Studies in the Theory of Balance~of~Payments Crises Studies in the Theory of Balance ... of· Payments Crises

By ALPO WILLMAN Lic.Soc.Sc.

Doctoral dissertation to be presented, by permission of the Facqlty of Social Sciences of the University of Helsinki, for public examination in Auditorlum III, Porthania, Hallituskatu 11-13, on July 3, 1992, at noon.

Helsinki 1992 ALPO WILLMAN

Studies in the Theory of Balance ... of... Payments Crises

SUOMEN PANKKI Helsinki 1992 B:46 ALPO WILLMAN

Studies in the Theory of Balance .. of.. Payments Crises

ISBN 951 .. 686 .. 316 .. 7 ISSN 0357 .. 4776 Bank of Finland Publications, Series B:46

OY TRIO .. OFFSET AB Helsinki 1992 Preface

This thesis consists of a summary and five previously published artic1es on the theory of balance-of-payments crises: Speculative Attacks on the Currency with Uncertain Monetary Policy Reactions, Economics Letters 25 (1987), 75-78. Balance-of-Payments Crises ånd Monetary Policy Reactions in a Model with Imperfect Substitutability between Domestic and Foreign Bonds, Economics Letters 26 (1988), 77-81. The Collapse of the Fixed Exchange Rate Regime with Sticky Wages and Imperfect Substitutability between Domestic and Foreign Bonds, European Economic Review 32 (1988), 1817-1838. Devaluation Expectations and Speculative Attacks on the Currency, The Scandinavian Joumal of Economics 91 (1989), 97-116. Why there is a Lower Bound on the Central Bank's Foreign Reserves, Finnish Economic Papers 4 (1991), 113-129. 1 started this study in the Research Department of the Bank of Finland in 1985-1986 while 1 was on temporary release from my normal duties in the department. Although aIot of work was done in later years, these artic1es would not have been written without this opportunity. 1 benefited immensely from the friendly and encouraging atmosphere at the Research Department and later at the Economics Department. My greatest debt of gratitude is to Juha Tarkka. Innumerable discussions with him helped me to become experienced in the present theme, and the importance of his inspiring comments in writing the four first artic1es, in particular, cannot be overstated. My official examiners were Jouko Paunio and Erkki Koskela, the latter of whom, besides commenting on the summary, made useful comments on the fifth artic1e. 1 am grateful to Malcolm Waters for checking and improving the language, Päivi Lindqvist and Tuula Naskali for typing and word processing, Sampo Alhonsuo and Vuokko Varis for the graphics, Piljo Föhr-Tolvanen for the final word processing and Malja Hirvensalo-Niini for organizing and supervising the printing work. The assistance of the Bank of Finland library is also highl y appreciated. Finally, 1 wish to thank my wife Lea and my children Eero, Esko and Timo for their understanding and patience during my research work.

Helsinki, April 1992

Alpo Willman

5 Contents

Page

Preface 5

1 Introduction 9

2 A Survey of the Literature on Balance-of-Payments Crises 11

2.1 A balance-of-payments crisis in the F-G model 11

2.2 Extensions of the analysis and further results 13 2.2.1 Balance-of-payments crises in intertemporal optimization models 15 2.2.2 The minimum level of net foreign reserves 17 2.2.3 Balance-of-payments crises and devaluation 19 2.2.4 Imperfect foresight and balance-of-payments crises 20 2.2.4.1 Stochastic models of balance-of- payments crises 20 2.2.4.2 Uncertainty in deterministic models of balance-of-payments crises 22 2.2.5 Stabilization policies and balance-of- payments crises 25 2.2.6 Capital controls, imperfect asset substitutability and balance-of-payments crises 27 2.2.7 Real output, prices and wages and balance-of- payments crises 28 2.2.8 Other extensions of the analysis 30

3 Sumrnaries of the Papers 31

3.1 Why there is a lower bound on the central bank's foreign reserves 32

3.2 Speculative attacks on the currency with uncertain monetary policy reactions 35

7 3.3 Balance-of-payments crises and monetary policy reactions in a model with imperfect substitutability between domestic and foreign bonds 37

3.4 Devaluation expectations and speculative attacks on the currency 39

3.5 The collapse of the fixed exchange rate regime with sticky wages and imperfect substitutability between domestic and foreign bonds 42

Notes 45

References 47

Appendix 52

8 1 Introduction

More or less recurrent balance-of-payments crises were an essential feature of the international monetary system during the era of the historical gold standard as well as during the fixed exchange rate period of the Bretton Woods agreement. In the post-Bretton Woods period, the situation has remained the same in this respect for currencies pegged to some "hard" currency or a basket of currencies. A balance-of-payments crisis can be defined as a situation where confidence in the fixed parity has disappeared, leading to a massive outflow of reserves from the central bank. If, in this situation, the central bank's opportunities or willingness to borrow abroad are approaching a limit, the onIy way it can be attempted to restore confidence is by a drastic tightening of monetary policy. If this does not succeed, the exchange rate must be either devalued or allowed to float freely. In the literature, crises and the speculative attacks associated with them are popularly regarded as the product of either irrationality or unfounded expectations about future prices; crises are unforeseeable and impossible to capture by a formal model. As regards financial crises in a broader sense this view is stilI true.1 Recently, however, there has been progress in this field, the roots of which are to be found in the literature on non-renewable resources.2 In an application of the Hotelling model to the gold market, Salant and Henderson (1978) proved that specu­ lative attacks must occur and are foreseeable even when agents be­ have optimally and have rational expectations about future prices; psychological factors are not needed to explain their occurrence. In their model, the government attempts to peg the price of gold or to defend a price ceiling by managing a stockpile. As gold is a non­ renewable and continuously depleting resource, the competitive equilibrium contains a speculative attack, Le. the entire government stock is suddenIy purchased by previously inactive speculators. The increase in private stocks is justified ex post by the increased yield on hoI ding stocks, for when price stabilization policy breaks down the price of the resource begins to rise, providing a capital gain that makes the holding of stocks more attractive. Salant (1983) extended this analysis to commodity markets - subject to supply uncertainty - and showed that the government's attempts to stabilize the prices of commodities by the use of buffer stocks also result in speculative attacks if speculators are present or, if speculators are absent, in upward price jumps. Flood and

1 See notes on page 45.

9 Garber (1984a) introduced the ideas of Saland and Henderson into the system of the historical gold standard. They showed that only in the presence of very specific, coincident monetary actions is the permanence of the gold standard guaranteed. Otherwise it, or any commodity-money scheme, is vulnerable to collapse. There is also an obvious analogy between the government's attempts to peg the price of non-renewable resources and the use of foreign reserves to peg an exchange rate. Krugman (1979) was the first to discover this analogy and to apply the speculative attack model to the foreign exchange market. By preserving essential elements of Krugman's non-linear analysis, Flood and Garber (1984b) presented a linear model (F-G model) in which the analytical solution was readily derived? Thereafter the F-G mode! has been used very widely in the literature on balance-of-payments crises. This is one reason why 1 start this overview by presenting the continuous-time version of the F-G model. Another reason is that most of the other contributions can be presented by reference to this model. In chapter 2, 1 review the literature on balance-of-payments crises. The model and basic results are presented in section 2.1, and extensions of the analysis and further results are summarized in section 2.2. The four artic1es making up this thesis are summarized in chapter 3.

10 2 A Survey of the Literature on Balance-of-Payments Crises

2.1 A balance-of-payments crisis in the F -G model

The F-G model is a single-good monetarist model of a small open economy with purchasing power parity. As prices are perfectly flexible domestic output is fixed at its full-employment level. There are no capital controls and investors are risk-neutral, implying uncovered interest~parity. Domestic money, domestic bonds and foreign bonds doIDinate foreign money, which_yields no mOlletru.:~llervic~s to domesticresidents; there­ fore, privatedomestic -resldents hold no foreign money. A special feature of the F-G model is that the structural form of the model is specified so that the dynamics of the model is determined by a linear differential equation. The structural equations are as follows:

M(t)/P(t) = et - ~ i(t); et and ~ > 0 (1)

M(t) = R(t) + D(t) (2)

D(t) = fl > 0 (3)

P(t) = P * (t)S(t) (4)

i(t) = i • (t) + Et[S(t)/S(t)] (5) where M(t), P(t), and i(t) are the domestic money stock, domestic price level and domestic interest rate, respectively. R(t) represents the official foreign reserves held by the central bank and D(t) domestic credit. S(t) is the spot exchange rate, i.e. the domestic money price of foreign money. Et denotes the expectation operator conditional on information available at time t. An asterisk (*) next to a variable indicates foreign, while a dot over a variable (.) indicates the time derivative. Both the foreign price level p' and interest rate i* are assumed to be constants. Equation (1) reflects the condition for money market equilibrium, the right-hand side representing the demand for real balances. As the goods market is in continuous equilibrium, real income is constant and inc1uded in the parameter et. Equation (2) expresses the money supply as the sum

11 of foreign reserves and domestic credit. Equation (3) states that domestic credit always grows at a positive, constant rate. Purchasing power parity and uncovered interest parity are imposed by equations (4) 'and (5), respective1y. In the rest of this section perfect foresight is assumed, so that the expectation operator Et in equation (5) can Qe ignored. Assume that initially the exchange rate is fixed at Sand everybody knows that it will remain at that 1eve1 until foreign reserves have been dep1eted to a 10wer bound of reserves. Without any 10ss of generality, the 10wer bound of reserves can be fixed at zero. It is also known that at the moment foreign reserves have been dep1eted to zero the central bank permanentIy withdraws ·from t.h.e foreign exchange market and allows the exchange rate to fioat. As in the fixed exchange ra,1yregime, $. (!l~"9)!~~~~tos~IsQtmoney balances M(tfremaLtlscg»stant and, on the basis of equations(1);~'(4) ~~~~'-o~ _ _ _ -'-'-.-::;_=_ ~_-.=:c_" ..-..-.,.." - and (5), iC equaIs bS, where b = (a - ~i * )P * > O. Now, equations (2)-(3) imp1y that R(t) = -]) :=: -fl. Hence, it is certain that at some point of time foreigri-reserves -will -hit the zero leve1, impI ying a regime shift from the fixed to the fioating exchange rate regime. The next step is to solve the mode1 (1)-(5) in the fioating exchange rate regime. In the literature with perfect foresight, the standard assumption is that the solution depends on1y on market fundamentals. Under this assumption and with R(t) = 0, the exchange rate is determined by the following relation: --

S(t) :=: (3f1,/b 2) + [D(O) + flt]/b (6)

where a =~P * > O.

As long as the fixed exchange rate regime prevails, equation (6) defines the .~ha~2.~_.c:~&ch~QK~~@:.te, i.e. the leve1 of the exchange rate which wouIcrprevail if, at any moment, the central bank exhausted its foreign reserve stock. After the collapse of the fixed exchange rate regime, equation (6) determines the actual exchange rate. A condition that connects the fixed exchange rate regime to the post­ collapse rate regime is that the exchange rate cannot jump discrete1y when the regime shift occurs. This is the continuity condition, which the perfect foresight solution must satisfy in order to be unique.4 It is easy to give an economic interpretation to this condition. If we den ote the collapse til].e by T, the discrete depreciation of the domestic currency, Le. S(T) > S, wou1d provide those specu1ators who attack foreign reserves at T with profits which would accrue at an infinite rate. Speculators competing to capture these profits have an incentive to pre-empt their competitors by purchasing all the reserves an instant

12 before T. This would imply that the regime shift would occur before T, which is in contradiction with the assumption that T is the collapse time. Hence, there cannot be discrete depreciation of the exchange rate at T. Discrete appreciation is also exc1uded. Otherwise it would be profitable for speculators to shift the composition of their portfolios from foreign to domestic bonds at T. This would increase the foreign reserves and the fixed excJlange rate would survive. Hence, the arbitrage condition is that S(T) = S. By applying this arbitrage con..siition and the result that in the fixed exchange rate regime M(t) = bS, the collapse time for the fixed exchange rate regime can be straightforWardly solved from equation (6). The solution is T =R(O)/,u -~. a/b. -There niusfbe a speculative attack on the currency at T, because without an attack foreign reserves would be depleted to zero at R(O)/,u, which is later than T. What causes this attack? Although there is no jump in the level of the exchange rate at T, there is a jump in the rate of depreciation of the currency. Until T, the depreciation rate is zero, thereafter ,u/b, as can be easily seen from equation (6). Uncovered interest parity implies that the doIIlestic interest !at(;IT1l!~L~~r, resulting in _

2.2 Extensions of the analysis and further results

The above analysis was able to rationalize perhaps the most prominent feature associated with real world balance-of-payments crises, Le. the sudden speculative stock shift from domestic money to foreign assets which results in the collapse of the fixed exchange rate. There is nothing irrational in this speculative attack, rather it is a prerequisite for the equilibrium of the model. However, owing to its simplicity, the F-G model is able to analyze only financial aspects of balance-of-payments crises. The only reason

13 why a balance-of-payments crisis occurs is because credit policy is too expansionary; the model ignores all real events which are often observed to occur simultaneously with the depletion of foreign reserves towards zero. Moreover, in the preceding analysis, the perfect foresight assumption abstracted many features often observable in real world balance-of-payments crises. For instance, the following list of stylized features associated with many real world balance-of-payments crises can be given: 1. When the fixed exchange rate collapses the regime shift is not typically from the fixed to the permanently floating exchange rate regime. Most often the currency is devalued and fixed at a new higher level or, before fixing it, the exchange rate is a110wed to float only temporarily.5 2. Typically, the collapse of the fixed exchange rate is preceded by a longer period during which the interest differential between the domestic and the foreign interest rate widens. This period' is assodated with an outflow,of capital which exceeds the domestic credit expansion. 3. Quite often the final speculative attack, which triggers the collapse of the fixed exchange rate, is preceded by periods of speculative capital outflows and, at least partial, refluxes of foreign reserves. 4. Balance-of-payments crises are often preceded by large trade and current account deficits, a fall in the growth of production and the appreciation of the real exchange rate. 5. The collapse of the fixed exchange rate is typically followed by a gradual depreciation of the real exchange rate and at least a temporary expansion in real output. 6. Governments continue announcing their intention to defend the fixed exchange rate right up until the time the exchange rate collapses. In addition, besides being unable to explain the stylized facts mentioned above, both the F-G model and Krugman's (1979) original model can be criticized for their lack of choice-theoretic foundations. Hence, it is natural that the analysis presented in the previous section has been extended in many directions: Obstfeld (1986a), Calvo (1987), Claessens (1989) and Willman (1991) have studied balance-of-payments crises in frameworks with choice-theoretic foundations; the legitimacy of the postulated assumption that there exists a minimum level of foreign reserves below which they cannot be depleted has been examined by Obstfeld (1986a), Buiter (1986 and 1987) and Willman (1991); Obstfeld (1984) studied the case of devaluations; uncertainty, which can appear in many forms, was introduced into the analysis by e.g. Flood and Garber (1984b), Obstfeld (1986b), Willman (1987, 1988a and 1989), Cumby and van Wijnbergen (1989), Penati and Pennacchi (1989), Otani (1989) and Claessens (1991); Wyplosz (1986), Park and Sachs (1987), Bachetta

14 (1990) and Agenor (1990) took into account the possibility of capital controls in the context of speculative attacks; Buiter (1987) studied the effects of foreign borrowing by the government and Frenkel and Klein (1989) the effects of stabilizing fiscal policy measures on the viability of the fixed exchange rate; van Wijnbergen (1991) analyzed stabilization policies which, through endogenous exchange-rate regime switches, ultimately restore stability; in addition to a selling attack, Grilli (1986) and Buiter (1989) took into account the possibility of a buying attack; the highly restrictive assumptions of the F-G model have been relaxed in many ways (Connolly and Taylor 1984, Flood and Hodrick 1986, Willman 1988b and 1989 and Blackburn 1988 and Agenor, Bhandari and Flood 1991); and there are also some empirical applications (Connolly 1986, Blanco and Garber 1986, Garber and Grilli 1986, Cumby and van Wijnbergen 1989 and Goldberg 1990). In the following 1 briefly discuss the main results of these studies.

2.2.1 Balance-of-payments crises in intertemporal optimization models

Both choice-theoretic models and the F-G model are full-employment, market-c1earing models. The advantage of choice-theoretic models based on intertemporal maxirnization over postulated models like the F-G model, however, is that their internai consistency is guaranteed and they capture feed-back mechanisms which may be lacking in postulated models. Choice-theoretic models do not change the basic result of the F-G model that an exchange rate regime shift from the fixed to the floating exchange rate regime is associated with a speculative attack on the· currency. However, some differences concerning the basic nature of speculative attacks may exist. The way in which the F-G model is postulated implies that balance-of-payments crises are pure financial market phenomena without any real economy effects; domestic output, the current account and, hence, domestic demand remain unaffected by the endogenous exchange-rate regime shift. Compared to the results obtained in choice-theoretic frameworks, these results accord with those obtained by Obstfeld (1986a) but strongly contradict those obtained by Calvo (1987). In Calvo's model private consumption was at a high leve1 and the current account in deficit in the pre-attack fixed exchange-rate regime. In the post-attack floating exchange rate regime, the level of consumption decreased and the current account moved into surplus. The stock shift :fJ:om d(H],estic J112!!llJ9 foreign assets in the context .2.Lt.he speculati"e attack !~~sulfed .from the ~~=-:-"-__ ' _ -="-~_.-.~- --_'="=:_-. _." '.'.' '.• _ -_~·o_7;:"'C---_·---::-"'-·O;-.;----~-;c;;:--""'"=--"""=",.,"';:..:,"''',.....

15 downward adjustment of thy representative household's consumptipn. Hence,'~iti" Calvo's model pure real- economic phenomena underlie speculative attacks. Besides contradicting the results implied by both the F-G model and Obstfeld's (1986a) analysis, thi~Je§.ult is agains(,the conventional wisdom. tll

16 Clover constraint with Baumol-Tobin transaction technology.6 Hence, according to this view real economic aspects would be associated with balance-of-payments crises. A choice-theoretic framework is also used in many other studies on balance-of-payments crises (see e.g. Park and Sacks 1987, Penati and Pennachi 1989, Frenkel and Klein 1989, Bachetta 1990, Claessens 1991, van Wijnbergen 1991 and Willman 1991). With the exception of Frenkel and Klein (1989) and Willman (1991), these studies empI oy essentially the same frameworks as described above. Frenkel and Klein (1989) extended the model to include not only optimizing households but also optimizing firms with profit taxes and slow adjustment of the capitai stock to its optimal level. Willman (1991) extended the transaction concept, which determines the demand for cash balances, to include financial market transactions. The main contributions of these studies are summarized in the following sections.

2.2.2 The minimum level of net foreign reserves

A crucial and at the same time problematic assumption in the literature on balance-of-payments crises is that there exists a minimum level of foreign reserves below which they are not allowed to be depleted. Although this seems to be true in the real world, there are difficulties in accepting this assumption as a given fact, because it is possible for a central bank to create new gross reserves by bOJrowing in the world capital market and, hence, negative net reserve positions are feasible. Accordingly, good theoretical foundations for the minimum acceptable level of official net fOIeign reserves would be desirable. The full rationalization of endogenous exchange-rate regime shifts would require that the minimum level of fOIeign reserves was also determined by the model. Unfortunately, typical intertemporal optimization models used in analyzing balance-of-payments crises have been unable to define the lower bound fOI official net foreign reserves. On the contrary, Obstfeld (1986a) showed that, in an idealized world with non-distortionary taxation, a central bank's net foreign reserves can become indefinitely negative without violating the solvency of the consolidated government sector, if the growth rate of domestic credit is below the world interest rate.!g._~~~ .. Ee.al~~~l.~".. ~~~~~ver,. taxation is distQrti.on~lY~.~9:L~h~nceL the.~ violation . of the.My.(;r~Q!.~e_ctor 's polvenSY~l!~~lll(3d a g()od expll!gatiQl1jyhy. Jlt

17 reserves is a permanent deficit in the primary budget deficit of the central government. Buiter (1986 and 1987) showed that if domestic credit expansion is used to monetize the central government's non-monetary debt, the central bank's net foreign reserves can be dep1eted towards minus infinity without any solvency prob1ems. Willman (1991) showed that the same is true if the credit expansion is injected into the economy via the private sector, i.e. the central bank buys bonds issued by the private sector. In these cases government sector solvency prob1ems do not arise if domestic bonds carry at 1east the same interest rate as net foreign reserves (i.e. the world interest rate); interest costs resulting from foreign borrowing by the central bank can be covered by interest income from domestic credit expansion without any pressures on taxation. Hence, exhaustion of the central bank's net foreign reserves shou1d result in a ba1ance-of-payments crisis only in the context of government sector solvency crises. This is hard1y true of many of the ba1ance-of­ payments crises experienced in the rea1 wor1d. On the contrary, there are many rea1 world examp1es where monetization of government debt has resulted in balance-of-payments crises (see e.g. the examp1es presented by Penati and Pennachi 1989). In addition, even in cases where a ba1ance-of-payments crisis is a manifestation of a government solvency crisis it is stilI an umeso1ved question as to why it shou1d be some leve1 of net foreign reserves c10se to zero which triggers balance-of-payments crises. Why is not possib1e for a central bank to become indebted like the government and the private sector? A solution to these prob1ems is provided by Willman (1991). He introduced into the mode1 by Obstfe1d (1986a) a cash-in-advance constraint on financial market transactions which causes additiona1 debt service cost for borrowers. In this framework the violation of sectoral solvency constraints is no 10nger dependent on the way in which domestic credit expansion is injected into the economy; with non­ distortionary taxation either the government or private sector's intertempora1 budget constraint is vio1ated, if domestic credit expansion exceeds a critica1 magnitude somewhat be10w the world interest rate, and hence, net foreign reserves cannot be dep1eted bound1ess1y. More important1y, Willman (1991) showed that the dep1etion of the central bank's net foreign reserves be10w zero, even temporarily, causes a we1fare 10ss. :tI~.gce, fOI ~ we1far~:maxi!p:iz;!p.g go,:ernmep.t !his_i~ a 10wer _bound on th~ ..(;~l1:tra1 bank's net f.<::>reig}!__ .reserves below w:~ich .!..h.~ ~~!ves~~;g~ not ~lowedto be depleted, eY~lltllollg!l, frQ111Jhe .llQinLQf _view <::>J ~

18 2.2.3 Balance-of-payments crises and devaluation

In the analysis of section 2.1 it was assumed that when a fixed exchange rate regime collapses the central bank abandons it for ever. Often, however, a central bank faced with a speculative attack devalues the currency, i.e. it withdraws from the foreign exchange market temporarily and repegs the exchange rate at a higher level after a transitional period of floating. Obstfeld (1984) studied this case. As agents can never expect a discrete jump in the level of the exchange rate, the asset market equilibrium requires that the exchange rate starts the transitory float from its pre-devaluation level and ends the float at the post-devaluation level. Now the general solution of the differential equation determined by equations (1)-(6) in section 2.1 is needed to define the path of the exchange rate during the transitory float. The post-devaluation level of the exchange rate offers the terminal condition with the help of which the arbitrary constant multiplying time exponential in positive root can be solved.7 As before, the pre-devaluation level of the exchange rate gives the condition from which the timing of the speculative attack can be calculated. Obstfeld (1984) showed in his paper that the date of the crisis and the size of the attack are well-defined functions of both the magnitude of the expected devaluation and the length of the transitional period of floating precedingthe establishment of a new exchange parity; the greater is the devaluation and the shorter is the transitional float the earlier a speculative attack occurs. If the transitional float is sufficiently brief, a speculative attack on the currency will occur as soon as the market realizes that the current exchange rate cannot be enforced indefinitely. Hence, in continuous time models with free international capital mobility, discrete devaluations are prec1uded. This is due to the fact that a discrete devaluation in a continuous time model could offer abnormal profit opportunities, i.e. profits accruing at an infinite rate, at the instant of devaluation. In this respect discrete time models deviate from continuous time. In discrete time, there are never infinite profit opportunities between periods and devaluing the currency between the two subsequent periods is possible. Viewed from this perspective, a devaluation with a short transitory period of free floating in a continuous time model seems to be the correct counterpart to a devaluation in a discrete time model. This interpretation also seems to cover real world devaluations which are typically implemented either so that the exchange rate is allowed to float freely for a while before repegging it or so that the foreign exchange

19 market is c10sed fOI a short period of time before the new exchange rate is announced. However, it is hard to believe that devaluations would be impos­ sible in the real world without at least a short transition period of free floating. A solution to this problem is offered by the fact that real world devaluations are never implemented under perfect foresight. Por instance, speculatOIs do not know in advance with certainty the level of foreign reserves which triggers the devaluation. Therefore the exact timing of the devaluation is unknown and, as Otani (1989) showed in a continuous time framework, the expected change in the exchange rate is never infinite. This makes discrete devaluations possible in continuous time models as well.

2.2.4 Imperfect foresight and balance-of-payments crises

Imperfect foresight can be incorporated into the models of balance of payments in many different ways. One can abandon the deterministic nature of the model and assume that the fundamentals of the model may be generated by some stochastic process. This is what Flood and Garber (1984b), Obstfeld (1986b), Grilli (1986) and Buiter (1989), among others, have done. Or, without necessarily abandoning the deterministic framework one can assume that there may be uncertainty about the reactions of the monetary authority (Willman 1987 and 1988a) OI about the level of foreign reserves that will trigger the abandonment of the fixed exchange rate (Krugman 1979, Cumby and van Wijnbergen 1989, Otani 1989 and Willman 1989).

2.2.4.1 Stochastic models of balance-of-payments crises

Flood and Garber (1984b) transformed the continuous time model of section 2.1 into a discrete time model and introduced a random disturbance into the credit expansion relation. Because of the §j:g~hg~tic nature of the er~QiL~xpansion, agents do not know with certainty when tneshadowexchange rate will exceed the fixed rate and, hence, t~~k perfect foresight about t1!,e::j:~Qllapse,Jime; instead of being certain the collapse'"time becomes a random variable. This, however, causes the model to track a generally encountered phenomenon; a ~eak. currency's forward, exchange rat~ may~ exceed the fixed rate, for' a long period of !illle, wJ1ich in t.he iiterat~~~ is!mown as the "peso probleniii:g'~~,-~'" Under certain circumstances, specuTätive attacks canbeself-fulfilling rather than inevitable results of unsustainable macroeconomic policies.

20 This was shown by Obstfeld (1986b). In the example he demonstrated, it was assumed that the public has the following expectations: if a collapse occurs at any time T, the central bank allows the exchange rate to float forever and switches the stochastic but non-inflationary domestic credit rule to an inflationary rule. With some probability, which is extrinsic, ,Pfivate agents believe that a run on the bank's reserves will take place, (i.e. they believe that everybody joins in the attack) ~f and only if, the size of the shock in the supply of domestic credit exceeds a certain minimum size. With probability one less the probability of a run, they believe that no run will take place. Under these circumstances there would be no "natural collapse" of the fixed rate, i.e. no collapse without a speculative attack. However, the public believes that, if the fixed exchange rate regime were to collapse, the exchange rate would start depreciating and the nominal interest rate would rise from its fixed exchange rate regime level since the central bank would change its non-inflationary domestic credit policy in favour of an expansionary one. It is now possible that if private agents expect a run to take place in a certain period, it will be profitable for them to participate in it, because the shadow exchange rate in that period exceeds the fixed exchange rate. If they do not expect a run, they will refrain from buying up the central bank's remaining reserves and this decision, too, will be validated because without a run reserves stay above their minimum threshold level. The events that trigger the belief that a run will occur can be totally extraneous. The reasoning is quite similar to that used by Diamond and Dybvig (1983) in their analysis of commercial bank runs.9 Grilli (1986) pointed out that an implicit assumption in Obstfeld's multiple equilibria result is that speculators are small and incapable of organizing in the face of profit opportunity. In that case it is necessary that the individual agents act simultaneously for the attack to be successful. Therefore, a speculator will attack only if he thinks that a general attack will take place. However, if speculators are large enough for such coordination of actions to be unnecessary for an attack to be successful, then equilibrium is unique and a speculative attack occurs whenever a profit can be made by attacking the currency. Grilli thinks it empirically reasonable to assume that speculators are large with respect to the magnitude of a speculative attack on the currency in the case of many small and medium-sized countries. Grilli (1986) and Buiter (1989) also showed .that in a stochastic environment a speculative attack need not be associated with any ( . particular trend in the monetary aggregates and, hence, the central bank may be forced to abandon the fixed exchange rate regime even though without the speculative attack it would have been viable.

21 The model employed by Grilli (1986) allowed for the possibility of speculative attacks, not only on a structurally weak currency but also on a structurally strong currency. For that purpose he specified two shadow exchange rates, one governing a selling attack and one governing a buying attack. In a two-country framework, Buiter (1989) transformed Grilli's two shadow floating exchange rates into a single shadow floating exchange rate between two barriers; a selling attack on one country's currency is a buying attack on another country's currency. In their frameworks Grilli (1986) and Buiter (1989) were able to produce forward premia for a structurally strong currency, in addition to forward discounts for a structurally weak currency. Buiter (1989) show ed further that, without management of primary deficits, policy aimed at stabilizing foreign reserves may lead to public debt destabilization; in a stochastic environment there may be difficulties in safeguarding both the fixed exchange rate system and fiscal solvency. The models by Obstfeld (1983), Grilli (1986) and Buiter (1989) are open to the criticism presented by Krugman (1989) and Krugman and Rotemberg (1990) known as the "g01d standard paradox". 1he Raradox occurs if a country's currencyappieciaies after itruiis out of göidor, ~qUlväleii11y, If a speculative~"åitacf§anhappen ollIyafteitnecOUiitry

Ilnaturanyll~filliSOuforreserves:-cP"C'C v -- -- -0 -c- ._- -,-- .. "' :&l1ter"ållac-GiiI1r~(I989) showed that the possibility of the II gold standard paradox" appearing is dependent on the nature of the stochastic process. Stochastic processes used in the three papers mentioned above were such that, with some parameter configurations, there is a possibility of a perverse (incorrect) attack at the lower boundary and/or at the upper boundary. They showed that this possibility does not appear in deterministic models nor in models where stochastic increments are always non-negative, e.g they are drawings from an exponential distribution as in Flood and Garber (1984b), nor in models with random walk without drift.

2.2.4.2 Uncertainty in deterministic models of balance-of-payments crises

A very obvious source of uncertainty in the context of balance-of­ payments crises concerns the minimum level of foreign reserves that triggers the exchange rate regime shift. This kind of uncertainty does not change the deterministic nature of the model. Uncertainty concerning the minimum level of foreign reserves can be defined in basically two different ways: (i) there is no specific minimum level of foreign reserves but s~~~~.PJ~llities for the collapse of

22 the fixed exchange rate are attached to each level of foreign reserves. The probability distribution can be constrained to some intervai of foreign reserves. The upper bound can be fixed or, as in Cumby and van Wijnbergen (1989), be equal to the present level of reserves. When foreign reserves are greater than the upper bound of that range, the probability of the fixed exchange rate collapsing is zero. The attainment of the lower bound, in turn, increases the probability of the collapse to unity. (ii) There is a specific threshold .level of foreign reserves which triggers the collapse of the fixeCT exchange"rate but that level of reserves is unknown to the public. The public knows only the subjective probability distribution of the threshold level of reserves. As shown by Willman (1989), the dynamics of speculative behaviour becomes quite different depending on which of the alternatives (i) or (ii) is chosen. If uncertainty is as in case (i), as is assumed by Cumby and van Wijnbergen (1989) and Otani (1989) in their studies, a steady rate of domestic credit expansion results in an accelerating. speclllativt; outflow of foreign reserves rather than a sudden~speculativeattack. This ls caused by the growing forward discount, which in the fixed exchange rate regime is known as the "peso problem". Hence, the "peso problem" is not only a property of stochastic models but it can also be generated by a deterministic model. In case (ii), discussed in a heuristic way by Krugman (1979) and Wyplosz (1986), there are repeated single period attacks on the· fixed exchange rate with zero probability of the collapse of the fixed exchange rate and a zero forward discount between the attacks. If the fixed exchange rate does not collapse, reserves are rebuilt immediately after the attack. This kind of behaviour results from the fact that investors know that the threshold level must be lower than the level to which the attack depleted reserves. There is some realism in both of these cases. A dedsion to abandon the fixed exchange rate is not made merely in the light of reserves; rather other considerations are also taken into account, the occurrence of which it is impossible to prepare for in advance. By assuming that there is no specific threshold level of foreign reserves, case (i) captures something of this, although in a very naive way. It is also realistic to assume some informational asymmetry between a central bank and the public. This is what case (ii) tries to capture. An unsuccessful attack provides speculators with new information on how low the central bank is willing to allow foreign reserves to be depleted. As Willman (1989) showed, these two approaches concerning the unknown threshold level of foreign reserves can be combined. In this case there are also successive attacks on the currency but now the probability of the collapse of the fixed exchange rate does not drop to

23 zero between subsequent one-period attacks and, hence, the forward discount may permanently deviate from zero. Owing to strong non­ linearities in the model, the dynamics of the speculative behaviour may become very complicated and the possibility of completely chaotic behaviour is not ruled out. There are some examples in the literature where it is attempted to estimate the collapse probabilities of the fixed (or the crawling peg) exchange rate regime from empirical data (see e.g. Blanco and Garber 1986, Cumby and van Wijnbergen 1989 and Goldberg 1990). The empirica1 results obtained in these studies seem promising. But, as the discussion above has shown, these results must be very sensitive to the way in which the threshold level of foreign reserves is defined.10 This is also what Goldberg (1990) found. Therefore further research along these lines might also help us to better understand, how the threshold level of reserves is determined. Besides uncertainty about the minimum level of foreign reserves, there can be another kind of uncertainty in deterministic models; there may be also uncertainty about the central bank's future policy reactions. Using a deterministic framework, Willman (1987) studied the case where a central bank announces that there will be no exchange-rate regime shift in the future; the present inflationary monetary policy rule will be changed so as to be consistent with the prevailing managed exchange-rate regime at the latest at the moment foreign reserves are depleted to zero. However, the public does not think this announcement is credible until it actually materializes. Willman (1987) showed that the effects of this kind of uncertainty on the size and timing of the speculative attack are dependent on the nature of the prevailing managed exchange rate regime, Le. if the exchange rate is perfectly fixed or if it is a target zone with at least an upper bound. In the latter case the policy announcement implies that there is a risk of a discrete appreciation and of a discrete depreciation of the exchange rate. The discrete appreciation occurs if the change in the monetary policy rule materializes. It is this two-sided risk of a discrete jump in the exchange rate which delays the timing of the speculative attack and lessens the size of the attack. In the case of a perfectly fixed exchange rate, no risk of discrete appreciation exists and a speculative attack is a pure one-sided bet. Owing to this property the expected shadow exchange rate equals the prevailing fixed exchange rate at exactl y the same point of time as in the case where no change in the· monetary policy rule is assumed. This implies that if the announcement concerning a future policy shift is not

24 perfectly credible, it has no effect on the timing and the size of the speculative attack. Although the interpretation of this result is c1ear-cut, it is difficult to accept it as a real world fact. Rather, as 1 show in the appendix of this survey, it is an artifact of the continuous time F-G model. In a discrete time framework the possibility that monetary policy will be changed in the future, so as to be consistent with the fixed exchange rate, delays the timing and reduces the size of the speculative attack. The reason for these differences in the results of continuous time and discrete time models is that in a discrete time model profits can never accrue at an infinite rate. Therefore in a discrete time model financial market equilibrium requires that in the period of the attack the expected change in the exchange rate decreases the demand for money so much that it suffices to deplete foreign reserves to zero; otherwise no attack occurs and the expected exchange rate equals the fixed rate. It is shown in the appendix that the greater is the probability that the monetary policy rule will be changed so as to be consistent with the fixed exchange rate the later the speculative attack occurs and the smaller is its magnitude. Such qualitative differences between the results of continuous time and discrete time models disappear, however, if imperfect substitutability between domestic and foreign assets is assumed and the level of domestic demand (and thus the current account) depends on the real interest rate. If now the monetary policy rule is to peg the stock of nominal money balances, then, as Willman (1988a) shows in a continuous time framework, uncertainty about monetary po"iicy reactions delays the timing and reduces the size of the speculative attack. This is due to the fact that there is no longer a one-sided bet for a discrete depreciation. Instead, there is the simultaneous risk of a discrete jump in the depreciation rate of the exchange rate and a discrete jump in the dmnestic interest rate.ll

2.2.5 Stabilization policies and balance-of-payments crises

Through its borrowing policy the government can affect the size of the central bank's foreign reserves. Hence, this raises the question as to what are the government's chances of using borrowing abroad to postpone a speculative attack on the currency and thereby to buy time for more fundamental policy corrections. This issue is studied by Buiter (1987), Garber and Grilli (1986) and Grilli (1990). Buiter (1987) broke down domestic credit expansion into its components: the primary deficit, interest payments and government lending. He showed that if foreign reserves are replenished by an open-

25 market sa1e just before the exchange rate regime wou1d collapse without rep1enishment, the date of collapse is postponed. However, if the rep1enishment takes place far enough in advance of the collapse date without rep1enishment and the interest rate on foreign reserves is below the world market rate, the date of collapse is actual1y brought nearer. This follows from the fact that, ho1ding the primary deficit constant, the rise in debt servicing costs causes domestic credit expansion to acce1erate and foreign reserves to be depleted at a faster rate than would be the case without rep1enishment. Moreover, the size of the attack is greater because the faster rate of domestic credit expansion also implies a faster rate of depreciation of the exchange rate in the post-attack floating exchange rate regime. However, if foreign reserves earn the world market rate of interest, the date of collapse will be postponed independent1y of the tirning of an open market sa1e. Buiter (1987) showed further that in a stochastic environment the rep1enishment of foreign reserves by a stock-shift open-market sa1e will 10wer the likelihood of an ear1y collapse but increase the likelihood of a collapse in the longer runo If foreign reserves earn the world market rate of interest, then an open-market sa1e will 10wer the likelihood of a collapse for all future periods in a stochastic environment. The application of the F-G mode1 by Garber and Grilli (1986) and Grilli (1990) to the Bellmont-Morgan Syndicate bond issue and its role in defending the gold standard in 1895 is related to this issue. They argued that when the U.S. Treasury borrowed abroad through the Syndicate to replenish its stock of gold reserves, this act of borrowing increased the viability of the gold standard in the sense that it reduced uniformly the probability of collapse after the loan was secured compared to what wou1d have been the case without the 10an. In this way, the government bought some "extra time", during which more fundamenta1 fisca1 correction was imp1emented. Quite sirnilar results were obtained by van Wijnbergen (1991) in a mode1 with choice-theoretic foundations. The aim in his paper was to study the prerequisites for achieving a sustainab1e reduction in infiation through an exchange rate freeze or crawling peg. He showed that restricting domestic credit growth to a rate that prevents reserve outfiows during a freeze is not sufficient to prevent a specu1ative attack; restrktions on the growth of interest-bearing debt are also required. If the crawling peg or exchange rate freeze ends through a speculative attack, the post-collapse infiation rate will exceed the infiation rate that prevailed before the start of stabilization. This would be the case even if reserves earned the world market rate of interest; the 10ss of foreign reserves wou1d imp1y that the government's net interest payments, and, hence, the government deficit to be financed through credit creation, wou1d be

26 greater in the post-stabilization period than in the pre-stabilization period. This is an open economy extension of Sargent's and Wallace's (1984) "Unpleasant Monetary Arithmetic".

2.2.6 Capital controls, imperfect asset substitutability and balance-of-payments crises

Many countries have resorted to temporary capitai controls when the domestic currency has come under pressure. The capital liberalization directive of the European Community also allows explidtly for temporary controls when a country faces balance-of-payments difficulties. The effects of temporary capital controls on the timing and dynamics of balance-of-payments crises have been studied by Wyplosz (1986), Park and Sachs (1987), Dellas and Stockman (1988) and· Bacchetta (1990). In the papers by Wyplosz (1986) and Park and Sachs (1987), it was shown that even in a continuous perfect foresight framework a regime shift from a fixed to a floating exchange rate regime is associated with a discrete jump in the exchange rate if capital controls are perfect. Hence discrete devaluations are also possible. This result is intuitively easy to understand because capital controls prevent speculators from exploiting the infinite profit opportunities otherwise associated with discrete devaluations. . An interesting feature of the model by Wyplosz (1986) is that the capital controls, when in force, are applied only to transactions by domestic residents whereas non-residents are free from restrictions. Holdings of domestic currency denominated assets by non-residents set the ceiling on the potential volume of speculative transactions. Non­ residents monitor reserve levels, inducing a speculative attack when reserve levels equal non-resident holdings of domestic currency. The currency is then devalued, setting off a new cyc1e, if credit expansion is continued. Using a choice-theoretic model Bacchetta (1990) showed that imposition of temporary capital controls may cause an attack on the currency just before the imposition of the controls. Similarly, Agenor (1990) in a stochastic framework with permanent capitai controls showed that the intensification of exchange controls in an effort to postpone the exchange rate realignment may, instead of delaying, actually precipitate the crisis. Capitai controls can also be imposed in the form of a tax on foreign interest earnings. Utilising the F-G model Agenor, Bhandari and Flood (1991) showed that the imposition of such a tax delays the timing of the collapse of the fixed exchange rate regime. Discrete devaluations are not,

27 however, possible in a continuous time perfect foresight framework unless the tax on foreign interest earnings is so high that there is no capital mobility. The assumption of imperfect capital substitutability gives results which in many ways resemble those obtained under the assumption of capital controls (see Willman 1988 a,b and Blackburn 1988). This is not surprising as imperfect substitutability between domestic and foreign assets may be a result of regulations constraining capital movements as well as of foreign exchange risk. 12 These models show that the smaller asset substitutability is the later the speculative attack occurs. Or, to put it another way, the level of a central bank's foreign reserves which suffices to safeguard the fixed exchange rate regime is much higher with free capitai mobility (high asset substitutability) than in a world in which capital movements are controlled. In addition, , imperfect substitutability between domestic and foreign bonds, as in Willman's (1988 a,b) models, implies that it is essentially the accumulating trade balance deficit which leads to the depletion of foreign reserves and eventually causes the balance-of-payments crisis. Because of the potential impact of government spending and wage settlements on external deficits, it is therefore not monetary policy alone, but rather the fiscal-monetary and incomes policy mix which is of importance in the analysis of the collapse process.

2.2.7 Real output, prices and wages and balance-of-payments crises

In section 2.2.1 it was shown that persistent (often growing) deficits in external balances, appreciation of the real exchange rate and decreasing domestic output are typical features of the process leading to a balance­ of-payments crisis. Choice-theoretic models and imperfect asset substitutability have already highlighted the role of the trade balance in the development of balance-of-payments crises but if price and real events of a potential crisis are also to be captured the single good approach must be abandoned. The assumption concerning wage and/or price rigidity also plays an essential role in explaining such events. The single-good assumption was relaxed by Connolly and Taylor (1984), Calvo (1987) and Willman (1988b) and Agenor, Bhandari and Flood (1991)P Both Connolly and Taylor (1984) and Calvo (1987) introduced a non-traded good into the mode!. Willman's (1988b) model contained two imported goods, Le. an imported final good and imported inputs, and one domestic final good. The domestic and foreign final goods were imperfect substitutes for each other.

28 Both Connolly and Taylor (1984) and Calvo (1987) showed that an increase in the rate of domestic credit expansion raises the relative price of non-traded goods, i.e. appreciates the real exchange rate, unti! the exchange rate regime shift occurs. At the instant of the regime shift there is a sudden permanent depreciation of the real exchange rate, which in Calvo's model was below the level prevailing before the acceleration in the rate of credit expansion. To obtain this result in the model of Connolly and Taylor (1984), one has to assume that the government's propensity to spend on traded goods is higher than that of the private sector. The models by Connolly and Taylor (1984) and Calvo (1987) were able to explain the appreciation of the real exchange rate preceding the exchange rate regime shift. An unrealistic feature in their results, however, is that the regime shift is accompanied by a precipitous depreciation of the real exchange rate, i.e. a discrete drop in the price of the non-traded good. In the model by Willman (1988b) and in a slightly modified model by Agenor et al. (1991), a sudden anticipated depreciation of the real exchange rate was not possible at the instant the exchange rate regime shift, occurred because wages were assumed t() be sticky. In the case of sticky, forward-looking wages~ domestlccosts and prices already started to rise and real output to decrease in the pre-attack fixed exchange rate regime in anticipation of the future rise in the exchange rate. In the post­ attack floating exchange rate regime the real exchange rate gradually converged towards the long-run equilibrium, which with a fixed real wage target was the same in both regimes. The model used by Willman (1988b) was a variant of the Mundell-Flerning model and, hence, real econornic activity was affected by monetary, fiscal and incomes policy. It was shown in this framework that, if the government tries to keep economic activity at too high aleveI, this leads inevitably to the collapse of the fixed exchange rate regime. However, conventional stabilization policy, which includes recurrent temporary policy changes in both the expansive and the contractive direction, is possible without giving rise to expectations concerning the exchange rate regime shift.

29 2.2.8 Other extensions of the analysis

Flood and Hodric (1986) also sought to study the real effects of anticipated exchange rate regime shifts. However, they approached the issue from an entirely different perspective from that presented above. Their aim was to compare the variability of real output in the permanently fixed exchange rate regime, in the fixed exchange rate regime which is anticipated to collapse and in the permanently floating exchange rate regime. They used a simple stochastic version of the Dombusch (1976) model in which real output could deviate from its naturallevelonly because ·of unanticipated demand disturbances. They found that real output is more volatile in a regime of colIapsing fixed rates than in a regime of permanently fixed rates, but no more volatile than under a pure flexible exchange rate regime. In addition to the variability of real output under different exchange rate regimes, Flood and Hodric (1986) studied optimal wage indexation. In typical wage indexation models, different permanent exchange rate regimes require different wage indexation policies, and each policy is a fixed function of the time-invariant stochastic structure of the economy. In the framework of Flood and Marion (1982), Flood and Hodric found that, unlike in other regimes, in a collapsing fixed exchange rate regime the optimal degree of wage indexation is state-dependent and thus time­ varying even though the stochastic structure of the economy is time­ invariant. An almost totally neglected topic in the balance-of-payments literature is how risk aversion affects the dynamics of balance-of­ payments crises and how risk premia behave in the course of a crisis. Typically, agents are assumed to be risk-neutraI. However, if there is uncertainty in the model, the size of a speculative attack also depends on risk aversion. This was shown by Willman (1989). In an extension of the F-G model with an endogenously determined risk premium, he showed that the greater risk aversion is the lower foreign reserves can be depleted before speculative capitaI movements start to play a major role. The size of the risk premium was elosely related to the probability of devaluation, which, in tum, was determined by foreign reserves. With low and high values of devaluation probabilities, the risk premium was elose to zero and with a devaluation probability of just over a haIf, the risk premium was at its maximum. The interpretation of this resuIt is that uncertainty in the model becomes the smaller the eloser to zero or to unity the devaluation probability iso With zero and unity devaluation probabilities, the model is transformed into the perfect foresight model with uncovered interest parity.

30 3 Summaries of the Papers

In the following 1 summarize the main results of the five artic1es making up this thesis. These artic1es investigate balance-of-payments crises from different perspectives and it is the intemal logic of the topic which determines the order in which the artic1es are presented, not the order in which they were published. They extend the literature on balance-of-payments crises in the following directions. The first artic1e considers the justification of the assumption that there exists a minimum level of foreign reserves below which they cannot Of are not allowed to be depleted. In the second artic1e the pre-attack exchange rate regime is generalized to be a target zone (including the fixed exchange rate as a special case). The collapse of this regime is studied in the F-G model by taking into account the possibility that, when the speculative attack on the currency occurs, the central bank tries to defend the prevailing exchange rate regime. This theme is continued in the third artic1e in which the assumption of uncovered interest parity incorporated in the F-G model is relaxed and, instead, domestic and foreign assets are assumed to be imperfect substitutes for each other. It is shown that the neutrality of the collapse of the fixed exchange rate regime with respect to uncertainty about monetary policy reactions no longer holds in this setting. The fourth article introduces uncertainty about the minimum level of foreign reserves which triggers a devaluation. Also, the assumption of risk-neutral agents is abandoned and the risk premium is allowed to be determined endogenously. In the fifth artic1e the framework of the analysis deviates most from the F-G model and it can be c1assed as belonging to the models of the Mundell-Fleming tradition; wages are sticky and domestic goods and domestic bonds are imperfect substitutes for their foreign counterparts. This framework is used to investigate the real effects of an anticipated exchange-rate crisis.

31 3.1- Why there is a lower bound on the ./ central bank's foreign reserves

(Alpo Willman, Finnish Economic Papers 4, 1991, 113-129)

A crucial assumption in the literature on balance-of-payments crises is that there exists a minimum leve1 of foreign reserves below whlch they cannot be depleted. It is thls assumption whlch gives rise to the analogy between balance-of-payments models and non-renewable resource models of speculative attack. However, the justification· fOI thls assumption can be challenged; a central bank facing a perfect world capital market can always create gross foreign reserves by borrowing and, hence, negative net reserve positions are feasible. Thus one can question whether there is any limit to the depletion of net fOIeign reserves. One can approach thls issue by asking (a) does there exist a lower bound on the net foreign reserves below whlch they cannot be depleted without violating sectOIal solvency constraints, or (b) is there some level of fOIeign reserves below which the central bank is unwilling to allow the net foreign reserves to be depleted although, from the point of view of solvency, further depletion would be possible? These issues have previously been studied in the light of the first question by Obstfeld (1986a) and Buiter (1986 and 1987). They considered whether a continuous domestic credit expansion results in the violation of the public sectOI solvency (or intertemporal budget) constraint. In thls respect their results are inconc1usive: if the credit expansion is used to finance a primary deficit in the government budget and taxation isdistortionary, the solvency constraint will be violated. However, if domestic credit is used to monetize the government's non-monetary debt, net foreign reserves can be depleted boundlessly without solvency problems. Thls is not in accordance with real world experiences; there are many examples in the real world where monetization of government debt has resulted in balance-of-payments crises (see e.g. Penati and Pennachl 1989). In this study the existence of a minimum level of net foreign reserves is examined in the light of both questions (a) and (b). For that purpose a choice-theoretic model with intertempOIal budget constraints is specified. The model comprises the household sector, the central government and the central bank. The two latter sectors face the .common consolidated government sector budget constraint. The time horizon of the representative household is infinite and the household is assumed to maxirnize a discounted sum of future instantaneous utilities.

32 A novel feature of the model is that a cash-in-advance constraint in financial market transactions is imposed.14 The transactions technology is such that debt-service costs must be paid in the form of cash. Because of this constraint, there is not only demand for interest­ bearing foreign assets but also demand for non-interest-bearing foreign 0 currency in the model. Hence, for instance, if either 0 the central government or the central bank has foreign currency-denorninated debt, part of the central bank's gross foreign reserves 0 must be in° the form of cash balances. In the fixed exchange rate regime there are three altemative uses for domestic credit: if it is injected via the central government sector, the credit can be used to finance the primary deficit in the government budget (the case of expansive fiscal policy) or to invest in foreign assets (the case of monetizing the govemment debt); if the credit is injected via the private sector, i.e. the central bank buys bonds issued by the private sector (the case of expansive monetary policy), this results in the accumulation of the foreign assets held by the private sector. As to the violation of sectoral solvency constraints with non-distortionary taxation, the cash-in-advance constraint in financial market transactions changes the conc1usions obtained by Obstfeld (1986a) and Buiter (1986 and 1987) in the following way: all three policy altematives become sirnilar in the sense that either the public sector or the private sector solvency constraint is violated, if the rate of domestic credit expansion exceeds a critical magnitude, which is somewhat below the world market interest rate. However, the depletion of the central bank's net foreign reserves towards rninus infinity is stilI possible in all three cases, if the rate of credit expansion is below the critical magnitude. More importantly, the cash-in-advance constraint in financial market transactions introduces the welfare aspect into the analysis. The depletion of the central bank's net foreign assets below zero .c even temporarily) causes a welfare loss. This is because the foreign currency denorninated cash balances needed for debt servicing are not interest-bearing. Hence, for a welfare-maximizing government the lower bound on the central bank's net foreign reserves is zero, below which the reserves are not allowed to be depleted although, from the point of view of solvency, this would be possible. It was also studied what kind of effects tl).e cash-in-advance constraint on financial market transactions has on the size and tirning of the speculative attack associated with the exchange rate regime shift. It was found that the cash-in-advance constraint introduces a kind of asymmetry into the model: the size of the speculative attack

33 · on the currency is greater and the exchange rate regime shift occurs earlier in the case where credit expansion is injected via the central government budget than in the case where the credit expansion takes place through the household sector. This asymmetry is due to the fact that the cash-in-advance constraint concerns the financial transactions between the private sector and the central bank but not the transactions made within the consolidated government sector.

34 3.2 Speculative attacks on the currency with uncertain monetary policy reactions

(Alpo Willman, Econonrlcs Letters 25, 1987, '75-78.)

A standard assumption in the literature on balance-of-payments crises is that the exchange rate is, with certainty, allowed to float freely once foreign reserves have been depleted to zero (or to some other a priori lmown minimum level). In such a setting, discrete devaluations in a continuous time framework are not possible, jf abnormal profit opportunities are to be prec1uded. In this study, uncertainty about the policy reactions of the central bank is introduced into the F-G model; with some probability greater than zero, the central bank will defend the prevailing exchange rate regime, Le. the monetary policy rule is changed to be consistent with the regime. This kind of situation can occur if the government amiounces that the present inflationary policy will be changed to be consistent with the prevailing exchange regime in the future, at the latest at the instant foreign reserves are depleted to zero, but the public has doubts about the credibility of the announcement until the change in the poUcy rule actually occurS. 15 The present exchange rate regime is either the perfectly fixed exchange rate or a target zone with at least an upper bound. If the exchange rate regime is the target zone, it is assumed that the exchange rate is at the upper bound of the zone. Since the present paper was written a large target zone literature on exchange rate dynamics has appeared (see e.g. Krugman 1987, 1991, Miller and Weller 1989, Flood and Garber 1989 and Krugman and Rotemberg 1990). AlI of these studies apply a stochastic framework and stochastic shocks play a key role in the target zone exchange rate dynamics. By contrast, the framework in the present study is deterministic and thus the results are unaffected by the new target zone literature. Perhaps the most striking result of the study is that, in the case of the perfectly fixed exchange rate, policy uncertainty has no effect on the timing or size of the speculative attack. This is so because, although the rate of increase in the shadow floating exchange rate is smaller than in the case without policy uncertainty, the shadow exchange rate coincides with the fixed exchange rate at the same point of time as in the case with perfect foresight and with unchanged monetary policy. Hence, although the announced change in the monetary policy rule does not materialize, the timing of the exchange rate regime shift is exactly the same as in the perfect foresight model.

35 If the policy change does materialize and the fixed exchange rate prevails, the demand for money before and after the attack must be equal. This implies that the speculative capital outfiow and the capitai infiow following the implementation of the change in monetary policy rule must occur at the same instant of time. However, if the original exchange rate regime is a target zone, there is a risk of discrete appreciation or discrete depreciation even in the case where the central bank decides to defend the target zone regime. In this case the rates of increase in the shadow exchange rate with and without policy uncertainty are the same but in the former case the shadow exchange rate coincides with the fixed exchange rate later than in the case with perfect foresight. Hence, policy uncertainty delays the timing of the speculative attack and reduces the size of the attack. In the limiting case, where the central bank defends the target zone with probability one, there is no speculative attack on the currency. When there is uncertainty about the policy reactions of the central bank, speculative attacks which exhaust foreign reserves are always associated with discrete jumps, either upwards or downwards. The upward jump (discrete depreciation) occurs if the central bank adheres to its earlier expansive monetary policy rule and allows the exchange rate to float freely. The downward jump (discrete appreciation) occurs if the monetary policy rule which is consistent with the target zone is adopted. However, at the moment of the speculative attack, uncertainty concerning the direction of the jump acts as a counterbalancing force so that the size of the expected jump is zero.

36 3.3 Balance-of-payments crises and monetary policy reactions in a model with imperfect substitutability between domestic and foreign bonds

(Alpo Willman, Economics Letters 26, 1988, 77-81.)

This study continues the theme of the artic1e presented in the previous section. It is shown that the neutrality property of the possible collapse of the fixed exchange rate with respect to uncertainty about monetary policy reactions does not hold if imperfect substitutability between domestic and foreign assets is assumed and the level of domestic demand (and, hence, the current account) depends on the real interest rate. The model framework used in this study is the same as in the study by Willman (1986) in which a model with imperfect substitutability between domestic and foreign bonds was specified. The central bank is assumed to control the nominal stock of money, which in the fixed exchange rate regime is equivalent to the pegging of the nominal interest rate. The net demand for foreign assets is a function of the uncovered interest differential as in a conventional portfolio-balance equation with constant absolute risk aversion.16 Unlike in the F-G model the timing of the speculative attack cannot be solved directly from the arbitrage condition that at the instant the exchange rate regime shift occurs the shadow floating exchange rate must equal the collapsing fixed exchange rate. This is because the solution of the model contains one stable root and, hence, an arbitrary constant multiplying the exponent in this stable root. Therefore the arbitrage condition that there cannot be anticipated jumps in the exchange rate does not suffice to accurately define the shadow floating exchange rate. The arbitrage condition is needed for solving the unknown constant, which, however, is still a function of the collapse time. An additional condition is needed for solving the exact collapse time, after which the shadow floating exchange rate is also fully defined as a function of time. The fact that just at the instant preceding the speculative attack the stock of foreign reserves must equal the stock shift in the net foreign assets which exhausts foreign reserves to zero gives this condition. The size of this attack, in turn, is determined by the upward jump in the rate of "depreciation of the exchange rate at the instant of the regime shift and by the interest sensitivity of the demand for net foreign assets. Hence, the less

37 interest-sensitive the demand for net foreign assets is (Le. the higher is the degree of the capital control), the smaller is the speculative attack. In this framework, uncertainty about the policy reactions of the central bank (Le. uncertainty as to whether the central bank adheres to the fixed exchange rate and reduces the supply of money consistent with it or if it allows the exchange rate to float freely with no monetary correction) affects the timing and the size of the speculative attack. This is because, unlike in the F -G model, a condition which ties the size of the speculative attack to the expected rate of depreciation is needed in defining the timing of the attack. Uncertainty about monetary policy reactions decreases the expectedrate of depreciation and, hence, the size of the attack. If the choice of the post-attack policy regime of the central bank is the freely floating exchange rate, then the regime shift is associated with a discrete jump in the depreciation rate of the exchange rateP If, instead, the money supply is changed so as to be consistent with the fixed exchange rate, then there is an upward jump in the domestic interest rate at the time of the attack. This rise in the interest rate brings the current account into equilibrium. Hence, the simultaneous risk of a discrete jump in the depreciation rate and a discrete rise in the domestic nominal interest rate explains why uncertainty about monetary policy reactions affects the behaviour of investors. This kind of trade-off between risks is lacking under the assumption of uncovered interest parity.

38 3.4 Devaluation expectations and speculative attacks on the currency

(Alpo Willman, The Scandinavian Joumal of Economics 91, 1989:1, 97-116.)

In addition to uncertainty about the policy reactions of the central bank, there is also another type of uncertainty concerning the behaviour of the central bank: uncertainty about the minimum (threshold) level of foreign reserves which triggers the exchange rate. regime shift. In his seminal paper, Krugman (1979) and later Wyplosz (1986) discussed this issue in a heuristic way. They assumed that there are two possible levels of minimum reserves: investors attach some positive probability to the higher one that it is the true threshold level and, if ex post, it tums out that it was not the threshold level, investors know with certainty that the lower one is the correct threshold level. Krugman reasoned that there will be a speculative attack which in the first phase exhausts foreign reserves to the higher critical minimum level. If it later appears that the central bank is willing to run down reserYes further, there will be a restoration of confidence with a reflux of reserves. No speculative capital movements appear until the second attack exhausts foreign reserves to the true threshold level. Cumby and van Wijnbergen (1989) also introduced unc.ertainty about the level of foreign reserves that triggers abandonment' of the exchange rate regime. The investors' prior distribution for the' central bank's critical value for reserves was assumed to be uniform over the upper and lower limits on the range of possible values for the critical level of reserves. They assumed that the upper !imit was the current­ level of reserves. The treatment of the upper limit makes the case of Cumby and van Wijnbergen qualitatively quite different from that of Krugman (1979) and Wyplosz (1986); in their case there is no possibility of the restoration of confidence without changes in the monetary policy pursued by the central bank. This study applies and extends the analysis of both of these cases. Speculative behaviour associated with devaluation expectations is studied' in a framework where the threshold level of foreign reserves is unknown to the public. Its value is determined in three altemative ways: (i) a new value is drawn from a known probability distribution at the end of each period; (ii) a value for the threshold level is drawn from the, distribution only once; and (iii) as a combination of these two, a new value is drawn with a probability greater than zero but smaller than one at the end of each period. Qualitatively, case (i)

39 corresponds to that of Cumby and van Wijnbergen (1987) and case (ii) to that of Krugman (1979) and Wyplosz (1986). Unlike in Krugman (1979) and Wyplosz (1986), the probability of devaluation is determined endogenously in the present study. The analytical framework is the discrete time version of the F-G model extended to take into account uncertainty concerning the minimum level of foreign reserves and the possibility that investors ar~ risk-averse. The method of analysis is the numerical simulation technique. With a devaluation risk and risk-averse investors, uncovered interest parity no longer holds. The domestic interest rate depends not only on the foreign interest rate and the "expected depreciation rate of the exchange rate but also on an endogenously determined risk preinium. There are drastic differences in speculative behaviour between the first two cases. In case (i), steadily growing domestic credit expansion results in a speculative outflow of capitai distributed over several periods of time. Hence, like disturbances in the domestic credit creation process' (Flood and Garber 1984b), this model is able to produce a forward discount under the fixed exchange rate system which is known as the "peso problem", In case (ii) there are repeated single-period attacks on the currency with zero probability of devaluation between the attacks. Hence, as in the discussions of Krugman (1979) and Wyplosz (1986), if the devaluation does not materialize, reserves are rebuilt in the period following the attack. This kind of behaviour results from the fact that investors know that the threshold level must be lower than the level to which the previous attack depleted reserves. In the combined case (iii), there are also successive attacks on the currency but now the probability of devaluation does not drop to zero between attacks and, hence, the forward discount may permanently deviate from zero. Although the model is perfectly deterministic in both case (iii) and case (ii), there is an important qualitative difference between these two cases. In case (ii) the exact timing of the next attack is easy to forecast; it occurs when foreign reserves are depleted passively down to the level which the previous attack a,ttained. In case (iii) the new attack can occur at any level of foreign reserves and, hence, the timing of the next attack can no longer be forecast. It is the strong non­ linearities present in the model which make the dynamics of the model very complicated .. With some parameter configurations the possibility of completely chaotic behaviour of the model is not exc1uded. As to risk aversion, it was found that the greater is risk aversion the lower foreign reserves can be depleted before speculative capital movements start to play a maj or role. It was also found that the risk

40 premium converged towards zero, the eloser to zero or unity the probability of devaIuation approached. The risk premium was at its maximum when the probability of devaIuation was somewhat greater than a haIf. One mmn conelusion of this study is that specuIative behaYiour is quite sensitive to the specification of the process which. produces the critical Iower level of foreign reserves. This has important implications when one tries to appIy a baIance-of-payments model to empirical data.

41

/' 3.5 The collapse of the fixed exchange rate tegime with sticky wages and imperfect substitutability between domestic and foreign bonds

(Alpo Willman, European Economic Review 32, 1988, 1817-1838.)

Models used in the analysis of balance-of-payments crises have been structurally very simple and, hence, many features which can be observed in the real world are not captured by these models. Typically, e.g. the F-G model, these models contain only one good, domestic and foreign bonds are perfect substitutes for each other and, most importantly, all markets are assumed to dear instantaneously. Therefore, the effects of stabilization policies in the traditional sense cannot be studied with models like the F-G model or with choice­ theoretic models. Models of the Mundell-Fleming type are more suitable for this purpose. In this study a variant of the Mundell-Fleming model is specified. There are two imported goods, Le. imported final goods and imported inputs, and one home good. The home good, which can also be exported, and the imported tinal good are imperfect substitutes for each other. Real output is demand-determined and depends on the relative price of the home and the imported final good, on the real interest rate and on the expansiveness of fiscal policy. The price of the domestic good is determined as a mark-up over variable costs, i.e. over the weighted average of wages and the price of imported inputs. Net exports depend negatively on the relative price of the domestic and imported final good and on the level of domestic demand. There are three assets in the model, Le. domestic money, domestic bonds and internationally tradeable bonds. The net demand for foreign bonds is a function of the uncovered interest differential, as in the model of section 3.3. Three alternative wage formation schemes are specified: a fixed nominal wage rate, sticky backward-looking wages and sticky forward- 100king wages. In the first case the rise in the price level permanently lowers the real wage rate whereas the two latter wage formation schemes imply an unchanged real wage rate in the long runo It is assumed -that fiscal policy becomes more contractive when prices increase. This is because government outlays are not fully indexed and/or an inflation correction is not made in the progressive

42 income tax scales. Monetary policy is assumed to be accommodating so that the target of the central bank is to peg the domestic interest rate. The aim of the study is to exarnine the dynamics associated with an endogenous regime shift from the fixed to the fioating exchange rate regime. Also, the effects of expansionary economic policies are studied when the fixed exchange rate regime is not· viable. In the model it is a deficit in the current account balance which depletes foreign reserves towards zero and, eventually, results in a speculative attack on the currency and in the. collapse of the fixed exchange rate regime. Hence, the higher are real wages, the more expansionary is fiscal policy, and the lower is the domestic interest rate the greater is the deficit in the current account and the faster foreign reserves are depleted. In addition to these factors, the timing of the collapse of the fixed exchange rate regime depends on the size of the speculative attack. From this point of view, the interest sensitivity of the demand for net foreign assets plays the most important role. Like the model presented in section 3.3 but unlike the F-G mode!, the timing of the speculative attack cannot be solved directly from the arbitrage condition that at the instant when the exchange regime shift occurs the shadow exchange rate must equal the collapsing fixed exchange rate. This is because the solution of the model contains at least one stable root and, hence, at least one arbitrary constant multiplying the exponents in these roots. The exchange market arbitrage condition is needed for solving one of these unknown constants. An additional condition is needed to solve the exact collapse time. The fact that just at the instant preceding the speculative attack the stock of foreign reserves must equal the stock shift in the net foreign assets which exhausts foreign reserves to zero. gives this condition. The size of this attack, in turn, is determined by the upward jump in the rate of depreciation of the exchange rate and the interest sensitivity of the demand for net foreign assets. After the fixed exchange rate regime has collapsed, the exchange rate fioats towards a new long-run equilibrium, which balances the current account. Qnly in the case with fixed nominal wages can the current account be balanced via real wage adjustment. In this case the adjustment process is associated with the rise in real output. In the cases of sticky .backward-Iooking and sticky forward-Iooking wages, real wages cannot permanently be lowered and, given a fixed nominal interest rate, it is fiscal policy which must be used to bring the economy into equilibrium. As the government budget is not fully indexed the depreciation of the currency results in the tightening of . fiscal poliey. Therefore real output must deerease in the long run,

43 although it increases temporarily after the colIapse of the fixed exchange rate. The temporary increase in output is due to the temporary fall in the real interest rate and the fact that wages adjust more slowly to the new long-run equilibrium than prices. The chief difference between the cases with sticky backward­ looking and sticky forward-Iooking wages is that in the latter case wages and the price of domestic goods start to rise and real output to decrease in the fixed exchange rate regime as soon as the collapse of the fixed exchange rate is anticipated. Hence, this implies that economic activity is at a low level when the fixed exchange rate collapses. As toprice behaviour, it resembles that {)f the price of non­ tradeable goods in the model of ConnolIy and Taylor (1984). However, in their model there was a steep fall in the price of non­ tradeable goods at the moment of the collapse of the fixed exchange rate. In thisstudy, owing to wage rigidity, the price of the domestic good does not falI. What happens is that, starting from this moment, the relative price of domestic goods with respect to the price of foreign goods starts to decrease. The relative price continues to fall untU parity is attained. The framework of this study also enables analysis of the effects of fiscal policy by taking into account the fact that the viability of the fixed exchange rate regime is dependent on the policy pursued. In the case of an expansionary policy measure, the possibility of the colIapse of the fixed exchange rate plays a role only if, before the implementation of the new policy, there is a current account deficit or the policy results in a current account surplus shifting permanently into deficit. In the first case expansionary policy hastens the timing of the regime collapse and in the latter case it results in the collapse of the fixed exchange rate, which otherwise would have been viable. If the wage formation scheme is forward-Iooking, expansionary policies cause inflationary pressures in the economy long before the colIapse of the exchange rate regime. However, conventional stabilization policy, . which inc1udes recurrent temporary policy changes in both the expansive and the contractive direction, is possible without giving rise to expectations conceming the exchange rate regime shift. This result is independent of the wage formation scheme.

44 Notes

1 For financial crises, see e.g. Kindleberger (1978), Kindleberger and Laffargue (1982) and Capie and Wood (1986).

2 The basic theory of exhaustible resources was developed by Hotelling (1931).

3 In his analysis Krugman used a slightly modified version of a model developed by Kouri (1976). A special feature of this model is that it includes only two assets: domestic and foreign money.

4 Calvo (1977) provides a geneml discussion of the asset-price continuity condition for continuous-time perfect foresight models. For the continuity condition's role in dynamic exchange mte theory, see Obstfeld and Stockman (1984).

5 See Edwards and Montel (1989), who analyzed the circurnstances preceding 20 major devaluation crises in 16 developing countries. The model they used in their analysis was able to capture many common features of balance-of-payments crises. However, unlike in papers presented in this review, they assumed perfect capital controis which excluded the possibility of speculative attacks on the foreign reserves.

6 For functional equivalence betweenliquidity costs and the utility of money, see Feenstra (1986). "-~C==

7 What Obstfeld (1984) actually did in his paper was to combine the case of the collapse of the fixed exchange rate regime and the case of the anticipated future retum into a fixed exchange rate. The latter case was studied earlier by Flood and Garber (1983) and Obstfeld and Stockman (1984).

. 8 A peso problem arises when market forecasts refiect the possibility of major events that occur relatively infrequently in the data set available to econometricians. Even though market expectations may by entirely reasonable ex ante, market forecasts appear biased and forecast errors appear serially correlated in the ex post sample. For the peso problem, see e.g. Obstfeld (1987) and the references given there.

9 Diamond and Dybvig (1983) present a stylized model of financial intermediation in which there are two equilibria: one in which agents have confidence in the solvency of financial intermediaries, and one in which lack of confidence leads to a runo Both equilibria involve self-fulfilling expectations because banks fail if, and only if, there is a runo

10 For a good review of these empirical studies, see the paper by Agenor et al. (1991).

11 There was a misinterpretation in the paper by Willman (1988a). It was argued that, if the announced change in the monetary policy rule does not materialize, there is a discrete jump in the exchange mte immediately after the attack. This is not, however, true. There is a discrete jump in the mte of depreciation not in level of the exchange mte. This is dealt with more closely in note 17.

12 For details, see note 16.

45 13 Actually, the single-good assumption was also abandoned in the studies by Flood and Hodric (1986), Blackburn (1988) and Goldberg (1991).

14 For more on the cash-in-advance constraint on financial market transactions, see e.g. Helpman and Razin (1985), Grilli and Roubini (1989) and Lucas (1990).

15 The present analysis is related to the question of the credibility of the central bank. Unlike in macroeconornic policy game literature, in the present analysis credibility is treated as an exogenous variable. (See e.g. Andersen and Risager 1991 for an analysis of credibility factors in the context of exchange rate management).

16 In a deterrninistic perfect foresight framework imperfect substitutability between domestic and foreign bonds cannot be based on risk aversion behaviour as no risk exists in that framework. However, imperfect substitutability between domestic and foreign bonds can also result from capital controls not being perfect; the controIs can be circumvented but not without costs. If we assume that the more one invests in foreign

bonds the loyver is the interest income on these investrnen~ we can write iF:: i* - (b/2)F, where iF is the effective interest rate on investment in foreign bond F and i* denotes the foreign interest rate. Parameter b measures the degree of capital control equal to zero in the case with no capital controls and to infinity in the case with perfect capital controls. Now, by denoting the domestic interest rate by i and maximizing the end-of­

period wealth, i.e. Wt+1 :: (l+i)(Wc FJ + (l+iF)Ft, S.t. iF :: i*-(b/2)Ft, we obtain Ft :: (1/b)(i*-i).

17 In the article by Willman (1988a) it was argned that in this case the exchange rate regime shift from the fixed to the fioating exchange rate is associated with a discrete jump in the exchange rate. There is, however, a mistake in this argumentation: there is no jump in the level of the exchange rate but rather a jump in the rate of depreciation of the exchange rate. This can be shown easily: at the time T when The speculative attack occurs the expect~d shadow exchange rate is deterrnined .!Is follows, E-!(T) = ltP '(T) + (l-lt)P, where p* indicates freely fioating and P the fixed exchange rate. lt is the collapsing probability of the fixed exchange rate. As_there cannot be a jump in the expected exchange rate at T, it follows that E-!(T) = P and hence, also P '(T) = P; i.e. there is no jump in the exchange rate. If the regime shift occurs, the expected rate of depreciation jumps from its pre-attack level ltP '(T) to P '(T).

46 References

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48 \

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49 Obstfeld, M., 1984, Balance-of-Payments Crises and Devaluation, Joumal of Money, Credit and Banking 16, 208-217. , Obstfeld, M., 1986a, Speculative Attack and the Extemal Constraint in a Maximizing Model of the Balance of Payments, Canadian Joumal of Economics XIX, 1-22.

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50 Willman, A, 1989, Devaluation Expectations and Speculative Attacks on the Currency, The Scandinavian Joumal of Economics 91, 97-116.

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51 Appendix

A speculative attack on the currency with an uncertain monetary policy reaction in a discrete time model

Willman (1987) showed that the timing of the speculative attack on the fixed exchange rate was not affected by uncertainty about the monetary policy reaction. In the following, 1 show that this result is an artifact of the continuous time F-G model; in a discrete time model the possibility that monetary policy will be changed so as to beconsistent with the fiscal exchange rate regime delays the timing of the speculative attack and reduces its size. The discrete time version of the F -G model presented in section 2.2 can be written as follows:

where M is the stock of money, D is domestic credit, R is the stock of foreign reserves, S is the spot exchange rate and Et denotes the expectation operator conditional on the information available at the beginning of period 1. _ In the fixed eXC2..hange .Jate regime, St = S, the stock of money is constant, i.e. Mt = M = bS, and foreign reserves are depleting as Rt = Ra - fA- 1. In the floating exchange rate regime with perfect foresight and where Rt = 0, equation (Al) implies that the exchange rate is determined by

(A4) St = 3fl/b + (Da + fA-t)/b.

If the regime shift from the fixed to the floating exchange rate regime occurs in the period following the period when -fA- < Rt ::;; 0, the collapse time of the fixed ex,Shange rate can be solved from equation (A4) with the condition St = S. Assume for notational simplicity that the speculative attack depletes foreign reserves exactly to zero. Thus the period of attack is

(AS) T = RalfA- + a/b.

52 Assume next that in the fixed exchange rate regime the central bank makes an announcement that in the near future, at the latest when Rt = 0, the expansive monetary policy rule (A3) will be changed to be consistent with the exchange rate, i.e. fl, = O. However, the public has doubts about the credibility of this announcement until the policy shift actually occurs; with probability :Tt the public believes that no change in the policy rule occurs and that· the exchange rate regime shifts from the fixed to the floating exchange rate at the moment -fl, < Rt < 0; with probability 1 - :Tt the monetary policy rule_ will be changed from fl, > 0 to fl, = 0 and the fixed exchange rate S will prevail for ever. As long as the announced change in the policy rule does not materialize, the shadow exchange rate se is obtained as a weighted average of the perfect foresight shadow e,!change rate St defined by equation (A4) and the fixed exchange rate S:

(A6) St e = :TtS t + (1 -:Tt)S ;0 < :Tt <1.

It is sJJaightforward to see that St e = S at the same point of time as St = S, i.e. at the time t = T. This result holds with all values of :Tt. In the continuous time version of the F-G model this would also be the timing of the speculative attack on the currency. This results from the arbitrage condition that in continuous time analysis there cannot be anticipated jumps in the exchange rate implying the depreciation of the exchange rate at an infinite rate. In discrete time changes in the exchange rate are never infinite between two subsequent periods and, hence, abnorm2l profit opportunities are prec1uded. Therefore the condition that St e = S does not guarantee the fulfi1ment of the money market equilibrium condition (A1). If the period of the attack is denoted by L, equation (A1) implies that the demand for money in the period preceding the attack and in the period of the attack is determined as follows: - - (A7a) M = D-c-l + R-c-l = bS - . - (A7b) M-c = D-C-l + fl, = bS - aEiS-c+l -S).

The size of the speculative attack is obtained by subtracting (A7b) from (A7a), i.e. - - (A8) M - M,; = R';_l - fl, = aEiS';+l - S).

In the period of the attac~ E,;S';+l = S';:l_and, hence, on the basis of equation (A6), E,;(S';+l - S) = :Tt (S-c+l - S)

53 Now by solving equation CA8) for the attacking period 1:, we obtain

CA9)

It can be seen from equations CAS) and CA9) that with 0 < n < 1, Le. the announcement by the central bank about the future change in the monetary policy rule does not enjoy fuli credibility, 1: > T and the timing of the speculative attack is delayed. 1n the limiting case of no credibility, Le. n = 1 the announcement of the central bank has no effect on the timing of the speculative attack and 1: = T. 1n the case of full credibility, Le. n = 0, there is no speculative attack on the currency and 1: = Ro/fl. _ Hence, we see that the shadow exchange rate St:1 can be above S for several periods beforeJhe attack. This does not, however, imply that Et St+1 would be above S before the period _ t = 1:. Until that period everybody knows that the difference St:1 - S is not great el!...ough to trigger the speculative attack and, hence, Et St+1 must equal to S.

54 Finnish Economic Papers- Volume 4-Number 2-Autumn 1991

WHY THERE IS A WWER BOUND ON THE CENTRAL BANK'S FOREIGN RESERVES*

ALPO WILLMAN

Bank of Finland, P.O. Box 160 SF-00101 Helsinki, Finland

This article examines the implications for the balance of payments of impos­ ing a cash-in-advance constraint on financial market transactions. 1 show that with a weljare-maximizing government this constraint introduces a lower bound on the central bank's net foreign reserves; the depletion of the net foreign reserves below zero results in a weljare loss. 1 further show that either the pri­ vate or public sector solvency constraint is violated, if the growth rate of domes­ tic credit expansion exceeds a critical magnitude, which is somewhat below the foreign interest rate. Unlike in Buiter (1987), the violation ofthe solvency con­ straint does not depend on ,the way the credit expansion is used. The timing and the size of the speculative attack associated with an anticipated exchange rate regime shift are, however, dependent on the way the credit expansion. is injected into the economy.

1. Introduction ing and, hence, negative net reserve positions for defending a fixed exchange rate regime are In the literature on balance-of-payments feasible. Thus, one can question whether there crises the assumption that there exists a mini­ is any lower bound on net foreign reserves. mum level of net foreign reserves, below Obstfeld (1986) showed that, in an idealized which they cannot be depleted, is of key im­ world, where taxes are not distortionary, a portance from the point of view of analysis.! central bank's net foreign reserves can become It is this assumption which leads to the anal­ infinitely negative without violating the con­ ogy between balance-of-payments models and solidated government· sector' s intertemporal non-renewable resource models of speculative budget constraint, if the growth rate of attack. However, the justification for this as­ domestic credit is below the world interest . sumption can be challenged; a central bank rate. Only if the growth rate of domestic credit facing a perfect world capitai market can al­ expansion exceeds the world interest rate is the .. ways create gross foreign reserves by borrow- solvency constraint violated and does there exist a limit to the central bank's externall)or­ rowing. * 1 would like to thank Erkki Koskela, Juha Tarkka Later Buiter (1986 and 1987) showe<;l that and an anonymous rejeree jor usejul comments. the depletion of net foreign reserves can con­

1 See, jor instance, Salant and Henderson (1980), tinue boundlessly even if the growth rate of Salant (1983), Krugman (1979), Flood and Garber (1984), domestic credit expansion is above the foreign Calvo (1987) and Willman (1987, 1988a, b). interest rate. This is because the fiscal authori- 113 Finnish Economic Papers 2/1991 - A. Willman ty can use its credit from the central bank to could be faced by the lender (at the time of accumulate foreign bonds. In this case the in­ purchasing assets), as in the papers by Grilli crease in domestic credit expansion results in and Roubini (1989) and Lucas (1990). How­ a decrease in the central bank's net foreign ever, for notational simplicity 1 assume for reserves without any effect on the consoli­ most of this paper, like Helpman and Razin dated government sector's net foreign debt (1985), that it is cash-in-advance constraints and solvency. faced by borrowers which determine domes­ Hence, only if the growth rate of domestic tie demand for foreign currency. Further, as credit expansion is above the foreign interest in all the cash-in-advance literature, 1 assume rate and the credit expansion is used to finance that the receiver of a payment (i.e. the receiv­ the government deficit is there a lower bound er of a debt service payment or the seller of on the level of the central bank's net foreign bonds) can adjust his/her portfolio at the in­ reserves. In this case the continuous depletion stant the payment is received. of the central bank's net foreign reserves 1 show in this article that the cash-in-ad­ results, sooner or later, in a balance-of­ vance constraint in financial market transac­ payments crisis. However, a balance-of-pay­ tions introduces a lower bound on the central ments crisis should be only a manifestation of bank's net foreign reserves. In addition, it is a government solvency crisis, originating from suggested that the criticallevel of the net for­ a primary government deficit (net of interest eign reserves is zero. The latter result is not, on debt). The reason why in the real world, however, due to the violation of the public sec­ balance-of-payments crises have also occurred tor solvency constraint but rather to the fact without the existence of persistent government that the depletion of the central bank's net budget deficit problems remains to be ex­ foreign reserves below zero decreases welfare. plained. This resuIt is not dependent on the purpose A conclusion which can be drawn on the ba­ for which the credit expansion is used. Hence, sis of Obstfeld's and Buiter's studies men­ a balance-of-payments crisis need not to be a tioned above is that the world which their ana­ manifestation of a publie sector solvency lytical framework describes is too idealized crisis,. (i.e. no kind of friction exists) to be able to Further, unlike in Buiter (1986 and 1987), provide a rational explanation for the exis­ a sufficiently large domestic credit expansion tence of the lower bound on the level of a cen­ always results in the violation of either the tral bank's net foreign reserves. With this con­ public sector or the private sector solvency sideration in mind, 1 extend their analysis by constraint. Finally, 1 show that the effects of assuming a cash-in-advance constraint for as­ the cash-in-advance constraint in financial set transactions. This introduces a form of transactions on the dynamics of the en­ friction into the mode!. dogenous exchange rate regime shift are quite . Although a cash-in-advance constraint is a minor. fairly conventional assumption in commodi­ The paper is organized as folIows. In sec­ ty market transactions, Helpman and Razin tion 2 1 derive a model framework with choice (1985) were the first to introduce this con­ theoretic foundations and in section 3 the in­ straint into financial market transactions.2 In tertemporal public sect~r and economy-wide particular, they analyzed the case of a debtor budget constraints are derived. In section 4 the country running a current account deficit importance of the cash-in-advance constraint whieh has to accumulate foreign currency in for the dynamics of the model is examined. advance in order to repay its foreign debt. In section 5 the issue of the lower bound on Equally well, a cash-in-advance constraint the centnil bank's net foreign reserves is dis­ cussed. The implications of the cash-in­ advance constraint in financial transactions 1 Helpman and Razin (1985) note that it was also for the endogenous shift from the fixed to the recognized in earlier monetary literature that money is absorbed in financial transactions [see e.g. Friedman floating exchange rate regime is studied in sec­ (1974)]. Later, Lucas (1990) points out that a significant tion 6. Finally, a few brief conclusions are proportion of money holdings is managed by financial offered in section 7. intermediaries, which suggests that cash is required for financial market transactions. 114 Finnish Economic Papers 2/1991 - A. Willman 2. The Model framework The representative household maximizes a discounted sum of future instantaneous utili­ 1 employ the full employment, single good ties. Direct instantaneous utility is obtained small open economy mode!. The good from the consumption of the single commodi­ produced and consumed is non-storable. The ty and from real money balances m. model includes two domestic sectors, Le. the To obtain a explicit solution, 1 follow Obst­ household sector and the government sector. feld (1986) and assume that the instantaneous utility function is logarithmic. Hence, the 2.1 The household sector household's utility at time t is:

The special feature of the model is that there 00 are two kinds of cash balances; cash balances (1) U (t) = f e-p(v-l) [log c (v) t needed for financial transactions and cash balances which satisfy the need for goods mar­ +ylog m(v)]dv; Oi-e. This =i*sFp-iD p+ s(y+ g-c) is because domestic debt issuance is always associated with an increase in non-interest­ where M is nominal money balances (equalling bearing cash balances N needed for debt serv­ sm), y is real income, g is real net lump-sum ice. transfers from the government to the represen­ Integrating (5) and imposing the no-Ponzi­ tative household and s denotes the exchange game condition, a(t)e-rt __ O as t-- 00, we rate. As 1 use a single good, small open econ­ note that under perfect foresight the follow­ omy model and the foreign price level is set ing budget constraint must hold: equal to unity, the exchange rate s also represents the domestic price level. Let us define real assets as a = m + n + (7) a(t)2= J e-{f(U)dU[[(I+13)r-i*lFp M~ + F p- dp, where m, n and dp denote real t values of the variables M, N and Dp. Equa­ + (r+e)m+c-y-g]dv tion (3) can now be written as: Without the term associated with the for­

5 Like the simple Clower constraint in goods market eign bonds Fp, equation (7) would be the fa­ transactions, when imposed in continuous time models, miliar condition that the present value of fu­ the cash-in-advance constraint for debt service payments ture expenditure (i.e. consumption, c, plus the in (2a and b) is only an approximation of actual payments. opportunity cost of holding money, (r + e) m) This does not, however, concern us here, because our would be smaller than or equal to the present results can be extended to a discrete-time-version of the value of future receipts, y+ g, plus the initial model. financial wealth a. However, this is also true 6 As all bonds traded are consols of infinite maturi­ in the present case, as the later analysis will ty, the debt-service costs include only interest payments. Allowing maturities of bonds to be finite rather than in­ show that the term multiplying Fp equals . finite, would, in general, make the qualitative results zero. presented in this paper quantitatively stronger but nota­ At each point of time t, the representative tionally the analysis would be more complicated. household chooses the paths of c, m and F p 7 The way in which (2b) is written implicitly assumes which maximize (1) subject to (7), given a(t), that the representative household does not both buy and (or equally subject to (5) and the no-Ponzi­ sell foreign bonds during the same intervaI 't. If we wanted game constraint). to take this possibility into account, the integral of sales Using the maximum principle, the Hamil­ of bonds during the period 't would have to be added to tonian to be maximized is: the right-hand side of the lower row of (2b). 8 A dot (') over a variable denotes its time derivative. In what follows, the dependence of variables on time will (8) H=log c+y log m+A[ra-(r+e)m- be deleted if it is not strictly necessary for c1arity. [(1 + 13)r-i*lFp +y+.g-c] 116 Finnish Economic Papers 2/1991 - A. Willman where t.. is the costate variable, which is inter­ Substituting the right-hand sides qf equa­ preied as the shadow value, in utility terms, tions (13) and (14) for c and m in (17) and us­ of real assets. The implied first-order condi­ ing (15), we end up with the relation: tions (at the interior maximum) are: (18) 1/t.. = [p/(l + y)](a + 1e- {r(u) du (9) oHloc= 1/c-t..=O t [y+ g]dv] = [p/(1 +y)]W (10) oHlom=ylm-t..(r+e)=O where W is the real wealth of the representa­ (11) oHloFp=A.[(1+~)r-i·]=O tive household at time t, when the optimiza­ tion calculations are done. On the basis of (12) i=pt..-oHloa= -(r-p)t.. (15), real wealth evolve~ over time as follows:

Equations (9) and (10) define the paths of (19a) W=(r-p)W, consumption, c, and money, m, in terms of the costate variable t..: or

(13) c(v) = 1/t..(v) (19b) W (v) = W (t)e(r-p)(v-t)

(14) [r(v) + e(v)] m (v) = y/t..(v) Now the consumption function (13) and the demand for money function (14) can be writ­ and (12) defines the path of t.. as follows: ten in the familiar form:

(20) c(v) = [p/(l + y)]W (v) (15) t..(v) = \e{ [p-r(u)]du = [p/(1 + y)] W (t) e(r-p)(v-t) where t..t denotes the value of t.. at time t, the point of time when optimization is carried out. (21) m(v) = [py 1(1 + y)]W (v)/(r + e) It c,an be seen that, if the rate of subjective = [py 1(1 + y)] [W (t)/(r + e)] e(r-p)(v-t) time preference is permanently greater (smaller) than the interest rate r, then t.. con­ The good produced and consumed is non­ verges towards infinity (zero) and c and m storable and, hence, real income y equals converge towards zero (infinity). Only if domestic output. With domestic output re­ p = r(t) is the path of consumption constant. maining at a constant, full employment level If, in addition, the rate of depreciation e is and private consumption being obtained as a constant, then the amount of money also re­ solution of the household's optimization mains unchanged in time. problem, net exports x are obtained from the On the basis of conditions (11) and (6), the identity following relation for the domestic interest rate is obtained: (22) x(t)=y(t)-c(t)

(16) i=(l-a)(r+e)=(1-a)[i'/(1+~)+e] 2.2 The governYrJent sector Equation (16) gives a modified uncovered Besides the household sector, our economy interest parity r~lation. With the cash-in­ inc1udes the central bank (monetary authori­ advance parameters a and ~ equalling zero, ty) and the central government (fiscal authori­ it reduces to the conventional uncovered in­ ty). The profit function of the central bank terest rate parity relation. is specified as follows: Now the costate variable t.. can also be solved in terms of the fundamentals of the (23) P = i(Dp + Dg) + i's(A - M~b) -i'sL + sR model. On the basis of (16), equation (7) where P denotes profit, Dg is the central reduces to: bank's credit to the central government, A is gross and R net official foreign reserves, M~b (17) a= J e- {r(u) dU[[(r+e)m+c-y-g] dv is the part of gross reserves which is held in 117 Finnish Economic Papers 2/1991 - A. Willman a non-interest-bearing form (cash balances), (26) sR+Dp+Dg=M+N +K, or and L is the central bank's foreign currency denominated debt. The central bank's profit (26') sR+ (1-a)Dp+Dg =M+K consists of the net interest income and capi­ taI gains (or losses) on the stock of net foreign where K is the central bank's own capital. K reserves resulting from changes in the ex­ evolves over time as follows: change rate. The central bank's gross foreign reserves A (27) K=sR equal the sum of net reserves and foreign cur­ rency denominated debt (R + L). The condi­ Equation (27) states that capital gains tion A~M~b~O must hold by definition. The arising from exchange rate movements are interest rate on gross reserves in interest­ added to the central bank's own capital. The bearing form as well as on the central bank's rest of the central bank' s profit is paid to the foreign currency denominated debt is the central government. Hence, the central bank world market rate i*. The cash-in-advance and the central government do not possess constraint in debt service states that part of separate budget constraints and the supply of the central bank's gross reserves may be in the money is determined by the relation: form of cash balances. The cash-in-advance constraint is of the following form:

(24) M~b~ -9L+0Fg, In the present model the role of the central government is to distribute lump-sum trans­ where fers to (or collect lump-sum taxes from) the 9= 0, if L=O and 0= 0 ,if Fg~O household sector. lO Its other revenues or out­ = -'d*, if L>O = -1"i*,if Fg 0, all terms, we obtain: gross foreign reserves are in the form of cash (30) balances,,i.e. A = M~b' and in (24) the differ­ R+ Fg + dp= i*(R + Fg) + idp-i*M~b + m ence, M~b - R, can be substituted for L. Now +n +(m+n-dp)e-g equation (24) can be transformed into the form: Finally, the balance-of-payments identity is obtained by expressing first the household sec­ tor budget identity (3) in real terms (i.e. de­ (25) M~b = [9/(1 + 9)] R -1'- [0/(1 + 9)] Fg flated by s) and then adding it to the consoli- The central bank's balance sheet is written as: /0 The model could easily also include government consumption (i. e. the distribution af goods ta tHe private sector via the government sector). However, in our sin­ 9 If we assume. that the cash-in-advance constraint gle good model the publie good should be a perfect sub­ also applies ta purchases afforeign bonds, and for sim­ stitute for the private good in the representative house­

plicity that k (t):5 0, then in the regime with Fg > 0 and hold's utility function and all the results presented in this Fg>O, Mc~~ -OL+Fg(t+T)-Fg(t). paper would remain unaffected. 118 Finnish Economic Papers 2/1991 - A. Willman

dated government sector budget identity (30). It states that the sum of the present dis­ Using identity (22), the baIance-of-payments counted vaIue of government deficits and the identity can be written in reaI terms as follows: drain on publie sector revenues caused by non­ . ... interest-bearing reserves and Iow-interest-rate (31) R=x-(Fg+Fp+M;) debt granted to the private sector plus the + i*(R + Fg +Fp -M~b) present vaIue of future resources appropriated by printing money shouId be smaller than or It is worth noting that equation (31) takes equaI to outstanding net publie assets. into account the fact that part of the foreign It is usefuI to aIso derive the intertemporal reserves, i.e. M~b' is in non-interest-bearing economy-wide budget constraint, which is ob­ form. tained by soIving forward the baIance-of­ payments identity (31). As for the govern­ 3. The intertemporal publie sector ment, it is assumed that the representative consumer does not finance consumption in­ and the economy-wide resource definitely through Ponzi schemes. If solved constraints for the present value of consumption, the in­ tertemporal economy-wide resource constraint All sectoraI flow budget identities and baI­ can be written as follows: ance sheet identities must hoId at any point of time. These identities impIy intertemporaI (35) f e-i'v c(v)dv

- adp - [i -i* -(1-a)e]dp lic sector intertemporal budget constraints (7) and (35). If, for instance, the intertemporal + i*M~b + g] dv publie sector budget constraint is not violated where T>t. We next assume that in the long but the economy-wide budget constraint is run the stock of assets heId by the pub lic sec­ violated, then the intertemporal private sec­ tor cannot be depIeted at a rate faster than the tor budget identity must also be violated. foreign interest rate, i.e.: (33) lim [R(T) + Fg(T) + dp(T)] e-i'T = 0 4. The behaviour of the model T-oo This is the no-Ponzi-game transversality The dynamics of the model presented in sec­ condition and it ruIes out situations where the tion 2 is simplified considerably if we make government deficit is financed through con­ a heroic but rather conventional assumption tinued and cumuIative borrowing. Now the in­ that the rate of subjective time preference p tertemporaI consolidated government sector equals the foreign interest rate i*.l1 budget constraint reduces to: Now the model can be written as follows: (34) R(t)+Fg(t)+dp(t)~ f e-i'v( -(m+me) t II This assumption has been used by e.g. Drazen and Helpman (1985), Obstfeld (1986), Calvo (1987), Claes­ - adp - [i -i* -(1-a)e]dp sens (1988) and Frenkel and Klein (1989). As an excep­ + i*M~b + g] dv tion to this practice, see e.g. Puumanen (1986). 119 Finnish Economic Papers 2/1991 - A. Willman

tween the rate of subjective time preference (36) W (0) = a(O) + f e- {r(u) du (y + gJ dv o (equalling i*) and the real rate of domestic financing costs. Equation (38) is the modified where uncovered interest parity condition, which states that real financing costs arising from a(O) = m(O) -(1-a)~ (0) + (1 + I3)Fp(O) domestic (defined by equation (6» and foreign = dg(O) + R(O) + (1 + I3)Fp(O) sources must be equal. Equation (39) defines the flow of consump­ and 13 = 0, if F p;::: 0 tion and equation (40) the stock of money = -'Li*, if Fp<0 demanded as shares of net life-time wealth. Equation (41) is the money supply equation (37) W(t)= (r-i*)W(t) (28) express ed in real terms. Equation (42) is the public sector flow (38) r=iI(l-a)-e=i*/(1 + 13), budget identity solved for the government lump-sum transfers to the house-hold sector. where a='Li It is obtained from (30) after inserting (41). Together with relations (44) and (45) deter­ (39) c(t)= [i*/(1 +y)]W(t) mining the domestic sector's foreign curren­ cy holdings, the balance-of-payments identi­ (40) m(t) = [i*y/(1 +y)]W (t)/[r+ e(t)] ty (43) closes the model. From the point of view of the dynamics of (41) m(t)=(1-a)dp(t)+dg(t)+R(t) the model it is crucial if the private sector is - [m(t) -(1-a)dp(t) -dg(t)] e(t) the net investor or the net debtor in the for­ eign financial markets. ln the first case, Le. (42) g(t) = dg(t) + dg(t)e(t)-Fg(t) Fp;:::O, the parameter 13=0 and r=i*. ln this + i*[R(t) + Fg(t) + dp(t)] + [i(t)'-i*] dp(t) interest rate regime equations (37) and (39) im­ ply that wealth W (t) and consumption c (t) re­ -i*M~b(t) main constant in time. As long as the depreci­ (43) R(t) = y -c(t) - [Fg(t) + Fp(t) + M;(t)] ation rate e in (40) is constant, this is also the case as regards the stock of money de­ + i* [R(t) + Fg(t) + Fp(t) - M~b(t)] manded. If the private sector is a net foreign bor­ (44) M;(t)=I3Fp(t);I3= O,ifFp;:::O rower, Le. Fp i* .12 time. It is worth noting that a(O) is predeter­ mined by the past history of asset accumula­ tion whereas the variable W (0) and the divi­ 12 With the nation's initia/ net stock of foreign assets sion of a(O)between its components can given, one can seefrom the intertempora/ economy-wide change at time t = 0, if information concern­ resource constraint (35) that in the high-interest-rate ing future developments in the variables on the regime r>i* and Fp0 and, right-hand side of equation (36) changes. hence, consumption is at a tower /eve/ than in the /ow­ interest-rate regime, In addition, on the basis of equa­ Equation (37) defines the change in net pri­ tions (39) and (40); m = [y/(r+ e)] cfrom which it is easy vate wealth as a function of the difference be- to see that m is a/so at a tower /eve/ in the high-interest- 120 Finnish Economic Papers 2/1991 - A. Willman

However, if the economy is initially in the point of view solvency, further depletion high-interest-rate regime, it cannot instantane­ would be possible? ously jump to the low-interest-rate-regime. These issues have previously been studied This is because at least part of the initial finan­ in the light of the first question by Obstfeld cial wealth is in the form of money, which (1986) and Buiter (1986 and 1987). They con­ must always be at a positive level. Hence, as sidered whether a continuous domestic credit foreign assets are the flexible component of expansion results in the violation of the pub­ financial wealth, it is quite possible that lic sector solvency (or intertemporal budget) Fp(O) 0, when R < 0, atherwise i~b = 0. 13 It is easy to see that the Domestic credit can be expanded either in right-hand side of (53) appraaches to minus the form of central bank credit to the central infinity if the growth rate af domestic credit government dg or in the form of credit to the 11 is above the foreign interest rate i', im­ household sector dp. Expansion in dg can be plying the violation af the intertemporal used either far investing in fareign assets F g economy-wide (and hence the private sector) or for lump-sum transfers to households. budget constraint. The explanation for this is Each of these cases is studied in turn. that the present value of the net 1ump-sum Case 1: the central government invests in transfers need ed ta cover the additional in­ foreign assets l;>y bprrowing from the cen­ terest payments becomes minus infinity, a1so tral bank, i.e. dg= Fg. reducing private wealth to minus infinity. The cash-in-advance canstraint in financial If the growth rate of domestic credit 11 market transactions is of importance in the is in the interval 0:511 < i*, the right-hand model as long as R(t»O (or L(t) =0). Far this side of (53) is finite, equalling y -Ili~ddg (0)/ reasan it is canvenient to se1ect the time t = 0 (i' -11). This implies that the maximum sus­ so that the net foreign reserves are depleted tainable level of consumption is the smaller to zero, i.e., R(O)=O. For notational sim­ the closer ta i' is 11. Further, the non-nega­ plicity 1 further assume that Fg(O) = dp(O) =0. tivity af consumption requires that 11 must As the demand for money is con.staqt in the be smaller than i' /(1 + x) where x = i~bdg (0)/ fixed exchange rate regime, R = dg and, y>O. hence, R(t) = -Fg(t). The public sector flow In additian, it can easily be seen from (53) budget identity (46) and the intertemporal and (52) that the level of private consumptian budget constraint (47) are simplified to: decreases if the net foreign reserves R(t) are

(50) g(t) = -i'M~b(t) J3 It can be seen that i;b = 0, if 't = 0 (i.e. no cash-in­ advance constraint). In this case consumption is unaf­ 00 fected by the depletion of the net foreign reserves towards (51) O~ J e-i'![g(t)-i'M~b(t)Jdt minus infinity, o 122 Finnish Eeonomie Papers 2/1991 - A. Willman

frozen at some negative level or if they are this is the same as in case 1, i.e. that the bur­ even temporarily below the zero level. From den caused by the central bank's additional the utility function (1), it can be seen directly interest payments on its foreign debt can be that, as m is a constant share of c, the reduced shifted to the household sector as increased level of consumption implies the lower level lump-sum taxes. of utility. Using the identity R(t) = m(O) - dg(t) = Hence, in summarizing we can conclude dg(O)[I-efl!] equation (56) reduces to that government borrowing from the central i bank to finance its investment in foreign as­ (57) O;:::dg(O)(I+[lim e(fl- ')T-l]J sets does not affect the solvency of the public T-<» sector. This is the result reported by Buiter Equation (57) repeats the result obtained by (1987). Given a cash-in-a~vance constraint i?­ Obstfeld (1986); the public sector solvency financial market transactlOns, however, thlS constraint is violated if the growth rate of the policy worsens the solvency of the private sec­ domestic credit j..t;::: i*, but it is not violated if tor whenever the net foreign reserves have j..tO. simplicity, assume that also Fg(t)=dp(t)= Case 3 (the ease of expansionary monetary Fp(O) =0. poliey): The central bank buys bonds dp is­ Now the public sector flow budget identity sued by the household sector. (46) and the intertemporal budget constraint 1 now assume that dg(t) = Fg(t) = 0 and that (47) can be written as follows: also R(O)=O. Hence, R(t)=(1-a)[dp(O)­ d (t)] where a = ,i. The increase in the (54) g(t)=dg(t)+i'R(t)-i'M~b(t) h~)Usehold sector's domestic credit is used for investments in foreign assets Fp so that (55) 0;::: 1e-i'![g(t)+i'M~b(t)]dt F = - Rand for the accumulation of cash o b~ances N needed for debt service. The pub­ lic sector flow budget identity (46) and inter­ Assume again that dg(t) = j..tdg(t) and insert temporal solvency constraint (47) can be writ­ g(t) defined by (54) into (55) to obtain: ten as:

(58) g(t) = i'R(t) + idp(t) -i'M~b(t) (56) 0;::: 1e-i'![j..tdg(t)+i'R(t)]dt o (59) dp(O);::: 1e- i'! [-adp(t)-(i-i')dp(t) The fact that the term i'M~b(t) disappears o from the right-hand side of (56) implies that + i*M~b (t) + g (t») dt the cash-in-advance constraint in financial market transactions has no effect on the pub­ where i-i*= -,i'2/(1 +,i')

the world market interest rate i'. Assuming there is no lower bound on the net foreign dp(t) = I!dp(t) and inserting (58) into (59), we reserves, if the growth rate of credit expan­ end up with: sion is below the foreign interest rate. Only if the growth rate of credit expansion exceeds co (60) dp(O);::: f e-i'l[ -ul!dp(t) the foreign interest rate is the solvency con­ o straint violated. Under all these policy alter­ + i'[dp(t) + R (t)]] dt natives, the welfare of the household sector remains unaffected. These results are in ac­ = dp(0)[1- U lim e(Jl-i')t] cordance with those obtained by Obstfeld T-co (1986) and Buiter (1987). It can be seen that the right-hand side of The cash-in-advance constraint in financial (60) equals the left-hand side, if I! i'. Hence, stated above in two important ways: Firstly, expansive monetary policy does not result in all three policy alternatives become similar in solvency problems for the government sector. the sense that either the public sector or the Rather, it tends to loosen the solvency con­ private sector solvency constraint is violated, straint (the case where I!;::: i'). This is so in if the rate of domestic credit expansion ex­ spite of the fact that the interest rate on the ceeds a critical magnitude, which is somewhat central bank's growing foreign debt is higher below the foreign interest rate. However, the than the domestic nominal interest rate which depletion of the central bank's net foreign the household sector pays on its debt to the reserves towards minus infinity is still possi­ central bank. The interpretation is that the ble in all three cases, if the rate of credit ex­ pace at which the net foreign reserves dimin­ pansion is below the critical magnitude. The ish is slower than the pace at which the house­ faster rate of credit expansion would not be hold sector's debt to the central bank grows. sustainable with a positive level of consump­ This is due to the fact that, owing to the cash­ tion. Secondly, the cash-in-advance constraint in-advance constraint in debt service pay­ introduces the welfare aspect into the analy­ ments, the leakage from the domestic credit sis; the welfare of households is decreased, if expansion is not complete; a fraction u = 't'i of the net foreign reserves are depleted below the it is used for accumulating cash reserves N. zero level. Especially the latter conclusion is impor­ The intertemporal economy-wide budget tant, because its implication is that a welfare­ constraint reduces into equation (53). Again maximizing government is unwilling to allow the right-hand side of (53) approaches minus the central bank's net foreign reserves to be infinity, if I! > i', implying the violation of depleted below zero. Hence, for a welfare­ the intertemporal private sector budget con­ maximizing government, the critical lower straint. Likewise, our conclusions concerning bound on the net foreign reserves is zero. the welfare effects of unduly expansive mone- . tary policy, which depletes the net foreign reserves below zero, are the same as in cases 1 and 2 and, hence, the depletion of the cen­ 6. The endogenous regime shift from tral bank's net foreign reserves below zero the fixed exchange rate regime to decreases the utility of the representative the jloating exchange rate regime household. We can conclude that without the cash-in­ 1 showed above that, with a cash-in-advance advance constraints in financial market trans­ constraint in financial market transactions, a actions, the sectoral solvency constraints set welfare-maximizing government does not al­ no lower bound on the central bank's net for­ low its official foreign reserves to be depleted eign reserves, if the domestic credit expansion below zero. Within the present framework, is used to accumulate either the central gov­ however, no argument can be found which ernment's (case 1) or the private sector's (case would motivate the government to continue 3) foreign assets. Likewise, in the second case, expansive credit policy and allow the exchange in which the credit expansion is used to fi­ rate to float after the net foreign reserves have nance lump-sum transfers to the households, been exhausted to zero; although consumption 124 Finnish Economic Papers 2/1991 - A. Willman would be unaffected by the exchange rate re­ tral bank's net foreign reserves reach zero. gime shift the increased rate of inflation would The aim is to examine what kind of effects the decrease the demand for money (see equation cash-in-advance constraint has on the size and (40), which on the basis ofthe utility function timing of the speculative attack on the curren­ (1) would decrease welfare.J4 Hence, instead cy associated with the exchange rate regime of allowing the exchange rate regime to shift, shift. 1 also discuss about the case in which the welfare-maximizing government would the cash-in-advance constraint is extended to stop credit expansion at the latest when the include purchases of foreign bonds. 1 start the net foreign reserves have been depleted to analysis by studying the behaviour of the zero. 15 model in the floating exchange rate regime. To fully rationalize the endogenous regime shift from the fixed exchange rate regime to the floating exchange rate regime, an addi­ 6.1 The behaviour of the model in tional argument is required which would moti­ the floating exchange rate regime vate permanentIy expansive credit policy. One In deriving the behaviour of the model in argument for such an inflationary policy could the floating exchange rate regime, 1 closely be that suggested by Barro and Gordon follow the exposition of Puumanen (1986). (1983); if the welfare-maximizing government Assume that the supply of money rule is treats expected inflation as exogenously given, M= J.tM. Hence, the growth in real money a positive rate of inflation can be a property balances is determined as follows: of the time consistent solution. Distortionary taxation could provide an­ (61) riJ./m=J.t-e other,argument for expansive monetary poli­ cy; inflation can also be treated as a tax. In Denote the right-hand side of the demand the literature on seigniorage it has been shown for money function (40) as that attaching a positive inflation rate to dis­ tortionary taxation can be part of the optimal (62) cf>(t) = i*yW(O)/[(1 +y)(i*+ e(t»] tax system. 16 Hence, the extension of the present framework to include distortionary Insert (62) into (40) and differentiate loga­ taxation might result in reasonable justifica­ rithmically. As r = i*, the real wealth in (40) tion for an expansive credit policy and the en­ is constant, Le. W (t) = W (0), and we obtain: dogenous exchange rate regime shift at the moment the central bank's net foreign reserves (63) 1>/cf>=J.t-e have been depleted to zero. Further analysis ofthis issue is, however, beyond the scope of Solve (62) for the depreciation rate e and this article. insert it into (63) to obtain: Instead 1 take expansionary credit policy as given and, on the basis of the analysis of the (64) 1> = (J.t +i)cf>-yi*W(O)/(i+y) previous section, 1 assume that the exchange rate regime shift occurs at the moment the cen- This linear differential equation has an un­ stable root (J.t + i* > 0) and the saddle-path

J4 The property that consumption is unaffected by the stability implies that, if the growth rate of the exchange rate regime shift is due to thefact that the util­ supply of money changes, there is an instan­ ity function (1) is separable in money and consumption. taneous jump in cf> to its new stationary value If it were inseparable, then consumption would also be defined as: affected by the exchange rate regime shift [see Claessens (1988)]. (65) cf>* = yi*W (O)/[(i + y)(J.t + i*)] 15 As long as the central bank's net foreign reserves are positive, it does not matfer from the point of view of welfare how expansive the domestic credit policy iso Hence, as in (63), 1> = 0 the depreciation rate This kind of policy would only change the time pattern e must equal the growth rate of the supply of of the lump-sum transfers without any effects on the money, Le. e= J.t. This implies that real money present value of these transfers. balances remain constant and that the level of 16 See Phelps (1973) and, for instance an overview by the exchange rate is determined as s (t) = Spaventa (1989) and the literature mentioned there in. M(t)/m(t).

-4 125 Firmish Economic Papers 2/1991 - A. Willman 6.2 The timing af the exchange rate where z_indicat~s the instant before the at­ regime shift and the size af tack. As R(z+) = 0, where z+ denotes the in­ the speculative attack stant after the attack, R (z_) equals the size of the attack. The speculative attack has Assume for notational simplicity that the caused the discrete jump in the depreciation central bank's own capital K(O)=O and that rate of the exchange rate from zero to Jl (also in the fixed exchange rate regime the exchange implying a rise in the domestic nominal in­ rate is set equal to unity. Now the money sup­ terest rate). As shown in the previous section, ply identity (28) can be written in stock form this is due to the fact that in the floating ex­ as follows: change rate regime the depreciation rate of the exchange rate equals the growth rate of the (66) M(t)=(I-a)Dp(t)+Dg (t)+R(t); a='ti supply of money M. It)s easyto see from (66) that in the floating exchange rate regime, with It can be seen that domestic credit can be R(t) = 0, M grows at the same rate as Dp, i.e. injected into the economy through two alter­ at the rate Jl. native channels, i.e. through central bank As the level of the exchange rate remains credit to the central government, Dg, or unchanged at time z, equation (40) implies through central bank credit to the household that the real as well as the nominal money sector, Dp. Assume first that the credit ex­ balances drop at time t = z from the level M (0) pansion is created via the household sector to the level [i*/(i*+Jl)]M(O). Now the stock and for simplicity that Dg(O) = O. Specify the of money immediately after the speculative at­ following money supply rule: tack can be written as follows:

(67) Dp(t) = Dp(0) e!1! (70) M(z+)= [i*/(i* + Jl)]M(O) In the fixed exchange rate regime the de­ = [1-'ti(z+)]Dp(z) mand for nominal money balances is constant, On the basis of equation (16), the domestic i.e. M(t)=M(O). Now, .on the basis of (66) nominal interest rate jumps from the level and (67), the central bank's net foreign i = i* 1(1 + 'ti*) to the level i = (i* + Jl)/[1 + 't reserves evolve in the fixed exchange rate re­ (i* + Jl)] at time t = z. This implies that: gime as follows: (71) iliz=i(z+)-i(z_) (68) R(t) =M(O) - [M(O)-R(O)]e!1! = Jl/Ul + 't (i* + Jl)] (1 + 'ti*)J Equation (68) states that within some finite As the denominator on the right-hand side time interval the net foreign reserves have been of (71) is greater than one, the nominal in­ depleted to zero and, hence, there is a regime terest rate rises less than the rise in the depreci­ shift from the fixed exchange rate regime to ation rate of the exchange rate. the floating exchange rate regime. Subtract (70) from (69) and solve for the The condition which connects the fixed ex­ size of the speculative attack to obtain: change rate regime to the floating exchange rate regime is that the exchange rate cannot jump discretely when the regime shift occurs. This is the continuity condition, which the per­ It can be seen that the size of the specula­ fect foresight solution of the model must sat­ tive attack is smaller than the stock-shift in isfy in order to be unique [see Calvo (1977)]. money balances, M, which the first term on Denote by z the timing of the regime shift. the right-hand side of (72) measures. The sec­ At time z, all the remaining net foreign ond term on the right-hand side of (72) is in­ reserves are exhausted to zero. On the basis troduced by the cash-in-advance constraint in of equation (66), the stock of money immedi­ financial market transactions. This constraint ately before the attack is: implies that the rise in the domestic nominal interest rate increases the cash balances, N, (69) M(z_)=M(O)= [1-'ti(z_)]Dp(z) need ed for debt service. This partly compen­ + R(z_) sates for the effect of the diminished demand 126 Finnish Economic Papers 2/1991 - A. Willman for money, M, on the size of the speculative 7. Conclusions attack. What if the cash-in-advance constraint is ex­ In the balance-of-payments crisis literature tended to include purchases of foreign bonds? the analysis of endogenous exchange rate re­ The analysis presented above would not gime shifts is based on the assumption that change at all. The only difference would be there is a lower bound on the central bank's that at the instant the attack and the exchange net foreign reserves below which the reserves rate regime shift occurs, the portfolio shift are not allowed to be depleted. Typically, the from money, m, into foreign bonds, Fp , lower bound of the reserves is postulated to would occur sequentially via foreign cash equal zero. Besides being analytically simple, balances M;; at time z domestic currency this assumption accords well with common­ would be changed into foreign cash balances sense wisdom. A worrying feature of this as­ and at time z + 't in the floating exchange rate sumption is, however, that even in models regime these money balances would be with choice theoretic foundations it is made changed into interest-bearing assetsY ad hoc. Using equations (66) and (68), the exact In this article the basic choice-theoretic timing of the exchange rate regime shift can framework was extended to include cash-in­ be solved from (72). We obtain: advance constraints in financial market trans­ actions. The most important result was that, (73) z = (l/Il)[log [i*M(O)/(i* + Il)] owing to the cash-in-advance constraint, the -log [M(O)-R(O)] depletion of the central bank's net foreign reserves below zero (even temporarily) causes -log [1-(1 +'ti*)'t~iz]] a welfare loss. Hence, for a welfare-maxi­ The last term on the right-hand side of (73) mizing government a lower bound on the cen­ is introduced by the cash-in-advance con­ tral bank's net foreign reserves below which straint in financial transactions and it is nega­ the reserves are not allowed to be depleted. tive. Hence, the cash-in-advance constraint This result does not depend on the way in delays somewhat the timing of the endogenous which the recipients of domestic credit (the exchange rate regime shift. central government or households) use the Assume next that the credit expansion is in­ credit nor is the violation of sectoral solven­ jected via credit to the central government D cy constraint required. Hence, a balance-of­ g payment crisis can occur independently of a and that Dp = O. It is easy to show that in this case the size of the speculative attack equals government sector solvency crisis. the stock shift in the amount of money M and, As to the sectoral solvency constraints, we hence, the term ~iz in (72) and (73) disap­ found that with the cash-in-advance constraint pears. This implies that the size of the specula­ in financial transactions either the household tive attack on the currency is greater and the or the government sector solvency constraint exchange rate regime shift occurs earlier than is violated if credit expansion exceeds a criti­ in the case where credit is expanded through cal magnitude, which was shown to be some­ what below the foreign interest rate. The way Dp • This is due to the fact that the cash-in­ advance constraint concerns the financial in which the credit expansion occurs does not transactions between the private sector and the matter, as far as this result is concerned. How­ central bank but not the transactions made ever, from the point of view of the timing and within the consolidated government sector. the size of the speculative attack on the cur­ Hence, in this respect, the cash-in-advance rency, it does matter whether the credit expan­ constraint in debt service introduces a kind of sion takes the form of central bank credit to asymmetry into the model. the private sector or to the central govern­ ment; if channelled via the private sector the regime shift occurs somewhat later than if credit is channelled via the central government 17 The fact that part of the nation 's financial resources is in non-interest bearing form for the period 1: decreases sector. the wealth of the representative household. This effect In order to fully rationalize on choice - is, however, taken into account at the instant the expan­ theoretic foundations an endogenous regime sive monetary policy rule is announced. shift from the fixed to the floating exchange 127 Finnish Economic Papers 2/1991 - A. Willman

rate regime (or recurrent devaluations), it is Puumanen, K. (1986), »Three essays on money, wealth still necessary to show why an expansive credit and the exchange rate,» Bank ofFinland Series B:41, Helsinki. policy is optima!. It was suggested that dis­ Salant, S.W (1983), »The vuInerability of price stabili­ tortionary taxation together with the cash-in­ zation schemes to speculative attack,» JournaI ofPo­ advance constraint in financial market trans­ litical Economy 91, 1-38. actions may offer one solution to this prob­ Salant, S.W. and Henderson, D. (1978), »Market antici­ lem. To prove this explicitly is a topic for fur­ pations of government policies and the price of gold,» JournaI of Political Economy 86, 627 -648. ther research. Samuelson, P. (1947), »Foundations ofeconomic analy­ sis, Harvard University Press, Cambridge, MA. Spaventa, L. (1989), »Seigniorage: old and new policy issues,» European Economic Review 33, 557 - 563. Willman, A. (1987), »Speculative attacks on the curren­ cy with uncertain monetary policy reactions,» Eco­ nomics Letters 25,75-85. Rejerences Willman, A. (1988a), »Balance-of~payments crises and monetary policy reactions in a model with imperfect Barro, R.J. and Gordon, D.B. (1983), »Rules, discretion substitutability between domestic and foreign bonds,» and reputation in a model of monetary policy,» Jour­ Economics Letters 26, 77 -81. naI of Monetary Economics 12, 101 -121. Willman, A. (1988b), »The collapse ofthe fixed exchange Blanchard, O.J. and Fischer, S. (1989), Lectures on mac­ rate regime with sticky wages and imperfect substitut­ roeconomics, MIT Press, Cambridge, Mass. ability between domestic and foreign bonds,» Euro­ Buiter, W.H. (1986), »Fiscal prerequisites for a viable pean Economic Review 32,1817-1838. managed exchange rate regime: a non-technical eclec­ tic introduction,» Working Paper No. 2041, National Bureau of Economic Research. Buiter, W.H. (1987), »Borrowing to defend the exchange rate and the timing and magnitude of speculative at­ tacks,» JournaI of International Economics 23, 221-239. Appendix 1: Calvo, G.A., (1977), »The stability of money and per­ fect foresight: A comment,» Econometrica 45, 1737- 1739. The internaI dynamics of the model Calvo, G.A. (1987), »BaIance-of-payments crises in a cash-in-advance economy,» JournaI of Money, Cred­ There are two possible interest-rate regimes it, and Banking 19, 19-32. in the model defined by eq~ations (36) - (45) Claessens, S. (1988), »Balance-of-payments crises in a per­ in section 4; the low-interest-rate regime, with fect foresight optimizing model,» JournaI of Inter­ . national Money and Finance 7, 363 - 372. r = i*, and the high-interest-rate regime, with Feenstra, R.C. (1986), »Functional equivalence between r=i*/(l-·d*). Below, 1 show that both re­ liquidity costs and the utility of money,» JournaI of gimes are feasible but that the internai dy­ Monetary Economics 17, 271-291. namics of the mode! pushes the economy to Flood, R.P. and Garber, P .M. (1984), »Collapsing exchange-rate regimes: Some linear exampIes,» Jour­ the low-interest-rate regime. naI of International Economics 17, 113. For notational simplicity, 1 keep as a bench­ Friedman, M. (1974), »A theoretical framework for mark case the case where dg(t) = d (t) = monetary anaIysis; in R. Gordon (ed.), Milton Fried­ Fg(t)=O. This implies that m(t)=R(t) and man 's monetary framework, The University of Chica­ go Press, Chicago. a(t)=R(t)+(1+/3)Fp(t) with /3=-Ti*, if Grilli, V. and Roubini, N. (1989), »Financial integration, F p< 0, and /3 = 0 otherwise. Assume also that liquidity and exchange rates,» Working Paper No. e = O. Now the public sector flow budget iden­ 3088, National Bureau af Economic Research. tity (42) implies that the lump-sum transfers Helpman, E. and Razin, A. (1985), »Floating exchange rates with liquidity constraint in financial markets,» g(t)=i*R(t). JournaI of International Economics 19, 99 -117. Assume that the prevailing interest rate re­ Krugman, P.R. (1979), »A model of balance-of-payments gime is r = i*, implying the steady state solu­ crises,» JournaI of Money, Credit and Banking 11, tion. The definition of the initial stock of 311-325. Lucas, R.E. (1990), »Liquidity and interest rates,» JournaI wealth (36) now gives: of Economic Theory 50, 237 - 264. Obstfeld, M. (1986), })Speculative attack and the exter­ (A1) W(0)=2 R(O)+Fp(O)+y/i* naI constraint in a maximizing model of the balance of payments,» Canadian JournaI ofEconomics XIX, 1-22. We see that the net foreign reserves appear Phelps, E.J. (1973), »Inflation in the theory of publie fi­ on the right-hand side of (A1) multiplied by nance,» Swedish JournaI of Economics 111, 67 - 82. two; the first time as a component of the ini- 128 Finnish Economic Papers 2/1991 - A. Willman tial money stock and the second time as the where b2 = (1 - (i*y)/[r (1 + y)]] > O. Next shift present vaIue of the lump-sum transfers. the initial point of time onwards from 0 to As R = m, the demand for money equation t 1• As wealth grows at the rate r - i*, we ob­ (40) gives in turn: tain:

(A2) W(O) = [(1 + y)/y] R(O)

Solve (A1) and (A2) for R(O) to obtain: Subtract (A6) from (A7) to obtain:

(A3) R(O)=[y/(l+y)] (Fp(O)+y/i*)

U sing the definition a = R + F p' equation which shows that a(t) grows with the growth (A3) can be solved for Fp(O): of wealth and, hence, a(t)/y grows avove the criticallevel set by equation (A4). (A4) Fp(O)=(1-y)a(O)-(y/i*)y The behaviour of the model can now be As (A4) was derived-under the assumption summarized as follows: If initially the share r = i*, the household sector must be a net in­ of financial wealth in total wealth is below a vestor in the foreign financial markets, i.e. criticallevel, the household sector is a net bor­ Fp(O»O. This is, however, possible only if rower in the foreign financial markets. Be­ a(O» [y/(l-y)]y/i*. If this condition is not cause of the cash-in-advance constraint in debt fulfilled, the economy must be in the high­ service, r=i'/(l-,i*), which is above the interest-rate regime r = i' /(1 - ,i') where subjective time preference rate. This results in Fp(O) <0. positive net saving, which is allocated between 1 next show that, aIthough initially the econ­ the money balances and foreign assets. The omy would be in the high-interest-rate regime, foreign debt of the household sector dimin­ the internaI dynamics of the model pushes the ishes and at some point of time the debt is paid economy to the low-interest-rate regime. backand the household sector tends to be­ The initial wealth is now: come a net investor in the foreign financial markets. At that moment, the interest rate drops so that r = i* and the economy settIes (A5) W(O)=a(O)+y/r+ f'" e-rt i*R(t) dt o on its steady state path where the stock of wealth, consumption and the stock of money where R(t)=m(t)=(yi'/[r(1+y)]JW(O) e(r-i')t. demanded stay constant. The stock of foreign After substituting this relation into (A5) and reserves also remains at a fixed level if there integrating the right-hand side, we obtain: are no changes in the supply of domestic credit. (A6) b2W(O) = a(O)+ y/r

129 Economics Leners 25 (1987) 75-78 75 North-Holland

SPECULATIVE ATTACKS ON THE CURRENCY WITH UNCERTAIN MONETARY POLICY REACTIONS

Alpo WILLMAN Bank 01 Finland, SF-OOIOl Helsinki, Finland

Received 22 May 1987 Accepted 24 July 1987

We show that balance-of-payments crises are accompanied by discrete shifts in the exchange rate if in the pre-crisis situation the exchange rate is constrained by set limits and there is uncertainty about monetary policy reactions.

L Introduction

In the recent literature concerning balance-of-payments crises it has been assumed that, once the foreign reserves have hit some criticallower bound, the central bank will, with certainty, abandon its earlier fixed exchange rate target and withdraw from the foreign exchange market. 1 With perfect foresight about the policy rules pursued by the central bank it has been shown that an exchange rate regime shift from a fixed to a floating exchange rate regime is preceded by a speculative attack on the currency and that the exchange rate regime shift occurs smoothly, Le., without a discrete jump in the exchange rate. However, there is nothing forcing the central bank to abandon the fixed exchange rate regime at the moment the foreign reserves have been exhausted to the criticallower bound. The central bank could equally as well change its monetary policy rule consistent with the fixed exchange rate target. In this paper we show that uncertainty about monetary policy reactions affects balance-of-pay­ ments crises, if in the pre-crisis situation the exchange rate is constrained by set limits (a target zone). In this case balarice-of-payments crises are accompanied by discrete shifts in the exchange rate.

2. The model and analysis

We employ the same linear continuous-time small open country model as Flood and Garber (1984) and Obstfeld (1984, 1986). By assurning full employment (exogenous production), purchasing power parity and uncovered interest rate parity the model can be written as

M(t)jS(t) = /3 - aE,S(t), /3, a>O, (1)

M(t) = D(t) + R(t), (2)

1 See e.g., Krugman (1979), Flood and Garber (1984), Connolly and Taylor (1984), Obstfeld (1984, 1986) and Grilli (1986).

0165-1765/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland) 76 A. Willman / Speculative attacks on currency where M(t) is the nominal stock of non-interest-bearing domestic high-powered money, D(t) domestic credit, R(t) the stock of foreign reserves of the central bank, valued in home currency, and S(t) the spot exchange rate. Et denotes the expectation operator conditional on the information available at time t. . Eq. (1) defines the demand for domestic real money balances as a decreasing function of the expected exchange depreciation. Eq. (2) defines the supply of money. If the exchange rate is allowed to float freely and assuming that the exchange rate depends only on market fundamentals, eq. (1) implies the following solution for the exchange rate:

S(t) = ljcx Joo e-/l(T-I}!a EtM( T) dT. (3) I

If, in turn, the exchange rate is fixed so that S(t) = S, eqs. (1) and (2) imply

M(t) = (3S, (4)

R(t) = (3S -D(t). (5)

Let us assume a target-zone-restricted exchange rate the upper bound of which is S. Assume further that, as long as the foreign reserves are above a critical lower limit which we assume to be zero, the central bank causes domestic credit to evolve over time according to the rule i>(t}=p.>O. (6)

Hence, if the exchange rate has moved to its upper limit S, eqs. (5) and (6) imply that the foreign reserves must decline at the rate p., i.e., R(t) = - p.. Of course, this is true if the exchange rate is fixed at any other level within the zone. This implies that any finite stock of foreign reserves is depleted to zero in finite time. Hence, at the moment the foreign reserves have been exhausted to zero, the central bank is forced to abandon the target zone or to change its monetary policy rule so that the condition R(t);;,. 0 is not violated. . In the following we assume that prior to the occurrence of a balance-of-payments crisis the exchange rate has moved to its upper limit S. It is further assumed that, providing that the monetary policy rule (6) is not changed, the exchange rate is allowed to float freely from the moment the foreign reserves have been depleted to zero. This alternative is expected to occur with the probability 'TT. With probability 1 - 'TT the central bank is expected to change themonetary policy rule consistent with the target zone with S as an upper limit, i.e., i>(t)=O. (7)

In this case too the exchange rate can equally as well be said to float freely. Before the occurrence of the policy regime shift, eq. (1) together with the altemative monetary policy rules (6) and (7), imply the following shadow floating exchange rate Se(t):

(8) A. Willman / Speculative attacks on currency 77

The shadow floating exchange rate at time t is the floating exchange rate expected to prevail by investors, if the target-zone-restricted exchange rate regime were to collapse at that instant, i.e., foreign reserves were to be dep1eted to zero. 2 The shadow floating exchange rate rises in time as D(t) grows at the rate p.. The policy regime shift occurs at the point in time when the shadow floating exchange rate equals the upper !imit of the target zone, i.e., Se(T) = S, where T indicates the point in time when the policy regime shift occurs. By this condition 'abnormal' profit or 10ss opportunities are precluded [see Flood and Garber (1984)]. Together with eq. (8), this condition implies the following 're1ation for the timing of the policy regime shift:

T= Ro//L - 'TT1X/f3, (9) where R o = f3S - D(O), equalling the stock of foreign reserves at t = O. It can be easily seen that the smaller 'TT is, i.e., the greater the probability is that the monetary policy rule after the policy regime shift will be (7), the 1ater the regime shift occurs. From eq. (6) it can be easily seen that the time needed to dep1ete the foreign reserves to zero without a specu1ative attack on the currency is Ro/p.. Hence, if 'TT = 0 there is no speculative attack on the currency. With 'TT = 1 eq. (13) reduces to the solution given by F100d and Garber (1984), i.e., T= Ro//L -1X/f3. This is the earliest possib1e point in time when the specu1ative attack and the policy regime shift can occur. What happens to the actua1 exchange rate at t = T? From that instant onwards there is no uncertainty in the mode1 and on the basis of eq. (3),

S(t)=1/IX[O e-/3(T-t)/aD(7') d7', t~ T, (10) t which, after integrating, can be expressed as

S(t) = D(T)/f3 + /L(t - T)/f3 + /L1X/f32 if b(t) = p., t ~ T,

=D(T)/f3 if b(t)=o, t~T. (11)

Eq. (11) implies that with no change in the monetary policy ru1e the exchange rate depreciates at a constant rate /L/f3 from T on. The adoption of the monetary policy ru1e b(t) = 0 implies, in turn, that the exchange rate will remain fixed at the 1eve1 D(T)/f3 from T on. 3 From eqs. (8) and (11) it can be seen that with 0 < 'TT < 1 there is a discrete jump in the exchange rate at t = T. The size of the jump is

S(T)-S=(1-'TT)/LIX/f3 2 if b(t)=/L; t~T,

if b(t) = 0; t~ T. (12)

The jump in the exchange rate can be upwards or downwards. The upward jump (discrete depreciation) occurs, if the central bank adheres to its earlier monetary policy ru1e b(t) = p. and allows the exchange to float free1y. The downward jump (discrete appreciation) occurs if the monetary policy ru1e (7), which is consistent with the fixed exchange target, is adopted.

2 The shadow floating exchange rate defined by (8) is !he relevant shadow rate only after exceeding the lower level of the target zone. . 3 For expositional reasons we assume that this level of the exchange rate is above !he lower limit of the zone. 78 A. Willman / Speculative attacks on currency

It is worth noting that, although investors foresee with certainty that there is a discrete jump in the exchange rate at t = T, the size of the expected jump is zero. This is due to the fact that there is uncertainty concerning the direction of the jump. This uncertainty, however, disappears, if the target zone within which the exchange rate is allowed to move is narrowed to zero. Under this regime of a perfectly fixed exchange rate, the preceding example is transformed as follows: with probability 7T the central bank is expected to permanently withdraw from the foreign exchange market, with no change in the monetary policy rule and with probability (1 - 7T) to change the monetary policy rule to D(t) = 0 retaining the fixed exchange target S(t) = S. The 'shadow floating exchange rate is in this case a weighted average of the freely floating exchange rate with the monetary policy rule D(t) = fL and the fixed exchange rate S. Hence we can write se(t)=7Tla{" e-/l(Hl/"[D(t)+fL('T-t)] d'T+(l-7T)S. (13) t

It is easy to see from (13) that with 0 < 7T

References

Connolly, Michael B. and Dean Taylor, 1984, The exact timing of the collapse of an exchange rate regime and its impact on the relative price of traded goods, Journai of Money, Credit, and Banking 16, May, 194-207. Flood, Robert P. and Peter M. Garber, 1984, Collapsing exchange-rate regimes: Some linear examples, Journal of International Economies 17, 1-13. Gril1i, Vittorio V., 1986, Buying and se11ing attacks on fixed exchange rate systems, Journai of International Economics 20, 143-156. Krugman, Paul, 1979, A model of balance-of-payments crises, Journai of Money, Credit, and Banking 11, Aug., 311-325. Obstfeld, Maurice, 1984, Balance-of-payments crises and devaluation, Journai of Money, Credit, and Banking 16, May, 208-217. Obstfeld, Maurice, 1986, Rational and self-fulfi1ling balance-of-payments crises, American Economic Review 76, March, 72-8l. Economies Letters 26 (1988) 77-81 77 North-Holland

BALANCE-OF-PAYMENTS CRISES AND MONETARY POLICY REACTIONS IN A MODEL WI'nI IMPERFECT SUBSTITUTABILITY BETWEEN DOMESTIC AND FOREIGN BONDS

Alpo WILLMAN Bank of Finland, SF-OOIOI Helsinki, Finland

Received 24 September 1987

We show that if domestic and foreign bonds are imperfect substitutes for each other and the exchange rate is fixed, then, unlike under the perfect substitutability assumption, uncertainty about monetary policy reactions affects the timing of balance-of-payments crises.

1. Introduction

In the literature on balance-of-payments crises it is shown that an anticipated shift from a fixed exchange rate regime to a floating exchange rate regime is associated with a speculative attack on the currency and that the regime shift occurs without any discrete jump in the exchange rate. 1 Willman (1987) showed that in a model with uncovered interest parity, uncertainty about monetary policy reactions in response to a speculative attack does not affect the results obtained under the assumption of perfect foresight. In this paper we show that, if the uncovered interest parity assumption is abandoned, then the size and the timing of the speculative attack is no longer insensitive with respect to uncertainty about monetary policy reactions. In addition, the regime shift from the fixed to the floating exchange rate regime, if it is realized, is associated with a discrete jump in the exchange rate.

2. The model

We employ a fuIl-employment smaIl open economy model with purchasing power parity. Domes­ tie bonds are' assumed to be non-tradeable and foreign bonds tradeable. Domestic and foreign bonds are imperfect substitutes for each other in the portfolios of domestic residents. Foreign reserves are assumed to be in the form of gold or foreign currency. FoIlowing the general convention, we ignore interest receipts as arguments of- disposable income and net foreign interest income in the current account balance. The current account B( t) in real terms can now be express ed as the difference between real domestic output and re~ private and government expenditure. Government expenditure is assumed to be constant. Under these assump­ tions the current account balance depends only on the real interest rate as foIlows:

B(t) = ao + al[r(t) - E,p(t)]; (1)

1 See, e.g., Krugman (1979), Flood and Garber (1984), Connolly and Taylor (1984), Obstfeld (1984), Calvo (1987) and Willman (1988).

0165-1765/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland) 78 A. Willman / Balance-of-payments crises where r(t) is the domestic nominal interest rate and P(t) is the domestic inflation rate. E, denotes the expectation operator conditional on information available at time t. We specify the following equations for the net demand for foreign assets and for the transact~ons demand for money:

F(t) = bo - bI[r(t) - E,P(t)]; bI > 0, (2) r(t) = Co - cI[m(t) - p(t)]; Co and CI > 0, (3) where F(t) is the real value of the stock of foreign assets held by domestic residents, m(t) the logarithrn of nominal money balances and p (t) the logarithm of the domestic price level. In specifying eq. (2) it was assumed that the foreign interest rate stays constant and that the foreign price level equals one. It follows from the latter assumption that the exchange rate can be identified with the domestic price level. Accordingly, in eq. (2) the domestic inflation rate P(t) measures the rate of depreciation of the domestic currency. The demand-for-money eq. (3) has been solved for the domestic nominal interest rate, which depends negatively on the logarithrn of the real money balances. The nominal money balances are assumed to be the exogenous control variable of the monetary authority. After writing the identity for the change in foreign reserves R(t), the model is c1osed.

R(t) = B (t) - P( t). (4)

3. The case of perfect foresight

In thls section our analysis proceeds in three stages. First, the behaviour of foreign reserves is studied in the fixed exchange rate reginIe. Second, the determination of the exchange rate in the flexible rate reginIe is examined. Third, the reginIe shift from the fixed to the floating exchange rate reginIe is studied. This is done under the assumptions of perfect foresight and that the fixed exchange rate regime is abandoned once the foreign reserves have been depleted to zero. Assume p(t) = P and that the central bank pegs nominal money balances at the level m(t) = mo. Inserting now eq. (3) into (1), we obtain

(5)

As we assume that mo > p + (ao + aIcO)/alcV there is a deficit in the current account, Le., Bo < 0. As also p(t) = 0, eq. (4) implies the following relation for foreign reserves:

R(t) = R o + Bot, (6) where R o is the stock of foreign reserves at t = 0. From (6) we see that the foreign reserves are depleted to zero and a balance-of-payments crisis is inevitable. In the flexible exchange rate regime, after setting R(t) = 0, eqs. (1)-(4) and the money supply rule m(t) = mo yield the following second-order equation for the exchange rate:

(7) A. Willman / Balance-of-payments crises 79

Assuming that the exchange rate depends only on market fundamentals, eq. (7) implies the following solution for the exchange rate:

(8) where k is an arbitrary constant multiplying time exponential in the stable root. Essential from the point of view of the timing of the regime shift is that when it occurs, the level of the exchange rate cannot jump discretely. This continuity condition precludes 'abnormal' profit or loss opportunities [see Flood and Garber (1984)]. Denote the time of the collapse of the fixed exchange rate regime by T. With p(T) = P as an initial condition, the arbitrary constant k can be solved and eq. (8) can be written in the form

t> T. (9)

Eq. (9) states that at t = T the exchange rate starts depreciating from the level p towards the level p - BO/alcl' which is the level of the exchange rate that brings the current account into balance. Although there is no jump in the exchange rate in connection with the exchange rate regime shift, there is a jump in the rate of depreciation of the exchange rate. At time T the depreciation rate jumps from zero to P(T) = - Bo/blCl > O. As can be seen from eq. (2), the rate of depreciation at t = T also causes a portfolio shift from domestic assets into foreign assets. This shift, which we denote by JJ.F(T), can be identified as a speculative attack and its size is JJ.F(T) = blP(T) = - BO/Cl' By defiuition the size of the speculative attack equals the amount of reserves lost in the attack. Eq. (6) defines the pre-attack level of reserves. By setting R(T) = JJ.F(T), the exact timing of the attack can be solved. We obtain

(10)

We see that in the case of perfect foresight our results support in many respects those obtained under the assumption of uncovered interest parity. The main merit in adopting the portfolio asset model of the balance of payments is that it introduces the current account balance explicitly into the analysis. The exact timing and the size of the speculative attack depend on all factors affecting the current account balance.

4. The case of uncertain monetary poIicy reactions

The central bank is not, however, forced to abandon the fixed exchange rate regime at the moment the foreign reserves have been exhausted to the criticallower bound. The continuation of the fixed exchange rate regime can be rescued by shrinking the supply of money sufficiently. In this section we study how this kind of uncertainty changes the results obtained in the preceding section. Assume that in the pre-attack fixed exchange regime the central bank follows the money supply rule m(t) = mo. However, prior to the moment the foreign reserves have been depleted to zero, there is uncertainty about whether the exchange rate will be allowed to float freely with no change in the money supply or if the money supply will be changed consistent with the fixed exchange rate regime, Le., m(t) = ml ~p + (ao - alcO)/alcl < mo. The former alternative is expected to occur with probability'TT and the latter alternative with probability 1- 'TT. 80 A. Willman / Balance-of-payments crises

The post-attack expected exchange rate, with expectations formed prior to the speculative attack, is now a weighted average of the freely floating exchange rate determined by eq_ (8) and the fixed exchange rate p, i.e.,

(11) where Tl- indicates the instant before aI).d Tt the instant after the attack. By using the continuity condition Etp(Tt) = p, the constant k can be solved. We obtain k = (BO/alcl) exp( -alTl/bl ). Now, by differentiating (11) with respect to time at t = Tl, we obtain EtP(Tt) = 'TTBo/blCI' Proceeding as in preceding section, this implies that the size of the speculative attack /J.F(TI) = -'TTBO/CI and the exact timing of the attack is

(12)

We see that if 'TT < 1 then Tl > T, and that the c10ser to zero is 'TT (i.e., the smaller is the probability that the exchange rate regime shift is' allowed to take place) the c10ser to zero is the size of the attack and the later it occurs. Assume next that the post-attack policy regime chosen by the central bank is the freely floating exchange rate with m(t) = mo. After the choice has been made there is no uncertainty in the model. This implies that the actual exchange rate is determined by eq. (8). As the stock of foreign assets F(t) has changed into a predetermined state variable, information about its size at t = TI provides us with the initial condition we need in solving k. At t = Tl the post-attack level of the foreign assets F(TI) = Fo + /J.F(Tl), where Fo is the stock of foreign assets in the pre-attack fixed exchange rate regime equalling Fo = bo - bl Co + bl cI (mo - p). On the other hand, eqs. (2), (3) and (8) imply .

(13)

Now as a solution for k we obtain

(14)

Eqs. (8) and (14) imply

(15)

Hence, with 'TT < 1 there is a jump in the exchange rate at t = TI. This jump is greater the c10ser 'TT is to zero. If, instead, the money supply is changed to be consistent with the fixed exchange rate p(t) = p, then there is an upward jump in the nominal interest rate at t = Tl , the size of which is /J.r(TI) = cI(mO - ml), implying an instantaneous capital inflow and the t\.rrning of the current account deficit into surplus. Hence, the simultaneous risk of discrete depreciation and a discrete rise in the domestic nominal interest rate explains why uncertainty about monetary policy reactions affects .the behaviour of investors. Under the assumption of uncovered interest parity this kind of trade-off between risks is lacking. A. Willman / Balance-of-payments crises 81

References

Calvo, Guillermo A., 1987, Balance of payments crises in a cash-in-advance economy, Joumal of Money, Credit, and Banking 19, Feb., 19-32. Connolly, Michael B. and Dean Taylor, 1984, The exact tinrlng of tbe collapse of an exchange rate regime and its impact on the reIative price of traded goods, Joumal of Money, Credit, and Banking 16, May, 194-207. Flood, Robert P. and Peter M. Garber, 1984, Collapsing exchange-rate regimes: Some linear exampIes, Joumal of Intemational Econonrlcs 17, 1-13. Krugman, PauI, 1979, A modeI of balance-of-payments crisis, JoumaI of Money, Credit, and Banking 11, Aug., 311-325. Obstfeld, Maurice, 1984, Balance-of-payments crises and devaluation, JoumaI of Money, Credit, and Banking 16, May, 208-217. Willman, Alpo, 1987, SpecuIative attacks on the currency with uncertain monetary policy reactions, Econonrlcs Letters 25, 75-78. Willman, Alpo, 1988, The collapse of the fixed exchange rate regime with sticky wages and imperfect substitutability between domestic and foreign bonds, European Econonrlc Review, forthconrlng. Scand. J. afEconamics 91 (1),97-116,1989

Devaluation Expectations and Speculative Attacks on the Currency

Alpo Willman * Bank of Finland, Helsinki, Finland

Abstract Balance-of-payments crises are studied in a framework where investors do not know the threshold level of foreign reserves, the attainment of which implies that the central bank abandons its fixed exchange target. Investors are alternative1y risk neutral or risk averse. It is shown that, depending on whether the threshold level is stochastic or fixed but unknown to investors, currency speculation reveals itself as, respective1y, a speculative outflow distri­ buted over a longer time period or a sudden speculative attack on the currency.

1. Introduction There is general agreement that a regime of fixed exchange rates can be sustained only if domestic economic policy isconsistent with it. An exces­ sively expansive monetary policy, for instance, leads to the depletion of foreign reserves, eventually forcing the central bank either to devalue the currency or allow it to float. However, changes in the foreign exchange target have typically been preceded by speculative attacks on the currency. It has been shown in the literature on balance-of-payments crises that speculative attacks on the currency do not contradict the assumption of rational behavior and that the timing of such attacks is foreseeable. 1 These results, however, are derived under the assumption that there is some binding threshold level of foreign reserves, known by everyone, below which foreign reserves are not allowed to be depleted. As long as foreign reserves are above this level, the central bank adheres with certainty to its fixed exchange rate target; but after reserves have been depleted to the

*Many discussions with my colleague Juha Tarkka and comments by an anonymous referee are gratefully acknowledged. ISee e.g. Krugman (1979), Flood and Garber (1984), Grilli (1986), Obstfe1d (1984, 1986a, b), Wyplosz (1986), Buiter (1987) and Calvo (1987).

Scand. J. of Ecollomics 1989 98 A. Willman critical threshold level, the"central bank either devalues the currency or allows it to float. It is doubtful, however, whether any such binding minimum level of foreign reserves exists. A central bank facing a perfect capital market can, at least in principle, create foreign reserves by borrowing. Thus negative foreign reserves are also feasible.2 But even if such abinding threshold level of foreign reserves did exist and was known by everyone, it is very likely that the exchange rate target would be changed well before reserves had been depleted to the binding minimum level. This implies that the actual threshold level can be much higher than the binding minimum leve1.3 Moreover, its size is not known by the public. In this paper speculative behavior associated with devaluation expecta­ tions is studied in a framework where the threshold level of foreign reserves is unknown to public. lts value is determined in three alternative ways: (i) a new value is drawn from a known probability distribution at the end of each period; (ii) a value for the threshold level is drawn from the distribution only once; and (iii) as a combination of these two, a new value is drawn with a probability greater than zero but smaller than one at the end of each period. Investors are alternatively risk neutral or risk averse. The foreign exchange risk is one-sided, Le., there is only risk of a de­ valuation. The paper is organized as follows: An equation for 'the domestic interest rate incorporating the riskpremium is derived in Section II. The processes which determine the probability of devaluation' are defined in Section III and a simple macromodel is specified in Section IV. The speculative behavior arising from devaluation expectations is studied in Section V. Because of the highly nonlinear nature of the model, this' is done by utilizing a numerical simulation technique.

II. Determination of the Domestic Interest Rate Assume a perfect international bond market where domestic and foreign bonds differ only in one respect: their currency denomination. The foreign exchange risk is the only source of uncertainty affecting the portfolio allocation between domestic and foreign bonds. The investors' utility function is assumed ta be of the form (1)

2 See the discussion by Obstfeld (1986a). 3 ~rugman (1979) and Wyplosz (1986) discuss, in a heuristic way, the case where there is a series of possible threshold levels of foreign reserves with probabilities associated with each of them. The analysis in this paper can be regarded as a formalization and extension of their discussion.

Scand. J. of Economics 1989 Devaluation expectations 99 where "W; +I is the noninonetary wealth at the beginning of period t + 1 and a is the measure of absolute risk aversion. The larger the value of a, the more risk averse the investors. Wealth at the beginning of period t + 1 is "W;+ I =(1 + rJH, +(1 + r~ + J}.1+1)F;, (2) where r, and r ~ denote interest rates on domestic and foreign bonds, respectively. H, is the amount of domestic bonds and F, the amount of foreign bonds in the portfolio of a domestic investor during period t. Both are expressed in domestic currency. J}., + I denotes the percentage change in the exchange rate between periods t and t+ 1.4 Substitute (2) into (1) to obtain u( "W;+l) = Uo - UI exp[ - a((l + rJH, +(1 + r~ + J}.,+I)F,)]. (3) The maximizing problem for expected utility in period tis now max E,{uo - UI exp[ - a((l + r,)H, + (1 + r~ + J}.,+I)F,)]), (4) H,F where E, refers to expectations formed at the beginning of period t. The first-order condition of (4) implies

E,[(l + r7 + J}.,+I) exp( - aW,+I)]= E,[(l + r,) exp( - aW,+I)]' (5) Typically, under a regime of fixed exchange rates, the foreign exchange risk is not symmetric. If foreign reserves are continuously depleting and expected to do so in the future as well, then devaluation wilI be much more likely than revaluation. In the following we assume that the probability of revaluation is practically zero and hence we specify

0 with probability n, (6) J}.,+I = [ 0 with probability 1- n,

where 0 ~ n, ~ 1 and 0 is the devaluation percentage, if the central bank decides to devalue. With perfect information about r, and r ~, equations (5) and (6) imply n/r~ + 0 - r,) exp{ - a[(l + r,) W, +(r~ + 0 - r)FJ} =(1- n,)(r, - r~)

X exp{ - a[(l + r,) W, + (r~ - r,)F,]} (7) which, for F" gives F, =(l/ao){log[n/~~ + 0 - r,)] -log[(l- n,)(r, - r~)]}. (8) We see that F, is determined only with values of r, such that r ~ ~ r, ~ r ~ + <5. In the case where r, < r ~ foreign bonds strictly dominate domestic bonds,

4 Ta be precise, /), I + I is (1 + r~) times the devaluatian percentage.

Scand. J. of Economics 1989 100 A. Willman and in the case where r/ < r: + 0 the situation is, of course, reversed. In both of these cases, the risk relevant from the point of view of portfolio diversification is eliminated and hence only the values of r/ in the interval [r:, r* + 0] are admissible. For our purposes in the rest of this paper, we prefer equation (8) to be normalized with respect to the domestic interest rate. We obtain r/=r/* +on/-Z/ (9) where Z/ = n/o[exp(aoF/)-l]/[exp(aoF/)+ nt!(l- n/)]; (10) Z/ is the risk premium. lt is easy to see that Z/ can obtain values only in the interval [0, onJ In the risk neutral case (a ...... O) Z/ ...... O and hence equation (9) implies uncovered interest parity. We also see that if n / ...... 0 or n / ...... 1 then Z/ ...... O. Hence, equation (9) also collapses to the equation of uncovered interest parity in the limiting case of perfect foresight (i.e., n/ can only obtain the values 0 or 1).

111. Probability of Devaluation In the literature on balance-of-payments crises, it is normally assumed that there is a fixed threshold level of foreign reserves known by everyone below which the central bank does not allow reserves to be depleted. The attainment of this threshold level implies either devaluation or a perma­ nent shift from a fixed exchange rate regime to a floating rate regime.s We retain the assumption of a threshold level of foreign reserves. However, to take into account the fact that investors are uncertain about how much of its potential reserves the central bank is willing to use to defend its fixed exchange rate target, we assume that the threshold level of foreign reserves is drawn from a probability distribution. Investors know the probability distribution and its moments, but they do not know the size of the threshold level drawn from that distribution. They only knowthat once foreign reserves have been depleted below the threshold level, the central bank will devalue the currency by 0 per cent at the beginning of the next period. We study the speculative behavior of investors in three informationally different cases. In the first case, the central bank does not possess any informational advantage over the public; it makes a draw from the same probability distribution at the end of each period. If the threshold level

5Willman (1987a and b) studies balance-of-payments crises by taking into account the possibility that the consistency between the fixed exchange rate target and economic policy is restored through a change in monetary poliey.

Scand. J. of Economics 1989 Devaluation expectations 101 drawn is greater than or equal to the level of the foreign reserves attained in the period in question, then there is a devaluation of the currency at the beginning of the next period. Otherwise, there is a new draw at the end of the next period. The process continues until the threshold level drawn is below the level of reserves attained. Assume that the probability distribution from which draws are made is truncated so that the threshold level can obtain values only in the interval

[ - 0() , R II]. This implies that during period t, the probability attached to the occurrence of devaluation at the beginning of period t + 1 is

0, if R,> R" [ II (11) :7i{ = 1- G(R,)/G(RII), if - 0() < R, ~ R where G refers to the distribution function and R, is the level of the foreign reserves in period t. With R, < RII we see that the eloser R{ is to RII, the eloser :7i, is to zero, and the further R, is below RII, the eloser :7i{ is to unity. In the case R LI --+ 0() equation (11 ) reduces to :7i, = 1 - G( R J In the second case, it is assumed that a threshold level is drawn only once (i.e., at the beginning of the game) from a known probability function. With reserves diminishing and without the occurrence of devaluation, this allows investors to leam that the threshold level drawn by the central bank is below the lowest level of the foreign reserves attained up until the beginning of the present period. From the point of view of investors, the situation is the same as if, at the end of each period, the central bank made a new draw from a probability distribution in which the truncation point changes with respect to time so that

R~' =min(R'_l,R~'_I)' (12) Equation (12) states that if, during the previous period, foreign reserves have been depleted below the level corresponding to the truncation point at the beginning of the previous period, then in the present period the truncation point equals the level of foreign reserves at the end of the previous period. Otherwise, the truncation point is the same as in the pre­ vious period. The probability that the currency will be devalued at the beginning of the next period is now .

if R,> R;' (13) if - 0() < R,~ R;' where R;' is determined by (12). There is an important difference between these two cases. In the first case, the threshold level is stochastic and hence depletion of foreign

Scand. J. of Economics 1989 102 A. Willman reserves does not provide investors with any information about the size of the threshold level. In the second case, changes in foreign reserves supply this kind of information. This is because investors now know that the threshold level is a fixed figure although, as in the first case, its actual size is unknown to them. Perhaps more realistic than either of these two cases is the case where investors do not know with certainty if the threshold level adopted by the central ba.Ik is fixed or reconsidered at the end of each period;This case is examined more closely in Section V.

IV. The Model The model presented by Flood and Garber (1984) is widely used in the literature on balance-of-payments crises. Our starting point is the same single-good, full-employment small open economy model. We extend the model in two directions: investors are allowed to be risk averse, and the threshold level of foreign reserves, which triggers devaluation, is unknown to investors.6 Now, the model can be written as

Mt/pt = bo - bj rt (14) Mt=Rt+Dt (15) Dt=Do+ flt fl>O (16) Pt=PtSt* (17) rt = r~ + Et(St+tfSt -1) - Zt (18) Et(St+ tfSt -1) = 1I:tO 0> 0 (19) Zt = 1I:to[exp(aoFt) -l]/[exp(aoFt)+ 1I:t/(1- 1I:t)] a~O (20)

Rt=Rt- j + ptTt-(Ft-Ft- 1) (21) Tt =co +cj[rt - E/Pt+t!Pt -1)] (22)

if R > R~' t (23) if - 00

(24a)

6 The model is, of course, open to the criticism that it has not been derlved from choice­ theoretic foundations with intertemporal budget constraints. Recently, however, the literature on balance-of-payments crlses has also been developed in that direction; see e.g. Calvo (1987), Drazen and Helpman (1987a and b) and Wijnbergen (1987).

Scand. J. of Ecollomics 1989 Devaluation expectations 103 or (24b) where M is the domestic money stock, r the domestic nominal interest rate, p the domestie price level, R the stoek of foreign exehange reserves, D domestic eredit, p* the foreign price level, s the spot exchange rate, r* the foreign nominal interest rate, Z the risk premium, :n: the probability that the eurreney will be devalued at the beginning of the next period, F the stoek of foreign assets held by domestic residents and T the trade balanee in real terms. Equation (14) defines the demand for money and equation (15) the supply of money. Equation (16) states that domestic eredit always grows at the positive eonstant rate fl and (17) defines purehasing power parity. Equations (18) and (20) are the interest rate and risk premium equations derived in Seetion II and equation (19) defines the uneonditional expeeted rate of devaluation. The balanee-of-payments identity defines the ehange in the foreign assets as the differenee between the trade balanee surplus and the ehange in the stoek of foreign assets. Equation (22) defines the trade balanee as a funetion of the real interest rate and equation (23) defines the probability of devaluation with the truneation. point of the distribution funetion G determined alternatively by equation (24a) or (24b).' We assume that p* and r* are eonstant and that as long as the central bank does not devalue, Sr =s. By solving R r from equations (14)-(19), we obtain (25) where f3 = bop* - blP* r* and eI = bIP*. We assume that both f3 and eI are positive. Equations (17)-(19) and (21)-(22) imply the following relation for the stoek of foreign assets Fr=ys-'YjsZt-(Rt-Rt-I)+Ft-l (26) where y= coP*+ c1p*r* and 'Yj = CIP*. The behavior of the model in the fixed exehange rate regime St =Sis now determined by equations (25), (26), (20), (23) and (24a) or (24b). v. Speculative Behavior with an Unknown Threshold Level of Foreign Reserves We now study the speeulative behavior of investors when the threshold level of foreign reserves is assumed to be unknown to investors, Le., it is drawn from a probability distribution. The distribution funetion is assumed to be normal with parameters m and 0 2. Beeause of nonlinearities

Scand. J. af Economics 1989 104 A. Willman in the model, it can only be solved numerically. In our numerical simula­ tion experiments, the normal distribution is approximated by Hastings' best approximation formula.? Hence, we define G(Rt) = ~[(Rt - m)/ a], where ~ refers to the normal distribution function with mean m and standard error a. In the fixed exchange rate regime S t = S = 1, the model (25), (26), (20), (23) and (24a or b) can be rewritten as (27)

Ft - Ft- 1 = Y- 'fjZt -(Rt - Rt- 1) (28)

Zt = önt[exp(aöFt) -l]/[exp(aöFt )- n t(l- n t )] a~O (29) 0, t (30) n = [ 1- ~[(Rt - m)/a]N[(R~ - m)/a]

R~=RU (31a)

R~=min(Rt-b R;/-l). (31b) In all of our simulation experiments, we assumed that f3 - Do = 11 0, a = 500, Ö = 0.1 and Jl = 1. The trend variable t obtained values 1, 2, 3, etc. (except in the combined case, where it obtained values 2, 3, 4, etc.). As a benchmark in choosing the value of m, we use thecase of perfect foresight, i.e., all the probability mass of the distribution function ~ is concentrated on the point m. This point must ,be equal .10 the level of reserves at the end of the period preceding the attack, as thjs is the period when the probability of devaluation jumps from zero to' one. If, in the perfect foresight case, the threshold level were zero, then equation (27) would imply that the pre-attack level of reserves is aÖ + Jl. Hence, we set m = aÖ + Jl = 51. We begin our analysis by assuming that investors are risk neutral while (i) the threshold level is stochastic and (ii) the threshold level is fixed but unknown to investors. N ext, we examine the speculative behavior of risk­ averse investors. The analysis is then extended so that investors are Ull­ certain if the new threshold level is drawn from a known normal distribution or if the threshold level at the beginning of the previous period is stili in force.

7If x- N(O, 1) and y= lxi, then Hastings' best approximation formula for the distribution function of normal distribution is

ifx;;'O ~(x) = {l -F(y) F(y) ifx

SCGnd. J. of Economics 1989 Devaluation expectations 105

Forelgn reserves wlth a - 40 Forelgn reserves wlth a - 20 Forelgn reserves wlth a - i

150~------~------~------~------r------,

20 40 60 BO 100 lima

Probabl1lty of devaluatlon wlth a - 40 Probabl11ty of davaluatlon wlth a - 20 Probabl11ty of davaluatlon wlth a - 1

1.2 - - 1.0 - ,-.-- L - / ~ 0,8 I - , - , 0.6 /! i- , J - / - ,/ 1 0.4 - - 1 - 0.2 A I r- r- // r- ..-' 1 0.0 ~ 20 40 60 BO 100 lima

Fig. 1. Currency speculation with risk-neutral investors (stochastic threshold level).

Scand. J. of Economics 1989 106 A. Willman

The Case ofa Stochastic Threshold Level and Risk-Neutral Investors We assume that investors are risk neutral (the risk premium Z/ = 0) and that the central bank draws a new threshold level of foreign reserves from a known normal distribution at the end of each period. 1n this case the behavior of the model in the fixed exchange regime s/ = 1 is determined by equations (27), (30) and (31a). The mode! was simulated with alternative values of a, i.e., with a= 40, a= 20 and a= 1. RU was set to equallOO; implying that as.long as foreign reserves are above this level, the probability of devaluation is zero. We see from Figure 1 that, independently of the size of a, the cumulative capital outflow caused by devaluation expectations is of equal size, which is ·50 in our examples. However, the smaller ais, the shorter the period over which speculative behavior is concentrated. 1n the case where a= 40, the specu­ lative outflow is distributed quite evenly over a long period of time. Table 1 shows that when a= 20 almost half of, and in the case where a= 1, practically all of the speculative capital outflow is concentrated in a single period. The speculative capitai outflow is greatest in the period where the increase in the probability of devaluation is greatest. This can be seen from Figure 1b, which illustrates the time paths of the probability of devaluation with alternative values of a. As investors are risk neutral, Figure 1b also shows the time paths of the interest rate differential r/ - r 7, so that with .n / = 0, also r/ - r7 = 0 and with .n / -> 1, the differential r/-r7--ö.

The Case of a Fixed but Unknown Threshold Level and Risk-Neutral Investors 1nstead of being stochastic, the threshold level of foreign reserves is now assumed to be fixed. 1ts value is drawn by the central bank from a known truncated normal .distribution before the foreign reserves have been depleted below the truncation point of this distribution. Hence, investors know that the threshold level is fixed but they do not know its values. If

Table 1. Maximum size of the depletion of foreign reserves and its timing with different valuesofa

Size ofthe a Period capitaloutflow

40 38 2.3 20 35 21.2 1 57 50.9 o 60 51.0

Scand. J. of Economics 1989 Devaluation expectations 107

A) a-40 Forelgn reservss Probebl1lty of devaluetlon 150 1.0 r r- 0.8 100 ~ r ...... r- 0.6 ~ r- 50 ~ i-- r- ~ 0.4 r r- o ~ r- 0.2 r r- r j -50 0.0 20 40 60 BO 100 20 40 60 BO 100 B) a-20 Tie Tie Forelgn ressrvss Probab1l1ty of devaluet10n 150 1.0 - :-:.... 0.8 100 ...... - ...... hr--.. 0.6 50 ...... - r , 0.4 r- !---'IJ' - o r 0.2 r r ,-- r- -50 0.0 20 40 60 BO 100 20 40 60 BO 100 Tie Tie C) a - 1 Forelgn ressrves Probebl1lty of dsvaluetlon 150 1.2 r r- r r- 1.0 f- 100 f.... 0.8 r ...... r- r r ~ 0.6 r- "-... f- 50 0.4 r r- r ~ r- 0.2 r ~ f- o 0.0 20 ~O' 60 BO 100 20 ~O 60 90 100 Tie Tie Fig. 2. Currency specu\ation with risk-neutral investors (fixed but unknown threshold level).

Scalld. J. of Ecollomics 1989 108 A. Willman. investors are risk neutral, equations (27), (30) and (31b) determine the behavior of the model. In equation (27) the risk premium z = O. Simulation experiments were carried out with the same values of a as above, Le., with a= 40, a= 20 and a= 1. The initial value of R;'-l was set to equal1 00. The results are shown in Figure 2. Unlike in the case of a stochastic threshold level, the probability of devaluation does not grow continuously as reserves diminish. Rather, there is a'series of speculative attacks, the duration of which is one period. Between the successive attacks, the probability of devaluation is zero. Only before the first attack does the probability of devaluation deviate from zero for a.longer period of time. However, during this period, its size is quite small (in our examples less than 0.1 at its maximum) and more clearly observable, the greater the standard error a. It can also be seen that the sizes of the successive attacks and the probabilities of devaluation associated with their occurrence grow as foreign reserves diminish. The maximum size of the attack is ao, which equals 50 in our examples. In addition, the time intervai between successive attacks increases as the size of the attack grows. This is due to the fact that the next attack occurs after the period where the foreign reserves have attained the level R{ < R;' + fl for the first time. The attack does not occur earlier because investors know that the threshold level of reserves is below R;', Le., the level to which reserves were depleted during the preceding attack. The increase in the magnitude of the successive attacks is due to the fact that the lower the truncation point of the probability distribution, the more concentrated the probability mass is in the neighborhood of the truncation point. Figure 2 also shows that the smaller a is, the earlier speculative attacks attain a size very close to their maximum size ao.

Speculative Behavior with Risk A version We now relax the assumption of risk neutrality. This implies that we need the risk premium equation (29) and equation (28) which determines the stock offoreign assets held by domestic residents. R~ is determined by equation (31a) if the threshold level of foreign reserves is stochastic and by equation (31 b) if it is fixed but unknown to investors: As above, we assume that RU in (31a) and the initial value of R~-l in equation (31b) equa1100. In the simulation experiments shown in Figures 3 and 4, parameters y, 'fJ and a were set to equal 0, 1 and 20, respectively. The initial value of Ft -1 was set to equal 110 and the risk aversion parameter a obtained, alternatively, values of 0.05 and 0.1. Simu­ lations were also run with different values of 'fJ, the size of which depends positively on the size of the interest sensitivity of the trade balance. Our simulation results showed that the size of 'fJ is significant for the timing of speculative behavior. The greater 'fJ was, the earlier speculative capital

Scand. J. af Ecanamics 1989 Devaluation expectations 109

Fore1gn reserves Fare1gn reurvee a = 0 _ _ _ _ a = 0.05 ---a = 0.05 - - - a = 0.1 ----a ~ 0.1

150,.----,----,---"'T'"--...,.----, 150 ...... 100 ...... 50 ~ ~ o ~"" .... ~- -50 20 .. 0 60 80 100 120 Probab111ty of devaluatlon Probabl11ty of devaluatlon a ~ 0 . ___ a ~ 0.05 - - - - a - 0.05 - - - a = 0.1 ----a-0.1

1.0 1.0 - 1'" , - 0.8 0.8 - I } - 0.6 0.6 - i: / - 0.4 0.4 - / e- I I~ 0.2 0.2 e- e- ..J - -~rl 0.0 LP' 0.0 -- 20 40 60 80 100 20 .. 0 60 80 100 120 Rlek pre.1U11 Rlek pre.lu. a = 0.05 ___ a = 0.05 ---- a=0.1 ----a=0.1

0.5 0.20 ~ ~ ., I 0.4 , I ~ II 0.15 0.3 I ~ I ~ .. I, 0.10 0.2 , /'"1: I , ~ 0.05 0.1 ~n ,. .... 0.0 i- V ~ 0.00 .... ~ 20 100 100 120

Fig. 3. Currency speculation with risk-averse Fig. 4. Currency speculation with risk-averse investors (stochastic threshold level). investors (fixed but unknown threshold level).

Scand. J. of Economics 1989 11 0 A. Willman movements started. As speculative behavior resembled that presented in Figures 3 and 4 in other respects, these simulations are not presented here. Figure 3 shows the case with a stochastic threshold level of foreign reserves. We see that the more risk averse investors are (Le., the greater a is), the later the speculative capitai outflow occurs. AIso, the shape of the time paths associated with the probability of devaluation deviates from that of the risk-neutral case (a = 0). The lowest panel of Figure 3 showsdevelopments in the risk premium divided by the size of devaluation 0. As expected, we see that the greater parameter a is, the greater the risk premium. Moreover, the risk premium grows initially but, after attaining its maximum value, it suddenly decreases and starts to converge towards zero. This kind of time pattern is due to the fact that, with probabilities of devaluation c10se to zero and c10se to unity, uncertainties concerning the occurrence of devaluation are small. In our examples, the maximum values of the risk premium are associated with probabilities of devaluation greater than 0.5. With a = 0.05 and a = 0.1, these probabilities are 0.62 and 0.73, respectively. . As regards the domestic interest rate, this kind of time pattern for the risk premium impiies that, up to the point the risk premium attains its maximum value, the interest rate differential between domestic and foreign interest rates grows more slowly than the probability of devaluation but thereafter approaches the size of the expected devaluation Oli/ fairly rapidly. lt can be seen from Figure 4 that, in the case of a fixed but uncertain threshold level of foreign reserves, an increase in risk aversion delays the occurrence of the first speculative attack on reserves. In our examples, when a = 0 (see Figure 3), the first speculative attack occurs in period 40, while when a = 0.05 and a = 0.1, the first attack occurs in periods 58 and 101, respectively. We also see that with low probabilities of devaluation, the time pattern of the risk premium is quite similar to that of uncon­ ditional expected devaluation li/O. However, the risk premium drops permanently to zero in our examples in connection with the first specu­ lative attack. This is due to the fact that, after the first attack, the probability of devaluation drops to zero and hence there is no uncertainty in the model. After foreign reserves have - without speculative capital movements - diminished to the level which they attain when the first attack occurred, the probability of devaluation suddenly jumps c10se to unity. In this case, there is practically no uncertainty in the model either, and hence the risk premium stays at zero.

The Combined Case So .far, speculative behavior associated with one-sided foreign exchange risk has been studied in two polar cases, Le., in the case where the

Scand. J. of Economics 1989 Devaluation expectations 111 threshold level of foreign reserves is a stochastic variable and in the case where the threshold level is fixed but unknown to investors. We now combine these two cases by assuming that, with probability 1 - k, the central bank chooses a new threshold level from a known normal distribu­ tion at the end of each period and that, with probability k, the threshold level is same as at the end of the preceding period. In this combined case, developments in the probability of devaluation are determined by a fairly complicated formula. This formula and its derivation are presented in the Appendix. In the fixed exchange rate regime, the behavior of the model is deter­ mined by equations (27)-(29) and (A3)-(A5). In simulations of the model the parameter k was set to equal 0,9, implying that a new threshold level is chosen with a probability of 0.1 at the end of each period and that the threshold level is the same as in the preceding period with a probability of 0.9. a was set to equa140 and the initial value of Fr _ 1 was set to equal zero. As in the case of a fixed but unknown threshold level, there are specu­ lative attacks on the currency, the duration of which is one period (see Figure 5). After a speculative attack foreign reserves do not, however, return to the level corresponding to the paths of zero devaluation expecta­ tions. This is due to the fact that the term (1- k);n;(R r) in equation (A3) is always greater than zero. The sizes of successive attacks do not grow steadily. Smaller attacks can occur between larger attacks. As a result, the timing of successive attacks is less regular than in the case of a fixed but unknown threshold level. When a speculative attack occurs, the pre-attack level of reserves can be well above the level which reserves attained during the previous attack. This is due to the fact that investors cannot be sure that the threshold level of reserves has not been changed since the reserves attained their previous absolute or local minimum level. We can also see that the behavior of the risk premium is quite different from what it was in the preceding cases. In our present example, the risk premium does not converge or drop to zero with low values of foreign reserves. In the case of speculative behavior with risk aversion, we found that values of the probability of devaluation close to zero or close to unity were assodated with values of practically zero for the risk premium. In the present case, the probability of devaluation never drops to zero and it can obtain values close to unity only during periods of speculative attack. This explains why the risk premium in Figure 5 is above zero all the time. It can be seen that the risk premium also rises during periods of speculative attack. This, in turn, reflects the fact that probabilities of devaluation assodated with these attacks are not close enough to unity.

Scand. J. of Economics 1989 112 A. Willman

Fora1gn reserves wlth speculat10n Farelgn reaervea wlthaut apaculatlan

150r-----~----~----,_----._----_r----_r----~

20 40 60 80 100 120 140 lime

Probabl1lty of davaluatlon Rlak pre.lu.

1.0 - - - 0.8 - - - 0.6 1- r-- l- 0.4 I- 1-- l- 0.2 I ~ - A.A ~L.- -{~V - ~/r I - ,,\,- _1\ J __ 0.0 --~:" ~\ 1- -- 20 40 60 80 100 120 140 lime Fig. 5. Currency speculation with risk-averse investors (the combined case).

Scand. J. of Economics 1989 Devaluation expectations 113

VI. Summary and Concluding Remarks Previous studies of balance-of-payments crises have shown that in a fixed exchange rate regime, speculative attacks on the currency can result from rational behavior. This result, however, was derived in a framework where investors have perfect information about the threshold level of foreign reserves, the attainment of which implies that a central bank abandons with certainty its fixed exchange rate target and devalues the currency or, alternatively, alIows it to float. Owing to this perfect information assump­ tion, a speculative capital movement occurs only at the point in time (or in the period) immediately preceding the shift in the exchange rate regime. Hence, it is not able to explain the empirical fact that speculative attacks on the currency quite often occur in anticipation of a devaluation which does not materialize or materializes much later.. In this paper we relaxed the assumption of perfect information about the threshold level of foreign reserves which triggers devaluation of the currency. Speculative behavior was studied under alternative assumptions about the process which determines the threshold level of foreign reserves. In alI cases, investors were alIowed to be risk averse. Hence, we were also able to study changes in the risk premium associated with speculative capitai movements. In the first case, the level was assumed to be stochastic, Le., a new value for the threshold level was obtained from the known probability distribu­ tion in each period. In this case the central bank did not possess any infor­ mational advantage over the public. In the second case, the threshold level of foreign reserves remained fixed over time. However, investors did not know its value. AlI they knew was the probability distribution which had generated the threshold level. The third case was a combination of the first two cases. With a certain probability, a new value for the threshold level of reserves was generated at the end of each period. If the new value was not drawn from the known probability distribution, then investors knew that it remained the same as at the beginning of the preceding period. There were drastic differences in speculative behavior between the first two cases. In the case of the stochastic threshold level of foreign reserves, steadily growing domestic credit expansion resulted in a speculative outflow of capitai distributed over several periods of time, whereas in the case of a fixed but unknown threshold level, there were repeated single period attacks on the currency with zero probability of devaluation between the attacks. Hence, in the latter case, if the devaluation did not materialize, reserves were rebuilt in the period folIowing the attack. No restrictive policy measures by the central bank were calIed for. This kind of behavior results from the fact that, as there is no devaluation, investors know that the threshold level must be lower than the level to which the attack depleted reserves. The probability of devaluation drops to zero and

Scand. J. of Economics 1989 114 A. Willman stays at that level until the reserves have depleted to the level attained during the preceding attack. Then there is a new attack, the size of which is greater than the size of the preceding attack. In the third case, which is a combination of the previous two, there were also successive attacks on the currency but now the probability of devalua­ tion did not drop to zero between attacks. Successive attacks did not steadily increase in size, unlike in the second case, and new attacks could also occur well above the level towhich foreign reserves were depleted by the previous attack. How did risk aversion affect speculative behaviour? We found that the greater risk aversion is, the longer it took - Le., the lower the level to which reserves fell - before speculative capital movements started to play a major role. We also found that the risk premium converged towards zero, the eloser to zero or unity the probability of devaluation approached. Our main conelusion is that speculative behavior is quite sensitive to the specification of the process which produces the critical lower level of foreign reserves. This may be stochastic, fixed but unknown to investors, or any combination of these two. One area for further research, which would reduce our ignorance about this matter, would be to inelude the utility function of the central bank in the analysis and study the problem in a genuine game-theoretic framework.

Appendix. Determination of the Probability of Devaluation in the Combined Case '

Denote the probabiIity af devaluation by n(Rt) if the threshold Ievel is drawn from a known normaI distrib)ltion at the end of each period and by n(Rt, R:') if the threshold IeveI is drawn from a known truncated normaI distribution with R:' as the truncation point. Assume that under the fixed exchange rate regime S t = s there has not yet been any speculative attack on the currency. This impiies that R t - 1 is the minimum value of R untiI the beginning of period t. The probability that the currency will be devalued at the beginning of the next period is now simply (Al) After the first attack the' situation becomes more complicated. Assume that there was an attack in the period t -1 - n and that R t _ 1 is the minimum vaIue of R in the ll intervaI [t - n, t -1]. n is an integer such that n ~ O. Now, with probability k , the threshold IeveI is the same in period t as in period t - n and, with probability (1- k"), a new threshold IeveI has been drawn. If the new value has been drawn, then at the beginning of period t, investors know that its value is smaller than R t - 1• The probability that the currency will be devalued at the beginning of the next period is now

ll n(t)=(I-k)n(Rt )+k[k 'n(Ru R:')+(I- k"')n(Rt, Rt - 1)]

Scand. J. of Economics 1989 Devaluation expectations 115 with

n = {n,_1 + 1, ifR;'-1

(A2)

Formula (A2) is not general, however, beeause it does not take into aeeount the possibility that, between the time interval t - n -1 and t - 1, there may have oeeurred smaller attaeks on the eurreney associated with alevei 'of reserves lower than R'_I but greater than R;'. Denote the absolute minimum level of foreign reserves by R;'o and all sueeessive loeal minima by R;" sueh that R~'-t ~ R;'i~ R'_I (i = 1, ... , q). This eondition states that an earlier loeal minimum eontains informa­ tion whieh is not inc1uded in later loeal minima only as long as it is smaller than all later Ioeal minima until period t. If we denote by no , and ni t the numbers of the periods elapsed sinee absolute and loeal minimum vaiues wer~ equal to the level of reserves at the beginning of the period, then the probability of devaluation ean be written in general form as " :rc(t) =(1- k):rc(R,) + k [k"Il":rc(R,; R;'O)+ i~1 k"I":rc(R,; R;")

with

ifR,_I R'-2 and R;''-I =R'".'.-I'

_ {no,'-l + 1, if R;'~l ~ R'_I no,,- °, if R;'''-I > R'_I

ni ,_1 + 1, if R;''-l ~R'-J n, = ' (A3) ',' {0, if R;''-J > R'_I

and q is any positive integer whieh is great enough to take into aeeount the number of all possible sueeessive Ioeal minimum values of reserves. It is easy to see that with R;'o= R;"= R'_I relation (A3) reduees to (A2) and that with R;'o< R"/= R'_I itreduees to (Al). After defining

:rc(R,) = 1- ~[(R,- m)/al (A4)

Scand. J. af Ecollomics 1989 116 A. Willman

if-co

References

Buiter, WilJem H.: Borrowing to defend the exchange rate and the timing and magnitude of speculative attacks. louma! of Intemationa! Economics 23, 221-39, 1987. Calvo, Guillermo A.: Balance of payments crises in a cash-in-advance economy. loumal of Money, Credit and Banking 19 (February), 19-32, 1987. Drazen, AlIan & Helpman, Elhanan: Stabilization with exchange rate management. Quarterly lauma! ofEconomics CII, 835-55, 1987a. Drazen, AlIan & Helpman, Elhanan: Stabilization with exchange rate management under uncertainty. Working Paper No. 2268, National Bureau of Economic Research, 1987b. Flood, Robert P. & Garber, Peter M.: Collapsing exchange-rate regimes; some linear examples. loumal of Intemational Economics 17, 1-13, 1984. Grilli, Vittorio, Y.: Buying and selling attacks on fixed exchange rate systems. loumal of Intemational Economics 20, 143-56, 1986. Krugman, Paul: A model of balance-of-payments crises. louma! of Money, Credit and Banking 11 (August), 311-25, 1979. Obstfeld, Maurice: Balance-of-payments crises and devaluation. louma! of Money, Credit and Banking 16 (May), 208-17, 1984. Obstfeld, Maurice: Speculative attacks and the external constraint in a maximizing model of the balance-of-payments. Canadian loumal ofEconomics XIX (February), 1-22, 1986a. Obstfeld, Maurice: Rational and self-fulfilling balance-of-payments crises. American Economic Review 76, (March), 72-81, 1986b. van Wijnbergen, Sweder: Fiscal deficits, exchange rate crises and inflation. Working Paper No. 2130, National Bureau of Economic Research, 1987. Willman, Alpo: Speculative attacks on the currency with uncertain central bank reactions. Economics Letters 25,75-8, 1987a. Willman, Alpo: Balance-of-payments crises and monetary policy reactions in a model with imperfect substitutability between domestic and foreign bonds. Mimeo, 1987b (forth- coming in Economics Letters). - Wyplosz, Charles: Capital controls and balance of payment crises. loumal of Intemational Money and Finance 5,167-179,1986.

First version submitted April 1987; final version received April 1988.

Scand. J. of Economics 1989 European Eeonomie Review 32 (1988) 1817-1838. North-Holland

THE COLLAPSE OF THE FIXED EXCHANGE RATE REGIME WITH STICKY WAGES AND IMPERFECT SUBSTITUTABILITY BETWEEN DOMESTIC AND FOREIGN BONDS

Alpo WILLMAN* Bank of Finland, SF-0010l Helsinki, Finland

Reeeived September 1985, finaI version reeeived Deeember 1986

The paper extends the reeent Iiterature on balanee-of-payments erises by allowing reaI effeets to be eonneeted with them. The eoIIapse of the fixed exehange rate regime and the dynamies associated with this regime shift are studied. ln addition, the effeets of a fiseaI expansion under the fixed exehange rate regime are studied, taking into aeeount the faet that the viabiIity of the regime is poIiey dependent.

1. Introduction A fixed exchange rate regime is sustainable only if domestic economic policy is consistent with it. Permanently more expansive monetary policy than elsewhere in the world results in a balance-of-payments crisis and the collapse of the fixed exchange rate regime. Since the study of Krugman (1979), a growing literature has emerged concerning balance-of-payments crises and the collapse of the fixed exchange rate system [see, e.g., Flood and Garber (1984), Connolly and Taylor (1984), and Obstfeld (1984, 1986a, b), Grilli (1986) and Buiter (1986)]. Common to this literature is the fact that links between real economic developments and balance-of-payments crises are not the focus of interest. This results from the assumption of perfectly flexible wages and prices and uncovered interest parity. In this paper we study balance-of-payments crises in a framework in which both fiscal and monetary policy measures have real effects. We assume that in the goods market domestic and foreign goods and in the asset market domestic and foreign bonds are not perfect substitutes for each other. We further assume that agents operating in the financial markets (including the foreign exchange market) are rational and that adjustments in these markets are instantaneous. In the g~ods market, however, adjustment is assumed to

*WhiIe writing this paper 1 have benefited from diseussions with Juha Tarkka. ln addition, 1 wouId Iike to thank an anonymous referee for heIpfuI eomments.

0014--2921/88/$3.50 © 1988, EIsevier Seienee PubIishers B.V. (North Holland) 1818 A. Willman, The collapse 0/ the fixed exchange rate regime be slow due to inertia in the labor market. In accordance with the practice adopted in the literature mentioned above we also assume that, once the foreign reserves have hit some critical lower bound, the central bank will with certainty withdraw from the foreign exchange market, i.e., there is a regime shift.from the fixed exchange rate to the floating rate regime.I The paper is arranged as follows. In section 2 the model and alternative wage formation schemes are presented. In section 3 the dynamics of the model are studied with· unchanged ·fiscal and monetary policy rules. In section 4 the effects of the anticipated and unantidipated policy changes are studied.

2. The model 2.1. The goods and financial markets Consider a small open economy which specializes in producing a single good but which consumes two goods: home and imported goods. The home good and the imported good are imperfect substitutes for each other. The economy is small in the sense that it faces a given foreign interest rate and given prices of imports. There are two tradeable assets, domestic and foreign bonds, which are imperfect substitutes for each other. There are two moneys, domestic and foreign money, which are nontradeable. The target of monetary policy is to peg the domestic. interest rate. This impiies a perfectly e1astic supply of money. Now the model may be written as

y(t) =lXo[s(t) - p(t)J -lXI [r(t) - p(t)J + g(t) , (1)

p(t) = 8w(t) + (1-8)s(t), (2)

g(t) =Yo -YIP(t), (3)

B(t) = fJo + fJI[S(t) - p(t)J - fJ2y(t), (4)

F(t) =

R(t) =B(t) - F(t). (6)

IIn practice, there are at least two kinds of uncertainties faced by investors. First there is no fixed critical lower bound to the foreign reserves. 1t would be more realistic to assume that the critical level of reserves is distributed acording to some probability ·distribuiton. Secondly, investors cannot be certain about the reactions of the monetary or fiscal authority the moment the reserves hit the criticallower bound. The consistency of the fixed exchange rate target can be attained through a change in monetary or fiscal poliey. The effects of this latter kind of uncertainty on the behavior of investors is studied in a paper by Willman (1986). A. Willman, The eol/apse af the fixed exchange rate regime 1819

All parameters in eqs. (1)-(5) are positive. ln addition, the size of the parameter 8 is restricted to the interval [0,1]. Variables y, s, p and w are the logarithms of the domestic output, the exchange rate, the domestic price level and the wage rate, respectively. As the price of imports in foreign currency is assumed to equal one, s also represents the log of the domestic currency price of imports. The variable g is the index of fiscal policy, which includes both net transfers and government expenditure. The variables rand r* are nominal domestic and foreign interest rates, respectively. In our analysis they are assumed to stay constant. B represents net exports, F net foreign assets held by domestic residents and R the stock of foreign reserves. B, F and R are expressed in terms of foreign currency. A dot over a variable indicates the time derivative. Eq. (1) is a conventional textbook IS curve. Eq. (2) is a markup pricing assumption: domestic goods prices are a weighted average of wages and the prices of imported inputs to production measured in domestic currency units . . Note that the relative price of imported final goods and imported inputs is assumed constant and for convenience it is set to equal one. Eq. (3) is based on the idea that some items in the government budget are constant in real terms and some items are constant in nominal terms, e.g.; interest payments on the public debt. A rise in the price level decreases the real value of the public debt and hence interest payments in real terms. As other items in the government budget are constant in real terms, a rise in the price level implies a tightening in fiscal policy. Eq. (4) states that net exports depend positively on the relative price of the imported and home goods and negatively on d.omestic activity. Eq. (5) is the portfolio-balance equation for the net. demand for foreign assets with constant absolute risk aversion [see, e.g., Dornbusch (1983)]. ldentity (6) defines the change in foreign reserves as the difference between net exports and the net private capital outflow. Interest income in the current account is neglected, because its inclusion would complicate the analysis without any essential effects on the results. To facilitate the analysis in the sections below we substitute eqs. (1)-(3) into (4) and obtain

(7)

where bo=f3o+lY.dJ2r-f32Yo; b1 =8(f31-IY.0f32-f32Yl); b2=f32Yl; b3=1Y.1f32' The sign of the parameters bo and b1 are ambiguous whereas the sign of the parameters of b2 and b3 are unambiguously positive. ln our analysis we assume that the parameter b1 is positive. If in eq. (3) the parameter Yl were zero, this assumption would equal the conventional Marshall-Lerner con­ .dition. Otherwise it is a somewhat stronger assumption than the Marshall­ Lerner condition. 1820 A. Willman, The collapse of the fixed exchange rate regime

2.2. Wage formation In this paper we study the collapse of the fixed exchange rate regime under three alternative assumptions on wage formation. The first is the assumption of a fixed nominal wage rate. In the other two formulas the nominal wage rate is rigid in the short run but flexible in the long runo However, while the second formula is Keynesian in the sen se that it is backward looking, the third formula is forward looking. We work with the following backward looking wage formation scheme '

t w(t)=j1. J e/L(t-t)p(T)d(r), j1.>0, (8) -00 which after differentiating with respect to time can be written in a more conventional form

w(t) = j1.[p(t) - w(t)]. (9)

Eq. (9) can be interpreted to imply that employees have a fixed real wage target, which in expression (8) is assumed to equal unity, but because of inertia in the labor market or lags in indexation of .wages to price movements the nominal wage rate cannot instantaneously adjust to changes in the price level [see, e.g., Sachs (1980)]. However, in the long run equilibrium w(t) = p(t). Our forward looking wage formation scheme is the overlapping contract ' model by Calvo (1983). The special feature in this .mode! is that the contract length is not fixed 'but varies among contracts. We write

00 w(t) = j1. J e/L(t-t)p(T) deT), (10)

In eq. (10) w(t) represents the contract wage rate of new and renewed contracts at the time t. If wåge contracts are made, as we assume, between employers and individual employees, then w(t)in (10) represents the marginal labor cost of production .and hence is the relevant wage concept in our price equation (2). Like eq. (8), eq. (10) also implies that in the long run equilibrium w(t) = p(t). Due to the forward looking nature of eq. (10) there is, however, an important difference between eqs. (8) and (10). In eq. (10), if a new policy affecting the future values of p(t) is announced at any time t, then w(t) is free to take any value, i.e., it can jump. In eq. (8) or equally in (9) w(t) is a predetermined variable. However, as long as the policy announced also A. Willman, The callapse af the fixed exchange rate regime 1821 remains unchanged in eq. (10) w(t) is continuous. This impIies that eq. (10) has finite right hand time derivatives at all points, Le., also at points where announcements of new policy changes are made. Differentiating (10) with respect to time we obtain

w(t) = ,u[w(t) - p(t)J. (11)

We can see that the dependence of a change in the wage rate on the present wage rate in the forward looking equation (10) is the reverse of what it is in the backward looking equation (8). After substituting eq. (2) into (9) and (11) we obtain

w(t) = m[s(t) - w(t)], (12a)

w(t):= m[w(t) -s(t)J, (12b) where m = ,u(1-0). Eq. (12a) corresponds to eq. (9) and eq. (12b) corresponds to eq. (11).

3. The dynamics of the model In this section our analysis proceeds in three stages. We first study the dynamics of wages and foreign reserves under a fixed exchange rate regime, secondly wages and exchange rate determination under a floating exchange rate regime, and thirdly the collapse of the fixed exchange rate regime and the dynamics of the model associated with the regime shift.

3.1. The dynamics ofwages andforeign reserves in thefixed exchange rate regime In the case of a fixed nominal wage rate we den ote w(t)=w, where w is constant. In the case of sticky backward looking wages the level of the wage rate depends on the whole history of s(t). If the fixed exchange rate regime s(t)=s covers the whole past history of s(t), as we assume, then from eqs. (8) and (2) also w(t)=s. However, this is not so in the case of sticky forward looking wages. In the fixed exchange rate regime eq. (12b) implies

(13) where Wo is the log of the wage rate at t=O. We can see from eq. (10) that Wo > s if it is known that the exchange rate will depreciate in the future. Let us denote by s' and s* the levels of the fixed exchange rates which in the case of the fixed nominal wage rate and in our two cases af the 1822 A. Willman, The collapse of the fixed exchange rate regime flexible wage rate, respectively, would produce B(t) =0. Eq. (7) implies s' = (b 1w-bo)/(b 1 +b2) and s*= -bo/b2 • In the fixed exchange rate regime the trade balance equation (7) can now be written in the form

(14a)

(14b)

(14c)

Eq. (14a) defines the trade balance in the case of the fixed nominal wage rate, eq. (l4b) in the case of sticky backward looking wages and eq. (l4c) in the case of sticky forward looking wages. If s' > s in eq. (14a) and s* > s in eq. (14b) then there is a deficit in the trade balance. If in eq. (14c) both s* and Wo are greater than s, then the trade balance is unambiguously in deficit. However, as we shall see later in section 3.3.3, Wo is dependent on sand s* and for Wo > s it is necessary that s* > s. Under the fixed exchange rate regime, eq. (6) reduces to

R(t)=B(t). (15)

After inserting (14a), (14b) and (14c), in turn, into (15) and solving we obtain

(16a)

R(t) =Ro +bz(s-s*)t, (16b)

mt R(t) =Ro + bz(s-s*)t + c(wo -s)(1-e ), (16c)

where c=bt!m+b3 0>0, and R o is the stock of foreign reserves at time t=O. In eqs. (16a) and (16b) the reserves diminish linearly whereas in (16c) the reserves diminish exponentially.

3.2. Wage and exchange rate dynamics in the flexible exchange rate regime When the exchange rate is floating, the change in the foreign reserves is zero and hence eq. (6) reduces to

B(t)=F(t). (17)

Differentiating eq. (5) with respect to time and inserting it and eq. (7) into (17), we obtain A. Willman, The collapse of the fixed exchange rate regime 1823

Together with alternative wage assumptions, eq. (18) defines the dynamics of the model in the flexible exchange rate regime. The case of the fixed nominal wage rate: 1n this case the system (18) collapses into

(19)

If we denote by Å.1 and Å. 2 the characteristic roots of the equation cor­ responding to the homogeneous part of the second order differential equation (19), we see that Å.1Å.2=-(b1+b2)N<0 and Å. 1+Å.2=-(1-8)b3N<0. Hence the model has the saddle-point property: there is one stable root (Å. 1) and one unstable root (Å.2)' The particular or steady state solution of eq. (19) is s'=(b1w-bo)/(b 1 +b2 ), which in section 3.1 was assumed to be greater than s. 1n the exchange rate literature with perfect foresight it has become standard to assume that the solution for the ·flexible exchange rate depends only on the market fundamentals. This implies that the arbitrary constant multiplying the time exponential in positive root is set equal to zero. The solution of (19) can be written in the form

(20) where A 1 is an arbitrary constant. As Å.1 is negative the exchange rate converges in the long run towards the long run equilibrium level s'. The case of sticky backward looking wages: 1n this case the pair of equations (18) and (12a) determines the exchange and wage dynamics. Utilizing trial solutions of the form s(t) = A e"t and w(t) = DeM, the homo­ geneous part of this two equation differential equation system can be transformed into

(21)

The characteristic equation of (21) is

(22)

1n eq. (22) the coefficients multiplying Å. 3 and Å. 2 are positive. Depending on the size of m the coefficient (b 1 + b2 - mb 3) can be negative or positive. The term mb 2 is positive. As shown in the appendix, this jnformation implies that 1824 A. Willman, The collapse of the fixed exchange rate regime eq. (22) has one positive (,13) and two negative roots (,11 and ,12). As the particular solution of (18) and (12a), we obtain s* = w* = - bo/b 2 , which on the basis of eq. (14) is greater than s. Eq. (21) also combines the arbitrary constants A and D so that D=mA/(m+Å.). By setting the arbitrary constant multiplying the time exponential in positive root to equal zero, the saddle­ point solution of eqs. (18) and (12a) can now be written in the form

(23)

(24)

We see that both s(t) and w(t) converge towards s* when t~oo. The case of sticky forward looking wages: Eq. (18), together with eq. (12b), defines the dynamics of the system. Following the method used in the backward looking case the homogeneous part of this system of two differential equations can be presented in the form corresponding to eq. (21). The only difference compared to eq. (21) is that the signs preceding the parameter m change. This impiies that in the present case the characteristic equation is of the form

(25)

We can see that the coefficients cp, (b 1 +b2 +mb3) and mb 2 are unambi­ guously positive. Only the sign of the coefficient muItiplying ,12 is ambiguous. This information implies that eq. (25) has one negative root (,11) and two roots with positive real parts (see appendix). The particular solution is the same as in the backward looking case, i.e., s* = w* = - bo/b 2 • The saddle­ point solution of eqs. (18) and (12b) is now

(26)

(27)

Eqs. (26) and (27) also imply that in the long run equilibrium s( t) = w( t) = s*.

3.3. The collapse of the fixed exchange rate regime and the dynamics of the model In section 3.1 we assumed that as a result of trade balance deficit there was a continuous decumulation of the reserves in the fixed exchangerate regime. In the introduction we also assumed that once the foreign reserves have been depleted to some critical lower bound there is a permanent shift A. Willman, The callapse af the fixed exchange rate regime 1825 from the fixed exchange rate regime to a floating exchange rate regime.2 For convenience we assume that the critical level of the foreign reserves is zero and known by everybody. An essential requirement for connecting the fixed exchange rate regime to the post collapse rate regime is that the exchange rate cannot jump discretely when the regime shift occurs. This is the continuity condition, which the perfect foresight solution of our model must satisfy in order to be unique [see Calvo (1977)]. Following Flood and Garber (1984), it is easy to give to this condition an economic interpretation. If we denote the collapse time by T, s( T) cannot be smaller than s. If s( T) < s the domestic currency would appreciate discretely at T. In this case it would he profitable for agents to sell foreign assets at the moment before the collapse. However, this would increase the foreign reserves and the fixed exchange rate would survive. The exchange rate cannot he greater than s either. An upward jump would provide those speculators who attack foreign reserves at T with profits which would accrue at an infinite rate. This gives speculators an incentive to attack prior to T. This, however, is in contradiction with the assumption that the collapse occurs at T. We can conclude that s(T)=s. We know on the basis of eqs. (16a), (16b) and (16c) that the stock of foreign reserves at T preceding the speculative attack is

R(T)=Ro+(b1 +b2)(s-s')T, (28a)

R(T) = Ro + bis-s*)T, (28b)

(28c)

We also know that at time T, when the exchange rate regime shift occurs, there is a jump in the rate of depreciation from zero' to s(T). If s(T) >0, this jump causes a portfolio shift from domestic assets to foreign assets. On the basis of eq. (5) the size of this shift is

LJF(T) =

It is this portfolio shift which depletes the foreign reserves to zero, and hence LJF(T) must be equal to R(T) as determined by eqs. (16a), (16b) or (16c). Eqs. (20), (24) and (26) define the rate of depreciation in the cases of fixed

20ne can argue that a central bank facing a perfect world capital market can always create foreign exchange reserves by borrowing. However, as Obstfeld (1986a) has shown, the government sector faces an intertemporal budget constraint analogous to that of the private sector, if the government (like individuals) is prohibited from incurring new debt to meet the interest payments due on its previous borrowing. In our case this would imply that, when the foreign reserves turn negative, the government should raise taxes to service the external debt it incurs. If this is not done the central bank is no longer able to peg the exchange rate. 1826 A. Willman, The collapse of the fixed exchange rate regime nominal wages, sticky backward looking wages and sticky forward looking wages, respectively. We next study the timing of the collapse of the fixed exchange rate and the dynamics of the model in each of thes~ three cases.

3.3.1. The case of the fixed nominal wage rate Using s(T)=s as the initial condition, eq. (20) can be written in the form

s(t) =(s-s') eA1 (t-T) +s', t;;; T. (30)

Differentiating (30) with respect to time at t = T we obtain

(31)

After substituting (31) into (29) and utilizing the condition LJF(T) =R(T) the collapse time can be solved, i.e.,

(32)

After stating 21 and s' in terms of the structural parameters of the model, eq. (32) can be written

(33)

It is easy to see that the smaller is bo and the greater are b3 and cjJ the sooner the collapse of the fixed exchange rate regime occurs. However, the signs of the partial derivatives 8Tj8b1 and 8Tj8b2 are not unambiguous. The dynamics of the model in the fixed exchange rate regime and in the post-collapse floating rate regime is shown in fig. 1. At time T the exchange rate starts depreciating and the domestic price level rising towards their long run equilibrium levels (fig. la). However, because of the fixed nominal wage rate the equilibrium price level is below the equilibrium exchange rate s'. The time path of the real wage rate, which is the mirror image of that of the domestic price level, is shown in fig. 1b. Until T the domestic production is constant but at time T there is a discrete upward jump in production (fig. lc). This is due to the jump in the real interest rate caused by the jump in the depreciation rate of the exchange rate. As a result of the relative price movements, domestic production continues to increase towards the new higher level. Figs. ld and le show how the foreign reserves at time T are depleted discretely at zero and the net purchase of foreign assets increases by the same amount. The stock of foreign reserves remains at zero from time T onwards while the stock of foreign A. Willman, The collapse of the fixed exchange rate regime 1827

Figure 1 a Figure 1 d

R s(t) s'

5

T T t

Figure 1 b Figure 1 e

w-p F

T t

T t

Figure 1 c Figure 1 f

y B

T

T Fig. 1. The collapse of the fixed exchange rate regime with a fixed nominal wage rate. assets reconverges to its pre-collapse level when the depreciation rate slows down to zero. As is shown in fig. lf, the trade balance deficit widens discretely at the moment of the speculative attack. This results from the jump in domestic production at time T. Thereafter the trade balance deficit diminishes towards zero as the exchange rate depreciates towards its long-run equilibrium level.

3.3.2. The case of sticky backward looking wages At the moment the exchange rate regime shift occurs there are now two 1828 A. Willman, The collapse of the fixed exchange rate regime predetermined variables, Le., the exchange rate and the wage rate. Using s( T) = w( T) = s as initial conditions, eqs. (23) and (24) can be written

t;?; T, (34)

t;?; T. (35)

Differentiating eq. (34) with respect to time at point t= T, we obtain

(36)

As s(T) in (36) is positive there is a speculative attack on the currency which instantaneously depletes foreign reserves at t = T. Eqs. (28a), (29), (36) and the condition R(T) =L1F(T) imply now that

(37)

In eq. (37) we know that ,,1,1 and ,,1,2 are negative, but we do not know exactly how they are related to the structural parameters of the basic model. We know, however, that they are independent of the parameter bo, which contains the monetary policy and fiscal policy shift parameters rand Yo, respectively. Since s*= -bo/b 2 it can be easily seen that 8T/8bo>O, Le., the more expansive monetary and fiscal policy is the earlier the exchange rate regime shift occurs. Subtracting (34) from (35) we obtain

t;?; T. (38)

It is easy to see that at time T eq. (38) equals zero and with t> T it is negative but converges to zero when t-H/J. Together with eq. (2) this implies that when the exchange rate regime shift occurs the domestic price level adjusts more slowly than the exchange rate to their new long-run equilibrium level s* (see fig. 2a). Hence the relative price of domestic goods in terms of importables decreases temporarily. The same is true for the real wage rate (see fig. 2b). Since there is no permanent change in relative prices, the long-run equilibrium is attained completely through fiscal contraction (see fig. 2c). As in the case of the fixed nominal rate, domestic production increases at the beginning of the floating exchange rate regime (see fig. 2d). These positive effects on domestic production are transmitted through the real interest rate A. Willman, The collapse af the fixed exchange rate regime 1829

Figure 2 a Figure 2 c

S,p,W 9

s*

~ ------

T T t

Figure 2 b Figure 2 d p-s y w-p

T t

~ ------

T . t Fig. 2. The coJlapse of the fixed exchange rate regime with sticky backward looking wages. and the relative priee ehannels. However, in the longer run these effeets die out and as a result of the fiseal eontraetion domestic produetion eonverges to a level which is lower than the level of produetion under the fixed exehange rate regime. 1n the fixedexehange rate regime the time paths of the foreign reserves and the trade balanee are similar to those presented in figs. ld and f. The variable F(t) also behaves as shown in figure le at time T and in the long runo However, in the medium term there may be some differenees between adjustment paths in these two eases.

3.3.3. The case of sticky forward looking wages With s(t) = sas an initial eondition, eqs. (26) and (27) ean be written in the form

s(t) =(s-s*) e·

w(t) = [m/(m-A,!)J(s-s*) eA1(t-T) +s*, t~ T. (40) 1830 A. Willman, The collapse time ofthefixed exchange rate regime

T

Fig. 3. The callapse time af the fixed exchange rate regime.

Differentiate (39) with respect to time at t= T to obtain

(41)

Eqs. (28c), (29), (41) and the condition R(T)=AF(T) imply now

(42)

Eq. (42) contains two variables which are determined endogenously in the model, i.e., T and wo. However, we know that at t= T the wage rate yielded by eq. (13), which ;defines movements in w(t) in the fixed exchange rate regime, and the wage rate yielded by eq. (40) must be equal. There is no jump in the wage rate at time T. The jump in the wage rate occurs earlier, i.e., immediate1y it becomes known that the fixed exchange rate regime will collapse. This is easy to see from eq. (10). We obtain mT wo=s+Å1(s-s*)e- /(m-Å1). After substituting this relation for Wo into (42) and dividing (42) with the terms (s*-s) we obtain

Solving T from eq. (43) cannot be done analytically. However, in fig. 3 the solution of (43) is presented graphically. The value of T corresponding to the intersection of the straight line Ro/(s*-s)+Å1 [cjJ-c/(m-Å1)]-b2 T and the mT curve [dd(m-Å1)] e- is the solution of eq. (43). As s* = -bo/b 2 and .Ål and c are independent of bo we can see that in fig. 3 only the location of the straight line is dependent on the size of bo. The smallel- is bo the further to the left is its location. As bo contains the monetary and fiscal policy parameters rand Yo this implies that the looser is the fiscal-monetary policy A. Willman, The collapse of the fixed exchange rate regime 1831

Figure 4 a Figure 4 d

p,w,S 9

g* ------5 F----j-J pItI

T T'

Figure 4 b Figure 4 e

w-p y p-s

y* ------

T t

Figure 4 c Figure 4 f

r-p B

T t

T t Fig. 4. The coIlapse of the fixed exchange rate regime with sticky forward looking wages. mix in the pre-collapse fixed rate regime the earlier the exchange rate regime shift occurs. Fig. 4presents' the dynamic time paths of the variables of the model. The time paths of R(t) and F(t) are exc1uded because they resemble those shown in fig. 1. However, instead of being linear as in fig. ld, the time path of R(t) is exponentially decreasing in the pre-collapse fixed exchange rate regime .. In fig. 4a, we can see how the wage rate and the price of domestic production already start to adjust towards the long-run equilibrium level s* in the fixed exchange rate regime. The rise in the nominal wage rate increases until T after which it starts to decrease. This can be seen by differentiating 1832 A. Willman, The collapse af the fixed exchange rate regime eqs. (13) and (42) twice with respect to time at t= T. The first time derivatives of these equations are equal but the second time derivative implied by (13) is 2 m 21(s-s*)/(m-2d, which is positive, and the second time derivative implied by (40) is m2I(s-s*)/(m-21), which is negative. Fig. 4b shows how the real wage rate w(t)-p(t) and the relative price p(t)-s(t) increase until T and thereafter start to converge towards zero. As a result of developments in inflation, the deviation of the real interest rate from the nominal rate of interest widens in the fixed exchange .rate regime (fig. 4c). At time T there is an upward jump in the rate of inflation and hence a downward jump in the real interest rate. From T onwards the inflation rate starts to converge towards zero and hence the real interest rate starts to converge towards the nominal rate of interest. Fig. 4d shows how, as a result of a rising domestic price level, the government 'budget policy becomes more restrictive. It is assumed in fig. 4e that the price effects on domestic production, which are transmitted through relative prices and the government budget, dominate those which are transmitted through the real interest rate channel. This implies that until T domestic production decreases. Because of the downward jump in the real interest rate, there is an upward jump in production at t= T. After that, if the real price effect is strong enough, domestic production may continue to grow for some time but later, reflecting the rising real interest rate and the fiseal contraction, production starts to deere ase. In the long run production decreases to the level which is consistent with the real wage and external balance requirements. As a result of the continuous loss of competitiveness, in fig. 4f the trade balance deteriorates in the pre-collapse fixed exchange rate regime. At time T, reflecting the jump in production, there is a further discrete deterioration in the trade balance. From T onwards the trade balance starts to converge to zero.

4. Macroeconomic policy and the collapse of the fixed exchange rate regime What are the effects of macroeconomic policy under the fixed exchange rate regime? The answers given in the literature to this question are deficient since they are based on analysis which does not take into account the fact that the viability of the fixed exchange rate regime is not independent of the policy pursued. However, the framework we have introduced in this paper takes this into account. In this section we examine the effects of expansionary macroeconomic policy in cases where the policy change is assumed to be permanent or temporary. In addition, the policy change can be unanticipated or antici­ pated. Because in our framework the effects of monetary and fiscal policy are A. Willman, The collapse ofthefixed exchange rate regime 1833

quite similar, we assume that it is the fiscal policy shift parameter Yo in eq. (3) which is changed.

4.1. The effects of a permanent fiscal expansion We examine first the effects of a permanent and unanticipated change in fiscal policy. If the reference path is chosen so that it is consistent with the fixed exchange rate target s(t) = 8, the effects of the change in Yo can be studied with the help of figs. 1, 2 and 4. If the nominal wage rate is fixed (fig. 1) or the wage formation scheme is backward looking (fig. 2) an unantici­ pated permanent rise in Yo implies that at the time of the implementation of the new policy there is an upward shift in g(t) and a downward shift in B(t) to the levels plotted in figs. 1 and 2. As a result of the trade balance deficit, the foreign reserves start to diminish leading to the collapse of the fixed exchange rate regime at t= T. Fig. 4 shows the dynamic adjustment paths of the variables of the model, when the wage formation scheme is forward looking. At the moment an unanticipated permanent change in Yo is announced and implemented (at time t=O in fig. 4) the variables w and p jump from the level sand the variables g, y and B from the levels g*, y* and zero, respectively, to the levels plotted in fig. 4. If the wage formation scheme is backward looking or the nominal wage rate is fixed, the effects of a permanent fiscal expansion are similar irrespective of whether it is unanticipated or anticipated. The announcement of the policy change, if announced before the implementation, has no effects on the variables of the model. From the point of view of the effects of the change in policy, it is the timing of the implementation of the new policy that matters. However, if the wage formation scheme is forward looking, there is an upward jump in the wage rate and in the price of the domestic production at the moment the permanent future fiscal expansion is announced. This can easily be seen from eq. (10). The announcement of the permanent fiscal expansions makes it known to everybody that the fixed exchange rate regime s(t) =8 will collapse and that thereafter the exchange rate will start to depreciate towards the level s*. Hence at the moment the fiscal expansion is announced the wage rate and the price of domestic production also jump . upward and start to rise towards s*. The dynamic adjustment paths of the variables w, p, s, g, y and B are shown in fig. 5. In fig. 5, the policy change is announced at t = 0, implemented at t = t 1 and the collapse time of the fixed exchange rate regime is T. At t=O the upward jump in p causes downward jumps in g, y and upward jump in B, i.e., the announcement of the future fiscal expansion affects economic activity contractively at the moment the fiscal expansion is announced. This contraction continues and through re1ative price develop- 1834 A. Willman, The collapse of the fixed exchange rate regime

w,P,s y

y*

p(t)

o T t o t

9 B

g* o

o T Fig. 5. The effects of an anticipated permllnent fiscal expansion. ments deepens until t = t 1> when the fiscal policy change is implemented. At that point of time g and y jump above their long-run equilibrium levels g*

and y* and the trade balance moves into deficit. From t 1 onwards the cause of development is similar to that in the case of an unanticipated fiscal expansion.

4.2. The efJects of a transitory fiscal expansion If the reference path is the fixed exchange rate equilibrium path (i.e., B(t) =0), then a transitory fiscal expansion need not lead to the collapse of the fixed exchange rate regime. That is the case if the initial stock of foreign reserves is great enough. All that happens is that at the time the fiscal expansion is implemented domestic production rises temporarily above its reference path, the trade balance temporarily shifts into deficit and, corres­ ponding to the cumulative trade balance deficit, the stock of the foreign reserves decreases to a lower level. The effects are similar irrespective of which one of our three wage formation schemes is assumed to be in force or whether the temporary fiscal expansion is anticipated or not. A. Willman, The collapse of the fixed exchange rate regime 1835

S,p y

T1 0 t o t 1 T1 To

9 B

t o t 1 T1 To 0 .,..----- ...... so(t) 1'/ i'

Fig. 6. The effects of an anticipated temporary fiscal expansion.

What are the effects of a temporary fiscal expansion if the reference path alsa leads to the collapse af the fixed exchange rate regime (i.e., B(t)

5. Summary and concluding remarks Previous studies on balance-of-payments crises have utilized highly simpli­ fied frameworks. In these studies no real effects are associated with balance­ of-payments crises. This is due to the assumption of perfectly flexible prices and wages. Since, in addition, the current account balance plays no role in these studies, the only reason for balance-of-payments crises is excessive supply of money. In this paper the assumption of imperfect substitutability between domestic and foreign bonds inplies that it is the accumulating trade balance deficit which causes the balance-of-payments crisis. Hence it is not just monetary policy but rather the fiscal, monetary and incomes policy mix which has importance in the analysis. We further assumed that domestic goods are not perfect substitutes for foreign goods and that nominal wages are not perfectly flexible. These assumptions connect economic policy with real economic developments and balance-of-payments crises. , In our analysis we assume.d' that at the moment the foreign niserves hit some criticallower bound known by everyone the central bank abandons the fixed exchange rate 'target and allows the exchange rate to float. The timing of the collapse of the fixed exchange rate regime and the dynamics related to this exchange rate' regime shift were studied under the three alternative assumptions of wage formation. Nominal wages were assumed alternatively fixed, sticky and backward looking or sticky and forward looking. Our model also, allowed us to study the effects of an expansionary economic policy in the fixed exchange rate regime while taking into account the' fact that the viability of the regime is policy dependent. We found that, if the initial stock of fore1gn reserves is great enough, a temporary policy change does not lead to the collapse of the fixed exchange rate regime. Hence, conventional stabilization policy, which includes recurrent temporary policy changes in both the expansive and the contractive direction, is possible without giving rise to expectations concerning the exchange rate regime shift. However, an expansive and permanent policy change, if it causes the trade balance deficit, resuts in the collapse of the fixed exchange rate regime. In addition, if the wage formation scheme is forward looking, this future shift in the exchange rate regime also causes inflationary A. Willman, The collapse ofthefixed exchange rate regime 1837 development from the moment the expansive policy change is announced. Inflation accelerates until the shift in exchange rate regime occurs.

Appendix: The signs of the characteristic roots The case of sticky backward looking wages: Rewrite the characteristic eq. (22).

(A.l) wh~re cP, b3(1- 8) +mcP and mb 3 are unambiguously positive but the sign of the term b 1 + b2 - mb 3 is ambiguous. Descartes' rule of signs implies that eq. (A.1) has one and only one positive real root. We denote it by ..1. 3, We next show that the real parts of two other roots ..1. 1 and ..1.2 are negative. Write the third degree polynomial (A.l) in the form

(A,1) and (A.2) imply that

..1. 1..1. 2 =mb2/..1. 3 cP >0,

..1. 1 +..1.2 = -b3(1-8)/cP-m-..1.3 <0.

Since ..1. 1..1.2> ° and ..1. 1 +..1. 2< ° the real parts of the roots ..1. 1 and ..1.2 are negative. The case of sticky forward looking' wages: Rewrite eq. (25)

(A.3) where cP, b1 +b2 +mb3 and mb 2 are unambiguously positive. Only the sign the term b3(1-8)-mcP is ambiguous. Descartes' rule ofsigns impiies that eq. (A,3) has one and only one negative real root, which we denote by ..1. 1, After transforming (A.3) into the form (A.2) we get

..1. 1..1. 2..1. 3 = -mb2 /cP,

..1. 1..1. 2 +..1.1..1. 3 +..1.2..1. 3= -(b1 +b2 +mb3)/cP, implying

..1. 2 ..1. 3 = -mb2 /..1. 1cP>0,

..1.2 +..1.2 = (mb 2/..1. 1 -b1 -b2 -mb3)/..1. 1cP>0.

We conc1ude that the real parts of the roots ..1. 2 and ..1.3 are positive. 1838 A. Willman, The collapse of the fixed exchange rate regime

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B:1 Valter Lindberg National Income in Finland in 1926-1938. 1943. 185 p. In Finnish with German summary.

B:2 Matti Leppo Der private und der öffentliche Anteil am Volkseinkommen. 1943. 104 p. In German.

B:3 T. Junnila The Property Tax as a Supplementary Tax on Funded Income. 1945. 183 p. In Finnish with English summary.

B:4 Mikko Tamminen The Fluctuations in Residential Building and Their Causes in the Towns of Finland during the Time of Independence. 1945. 281 p. + appendix. In Finnish with English summary.

B:5 T. Junnila - G. Modeen Taxation of Physical Persons in Finland in 1938 and 1945. 1945. 82 p. In Finnish.

B:6 Heikki Valvanne Taxation of Corporations in Finland in 1938-1945. 1947. 105 p. In Finnish.

B:7 Yngvar Heikel Development of the Industry of Finland in 1937-1944. A Research on the Basis of the Balances of the Industrial Companies. 1947. 158 p. In Swedish with English summary.

B:8 T. Junnila Inflation. VoI. 1 - Inflation, Its History, and How It is Explained by the Theory of the Value of Money. The Inflation in Finland in 1939-1946. 1947.304 p. In Finnish.

B:9 Mikko Tamminen Foreign Exchange Rates and Currency Poliey. VoI. 1. 1948. 218 p. In Finnish.

B:10 Heikki Valvanne State Income and Expenditure; Thrnover on Cash Account. A Research PIan and Its Application on the Years 1945-1947). 1949. 117 p. In Finnish.

B:11 K. O. AIho The Rise and Development of Modern Finnish Industry in 1860-1914. 1949. 240 p. In Finnish.

B:12 Reino Rossi The Interest Rate Policy of the Bank of Finland in 1914-1938. 1951. 327 p. In Finnish with EngIish summary.

B:13 Heimer Björkqvist The Introduction of the Gold Standard in Finland in 1877-1878. 1953. 478 p. In Swedish with English summary.

B:14 Ole Bäckman Clearing and Payments Agreements in Finnish Foreign Trade. 1954. 92 p. In Finnish. B:15 Nils Meinander The Effect of the Rate of Interest. 1955. 310 p. In Swedish with English summary.

B:16 Veikko Halme Exports as a Factor in the Trade CycIes of Finland in 1870-1939. 1955. 365 p. In Finnish with English surnmary.

B:17 Reino Rossi The Finnish Credit System and the Lending Capacity of the Banks. 1956. 191 p. In Finnish.

B:18 Heikki Valvanne Budget Balauce in the Macroeconomic Theory of Budgetary PoIicy. 1956. 194 p. In Finnish with English summary.

B:19 Heimer Björkqvist Price Movements and the Value of Money in Finland during the Gold Standard in 1878-1913. A Structural and Business Cycle Analysis. 1958. XII + 391 p. In Swedish with EngIlsh summary.

B:20 J. J. Paunio A Study in the Theory of Open Inflation. 1959. 154 p. In Finnish and English ..

B:21 Ahti Karjalainen The Relation of Central Banking to Fiscal PoIicy in Finland in 1811-1953. 1959. 183 p. In Finnish with English surnmary.

B:22 Pentti Viita Factor Cost Prices in Finnish AgricuIture and Industry Compared with International Market Prices in 1953-1958. 1959. 155 p. In Finnish with English summary.

B:23 Jaakko Lassila National Accounting Systems. 1960. 92 p. In Finnish.

B:24 Timo Helelä A Study on the Wage Function. 1963. 186 p. In Finnish with English summary.

B:25 Jaakko Lassila The Behaviour in Commercial Banks and Credit Expansion in InstitutionaIly Underdeveloped Financial Markets. 1966. 172 p. In Finnish with English summary.

B:26 Lauri Korpelainen The Demand for Household Furniture and Equipment in Finland, 1948-1964. 1967. 139 p. In Finnish with English surnmary.

B:27 Henri J. Vartiainen The Growth in Finnish Government Revenue due to Built-in Flexibility and Changes in Tax Rates, 1950-1964. 1968. 216 p. In Finnish with English summary.

B:28 Pertti Kukkonen Analysis of Seasonal and Other Short-Term Variations with Applications to Finnish Economic Time Series. 1968. 136 p. In English.

B:29 Markku Puntila The Assets and Liabilities of the Banking Institutions in Finnish Economic Development, 1948-1964. 1969. 116 p. In Finnish with English summary.

B:30 J. J. Paunio A Theoretical Analysis of Growth and Cycles. 1969. 80 p. In English. B:31 Ahti Molander A Study of Prices, Wages and Employment in Finland, 1957-1966. 1969. 119 p. In English.

B:32 Kari Nars Foreign Exchange Strategies of the Firm. A Study of the Behaviour of a Sample of Finnish Companies under Exchange Rate Uncertainty 1970-1977. 1979.214 p. In Swedish with English summary ISBN 951-686-054-0, and in Finnish ISBN 951-686-063-X

B:33 Sixten Korkman Exchange Rate Poliey, Employment and External Balance. 1980. 133 p. In English. ISBN 951-686-057-5.

B:34 Peter Nyberg Emigration, Economic Growth and Stability. A Theoretical Inquiry into Causes and Effects of Emigration in the Medium Term. 1980. 135 p. In Swedish with English summary. ISBN 951-686-058-3

B:35 Hannu Halttunen Exchange Rate Flexibility and Macroeconomic Policy in Finland. 1980. 189 p. In English. ISBN 951-686-064-8

B:36 Sirkka Hämäläinen The Savings Behaviour of Finnish Households. A Cross-Section Analysis of Factors Affecting the Rate of Saving. 1981. 171 p. + appendices. In Finnish with English summary. ISBN 951-686-074-5

B:37 Urho Lempinen Optimizing Agents, Exogenous Shocks and Adjustments in the Economy. Real and Nominal Fluctuations in Economies with a Wage Rigidity. 1984. 271 p. In English. ISBN 951-686-100-8

B:38 Heikki Koskenkylä Investment Behaviour and Market Imperfections with an Application to the Finnish Corporate Sector. 1985. 279 p. + appendices. In English. ISBN 951-686-110-5

B:39 Esko Aurikko Studies of Exchange Rate Policies and Disequilibria in the Finnish Economy. 1986. 153 p. In English. ISBN 951-686-115-6

B:40 Olavi Rantala A Study of Housing Investment and Housing Market Behaviour. 1986. 117 p. In English. ISBN 951-686-116-4

B:41 Kari Puumanen Three Essays on Money, Wealth and the Exchange Rate. 1986. 143 p. In English. ISBN 951-686-119-9

B:42 Tuomas Sukselainen Price Formation in the Finnish Manufacturing Industry in 1969-1981. 1986.399 p. In Finnish with English summary. ISBN 951-686-124-5

B:43 Ilmo Pyyhtiä The Revision and Realization of Investment Plans in the Finnish Manufacturing Industries in 1964-1986. 1989.290 p. In English. ISBN 951-686-220-9

B:44 Christian C. Starck Foreign and Domestic Shocks and Fluctuations in the Finnish Economy 1960-1988. 1990. 232 p. In English. ISBN 951-686-241-1

B:45 Jouko Vilmunen Labour Markets, Wage Indexation and Exchange Rate Poliey. 1992. 159 p. In English. ISBN 951-686-307-8

B:46 Alpo Willman Studies in the Theory of Balance-of-Payments Crises. 1992. 122 p. In English. ISBN 951-686-316-7