Abstract When the monetary authorities wish to target the value of a foreign currency, they may coordinate their intervention in the money and foreign exchange markets. Changes in the monetary aggregates aﬀect the fundamental value of a currency, while intervention in the foreign exchange market signals the central bank’s objective. Using a formal analysis, we propose a rationale for the dynamics of intervention by the G-5 in the mid and late eighties, explaining when foreign exchange intervention may be an useful instrument of policy making.

JEL Nos.: D82, G14 and G15. Keywords: Foreign Exchange Intervention, Monetary Policy, Signalling.

∗

Department of Accounting and Finance, London School of Economics, Houghton Street, London WC2A 2AE, UK; tel.: ++44-(0)20-7955-7230; e-mail: [email protected] † Department of Economics and Land History, Gabriele D’Annunzio University, Viale Pindaro 42, 65127 Pescara (Italy), tel. ++39-085-453-7647; e-mail: [email protected]

1

Introduction The eﬀectiveness of central bank intervention in the market for foreign exchange is argument of signiﬁcant disagreement among academics and practitioners. The experience of foreign exchange intervention in the past few years has increased such lack of consensus and has created further confusion. In fact, reading ﬁnancial newspapers it is not diﬃcult to ﬁnd headlines like: “Central banks fail to stem dollar’s decline”, or “Dollar still declining despite bank’s support, or “Why intervention by central bank failed to save the dollar”, closely followed by others like: “Central banks celebrate victory over markets”, or “Dollar soars on bank support”.1 Such headlines are then accompanied with comments in line with completely opposite theses on the relevance of foreign exchange intervention. For instance, you can read statements like the following: “Central banks would be hard pressed if they tried to provide a more eloquent testimony to the limits of governmental power at the hands of market forces. Their eﬀorts at intervention have been so ineﬀectual as to give this key monetary policy tool a bad name”. Others would read as follows: “The banks’ success this week has been indisputable. Intervention seeks to change expectations, and then to reinforce those changes”, or “The central banks’ action was seen as a powerful signal that policymakers favour a signiﬁcantly stronger dollar”.2 Such a confusion in the press simply reﬂects the absence of a general consensus among researchers on the scope and eﬀectiveness of foreign exchange intervention. In eﬀect, in the early eighties general opinion among researchers and policymakers was that foreign exchange intervention was ineﬀectual and that central banks should abstain from intervening. In line with this view, most central banks followed an “hands-oﬀ” policy in the foreign exchange market. Such a policy was interrupted in the mid-eighties by a series of individual and coordinated intervention operations carried out by the monetary authorities of the G-5 countries. The relative success of these operations has suggested researchers, such as Dominguez and Frankel (1993a), that foreign exchange intervention might be useful to condition market’s expectations and exchange rates. Anyhow, despite the resumed activity of the monetary authorities of most industrialised countries and the interest of researchers, it is still unclear when and how intervention operations have a signiﬁcant impact on the value of foreign currencies. In this paper we propose a theoretical analysis of coordinated monetary and foreign exchange intervention, using a simple monetary model of exchange rate determination. Assuming that the central bank aims at targeting the value of foreign exchange, we can show that it might ﬁnd it useful to accompany its open market operations with purchases and sales of foreign currencies. 1 2

Financial Times, 29/30 April, 16 and 19 August 1995. Ibidem.

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Indeed, when market participants are uncertain on the target level for the exchange rate, this level can be eﬀectively signalled by intervention operations in the foreign exchange market. The model we propose not only gives a clear signalling role to foreign exchange intervention, but it is also designed to represent the activity of the G-5 in the mid and late eighties. On the basis of the accounts of Funabashi (1988) of the meetings of the G-5 in those years, we know that between 1985 and 1987 the monetary authorities of the major industrialised countries deﬁned some target exchange rates for the three principal currencies. Since these target levels could not be oﬃcially announced, the central banks of the G-5 countries agreed on some coordinated foreign exchange intervention to signal such values and supplement their operations in the money market. An interesting conclusion of our analysis is that foreign exchange intervention might be useful in eliminating a problem of “overshooting” of the monetary policy. Indeed, within the monetary approach we know that a central bank needs to reduce the monetary growth to revalue its domestic currency. If the new desired value of the exchange rate cannot be announced, an even stricter monetary policy will be required, determining a phenomenon of overshooting of the monetary aggregates. When foreign exchange intervention accompanies open market operations such a problem can be mitigated, since the monetary authorities can use the former to reveal its objectives. While our analysis shows that foreign exchange intervention might be a valid supplementary instrument of policy making, it also suggests that this instrument might introduce elements of instability in the forms of multiple equilibria. In the present context, a rational expectation equilibrium will be formed by a system of beliefs of the market participants on the activity of the monetary authorities and a corresponding set of optimal intervention operations in the foreign exchange and money markets on the part of the central bank. We will see that for the same choice of the parameters of the model, there might be more than one set of market beliefs and intervention operations that forms a rational expectation equilibrium. We suspect that this result is not speciﬁc of the model presented. If this is true, it adds an important dimension to the problems central banks should face and suggests further complications to the evaluation of the eﬀectiveness of foreign exchange intervention. The paper is organised as follows. In Section I, we present a simple monetary model of exchange rate determination, that is used as an analytical framework for the study of coordinated monetary and foreign exchange intervention. General assumptions on the objectives of the monetary authorities and the behavior and information of market participants are discussed. In Section II, we study the implications of intervention operations conducted only in the money market, outlining the problem of overshooting of the monetary policy. In Section III, we consider coordinated intervention operations in both the money and foreign exchange markets. Problems of existence and multiplicity are discussed in details. In Section IV, we apply the model, carrying out a compara3

tive analysis exercise. We also discuss the consequences of the various scenarios on the volatility of exchange rates and interest rates. Section V proposes some concluding remarks and discusses possible extensions. In the Appendix brief proofs of the Lemmas and Propositions of the paper are given.

I. A Monetary Model of Exchange Rate Determination A. Basic Framework Let us consider the following very simple version of the monetarist model, deﬁned in discrete time for a small economy: mt − pt = −ψit ,

(1)

it = set+1 − st ,

(2)

s t = pt .

(3)

Here the variables mt , pt , it and st indicate respectively the money supply, the price level, the domestic nominal interest rate and the spot exchange rate, that is the price of a foreign currency in units of the domestic one. All the variables, apart from the interest rates, are expressed in logs. The superscript e stands for the expectation operator at time t and ψ is a non-negative parameter, representing the semi-elasticity of the money demand to the interest rate. Deﬁned in this way, equations (1)-(3) characterize the simplest possible monetarist model of exchange rate determination. In fact, equation (1) corresponds to an equilibrium condition for the domestic money market, in terms of real balances and nominal interest rate. Equation (2) is the classical Fisher open or uncovered interest rate parity, that relates exchange rate expectations to the interest rate diﬀerential. Finally, equation (3) is the purchasing power parity, that links the spot rate to the price level. We use this formulation because of its simplicity. We will see, in fact, that its reduced form presents a unique state variable. On the other hand, monetarist models of exchange rate determination have been strongly questioned on an empirical ground. In particular, there is ample evidence against the purchasing power parity assumption. In eﬀect, a sticky-price version of this model would be more realistic. However, if we abandoned the ﬂexible-price formulation, we would obtain a reduced form of the model with two state variables. This would signiﬁcantly complicate our analysis. Since we just desire to show the potentiality of coordinated monetary and foreign exchange intervention, we concentrate on a very simple formulation.

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Assuming market expectations are formed rationally, very simple manipulations of equations (1)-(3) produce the following reduced form for the spot rate: st = (1 − γ)mt + (1 − γ)

∞

γ j Et [mt+j ],

(4)

j=1

where γ = ψ/(1 + ψ) and Et indicates expectations formed at time t by private investors in the foreign exchange market. This equation indicates that the unique forcing process for the exchange rate is the money supply and that the current spot rate will depend on the market expectations of the future dynamics of the monetary aggregates. As it is commonly known, this is the consequence of the forward-looking nature of exchange rate determination, that is brought about by the Fisher open. To determine the dynamics of the spot rate we now need to specify the process governing the money supply. In this respect we assume the following: m m 2 mt ≡ um t + t , where t ∼ N (0, σm ).

m Here, the term um t is a variable under the control of the central bank, while t corresponds to idiosyncratic shocks in the money market. The central bank can in fact employ open market operations to control the monetary aggregates. Anyhow, given the wide range of ﬁnancial instruments that can be used to create liquidity, a complete control is not possible and the money supply will always contain a random component. We will see that the volatility of this random component, 2 , is a key factor in determining the use of foreign exchange intervention. In this respect, we will σm argue that the greater the liberalisation of ﬁnancial markets, the smaller the degree of control of the central bank on the monetary aggregates, the greater the scope for foreign exchange intervention.

To see in which way the monetary authorities will utilize their control over the money supply to condition the spot rate, we now need to introduce some hypotheses on the objectives of their intervention.

B. Exchange Rate Targeting In what follows we will assume that the monetary authorities intervene in order to target the value of the spot rate. In particular, this target level in period t is some value θt . For generality we assume that θt might change over time, according to the following dynamics: θt = θt−1 + θt , where θt ∼ N (0, σθ2 ).

(5)

The reasons why a central bank may decide to target the value of its currency are several. In the present context of a simple monetarist model, the main goal can be that of stabilising the price level. In general, another important reason might be the desire to reach a balance of the current 5

account. However, to justify this assumption we can simply look at the experience of the foreign exchange policy of the G-5 countries in the mid eighties. From the accounts of Funabashi (1988) on the process of policy making of the G-5 in the period starting with the Plaza meeting in September 1985 and ending with the Louvre meeting in January 1987, we know that reference levels for the exchange rates of the three main currencies, the US dollar, the deutsche mark and the yen, were set and repeatedly adjusted in those years. In particular, at the Plaza meeting baseline rates were set at 214-218 yen to the dollar and 2.54-2.59 deutsche marks to the dollar. In January 1987, after a strong devaluation of the dollar, these values were moved to 153.50 yen to the dollar and 1.8250 deutsche marks to the dollar. In April 1987, the G-5 were forced to shift the value for the yen/dollar exchange rate to 150. Assuming that the monetary authorities can announce in any period t the target level θt , their optimal intervention policy in the money market is simply to ﬁx um t = θt . Under the assumption of rationality of the market participants, we have the following equilibrium spot rate: st − θt = (1 − γ)m t .

(6)

In other words, when the objectives of the monetary authorities are known to all market participants, the exchange rate will diﬀer from the target level only because of exogenous shocks to the money supply. In this situation of complete information on the objectives of the central bank, there is nothing that foreign exchange intervention can achieve. However, we can consider situations in which the target level cannot be revealed. Stein (1989) proves that in a sticky-price model precise announcements on exchange rate targets may not be credible, because of problems of time-consistency. In other cases such announcements cannot be made because several monetary authorities cannot commit to defend these target levels or cannot agree on the shares of intervention. The experience of policy making in the mid eighties conﬁrms this assumption too. As we said, reference rates were set at the Plaza and at the Louvre, but oﬃcial communiqu´es on the decisions of the G-5 were imprecise. The reference rates were not disclosed and vague statements were passed to the press. At the Plaza the G-5 declared that “some further orderly appreciation of the main non dollar currencies against the dollar is desirable”, while at the Louvre the agreed that “their currencies [were] within ranges broadly consistent with underlying economic fundamentals”. We can substantially recognise two reasons for these understatements. First, the G-5 countries did not share the same level of commitment to the agreed target levels: some strongly wanted to stabilise the dollar; others desired to keep loose hands and substantially free-ride. Second, for internal reasons most governments, notably the Reagan administration, could not openly endorse a policy of foreign exchange intervention. According to the ideology dominant in those years in the 6

US administration, market forces should be let free to determine the equilibrium exchange rates. When the target level is not common knowledge the determination of the equilibrium exchange rate is not longer simple and foreign exchange intervention might have a role in signalling the objectives of the monetary authorities. Even if foreign exchange intervention is not employed, the optimal intervention policy in the money market will change. In fact, if the monetary authorities ﬁxed um t equal to θt , the expected spot rate would diﬀer from the target level. However, as discussed by Lewis (1988), the market participants will learn over time the parameters of the process governing the money supply. To explain how this will inﬂuence the policy of intervention of the central bank we need to introduce some further notation and assumptions. In the next Section we will consider the possibility that intervention is concentrated in the money market. Intervention in the market for foreign exchange will be introduced in Section III.

II. Monetary Intervention to Target the Spot Rate A. An Equilibrium Model Suppose that the central bank in any period t possesses the following cost function: Ct =

∞

β j ct+j ,

(7)

j=0

where

ct+j = qm (mt+j − θt+j )2 + (st+j − θt+j )2 ,

0 < β < 1, qm ≥ 0 and θt+j is given by equation (5). Using this formulation we assume that the monetary authorities intend to target the exchange rate, but that they also face a cost, when qm > 0, if the money supply deviates from the optimal level associated with a world of complete information. To understand the role of the ﬁrst term in the deﬁnition of ct+j consider that a level of mt+j smaller (greater) than θt+j might be interpreted as a signal of a very tight (loose) monetary policy. The central bank may not be willing to accept such a reputation and consequently it will try to use a more conservative policy. Moreover, within a sticky-price framework deviations from the optimal target level might induce undesired eﬀects on the output level. Finally, very restrictive or expansive monetary policies might severely disturb the functioning of ﬁnancial markets. We can now give the following Deﬁnition of equilibrium. ∞ Definition 1 A sequence of couples, {um t , st }t=0 , is an equilibrium if in any period t the central

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bank chooses its open market intervention, um t , solving the following programme Etb [Ct ] min m ut

s.t.

st = (1 − γ)mt + (1 − γ)

∞

γ j Et [mt+j ],

j=1

m mt ≡ um t + t ,

and the spot rate, st , solves equation (4) under the hypothesis of rational expectations on the part of the market participants. Before we can solve the central bank’s programme and ﬁnd an equilibrium spot rate we need to introduce the following Assumption. m Assumption 1 In period t the central bank chooses um t before observing the monetary shock, t . On the contrary, the market participants ﬁrst observe the money supply, mt , and then set their expectations on the future money supply, {mt+j }∞ j=1 . They also know the form of the cost function, Ct , but do not observe the target level, θt . They know that this value is generated according to equation (5) and possess some initial belief on θ0 . In particular, they assume that:

θ0 ∼ N (θ0e , Σ0 ). The market participants’ initial beliefs are common knowledge. We are now ready to present the following Lemma, that characterises all the linear equilibria of this model. Lemma 1 Under the hypotheses of Assumption 1, at time t the optimal intervention policy of the central bank and the spot rate in a linear equilibrium will be as follows: m e um t − θt = kt (θt − θt−1 ),

(8)

st − θt = (1 − γ)(mt − θt ) + γ(θte − θt ),

(9)

where θte is the expected target level given the information possessed by the market participants at time t. These expectations are up-dated using the information private investors obtain from observing the money supply dynamics, in that: e e + λm θte = θt−1 t (mt − θt−1 ).

(10)

e ), respects the following expression: Finally, the value function of the central bank, Vt (θt − θt−1 e e ) = πt (θt − θt−1 )2 + δt . Vt (θt − θt−1

Explicit formulas for the coeﬃcients, ktm , λm t , πt and δt are given in the Appendix. 8

(11)

This Lemma suggests that in equilibrium the central bank ﬁxes um t away from the ideal value θt .3 Since private investors do not have complete information on the objectives of the central bank, its intervention will be a function of the diﬀerence between the actual target level, θt , and the e . On the other hand, market participants will use the sequence of deviations expected one, θt−1 e }∞ , to up-date their expectations of the of the money supply from its expected value, {mt − θt−1 t=1 central bank objective and the equilibrium spot rate. Indeed, if the volatility of the target level, σθ2 , were nil it is not diﬃcult to prove that the spot rate eventually converges to its fundamental value given in equation (6).

B. Steady States The equilibria we presented in Lemma 1 can converge to long-run steady states. Since the model is stochastic, we deﬁne a steady state as a stationary equilibrium.

Definition 2 (Steady State) Any equilibrium characterised by equations (8)-(11) represents a e , has a stationary distribution. steady state when the state variable of the model, θt − θt−1 In practice, we have a steady state if the coeﬃcients, ktm , λm t , πt and δt given in Lemma 1 are time-invariant. In this case, in fact, the distribution of θt+1 − θte conditional on the information possessed by the market participants at time t is Normal with mean zero and constant variance across periods. While in general the existence and unicity of steady states is simple to prove in standard linear quadratic dynamic programming problems,4 this is not longer the case here, because the coeﬃcients e are not constant. These values depend on the expectations of the dynamic system governing θt −θt−1 of private investors and the intervention policy of the monetary authorities and can vary over time. The coeﬃcients ktm , λm t , πt and δt respect a complicated non-linear system of recursive equations, whose solution can generally be obtained only numerically. Nevertheless, we can still prove the following important Proposition regarding the characteristics of the stationary equilibria.

Proposition 1 (Overshooting) When a linear steady state exists, the monetary intervention overshoots the optimal quantity θt in that the time-invariant coeﬃcient km is non-negative. This Proposition simply recognise that when the monetary authorities wish to revalue (devalue) the domestic currency, but cannot reveal their intentions, they will have to implement a stricter 3

Notice that in equilibrium the optimal intervention policy at time t depends only on pre-determined variables. Thus, we have a time consistent equilibrium. See Backus and Driﬃl (1986). 4 See Whittle (1990).

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(looser) monetary policy than that required under complete information, as they will have to adjust for the inertia in the learning process of the market participants. This Proposition is the key result of this Section, since it suggests that the monetary authorities might ﬁnd it more convenient to use a supplementary instrument to signal their target level. While we cannot prove analytically the existence of steady states for all choices of the parameters of the model, there are important cases in which this is possible. In particular, corollary of Lemma 1 is the following Proposition.

Proposition 2 The dynamic system possesses unique stationary equilibria if either qm = 0 or σθ2 = 0. • If qm = 0, that is if the monetary authorities are only concerned with targeting the exchange rate, the coeﬃcients km and λm are the unique positive solutions of the system of equation: km = λm = where ∆ =

γ(1 − λm ) , (1 − γ) + γλm σθ2 ∆ − (1 + km )2 , 2 2σm

2 /σ 2 . The other values are as follows: (1 + km )2 + 4σm θ

π = 0 and δ =

[(1 − γ) + γλm ]2 2 σm . 1−β

• If σθ2 = 0, that is if there are not changes to the target level, the coeﬃcient km is equal to γ(1 − γ)/[(1 − γ)2 + qm ], the coeﬃcient λm is nil, while: π=

qm + (1 − γ)2 2 γ 2 qm and δ = σm . (1 − β)[(1 − γ)2 + qm ] 1−β

These two cases represent two polar situations. In the ﬁrst, the central bank will follow a bold monetary policy that guarantees the target of the exchange rate. In fact, for qm = 0 equation (9) collapses to: st − θt = [(1 − γ) + γλm]m t . Apart from an idiosyncratic shock, the spot rate does not diﬀer from the target level, explaining why the coeﬃcient of the value function is nil. In the second situation, instead, the target level is ﬁxed once for ever. Then, private investors will eventually learn θ. In the limit the equilibrium collapses to that with complete information. Indeed, the spot rate converges to equation (6) and since θ = θ e the optimal value of the control um t becomes θ. By continuity we can infer that in the neighbourhood of qm = 0 and σθ2 = 0 there still exist unique steady states. Moreover, numerical analysis of the equilibria suggests that this result can be

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extended to large portions of the parametric space. Using a numerical procedure to ﬁnd stationary equilibria, we always obtained a unique steady state for any choice of the parameters of the model.

C. Converge to the Steady State The analysis of the converge process to a steady state is complicated by the learning process of the market participants. The coeﬃcient λm t , that measures the informativeness of the money process m on the target level, θt , depends on kt in a complicated way. Applying the projection theorem for Normal distributions we ﬁnd that: = λm t

(1 + ktm )(Σt−1 + σθ2 ) , 2 (1 + ktm )2 (Σt−1 + σθ2 ) + σm

2 Σt = [1 − (1 + ktm )λm t ](Σt−1 + σθ ),

(12) (13)

where Σt ≡ Et [(θt − θte )2 ], that is Σt is the conditional variance of the target level given the information private investors possess in period t. Given these formulae, the dynamics of the coeﬃcients m m λm t , kt and of Σt turns out to be very complex. In eﬀect, if the coeﬃcient λt were time invariant, it would be possible to prove the existence of a single convergence path to a unique steady state using standard results for linear quadratic dynamic programming problems. Likewise, if ktm were constant over time, we could easily prove the convergence of the conditional variance Σt to a m constant level as well. Since during the convergence process both λm t and kt can vary, we cannot appeal to these standard results and a numerical procedure will be required. However, studying the non-linear system of recursive equations that gives the coeﬃcients ktm , λm t , πt and the conditional variance Σt , we come to the following conclusions concerning the convergence process.

Lemma 2 Suppose the dynamic system possesses a unique stationary equilibrium. Then, in equilibrium, the dynamic system will never collapse into such a steady state in a ﬁnite number of periods. Moreover, for any initial condition, there can be more than one convergence path.

In other words, while convergence to a steady state is not guaranteed, we cannot even exclude the possibility that more than one convergence path exist. On the other hand, along such paths the distance between the equilibrium coeﬃcients and their stationary values is always positive. A numerical procedure has been used to study the dynamics of the model coeﬃcients given an initial conditional variance Σ0 . The analysis of their dynamics suggests that the coeﬃcients ktm , λm t , πt and the conditional variance Σt converge to a steady state very quickly, but also conﬁrms that there can be more than one equilibrium convergence path, in that private investors and monetary authorities coordinate on diﬀerent sets of expectations and monetary intervention operations.

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In Figure 1 we represent the dynamics of the coeﬃcients ktm , λm t , πt and the conditional variance Σt . As one can see, the diﬀerence between Σt and its long-run value is almost immediately minimal. In eﬀect, the monetary authorities intervene heavily in the ﬁrst periods to reduce the uncertainty of the market participants on θt . Since this pattern is replicated for many choices of the model parameters, we can generalize the conclusion that the overshooting of the money supply is particularly strong in the ﬁrst stage of any convergence process. We can now see how these conclusions change if the monetary authorities decide to supplement their intervention in the money market with sterilised operations in the market for foreign exchange.

III. Introducing Foreign Exchange Intervention A. Sterilised Intervention in the Foreign Exchange Market There is ample evidence that central banks intervene frequently in the market for foreign exchange, purchasing and selling foreign currencies (Edison (1993)). Researchers agree that most of these operations are sterilised (Dominguez and Frankel (1993a), Catte el al. (1994), Watanabe (1994)), in that open market operations are run simultaneously to oﬀset any change in the money supply. Following Mussa (1981), economists now recognise that such operations may signal changes in the monetary policy of a central bank. A series of empirical papers (Klein and Rosengren (1991), Dominguez (1992), Domiguez and Frankel (1993a,1993b), Watanabe (1994), Lewis (1995), Kaminsky and Lewis (1996)) ﬁnds that foreign exchange intervention aﬀects exchange rates, conﬁrming this thesis. To understand how sterilised intervention can inﬂuence exchange rates without modifying their fundamentals, we need to consider the structure of the foreign exchange market. This is a dealership market in which a multitude of dealers, market makers, trade several currencies among themselves and with external customers, both directly and through brokers. The foreign exchange market is not centralised and lacks transparency, in that its participants cannot observe all transactions among other traders. However, a partial consolidation exists as market makers are attached to electronic services (Reuters 2000-2, Minex, EBS) on which inter-dealer transactions completed through a broker are reported. This permits a partial diﬀusion of information on the ﬂow of orders of individual dealers to the rest of the market.5 Given this lack of transparency, when a central bank intervenes in the foreign exchange market, placing market orders with one or more dealers, the exact dimension of its operations will not be revealed to all market participants. In eﬀect, the imperfect transmission of information on the 5

See Biais (1993) and O’Hara (1995) for formal deﬁnitions of transparency and discussions of related issues.

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individual trades in the foreign exchange market implies that a typical representative dealer will receive a noisy signal on the operations of the monetary authorities, when these sell or buy foreign exchange.6 In other words, if in period t the central bank trades the quantity uxt of a foreign currency, we will assume that the representative market participant will receive an imprecise signal, xt , where: xt ≡ uxt + xt , and xt ∼ N (0, σx2 ). Notice that by announcing its foreign exchange operations the central bank can reduce the degree of uncertainty of the market participants on its activity, reﬂected in the variance of the error term xt . Anyhow, as a common practice most monetary authorities conceal the size of their operations. Dominguez and Frankel (1993a) have questioned this practice, claiming that it reduces the eﬀectiveness of the foreign exchange operations. Instead, announcing its operations and goals, the central bank would be able to inﬂuence the exchange rates in the desired way with very little eﬀort. On the other hand, problems of time consistency may reduce the credibility of such announcements. We will see in what follows that the variance σx2 is a key factor in the decision of the monetary authorities on sterilised intervention. In particular, the smaller σx2 the more eﬀective will be foreign exchange intervention and the larger the scale of the operations in the foreign exchange market. While announcing the value of uxt might not be credible, any intervention that made the market for foreign exchange more transparent would be useful, because it would enhance the performance of sterilised operations.

B. Equilibria with Monetary and Foreign Exchange Intervention We can reconsider the optimisation problem of the central bank, introducing the possibility of intervention in the foreign exchange market. In particular, let us modify equation (7) in the following way: Ct =

∞

β j ct+j ,

(14)

j=0

where

ct+j = qm (mt+j − θt+j )2 + qx (uxt+j )2 + (st+j − θt+j )2 ,

0 < β < 1, qm , qx ≥ 0 and θt+j is given by equation (5). In this way we introduce a third component in the cost function of the monetary authorities, as operations in the foreign exchange market are expensive in terms of foreign reserves employed. Moreover, central banks are generally 6

See Bhattacharya and Weller (1997) and Vitale (1998) for analyses of the signalling role of sterilised intervention based on more detailed models of the micro structure of the foreign exchange market.

13

quite conservative and prefer not to intervene in the foreign exchange market. We now need to modify our initial Deﬁnition of equilibrium. x ∞ Definition 3 A sequence of triples, {um t , ut , st }t=0 , is an equilibrium if in any period t the central x bank chooses its open market operation, um t , and sterilised intervention, ut , solving the following programme

min Etb [Ct ]

x um t ,ut

s.t.

st = (1 − γ)mt + (1 − γ)

∞

γ j Et [mt+j ],

j=1

m mt ≡ um t + t ,

xt ≡ uxt + xt , and the spot rate, st , solves equation (4) under the hypothesis of rational expectations on the part of the market participants. We also need to modify Assumption 1 as follows: x Assumption 2 In period t the central bank cannot observe the shocks, m t and t , before choosing its intervention operations. It will observe mt only ex-post. On the contrary, the market participants observe mt and xt before setting their expectations on the future money supply, {mt+j }∞ j=1 . The hypotheses on their knowledge of the cost function are as in Assumption 1.

We are now ready to introduce the following Lemma, that characterises all the linear equilibria of this extended model. Lemma 3 Under the hypotheses of Assumption 2, at time t the optimal intervention policy of the central bank and the spot rate in a linear equilibrium will be as follows: m e um t − θt = kt (θt − θt−1 ), e ), uxt = ktx (θt − θt−1

st − θt = (1 − γ)(mt − θt ) + γ(θte − θt ).

(15) (16) (17)

The expected target level is up-dated using the information private investors obtain from observing the money supply dynamics and the signal xt , in that: e e x + λm θte = θt−1 t (mt − θt−1 ) + λt xt .

(18)

e ), respects the same formulation of Lemma Finally, the value function of the central bank, Vt (θt −θt−1 x 1. Explicit formulas for the coeﬃcients, ktm , ktx , λm t , λt , πt and δt are given in the Appendix.

14

Lemma 3 provides a rationale for foreign exchange intervention. In fact, the monetary authorities now use foreign exchange intervention to signal their target level for the exchange rate. Then, e , market participants will ﬁlter both the money supply deviation from its expected value, mt − θt−1 and the signal xt to up-date their expectations of the target level. In eﬀect, reconsidering the experience of foreign exchange policy in the mid eighties, we see that in January and February 1985 foreign exchange operations on the part of the Bank of Japan, the Fed and Bundsbank were used in support of the looser monetary policy in the United States. In September 1985, after the Plaza meeting, the G-5 undertook coordinated intervention operations both in the money and foreign exchange markets to strengthen the devaluation process of the US currency. The possibility of using this supplementary signalling mechanism has a feed-back eﬀect on the x monetary policy, in that the optimal value of um t clearly depends on ut . In this respect, simply x studying the coeﬃcients ktm , ktx , λm t , λt , πt and δt of Lemma 3 we can prove the following important Proposition.

Proposition 3 When the cost of intervention in the foreign exchange market is nil, i.e. qx = 0, the monetary policy does not overshoot, in that um t = θt . Proposition 3 suggests that when the central bank wishes to revalue (devalue) the domestic currency, but cannot reveal its goal, it is not necessarily forced to implement a very strict (loose) monetary policy, in that foreign exchange intervention might be used to signal its target. In eﬀect, the condition qx = 0 represents an extreme case, as the monetary authorities will be ready to buy or sell inﬁnite quantities of the foreign exchange. Anyway, even for qx > 0 intervention in the foreign exchange market will permit the central bank to reduce the value ktm and hence the overshooting of the money supply. To study in more details this relation between monetary and foreign exchange intervention we consider the steady states of the model.

C. Monetary and Foreign Exchange Intervention in a Steady State As for the dynamic system studied in Section II, a steady state corresponds to a stationary equix librium. This is brought about if the coeﬃcients ktm , ktx , δt , λm t , λt and πt are time-invariant. Notice that also in this generalised model we do not have a result that guarantees the existence of stationary equilibria. However, there are still important cases in which such existence can be proved.

Proposition 4 The dynamic system possesses unique stationary equilibria if either qm = 0 or σθ2 = 0 or qx = 0. In the ﬁrst two cases the steady states correspond to those of Proposition 2. 15

For qx = 0, instead, kx is inﬁnite, km = 0, δ = [qm + (1 − γ)2 ]/(1 − β), while λm , λx , Σ and π are equal to zero.

In other words, in the ﬁrst two cases there is not need for foreign exchange intervention. Indeed, for qm = 0 the monetary authorities prefer to signal the target level only using open market operations. For σθ2 = 0, in the long-run the ﬁxed target level is common knowledge and foreign exchange intervention has no use. In this second polar situation intervention operations in the foreign exchange market can only accelerate the process of convergence to the stationary equilibrium. For qx = 0, on the other hand, there is not limit to foreign exchange intervention and with very large operations the monetary authorities can reveal the value of θt in any period t. By continuity we can infer that in the neighbourhood of qm = 0, qx = 0 and σθ2 = 0 there still exist steady states. Furthermore, using a numerical procedure we could ﬁnd steady states for all the choices of the model parameters. However, diﬀerently from the results obtained in Section II, there are parameters conﬁgurations for which more than one stationary equilibrium exist. Though quick, several convergence paths for any steady state paths are also possible, as in the dynamic system studied in Section II. Steady states for this extended model were calculated using a numerical procedure for diﬀerent choices of the parameters of the model. In Figure 2 plots of the steady state values of the coeﬃcients 2 , km , kx , the conditional variance Σ and the cost Ct against the variance of the monetary shock, σm e = 1. Figure 2 shows are presented for a particular choice of the other parameters and for θt − θt−1 2 7 2 that two stationary equilibria are possible for σm relatively large. For σm small, instead, the model possesses a unique stationary equilibrium. When the variance of the monetary shock, m t , is relatively small with respect to that of the x error term t , signalling through foreign exchange intervention is not convenient and the monetary authorities refrain from trading in the foreign exchange market, as kx = 0. The stationary equilibrium collapses to that without foreign exchange intervention. However, when the central bank does not possess a strict control of the monetary aggregates, a steady state with foreign exchange intervention exists. This is always accompanied by a steady state without operations in the market for foreign exchange. In eﬀect, if market participants do not believe the monetary authorities are present in the foreign exchange market, these have no reason to intervene in this market and the private investors’ beliefs are self-fulﬁlling. Figure 2 indicates that multiple equilibria with and without foreign exchange intervention are 7

In eﬀect, the steady states are three in these cases. In fact, if kx and λx are non-negative, we can change their signs without aﬀecting the other coeﬃcients and obtain another stationary equilibrium with equivalent properties. We disregard these stationary equilibria, because they imply a schizophrenic behavior on the part of the central bank.

16

possible. This clearly underlines a diﬃculty that emerges when the monetary authorities consider the opportunity of intervening in the foreign exchange market. Since in general both the conditional variance, Σ, and the cost of the central bank are smaller in the stationary equilibria with foreign exchange intervention, the monetary authorities and the market participants will prefer these equilibria. In eﬀect, using two diﬀerent instruments to signal the exchange rate target, the monetary authorities are more eﬀective in targeting the value of foreign exchange. On the other hand, a smaller conditional variance, Σ, corresponds to a smaller uncertainty on the exchange rate on the part of ﬁnancial investors. In this respect, there might be mechanisms that permit the central bank and the private investors to coordinate their strategies toward the best outcome, but we cannot rule out an indeterminacy of the equilibrium.

IV. The Eﬀects of Monetary and Foreign Exchange Policy A. An Example: the Plaza To validate our analysis of foreign exchange policy we should calibrate our model using actual data on central bank intervention in the money and foreign exchange markets. However, this kind of exercise proposes a series of problems. First, data on central bank intervention is generally not available. Central banks prefer not to make available data on their open market operations and on their foreign exchange intervention and in fact most empirical studies on these issues have been conducted within their research departments. Moreover, our model does not capture all aspects of foreign exchange policy. Intervention operations are not only aimed at targeting exchange rates. Other objectives include smoothing interest rates, balancing trade accounts, stimulating the domestic economy, etc. Finally, the monetarist model we use represents a gross simpliﬁcation of the complex mechanisms of determination of exchange rates. On the other hand, we can consider an application of our model to study its implications with respect to the Plaza agreement of September 1985. The Plaza agreement probably represents the most important attempt on the part of the G-5 to manage exchange rates. In fact, after a long and complex negotiation, a comprehensive package of policy measures was arranged by the ﬁnance ministers and the governors of the most important industrialised countries in order to stabilise the value of the US dollar. Since mid 1984 interest rates in the United States had started decreasing. With the help of intervention operations in the foreign exchange market by the Bundesbank, the Fed and the Bank of Japan, the US dollar began devaluing. In few months the exchange rates with the German and Japanese currencies fell from 3.50 deutsche marks per dollar and 265 yen per dollar to 2.75 and 235 respectively. However, in the weeks preceding the Plaza meeting, the dollar restarted appreciating. 17

Pressure for protectionist action in the United States forced the G-5 to try to bring down the value of the US currency. In the non-paper conﬁdentially forged at the Plaza the G-5 decided to devalue the US dollar by about 10-12 percent from the levels of 240 yen per dollar and 2.85 deutsche marks to the dollar. A code of intervention in the foreign exchange market was agreed upon: intervention should be conducted within the following six weeks and single operations should be set on a daily basis according to market conditions. The total scale of intervention was ﬁxed at $18 billion alongside a daily limit for single ﬁnancial centres of $400 million. The actual total scale of intervention was $10.2 billion. Simultaneously, interest rates in the United States continued along their downward trend in the weeks following the meeting, while the discount rate was raised in Japan. On Monday 23 September, the day after the meeting, the US dollar fell by about 4% as a consequence of un-precedented large intervention operations and of the announcement eﬀect of the meeting’s communiqu´e. By mid October the dollar had devalued by 13 percent with respect to the yen and by 10 percent with respect to the deutsche mark, signing the success of the G-5’s action.8 In our analysis we have not considered issues of coordination among several countries, that were fundamental in the formulation of the Plaza agreement and in its implementation. However, in Figure 3 we see that our model permits replicating a dynamics of the intervention operations and of the exchange rate similar to that experienced after the Plaza. In fact, the initial overvaluation of the home currency, i.e. the US dollar, of 10 percent against a bundle of foreign currencies, i.e. the yen and the deutsche mark, is almost immediately eliminated, as one can notice from the right-bottom panel. This is brought about through massive operations in the foreign exchange market in period 1, i.e. in the ﬁrst of the six weeks during which intervention is concentrated. The overshooting of the monetary policy is limited and in eﬀect after the Plaza meeting the interest rates of the G-5 did not move dramatically. Some diﬀerences exist between the simulation of our model represented in Figure 3 and the real events that followed the Plaza. As documented by Dominguez and Frankel, intervention operations in the market for foreign exchange were smoother than those in the left-top panel of Figure 3, but the actual devaluation of the US dollar does not diﬀer signiﬁcantly from that presented in Figure 3.

B. Comparative Analysis In Section III we stressed that multiple stationary equilibria may exist. In eﬀect, in Figure 2 we observe a phenomenon of bifurcation of the steady state values of the coeﬃcients km , kx , and the variance Σ when we increase the variance of the idiosyncratic shock to the monetary base. This 8

See Funabashi (1988) and Dominguez and Frankel (1993a) for more details on the Plaza meeting and its aftermath.

18

clearly indicates that the characteristics of the equilibria of the model depend crucially on the parametric conﬁguration. A comparative analysis exercise might be useful in understanding how the decision of intervention in the foreign exchange market is inﬂuenced by the market conditions and the preferences of the monetary authorities. In Table 1 we report the results of a simple comparative exercise. Using our numerical procedure to calculate the steady state coeﬃcients and concentrating on the equilibria with foreign exchange intervention, we derived the elasticities of 2 and σ 2 . these coeﬃcients with respect to the parameter γ, qm , σm θ We can draw several conclusions from Table 1. First, consider the eﬀect of a rise in γ, that reﬂects an increase in the semi-elasticity of the money demand to the interest rate, φ, in equation (1). This reduces the aggressiveness of the monetary authorities, since now the same change in the money supply will have a smaller impact on the interest rate and, through the Fisher open (2), on the market expectations of devaluation of the domestic currency and on the current spot rate. The fall in the intensities of intervention, km and kx , explains the reduction in the coeﬃcient of the value function π and the rise in Σ. 2 and σ 2 , is crucial in the decision of intervention The ratio qm /qx , alongside that between σm x in the foreign exchange market. In fact, a phenomenon of bifurcation similar to that represented in Figure 2 emerges with respect to qm . When the cost of the monetary overshooting is large with respect to that of intervention in the market for foreign exchange, the central bank ﬁnds it convenient to signal its objective by selling or purchasing foreign currencies. Not surprisingly, when qm augments the intervention activity of the monetary authorities shift towards the foreign exchange market. A similar conclusion can be drawn when the central bank’s control on the 2 rises. monetary aggregates deteriorates, i.e. when σm

The parameter σx2 is a measure of the transparency of the foreign exchange market and of the ability of the market participants to detect the intervention operations of the monetary authorities in the foreign exchange market. Any institutional change to the organisation of this market that made it more transparent or any measure that increased the visibility of the central bank’s intervention operations, would reduce σx2 making foreign exchange intervention a more valid instrument of policy making. In support of a low value for σx2 the experience of the intervention operations of the G-5 after the Plaza meeting suggests that private investors were well aware of the new intervention policy. In those days, ﬁnancial newspapers reported ﬁgures on the scale of daily intervention that were quite precise, despite the central banks’ operations remained concealed.

C. Exchange Rate Volatility and Interest Rates Even if the volatility of the exchange rate and of the interest rates does not enter in the formulation of the cost function we use, there is ample evidence that central banks intervene in the money and

19

foreign exchange markets in order to smooth exchange rates and interest rates. It is not diﬃcult to calculate the conditional volatilities of the exchange rate, Vt−1 (st )f , and of the interest rate diﬀerential, Vt−1 (it − i∗t )f , when the central bank intervenes in both the foreign exchange and the money market.9 Similar volatilities, Vt−1 (st )m and Vt−1 (it − i∗t )m , can be obtained when only open market operations are carried out. We can then deﬁne the indexes Vˆt−1 (st ) and Vˆt−1 (it − i∗t ), given by the ratios f m (.). Simple numerical analysis of these (.)/Vt−1 between these conditional variances: Vˆt−1 (.) ≡ Vt−1 indexes indicates that foreign exchange intervention reduces the volatility of exchange rates, while augmenting that of the interest rate diﬀerential in a steady state. Indeed, using the benchmark values of Table 1 we obtain Vˆt−1 (st ) = 0.777 and Vˆt−1 (it − i∗t ) = 1.954, where Vt−1 (st )m = 18.13 and Vt−1 (it − i∗t )m = 1.20. We should have expected such results. In fact, the greater stability of the exchange rate is obtained with a more aggressive intervention policy by the central bank. This will have a greater impact on the expectations of future movements of the exchange rate on the part of private investors. Since the Fisher open holds, the volatility of the interest rate will rise.10 In Table 2 we report the elasticities of the indexes Vˆt−1 (st ) and Vˆt−1 (it − i∗t ) with respect to the 2 and σ 2 . The signs of the elasticities of V ˆt−1 (st ) correspond to those of Σ in parameters γ, qm , σm θ Table 1. This signals that changes in Σ, the conditional volatility of θt , are matched by equivalent changes in the volatility of st in a steady state. The eﬀects on the forward discount are diﬃcult to interpret and are probably due to the complex dynamics of the exchange rates’ expectations.

V. Concluding Remarks and Extensions In this paper we have tried to shed some light on sterilised foreign exchange intervention, considering the possibility that the monetary authorities coordinate their intervention operations in the money and foreign exchange markets in order to target the exchange rate. When new target levels cannot be announced, market participants learn slowly the new policy stance. Using a formal framework, 9

It is a matter of algebra to show that Vt−1 (st )f and Vt−1 (it − i∗t )f respect the following expressions in a steady state: Vt−1 (st )f

=

2 [(1 − γ) + γλm ]2 σm + γ 2 (λx )2 σx2 + {(1 + km )[(1 − γ) + γλm ] + γkx λx }2 (Σ + σθ2 ),

Vt−1 (it − i∗t )f

=

2 (1 − γ)2 (1 − λm )2 σm + (1 − γ)2 (λx )2 σx2 + (1 − γ)2 [(1 + km )(1 − λm ) − kx λx ]2 (Σ + σθ2 ).

Similar expressions can be obtained for Vt−1 (st )m and Vt−1 (it − i∗t )m . 10 A similar trade-oﬀ is hidden behind the honeymoon eﬀect of a credible currency band considered in the target zone literature.

20

we show that in this case the monetary authorities might be forced to implement an undesired monetary policy, unless foreign exchange intervention is used to signal their objective. To study the implications and the relevance of foreign exchange intervention, we employ a monetarist model of exchange rate determination, in which a central bank with an inﬁnite horizon minimises the expected value of a loss function over time, while private investors learn its objective using Bayes’ rule. A concept of equilibrium with rational expectations is used and a time-consistent policy is derived. A clear link between the central bank’s preferences, its intervention operations, the market participants’ expectations and the equilibrium exchange rate is underlined. The model we use is simple and assigns a clear signalling role to sterilised intervention in the market for foreign exchange. To obtain simple and easy to interpret relations between the intervention policy and the exchange rates an elementary analytical framework is considered. In eﬀect, monetarist models oﬀer a very crude representation of exchange rate determination, while stickyprice formulations would be more realistic and richer. In particular, they would permit analysing the impact of the intervention policy on the output level. Indeed, when prices are sticky, if a stricter monetary policy is not anticipated an undesired contraction of the output level would result. If foreign exchange intervention were employed to signal this new policy, such a negative eﬀect would be reduced. Therefore, within a sticky-price formulation, the scope for sterilised intervention would be even larger. However, a richer model would be obtained at the expense of its tractability: simple and easy to understand equilibria could not be obtained and only a numerical analysis could be considered. Other issues that deserve some considerations concern possible conﬂicts of interests and coordination problems. These conﬂicts might emerge among several governments, as in the case of policy management of the G-5, or even between institutions in the same country. Indeed, in most countries monetary and foreign exchange policy fall under the jurisdiction of diﬀerent bodies. For instance, in the United States the Fed controls monetary aggregates, but the Treasury governs foreign exchange intervention. A similar distinction is present in Japan and in other countries. Thus, we can conceive situations in which the two bodies do not agree on the policy objectives and in eﬀect in the United States there have been instances in which the Fed criticized the Treasury for having sent wrong signals to the market. These issues are quite interesting and constitute possible arguments of future extensions. Anyhow, our analysis permits reproducing the intervention policy of the G-5 in the eighties, underlining the possible use of foreign exchange intervention. Since the relevance of sterilised intervention represents an important subject of discussion among academics and practitioners, we claim that this paper constitutes an useful contribution and a ﬁrst attempt to disentangle the controversy over the eﬀectiveness of foreign exchange intervention using a formal approach. 21

VI. Appendix Proof of Lemma 1. m e To prove this Lemma, assume ﬁrst that ∀ t um t = θt + kt (θt − θt−1 ), as in equation (8). A simple application of the projection theorem for Normal distributions will give equation (10), where λm t respects equation (12) and the conditional variance of θt given the market information at time t is as in (13). Since the monetary policy is assumed to follow equation (8) in every period, plugging equation (10) into equation (4), we obtain the expression in equation (9) for the spot rate.

Suppose, now, that the market expectations of the target level are up-dated using equation (10) so that (9) holds. Assume also that the value function at time t + 1 is as in equation (11). Using equations (9) and (10), the following will hold: st − θt

=

m e m m [(1 − γ) + γλm t ](ut − θt ) − γ(1 − λt )(θt − θt−1 ) + [(1 − γ) + γλt ]t ,

θt+1 − θte

=

e m m m θ (1 − λm t )(θt − θt−1 ) − λt (ut − θt ) − λt t + t+1 .

The optimal level of um t will be the argmin of Etb [ct + βVt+1 (θt+1 − θte )].

(19)

Notice that the monetary authorities observe mt and st at the end of any period. Since the value of is common knowledge, in equilibrium the central bank will always know θte at the end of the period by observing mt . Plugging the expressions for st − θt and θt+1 − θte in (19) and solving this simple optimisation problem, we ﬁnd that um t respects equation (8), where:

θ0e

ktm =

m m (1 − λm t ){βπt+1 λt + γ[(1 − γ) + γλt ]} . m m 2 2 [(1 − γ) + γλt ] + βπt+1 (λt ) + qm

(20)

Plugging the optimal monetary policy into equation (19), we can check that Vt is as in equation (11) with πt and δt given as follows: πt δt

m m 2 m 2 m m m 2 = {[(1 − γ) + γλm t )]kt − γ(1 − λt ]} + qm (kt ) + βπt+1 [1 − λt − λt kt ] ,

= {qm + [(1 − γ) +

2 γλm t )]

+

2 2 βπt+1 (λm t ) }σm

+

βπt+1 σθ2

+ βδt+1 .

(21) (22)

In synthesis we see that the optimal policy of the central bank and the expectations of the market participants are mutually consistent. This proves that Lemma 1 characterises equilibria of the model with rational expectations and an equilibrium exchange rate. 2

Proof of Proposition 1. m When the dynamic system settles in a steady state, the coeﬃcients ktm , λm t , πt and the conditional variance Σt are time-invariant. Imposing the condition of time-invariance we ﬁnd that a steady state exists if there exist k m , λm and π that solve the following non-linear system.

km

=

λm

=

π

=

(1 − λm ){βπλm + γ[(1 − γ) + γλm ]} , [(1 − γ) + γλm ]2 + βπ(λm )2 + qm 2 σθ2 σm m m 2 (1 + k ) + 4 2 − (1 + k ) , 2 2σm σθ qm (k m )2 + {[(1 − γ) + γλm )]k m − γ(1 − λm )]}2 . 1 − β[1 − λm − λm k m ]2

22

(23) (24) (25)

Since the cost function is non-negative, we know that the coeﬃcient π must be positive. Suppose we select a positive value for π and let us consider only equations (23) and (24) that deﬁne respectively k m as a function of λm and λm as a function of k m for a given π. A graphical analysis of this two functions shows that they possess a unique intersection in the space (λm , k m ) for 0 < λm < 1 and k m > 0. Given these two values, if inserting them in equation (25) we obtained the initial value of π we would ﬁnd a steady state. We cannot prove there is a positive value of π for which this happens, but we see that in this case k m is positive. 2

Proof of Proposition 2. Suppose qm = 0. Here we can prove that a solution to the system of non-linear equations (23) to (25) exists such that π = 0. In fact, in this case equations (23) and (24) collapse to the expressions given in the Proposition. On the other hand, plugging k m in equation (25) gives exactly π = 0. Analogously, it is immediate to obtain the expression for δ from the expression given in the proof of Lemma 1. Suppose then that σθ2 = 0. Here in the limit Σt will converge to zero, as the target level will be revealed. This also implies that in the long run λm = 0. Solving for π and k m in the system (23) to (25) and calculating δ we ﬁnd the expressions given in the Proposition. 2

Proof of Lemma 2. The proof of the ﬁrst part of the Lemma is obvious. Suppose the dynamic system reaches a stationary equilibrium at time t + 1. We can solve backward the recursive system of equations (12), (13), (20) and (21). m m , λm We ﬁnd that ktm = kt+1 t = λt+1 , πt = πt+1 , Σt−1 = Σt , that it the stationary equilibrium is already reached in period t. Moving backward, we would ﬁnd that the dynamic system should be in a stationary equilibrium at time 0, a contradiction. The proof of the second part of the Lemma is more complicated. Suppose at time t + 1 the coeﬃcients m , λm kt+1 t+1 , πt+1 , and the conditional variance Σt are given. We can solve backward the recursive system of equations (12), (13), (20) and (21) under the initial condition for the conditional variance, Σ0 . Consider, then, that we can transform (12) as follows: λm t =

αm t Σt m where αm t ≡ 1 + kt . 2 σm

Plugging equation (23) into this expression for λm t , we obtain after some manipulation the following equation: 3 m 2 m at (λm t ) + bt (λt ) + ct λt + dt = 0,

(26)

where the coeﬃcients at , bt , ct and dt depend on β, γ, πt+1 and Σt . While at , bt are always positive and dt is always negative, ct might be either positive or negative. In the ﬁrst case, there exists only one root, positive, of the equation and hence a unique value for λm t that solves backward the recursive system of equation. For ct negative, however, this equation can have three roots, one positive and two negatives. Therefore, there can be more than one solution to the recursive system of equations (12), (13), (20) and (21), as shown in Figure 1. 2

Proof of Lemma 3. m e The proof of this Lemma is very similar to that of Lemma 1. Aassume ﬁrst that ∀ t um t = θt + kt (θt − θt−1 ) x x e and ut = kt (θt − θt−1 ), as in equations (15) and (16). The application of the projection theorem for Normal

23

x distributions will give equation (18), where λm t and λt are as follows:

λm t

=

λxt

=

2¯ αm t σx Σt , x 2 )Σ 2 σ2 ¯ t + σm + (kt )2 σm x 2 ¯ Σt ktx σm 2 σ 2 + (k x )2 σ 2 )Σ ¯ t + σ2 σ2 , ((αm ) x m m x t t 2 2 ((αm t ) σx

¯ t ≡ Σt−1 + σ 2 and αm ≡ 1 + k m . The conditional variance of θt given the market information at where Σ t t θ time t is then: m x x ¯ Σt = (1 − αm t λt − kt λt )Σt . Since the intervention policy is assumed to follow equations (15) and (16) in every period, plugging equation (18) into equation (4), we obtain the expression in equation (17) for the spot rate. Suppose, now, that the market expectations on the target level are up-dated using equation (18). Assume also that the value function at time t + 1 is as in equation (11). Using equations (17) and (18), the following will hold: st − θt

=

m e x x x m m [(1 − γ) + γλm t ](ut − θt ) − γ(1 − λt )(θt − θt−1 ) + γλt (ut + t ) + [(1 − γ) + γλt ]t ,

θt+1 − θte

=

e m x x m m x x θ (1 − λm t )(θt − θt−1 ) − λt (ut − θt ) − λt ut − λt t − λt t + t+1 .

x The optimal levels of um t and ut will be the argmin in equation (19). Notice that ex-post, by observing mt and st , the monetary authorities can determine in equilibrium the expected target level, θte . Plugging the expressions for st − θt and θt+1 − θte in (19) and solving this simple optimisation problem, we ﬁnd that um t and uxt respect equations (15) and (16) respectively, where:

ktm

=

ktx

=

∆ ≡

1 2 m qx (1 − λm (27) t )[γ(1 − γ) + (γ + βπt+1 )λt ], ∆ 1 x [qm + βπt+1 (1 − γ)2 ](1 − λm (28) t )λt , where ∆ 2 m 2 2 x 2 qm qx + (γ 2 + βπt+1 )(λxt )2 qm + {[(1 − γ) + γλm t ] + βπt+1 (λt ) }qx + (1 − γ) βπt+1 (λt ) .

Plugging the optimal intervention policy into Bellman’s recursion, we can check that Vt is as in equation (11) with πt and δt given as follows: πt

=

m x x m 2 {[(1 − γ) + γλm t )]kt + γλt kt − γ(1 − λt )]} +

qm (ktm )2 δt

=

+

qx (ktx )2

{qm + [(1 − γ) +

m x x 2 + βπt+1 [1 − − λm t kt − λt kt ] , 2 m 2 2 2 γλm t )] + βπt+1 (λt ) }σm + βπt+1 σθ +

(29)

λm t

(γ 2 + βπt+1 )(λxt )2 σx2 + βδt+1 .2 (30)

Proof of Proposition 3. The proof is straightforward. For qx = 0 in (27) ktm is nil. 2

Proof of Proposition 4. The proof of the ﬁrst part is obvious. For qm = 0 the central bank does not have a reason to use foreign exchange intervention, while for σθ2 = 0 in the long-run, there is nothing to signal. For qx = 0 there is nothing that prevent the monetary authorities to sell or purchase inﬁnite quantities of foreign exchange. Now, for k x ↑ ∞, λx k x ↑ 1 and Σ ↓ 0. Since qx = 0 k m = 0. Moreover, as k x ↑ ∞ λm , λx and π ↓ 0, while δ → [qm + (1 − γ)2 ]/(1 − β). 2

24

References Backus, D. and J. Driﬃl, 1986, The consistency of the optimal policy in stochastic rational expectations models, CEPR discussion paper no. 124. Bhattacharya U., and P. Weller, 1997, The advantage of hiding one’s hand: Speculation and central bank intervention in the foreign exchange market, Journal of monetary economics 39, 251-77. Biais, B., 1993, Price formation and equilibrium liquidity in fragmented and centralized markets, Journal of ﬁnance, 48, 157-84. Catte, P., G. Galli and S. Rebecchini, 1994, Concertated interventions and the dollar: An analysis of daily data, P. Kenen, F. Papadia and S. Saccomanni, eds., The international monetary system (Cambridge University Press, Cambridge). Dominguez, K.M., 1992, Exchange rate eﬃciency and the behavior of international asset markets (Garland, New York). Dominguez, K.M. and J.A. Frankel, 1993a, Does foreign exchange intervention work? (Institute for International Economics, Washington, DC). Dominguez, K.M. and J.A. Frankel, 1993b, Foreign exchange intervention: An empirical assessment, in J.A. Frankel, ed., On exchange rates (MIT Press, Boston). Edison, H.J., 1993, The eﬀectiveness of central bank intervention: A survey of the literature after 1982. University of Princeton department of economics special papers in international economics no. 18. Funabashi, Y., 1988, Managing the dollar: From the Plaza to the Louvre (Institute for International Economics, Washington, DC). Kaminsky, G.L. and K.K. Lewis, 1996, Does foreign exchange intervention signal future monetary policy?, Journal of monetary economics 37, 285-312. Klein, M.W. and E. Rosengren, 1991, Foreign exchange intervention as a signal of monetary policy, New England economic review, May/June, 39-50. Lewis, K.K., 1988, The persistence of the “Peso Problem” when policy is noisy. Journal of international money and ﬁnance, 5-21. Lewis, K.K., 1995, Are foreign exchange intervention and monetary policy related and does it really matter?, Journal of business 68, 185-214 Mussa, M., 1981, The role of oﬃcial intervention (Group of thirty, New York). O’Hara, M, 1995, Market microstructure theory (Blackwell, Oxford). Vitale, P, 1998, Sterilised central bank intervention in the foreign exchange market, Journal of international economics, forthcoming.

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Watanabe, T, 1994, The signaling eﬀect of foreign exchange intervention: The case of Japan, R. Glick and M. M. Hutchison, eds, Exchange rate policy and interdependence: Perspectives from the Paciﬁc Basin (Cambridge University Press, Cambridge). Whittle, P., 1990, Risk-sensitive optimal control (John Wiley & Sons, New York).

26

Table 1: Steady States Comparative Studies

Σ

km

kx

π

γ

0.3536

-0.0441

-0.2061

-0.4083

qm

-0.1352

-0.8159

0.0864

0.1437

2 σm

-0.0313

-0.1167

0.0684

0.1248

σθ2

0.0724

-0.0629

-0.0062

-0.0203

Parameter

Note: entries are the elasticities of the conditional variance, the intensities k m , k x and the value 2 function coeﬃcient π, with respect to γ, qm , σm , 2 σθ . Benchmark values are: γ = 0.5, qm = 1, 2 = 10, σθ2 = 10, qx = 0.25, σx2 = 1. β = 0.9524, σm

27

Table 2: Volatilities Comparative Studies

Vˆt−1 (st )

Vˆt−1 (it − i∗t )

γ

0.0113

-0.2592

qm

-0.0328

0.1324

2 σm

-0.1799

-0.2859

σθ2

0.1262

0.3547

Parameter

Note: entries are the elasticities of Vˆt−1 (st ) and Vˆt−1 (it − i∗t ) with

2 , σθ2 . respect to the parameters γ, qm , σm Benchmark values are as in Table 1.

28

Figure 1: Convergence Paths to Steady States

29

Figure 2: Steady States with Money and Foreign Exchange Intervention

30

Figure 3: Exchange Rate Dynamics and Intervention Policy: An Example

31