be relevant for the cases of Italy and the UK in 1992) and expectations driven crises (as may be relevant for the case of France in 1993).

Acknowledgements We gratefully acknowledge many valuable comments on earlier versions of this paper from: Soren Bo Nielsen, Maurice Obstfeld, Paolo Pesenti, Niels Thygesen and Eric van Wincoop, two anonymous referees and participants at seminars at Essex University, Copenhagen Business School, Aarhus University, Strathclyde University, the Bank of England, IGIER, the European University Institute, Bordeaux University and Southampton University.

Appendix A

Derivation of Eq. (7) Assume the current time is time t and consider the expected value of the spot exchange rate at time t9, which is some fixed future date. Formally Et [s(t9)] 5 h(´, t9 2 t)

(A1)

Applying Ito’s lemma to the function h(´, t92t) yields the following

s2 Et [dh(´, t9 2 t)] 5 2 ht (e, t9 2 t) 1 ] h´´ (´, t9 2 t) 2

(A2)

Svensson (1991) shows that, by the law of iterated expectations, the expected change in h(´, t92t) over a small interval of time, dt, must be zero, i.e. Et [dh(´, t92t)]50. So 2

s ht (´, t9 2 t) 5 ] h´´ (´, t9 2 t) 2

(A3)

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361

Replacing t92t with t yields Eq. (7).

Appendix B

Derivation of Eq. (11) To derive the solution to Eq. (7) it is useful to perform a change of variables. Define x5 ´ 2 ´ eL and R5 ´ eU 2 ´ eL and thus define a function (R 2 x)b xb u(x, t ) 5 h(x 1 ´ eL , t ) 2 ]]] ´ Le 2 ] ´ Ue . Rh Rh

(A4)

It is straightforward to check that this new function satisfies the following partial differential equation

s2 ut (x, t ) 5 ] u xx (x, t ) 2

(A5)

with boundary conditions (R 2 x)b xb u(x, 0) 5 2 ]]] ´ eL 2 ] ´ eU 0 # x # R Rh Rh

(A6)

u(0, t ) 5 0 0 # t # `

(A7)

u(R, t ) 5 0 0 # t # `

(A8)

and

Guenther and Lee (1988) (pp. 144–5) show that the solution to this equation is given as follows

s n p t np x O c expS 2 ]]] D D sinS]] R 2R `

u(x, t ) 5

2 2

n

2

(A9)

2

n51

where R

2 cn 5 2 ] R

(R 2 x)b xb np x E S]]] ´ 1 ] ´ D sinS]]D dx Rh Rh R e L

e u

(A10)

0

Reversing the change of variable yields

S

D S

` np (´ 2 ´ eL ) b s 2 n 2 p 2t ]]]] h(´, t ) 5 ] ´ 1 cn exp 2 ]]]] sin h (´ eU 2 ´ eL ) 2(´ Ue 2 ´ Le )2 n 51

O

which is Eq. (11) in the main text, and

D

(A11)

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F.G. Ozkan, A. Sutherland / Journal of International Economics 44 (1998) 339 – 364 ´ eU

2b cn 5 2 ]]]] h(´ eU 2 ´ eL )

np (´ 2 ´ ) E ´ sinS]]]] d´ (´ 2 ´ ) D e U

´ eL

e L e L

(A12)

which, after carrying out the integration, yields Eq. (12) in the main text.

Appendix C

Derivation of Eq. (14) The definition of welfare (Eq. (1)) can be rewritten in the following form V(´) 5 W(t) t 1Dt

5 Et

E [Z 2 y (t)]e 2

`

2d (t 2t )

dt 1 e

2dD t

t

Et

E [Z 2 y (t)]e 2

2d (t 2t 2 D t )

dt

t 1Dt

(A13) where the integral has been split into two. The first integral is over the short interval of time Dt. The second integral is over the remainder of the infinite horizon. Taking the limit of Eq. (13) as Dt tends to d t (an infinitesimal period of time) yields V(´) 5 (Z 2 y 2 )dt 1 (1 2 d dt)Et [V(´) 1 dV ]

(A14)

where dV is the change in V over time interval dt. Rearranging yields Et [dV ] 5 (2Z 1 y 2 )dt 1 d V(´)dt

(A15)

(where the term in dtdV has been cancelled because it disappears in the limit). Applying Ito’s lemma to the function V(.) also yields an expression for Et [dV ] as follows

s2 Et [dV ] 5 ] V 0(´)dt 2

(A16)

As noted in the main text, V is implicitly also a function of the expected trigger points. However, throughout the analysis of the policymaker’s optimising problem it is assumed that the policymaker takes the expected trigger points as given. It is therefore not necessary to take account of the implicit arguments ´ eU and ´ eL in applying Ito’s lemma. Equating (A15) and (A16) yields the following

s2 ] V 0(´) 5 d V(´) 1 y 2 2 Z 2 which yields Eq. (14) in the main text after substituting for y.

(A17)

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363

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