## Abstract

Alarge number of countries maintain separate exchange rates for commercial and financial transactions. In these so-called dual exchange rate regimes, the central bank usually intervenes only in the market for commercial transactions to maintain a stable exchange rate for exports and imports. The absence of official intervention in the market for financial transactions implies that the financial exchange rate has to move to ensure capital account equilibrium. Dual exchange rates are therefore often used by countries with balance of payments difficulties as a substitute for direct capital controls.

**A**large number of countries maintain separate exchange rates for commercial and financial transactions. In these so-called dual exchange rate regimes, the central bank usually intervenes only in the market for commercial transactions to maintain a stable exchange rate for exports and imports. The absence of official intervention in the market for financial transactions implies that the financial exchange rate has to move to ensure capital account equilibrium. Dual exchange rates are therefore often used by countries with balance of payments difficulties as a substitute for direct capital controls.

Dual exchange rate regimes have been discussed extensively in the economic literature, where it has been emphasized that the separation of the goods and financial markets that can be achieved with different exchange rates for commercial and financial transactions may be useful in insulating the goods markets of the home economy from the effects of disturbances in the financial markets.^{1} For the most part, however, the existing models of the effects of dual exchange rates have not challenged the assumption that the authorities succeed in separating the two markets or, equivalently, that private arbitrage activity has no important consequences.^{2} In contrast, this paper develops a model of the scale of private arbitrage activity and focuses on the important consequences of such activity for the viability and effectiveness of dual exchange rate regimes.

The “leakage” from the commercial to the financial exchange market is not just an interesting phenomenon of little practical significance. Bhandari and Decaluwe (1987) refer to the evidence available indicating that for Belgium the estimated trade balance functions show a statistically significant response to the exchange rate differential. Lanyi (1975) also emphasizes the importance of these leakages and describes the instruments used by traders to arbitrage between the two foreign exchange markets.

Taking private arbitrage activity into account leads to the conclusion that dual exchange rates (as well as capital controls) could succeed only temporarily in dampening the effects (on the domestic goods market) of shocks to financial or other markets. To offset such effects permanently, the authorities would have to induce a steadily increasing differential between the two exchange rates. But a steadily increasing differential would also lead to a steadily increasing incentive for private operators to circumvent the regulations that separate the two markets by buying foreign exchange at the lower rate (usually the controlled or commercial rate) and selling it at the higher rate (usually the free or financial rate). Such arbitrage activity would expand as the differential between the two exchange rates widened and would thus limit the size of the differential that the authorities could support.

For the same reason, these considerations suggest that any differential between the two exchange rates would tend to disappear over time unless the authorities were prepared to offset continuously the effects of private arbitrage activity by adjusting one of the exchange rates or by using domestic credit policy.^{3} Data from Belgium, which has had a dual exchange rate since 1946, confirm that the differential between the commercial and the financial exchange rates has usually been very small (Figure 1). However, substantial differentials have emerged during periods of tension in the financial markets (especially when a readjustment of the parities inside the European Monetary System (EMS) was expected). A similar pattern can be observed in the Mexican two-tier exchange market (Figure 2). The discount of the peso on the free exchange market (relative to the controlled exchange rate) started at nearly 100 percent at the end of 1982 and then declined continuously until June 1985.

By combining the simple model of arbitrage activity with an equally simple macroeconomic model, this paper also analyzes the consequences of various shocks such as changes in the international interest rate or devaluations of the commercial exchange rate. The analysis suggests that an unanticipated devaluation of the commercial rate would lead to a jump in the financial rate that could either overshoot or undershoot the amount of the devaluation and might thus increase or decrease the differential. The analysis also suggests that the anticipation of a future devaluation of the commercial rate would lead to an immediate depreciation of the financial rate and would thus give rise to a differential between the two rates; this differential would be eliminated, or at least diminished, only when the devaluation of the commercial rate actually took place. This result seems intuitively plausible because the financial rate is a forward-looking variable.

Another more surprising result is that an increase in the degree of separation of the two markets, as represented by the costs for potential arbitrageurs, would increase the effects of disturbances in the financial markets (for example, changes in the international interest rate) on the financial exchange rate and thus, given the commercial exchange rate, on the exchange rate differential. If the aim of a dual exchange rate regime is to insulate the goods markets from disturbances in the financial markets, this result implies that the price for this insulation might be a higher degree of variability of some shadow prices in the financial markets and an increased deadweight loss because of the increase in private arbitrage and evasion activity.

^{Figure 2.}

**The Mexican Two-Tier Exchange Market: Percentage Discount on the Free Peso**

Citation: IMF Staff Papers 1988, 003; 10.5089/9781451972986.024.A003

Note: The discount is computed as the natural logarithm of the ratio of the peso price of one U.S. dollar on the free exchange market to the peso price of one U.S. dollar on the controlled market.^{Figure 2.}

**The Mexican Two-Tier Exchange Market: Percentage Discount on the Free Peso**

Citation: IMF Staff Papers 1988, 003; 10.5089/9781451972986.024.A003

Note: The discount is computed as the natural logarithm of the ratio of the peso price of one U.S. dollar on the free exchange market to the peso price of one U.S. dollar on the controlled market.^{Figure 2.}

**The Mexican Two-Tier Exchange Market: Percentage Discount on the Free Peso**

Citation: IMF Staff Papers 1988, 003; 10.5089/9781451972986.024.A003

Note: The discount is computed as the natural logarithm of the ratio of the peso price of one U.S. dollar on the free exchange market to the peso price of one U.S. dollar on the controlled market.The long-run ineffectiveness of the controls that have to support the dual exchange rate regime is also illustrated by the discussion of the effects of monetary policy: with a fixed commercial exchange rate, no independent monetary policy is possible in the long run. Any increase in domestic credit would be offset in the long run by an equal loss of reserves. In terms of the socalled offset coefficient this implies that the impact offset coefficient is different from (negative) unity, but the long-run offset coefficient is always equal to unity.

By considering the relevance of private arbitrage activity, the paper also highlights the importance of two points that are often overlooked in policy discussions about dual exchange rate regimes. The two points can be developed by first considering the proposition that, in the absence of private arbitrage activity, a dual exchange rate regime is operative (in the sense that it induces a different capital and current account than would a unified floating or fixed exchange rate) only if the expected rates of change of the two exchange rates differ. This proposition can be shown to hold in a variety of models and has been emphasized recently by Adams and Greenwood (1985) and Frenkel and Razin (1986). The intuition behind the proposition is that the savings and investment decisions that determine the capital account depend on the expected intertemporal terms of trade, which in turn are a function of the rates of change of the two exchange rates rather than their levels. But the proposition must be modified in the presence of private arbitrage activity, as in the model developed in this paper; in that case a dual exchange rate regime can also affect the equilibrium current and capital accounts when the differential between the two rates is constant because of the arbitrage activity induced by the differential.

A first corollary of this modified proposition, given the tendency of the differential between the two exchange rates to disappear over time, is that a dual exchange rate regime cannot effectively constrain the (cumulative) capital or current account in the long run. Any effect that is obtained when the differential between the financial and the commercial rates is widening would be offset by an opposite effect when the differential is falling back to zero.

A second corollary of the modified proposition, contrary to the arguments of Adams and Greenwood (1985) and Frenkel and Razin (1986), is that in the presence of incomplete market separation no exact equivalence exists between a regime of capital controls that attempts to insulate domestic interest rates from international interest rates and a regime of dual exchange rates. Private arbitrage activity reacts differently to the two regimes. Under capital controls, the incentive for private arbitrage activity exists only if the capital controls are effective in creating a differential between international and domestic interest rates. Under dual exchange rates, however, the incentive for private arbitrage activity exists even if the differential between the two exchange rates is constant and there is no differential between international and domestic interest rates.

A more technical result of the paper is that taking private arbitrage activity into account serves to determine the *level* of the financial exchange rate even in the context of models in which it would otherwise be indeterminate, such as the basic Dornbusch (1976) model. In most other models of dual exchange rates, the level of the financial exchange rate is determined by considerations of wealth effects. When private arbitrage activity is introduced as a function of the difference between the levels of the financial and the commercial rates, however, the level of the financial rate is determined even in the basic Dornbusch model.

Section I sets up the model that links arbitrage flows to the exchange rate differential, and Section II then incorporates this model of arbitrage flows into a simple general equilibrium model. Section III describes the effects of anticipated (future) and unanticipated devaluations of the commercial rate. It shows that the differential between the financial and the commercial exchange rates could either increase or decrease in response to an unanticipated devaluation, but that an anticipated future devaluation would always have an unambiguous effect on the differential by leading to an immediate devaluation of the financial exchange rate. Section IV describes the effects of anticipated and unanticipated changes in monetary policy. Given the fixed commercial exchange rate, monetary policy consists in changes in the domestic credit of the central bank; accordingly, this section shows that an increase in domestic credit, whether anticipated or not, would lead to an immediate depreciation of the financial rate. The magnitude of this depreciation is an increasing function of the degree of separation of the two exchange markets. Section V contains some concluding remarks.

## I. A Model of Arbitrage Activity in Dual Exchange Rate Regimes

Under a dual exchange rate regime, the exchange rate applicable to current account transactions (the commercial rate) may differ from the exchange rate applicable to capital account transactions (the financial rate).^{4} The commercial rate is usually fixed by the authorities, whereas the financial rate is sometimes set in a free market and sometimes set by the authorities. Because dual exchange rate regimes have mostly been used by countries trying to limit capital outflows, the financial rate, whether determined by a free market or by the authorities, has usually priced foreign exchange at a premium compared with the commercial rate.

Dual exchange rate regimes and capital controls can be regarded as analogous, since under a dual exchange rate the authorities can either set (a path for) the financial rate to achieve a given capital account target or impose controls on capital movements and let the financial rate be determined in the market. Often the implicit target for the capital account (excluding any change in official reserves) is zero; in this case the authorities maintain the commercial rate at a certain level, but the financial rate is determined in a free market without government intervention in that market. This situation occurs in Belgium, where the authorities maintain the commercial franc within the limits imposed by the EMS, whereas the financial franc floats freely.

The existence of a dual exchange rate offers arbitrage opportunities to economic agents in the sense that they would like to buy foreign exchange at the lower rate and sell it at the higher rate. Because the financial rate usually prices foreign currency at a premium compared with the commercial rate, it is henceforth assumed that the arbitrage opportunity consists in buying foreign exchange at the commercial rate and selling it at the financial rate. Given this arbitrage opportunity, all dual exchange rates have to be complemented by a set of rules that define current account (that is, commercial) and capital account (that is, financial) foreign exchange transactions.

One way to circumvent the rules designed to keep the two markets separate would be for an exporter to underinvoice foreign clients and invest the unrecorded payments in foreign financial assets.^{5} The proceeds from the sale of these assets could then be repatriated at the financial rate. An importer would correspondingly let himself be over-invoiced and would use the foreign exchange bought at the commercial rate to acquire foreign assets, which could likewise be repatriated at a profit. It is assumed here that importers and exporters incur costs each time they over- or underinvoice because each transaction (export or import) is subject to the control of the enforcement agencies.^{6} (Costs might also be incurred in repatriating the profits at the financial exchange rate.) These costs might take the form of penalties assessed by the enforcement agencies and of side payments that have to be made to foreign suppliers and clients to induce them to collaborate in the over-or underinvoicing. The potential arbitrageur maximizes the difference between the arbitrage profits and his costs, given by *g(y)*:

where y_{t} represents the amount of over- or underinvoicing that the arbitrageur undertakes, measured as the amount of domestic currency that the arbitrageur effectively uses to buy foreign exchange at the fixed commercial rate, *E _{t}*. (Both these exchange rates are in terms of domestic currency per unit of foreign currency.) It is apparent that equation (1) is not an intertemporal problem because the arbitrage opportunity, which arises if

*(E*/

_{t}^{7}and the costs occur at the same time. The arbitrageur, therefore, maximizes profits when the marginal cost of increasing the amounts over- or underinvoiced is equal to the proportional discount of the financial exchange rate:

An interior equilibrium exists only if *g˝(y _{t})> 0:* that is, if

*g(·)*is convex or has increasing marginal costs. The assumption of increasing marginal costs can be defended in two ways. First, in practice dual exchange rate regimes have not collapsed immediately on account of large-scale evasion activity, contrary to what might be expected if evasion activity showed decreasing marginal costs. Second, it seems plausible that in reality importers and exporters would find it increasingly difficult to justify their official prices or terms of payment as they increased the scale of their over- or undervoicing.

These considerations suggest that, as long as the financial rate is at a discount, funds will flow to take advantage of the arbitrage opportunity, but the magnitude of the arbitrage flows will depend on the exact form of the “cheating” function *g(y _{t})*. A small positive exchange rate differential might be compatible with no arbitrage flows if the marginal cost of even small amounts of arbitrage is positive; formally, the threshold at which arbitrage flows will stop is given by

In the remainder of the paper, it was convenient to use a specific form of *g(·)* that yields a linear relationship between the discount and the scale of arbitrage flows. Such a linear relationship results if the arbitrage cost function is quadratic, that is,

This specific functional form implies that arbitrage flows stop only if the discount disappears. This assumption seems appropriate for a country such as Belgium, which lies in the heart of Europe, is well integrated in international markets (exports are equal to 75 percent of gross national product (GNP)), and thus has considerable scope for arbitrage flows through over- or underinvoicing. The assumption also seems consistent with the fact that in Belgium, during periods of calm in the foreign exchange markets (presumably associated with few or no arbitrage flows), the discount on the financial rate has always been close to zero.

## II. The General Equilibrium

This section discusses the general equilibrium consequences of the model of arbitrage activity developed here in the context of a streamlined macroeconomic model. The model is kept as simple as possible to illustrate the main channel through which the arbitrage flows affect conditions in the financial markets.

The model consists of three equations. The first equation describes the trade balance and thus the intervention activity of the government, which purchases any excess supply of foreign exchange resulting from commercial transactions or provides for any excess demand, to keep the (nominal) commercial rate constant. The first important point about the model is that the arbitrage flows induced by the dual exchange rate regime influence the intervention activity of the government because the overinvoicing of imports (underinvoicing of exports) leads to an increased demand for foreign exchange on the commercial market. The “true” trade balance is a function of the (commercial) real exchange rate, denoted by (*D(F _{t})*, needed to keep the commercial exchange rate at

In this expression, *e _{t}* and

*p*represents the logarithm of the price level. The second expression on the right-hand side of equation (5) is equivalent to equation (4) for small values of the discount.

_{t}The financial market for foreign exchange is not controlled; this implies that risk-neutral arbitrageurs ensure that uncovered interest rate parity holds.^{8} The domestic interest rate, *i*, is therefore equal to the foreign interest rate, denoted by *i**, plus the expected rate of depreciation, *E[D(e _{t})]*. Combining this expression with a simple money demand function yields the second equation of the model

where *F* represents the foreign assets of the central bank, *F +**M* represents the money stock, and λ is unity divided by the semi-elasticity of money demand. *E[D(e _{t})]* represents the expected rate of change of the financial exchange rate; in this perfect-foresight model no distinction is made between actual and expected values. The notation

*E(·)*is therefore suppressed for the rest of the paper. The model is closed with a conventional price adjustment rule:

The simple model consisting of equations (5) through (7) has three dynamic variables: the price level, *p _{t}*; the foreign assets of the central bank,

*F*; and the financial exchange rate,

_{t}*e*

_{t}. These variables are functions of the commercial exchange rate,

^{9}

It is also apparent from equations (5) and (6) that without the arbitrage flows the level of the financial exchange rate is not determined. Without arbitrage flows (that is, if Φ is equal to infinity), the model would determine only the rate of change of the financial exchange rate. This result has already been noted in Dornbusch (1976); in general it depends on the absence of wealth effects, which are used in Dornbusch (1976,1985) and Frenkel and Razin (1986) to determine the level of the financial exchange rate.

The net foreign assets owned by the private sector have not been mentioned so far because they are determined by the exchange rate differential but do not influence any of the endogenous variables. Under the assumption that interest payments have to go through the financial foreign exchange market, the evolution of the net foreign assets of the private sector is given by the condition that the capital account plus interest receipts has to balance. Denoting the net foreign assets of the private sector by *A _{t}* this implies

Because the variable *A _{t}* does not enter the system of equations (5)-(7), the net foreign assets are simply determined by the initial value of

*A*and the path of the exchange rate differential is as determined by the system of equations (5)-(7). Note that without leakage—that is, if Φ is equal to infinity—the net foreign assets of the private sector would have to fall toward zero over time.

_{t}The viability of the dual exchange rate regime depends on the objectives of the government. If the government is concerned only with the level of the differential between the financial and commercial rates, it can maintain any constant value of the differential if it is prepared to offset the impact of the arbitrage activity by continuously reducing domestic credit, _{t} - *D* (e_{t}) = 0.^{10}

Despite the extremely simple macroeconomic relationships used here, the general equilibrium is characterized by a system of three differential equations in the variables *F _{t} e_{t}* and

*p*. The only other model known to the author that takes illegal leakage into account, Bhandari and Decaluwe (1987), also arrives at a third-order system, in the same variables (the financial exchange rate, the stock of reserves, and the price level). But Bhandari and Decaluwe were unable to obtain an analytical solution because their system is not recursive. They therefore had to limit themselves to numerical examples. In contrast, the present frame-work can be solved analytically, as can be seen by writing equations (5) through (7) as

_{t}^{11}

The system has three roots, given by

μ_{1} = -α

The first, stable root (μ_{1} = -α) determines the path of the price level, which then acts as an exogenous forcing variable in the remaining dynamic system of *F _{t}* and

*e*because the complete system of equations (5)-(7) is recursive. The dynamic subsystem consisting of Fand

_{t}*e*has two roots (μ

_{2}, μ

_{3}) of opposite sign; thus, the subsystem exhibits the usual saddle-path instability. The absolute value of these roots is determined only by the parameters that characterize the financial markets, Φ and λ. The complete solution of the model that also uses the initial condition is presented in the Appendix.

## III. The Effects of Devaluations of the Commercial Exchange Rate

In most dual exchange rate regimes, the path of the nominal commercial exchange rate is fixed by the authorities; in many cases the authorities keep this exchange rate fixed for some time and devalue by discrete amounts whenever it is judged unavoidable because of developments in the trade account. This section, therefore, analyzes the effects of an unanticipated devaluation of the commercial rate,

The reaction of the financial exchange rate to changes in the commercial exchange rate is of particular interest for Belgium because the commercial franc is fixed within the EMS (neglecting the ±2.25 percent bands) in relation to the deutsche mark and other European currencies. From the beginning of the EMS until September 1986 there were ten realignments, six of which involved changes in the central parity of the Belgian commercial franc in relation to the deutsche mark.^{12} Figure 1 shows that around the dates of the realignments, the financial franc was at a discount. The discount on the financial franc usually rises in periods of tension in the foreign exchange markets—that is, before each EMS realignment—and even before those realignments at which the parity of the Belgian commercial franc was not adjusted. During 1981 and 1982 the discount sometimes reached 15 percent and oscillated in general between 7 percent and 10 percent. This pattern in the financial exchange rate of the Belgian franc can be explained in terms of the model presented here, as will be shown in the remainder of this section, which discusses the reaction of the financial exchange rate to unanticipated and future anticipated changes (devaluations) of the commercial rate.^{13}

An unanticipated change in * _{t}*, equal to

^{14}

Whether the financial exchange rate would jump by more, less, or the same amount as the commercial exchange rate depends on the relative magnitudes of β and *αM* (assuming that the economy was initially at its steady-state equilibrium). The financial rate would jump by the same amount as *D**(e _{t})* = 0) only if the devaluation of the commercial rate had no effect on real balances. This result could occur only if the effects of the rising price level on the demand for money were exactly offset by the increased inflows through the trade account. (Note that

*αM*describes the effect of the rising level of the logarithm of prices on the real demand for money (not expressed as a logarithm), and β describes the effect of the devaluation on trade flows.) In this special case, the potential for arbitrage activity has no impact because no difference emerges between the financial and the commercial rates.

By contrast, if the trade account reacts strongly to a devaluation of the commercial rate (that is, if β is relatively high), as would be expected in Belgium where the proportion of tradables in GNP is very high, the financial exchange rate would overshoot its long-run equilibrium. In this case arbitrage activity would emerge and would influence both the initial jump of the financial rate and its entire future path.^{15} Because the (absolute value of the) root is a decreasing function of the parameter Φ that describes the severity of the controls used by the authorities to separate the two exchange rate markets, it is not apparent from equation (11) whether an increase in the severity of the controls would tend to reduce or increase the over- or undershooting of the financial exchange rate. However, equations (27) and (28) in the Appendix prove that an increase in the severity of the controls would tend to increase the observed differential between the financial and the commercial exchange rates after unanticipated devaluations.

It has often been argued that if the financial exchange rate is at a discount, the authorities should devalue the commercial rate to eliminate the differential and thus the incentive for arbitrage activity. However, the above results imply that sometimes an unexpected devaluation might increase the differential between the financial and commercial rates.

In equations (5) and (7) the equilibrium real exchange rate, *k*, has the same effect (but with the opposite sign) on inflation and the trade balance as does the commercial exchange rate, *Δe _{t}* in response to an unanticipated depreciation of the equilibrium real exchange rate is equal to

If equations (11) and (12) are compared, they show that the same parameters that determine whether the financial exchange rate over- or undershoots in response to an unanticipated devaluation also determine whether the financial exchange rate appreciates or depreciates in response to an unanticipated depreciation of the equilibrium real exchange rate. This result implies that if β > αM, an unanticipated depreciation of the equilibrium real exchange rate would lead to an overshooting of the financial exchange rate (that is, an immediate discrete depreciation followed by a continuous appreciation that leads the financial exchange rate back to its original level). The similarity between equations (11) and (12) also implies that an increase in the severity of the controls would tend to increase the magnitude of the over- or undershooting.

Dual exchange rate regimes are supposed to isolate the real sector from the effects of disturbances in the financial sector. It is interesting to note that this result implies that a disturbance in the real sector (that is, a change in the equilibrium real exchange rate) has effects on the financial sector (that is, on the financial exchange rate and therefore on interest rates).

In contrast to the effects of a current unanticipated devaluation of the commercial rate, it is clear that a future anticipated devaluation has to raise the discount on the financial rate at the time the market begins to expect the future devaluation of the commercial rate. By how much the financial rate has to jump at the time the news about the future devaluation reaches the market is determined by the condition that no jump in the financial rate should occur at the time the commercial rate is actually devalued (provided that the amount and the timing of the devaluation were correctly anticipated by the market). However, the differential that arises as the financial rate depreciates tends to lead to a loss of reserves because of the arbitrage flows. This loss of reserves implies (see equation (6)) that the expected rate of devaluation has to increase, and the financial exchange rate will therefore continue to depreciate increasingly until the devaluation of the commercial rate takes place. At this time, the arbitrage flows should stop, or may even be reversed, depending on how far ahead the devaluation was anticipated and on the value of (β - α*M)*. In Belgium, the differential cannot be negative, and the initial jump of the financial rate is therefore determined also by the condition that the financial rate is expected to be at least equal to the new (devalued) commercial rate. But if β is so much larger than *αM* that an unanticipated devaluation would result in overshooting, the financial rate might remain at a discount even compared with the new commercial rate after the devaluation. In this case the differential would only be reduced by the impact of the devaluation of the commercial rate, and the differential would disappear only gradually over time.

The results of this section may be compared with those obtained in the usual no-leakage models of the literature on dual exchange rate regimes. In these models (see, for example, Lizondo (1984)), a steady-state exchange rate differential usually exists that is different from zero and that is determined by wealth effects for the reasons mentioned above. The steady-state exchange rate differential is therefore not affected by the level of the commercial rate. In Lizondo (1984), a devaluation of the commercial rate will therefore initially reduce the exchange rate differential (a result that is possible but not guaranteed in the present model), but the financial rate has to continue to depreciate until the former exchange rate differential is re-established. The results described in this section are therefore similar to the results of all those models that contain a steady-state differential that is not affected by the level of the commercial exchange rate; the essential difference remains that in this model the steady-state differential is equal to zero.

## IV. The Effects of Monetary Policy on the Exchange Rate Differential

The imposition of a dual exchange rate regime is often justified by the authorities’ desire to insulate the domestic economy from the effects of disturbances in international financial markets. This section therefore considers the effects of an increase in the international interest rate, *i**, on the exchange rate differential and on the domestic interest rate. A further reason for the imposition of dual exchange rate regimes has often been the authorities’ desire to acquire some independence for the conduct of monetary policy without having to let the exchange rate float freely. This section therefore considers the effects of anticipated and unanticipated increases in domestic credit on the exchange rate differential and calculates the extent to which reserve flows would offset any change in domestic credit.

For the purpose of the discussion, it is convenient to begin with an examination of the effects of an expansionary monetary policy—that is, an unanticipated increase in domestic credit. The long-run effects of such a policy in the context of the model used here can be seen immediately from the condition that domestic credit of the steady-state money supply is determined by the fixed commercial exchange rate. The increase in domestic credit must therefore be offset, in the long run, by an equivalent loss in reserves. Since the behavior of prices and thus the trade account is also determined by the fixed commercial exchange rate, the reserve loss can be caused only by arbitrage activity, which in turn has to be induced by a differential between the financial and commercial exchange rates. Formally, this condition can be shown by computing the initial jump in the financial exchange rate, Δe_{t}, as a function of the change in domestic credit, Δ^{16}

This expression implies that the initial jump in the financial exchange rate is an increasing function of the parameter Х that characterizes the degree of separation of the two exchange rate markets. A higher degree of separation (in the sense of a high cost for potential arbitrageurs and a high value of ϕ implies a larger initial jump in the financial exchange rate because the required outflow of reserves can take place only if there is a large differential that causes the arbitrage flows in spite of the high costs associated with them. After the initial discrete depreciation of the financial rate, the rate of depreciation becomes negative (that is, the financial rate appreciates). It continues to appreciate at a decreasing rate until the differential disappears.

These results may also be compared with those of the no-leakage models in the literature, such as Cumby (1984). In that paper a monetary expansion is not fully offset by an outflow through the balance of payments because a monetary expansion leads to a permanently higher financial exchange rate—which implies a higher real value of the net foreign assets held by the private sector and, through portfolio effects, a higher demand for money. Such a result would not be possible in the context of this model even if it did include portfolio effects because the leakage ensures that in the long run the financial and commercial rates have to be equal.

An equivalent result can be easily obtained for the effects of a change in the international interest rate. The money market condition of equation (6) implies that a change in domestic credit of Δ*DC* is exactly equivalent to a change in the international interest rate of Δ*(M/λ)i**. Using this fact and equation (12) implies immediately that an unanticipated increase in the international interest rate has to lead to a jump in the financial exchange rate equal to

The adjustment pattern after the initial jump is then parallel to the one following an increase in domestic credit: the initial differential disappears over time as the financial rate appreciates (at a decreasing rate) until it is equal to the commercial rate. The reserve flows induced by this differential reduce the domestic money supply and thus raise the domestic interest rate until it is equal to the international interest rate. This result implies that dual exchange rates cannot protect domestic financial markets from the effects of disturbances in international financial markets.

The effectiveness of monetary policy has often been discussed in terms of the so-called offset coefficient, which measures by how much any given change in domestic credit is offset by reserve flows. In this framework, the offset coefficient is a function of time, as can be shown by computing the change in reserves induced by a given change in domestic credit (see the Appendix for details of computations):

Given the continuous time formulation, the impact coefficient (that is, for *t* = 0) is equal to zero. However, over time the (absolute value of the) offset coefficient rises and goes to (negative) unity in the long run (that is, as *t* goes to infinity).

The use of dual exchange rate regimes has often been advocated on the grounds that a dual exchange rate isolates goods markets from the effects of disturbances in financial markets. In this model, the goods market (characterized by the level of the real exchange rate) is indeed isolated from the effects of disturbances in the financial markets, but because the effects of disturbances in financial markets cannot be eliminated, they show up in the shadow price associated with the differential between the financial and the commercial exchange rates. If the cost of moving from one market to the other is high, this shadow price has to move by more for a given disturbance, because only in this way can the shadow price induce the flows of funds that ultimately neutralize the effects of the original disturbance. Because the deadweight loss associated with the arbitrage flows is an increasing function of the differential between the two exchange rates, a better separation of the two markets might not improve welfare. In general, this consideration thus contributes another argument against the use of dual exchange rate regimes.

The effects of an anticipated future increase in domestic credit can be derived from the requirement that no discrete jump in the financial exchange rate should occur at the time that the increase in domestic credit actually occurs. This requirement implies that the financial exchange rate has to jump at the time the news of the future increase in domestic credit policy is received. The exchange rate differential that is then created leads immediately to capital outflows, the money supply starts to fall, and the rate of depreciation of the financial exchange rate becomes positive. The financial rate will then continue to depreciate increasingly until the anticipated increase in domestic credit actually occurs. At that point in time, the depreciation of the financial rate becomes negative (that is, the financial rate appreciates), and it continues to be negative until the exchange rate differential disappears.

Because the domestic interest rate is equal to the international interest rate, *i**, plus the expected rate of depreciation of the financial rate, the results on the effects of monetary policy have important consequences for the viability and usefulness of dual exchange rate regimes. If the aim of the dual exchange rate regime is to keep domestic interest rates below international interest rates, the results shown in this section suggest that an increase in the money supply (that is, an increase in domestic credit) could achieve this objective only if it comes as a surprise; but even then the effects would be only temporary. If a one-time expansion of domestic credit was anticipated, it would have the undesired effect of raising domestic interest rates initially. A continuously expansionary domestic credit policy that was correctly anticipated would have no effects on domestic interest rates in the long run either. Such a policy would lead to a constant premium of the financial exchange rate that was just large enough to induce arbitrage outflows that neutralized the effect of the domestic credit expansion. Such a situation would be viable, but it would also imply that the only effect of the dual exchange rate system was to offer arbitrageurs the opportunity to make profits.

Similarly, if in response to an increase in international interest rates the authorities tried to neutralize the effects of arbitrage flows on domestic interest rates by increasing the rate of domestic credit expansion, the differential between the two exchange rates would widen, thus increasing the arbitrage flows. Any attempt by the authorities to conduct an independent money supply policy would then lead to an unstable spiral of depreciations of the financial rate and increasing arbitrage flows. A dual exchange rate regime (with a fixed commercial exchange rate) imposes, therefore, essentially the same constraints on the conduct of monetary policy as a unified fixed exchange rate.

## V. Concluding Remarks

This paper has analyzed the consequences of incomplete market separation in dual exchange rate regimes by developing an explicit model of the arbitrage flows that occur when the two exchange rates differ. While it has often been argued informally that the differential between the two exchange rates should not be allowed to become too large, it appears that the consequences of arbitrage flows in response to the exchange rate differential have not previously been analyzed formally within a macro-economic framework.

The main result of the paper is to show that these arbitrage flows would eliminate the exchange rate differential over time. This result is in contrast to the existing literature in which there is no force that reduces the exchange rate differential to enable it to settle down to any value in the long run. However, the data on the Mexican dual exchange rate market show that, in the absence of new shocks, the discount on the financial exchange rate tends to decline over time. The experience of the Belgian dual exchange rate regime, which has been in place for almost forty years, also shows that the discount on the financial rate tends to stay close to zero except in times of turbulence when devaluations of the commercial rate are expected. The data from these two countries thus support the framework proposed here.

An important corollary of the tendency of the two exchange rates to converge is that any attempt by the authorities to neutralize permanently the effects of these arbitrage flows would lead to an unstable situation. The use of dual exchange rate regimes to offset permanently the effects of permanent shocks should therefore not be attempted.

By using a simple macroeconomic model, the paper also analyzed the effects of various disturbances on the exchange rate differential. An unanticipated devaluation of the commercial exchange rate (or a fall in the long-run equilibrium real exchange rate) would lead to an overshooting of (and thus to a discount on) the financial exchange rate if the trade balance was sufficiently elastic with respect to changes in the commercial exchange rate. An unanticipated increase in domestic credit would always lead to a discount on the financial exchange rate; moreover, this discount is an increasing function of the degree of severity of the regulations that are used to keep the two exchange markets separate.

Because arbitrage flows link the two exchange rates, the financial exchange rate, which is forward looking, reacts to the anticipation of future disturbances. A future devaluation of the commercial rate or a future increase in domestic credit would therefore lead to an immediate discount on the financial rate. This result is confirmed by the data for Belgium, which show that the discount on the financial rate tends to increase sharply before each devaluation of the commercial franc inside the EMS.

### APPENDIX: Derivations

The system of equations (5)-(7) in the text is recursive; the path of prices is determined by equation (7) alone and acts as an exogenous, forcing variable in the reduced system of equations (5) and (6). The particular solution for the two variables *F* and *e* therefore, contains not only a constant but also a term in [exp (–αt)]. The sum of the particular and the homogenous solutions for *F _{t}* and

*e*can therefore be written as

_{t}where *A*, *B*, *C*, and *D* are constants to be determined by the initial conditions. Equation (17) already uses the result that the stationary state implies that ē =

The Reaction of the Financial Rate

to a Devaluation of the Commercial Rate

To calculate the jump in the financial exchange rate that results from an unanticipated devaluation of the commercial exchange rate by Δ

*p*= Δ

_{t}^{-αt}+

Solving equation (18) for *B* and substituting into equation (19) yields

where the second equality sign comes from the result that μ = ±(λ/*M*ϕ)^{1/2}. At the time of the jump in the commercial exchange rate, the money supply is given because it is a slowly adjusting variable. Since the system is assumed to start from a stationary state, this implies that ė_{0}_{0} = 0. Using this initial condition in the complete solution for the exchange rate (17) yields

Combining equation (22) with equation (21) implies that the exchange rate at time zero is given by

or, since the financial rate was equal to

Effects of a Change in the Equilibrium Real Exchange Rate

The effects of a change in the equilibrium real exchange rate from *k*. The only difference is in equations (23) and (24); the new steady rate for the financial rate is still given by

The initial jump in the financial exchange rate is therefore given by

The Effects of Domestic Credit Policy

The effect of an unanticipated jump in domestic credit of Δ*p =**(F-*

The initial jump of *e _{t}* is equal to C; this implies that

Using μ=(λ/*M*ϕ)^{2} yields equation (12) of the text.

Effects on the Initial Jump of the Financial Rate of Making Evasion Activity More Costly

The effect of a change in the parameter ϕ on the absolute value of the initial jump in the financial exchange rate that occurs in response to an unanticipated devaluation of the commercial exchange rate can be calculated from text equation (10):

where the term (α/μ) in the denominator of equation (10) has been substituted using the result μ =(λ/ϕ*M*)^{1/2}. The expression above can be simplified to

which is always positive.

### REFERENCES

Adams, Charles, and Jeremy Greenwood, “Dual Exchange Rate Systems and Capital Controls: An Investigation,”

(Amsterdam), Vol. 18 (February 1985), pp. 43- 63.*Journal of International Economics*Bhandari, Jagdeep S., and Bernard Decaluwe, “A Stochastic Model of Incomplete Separation Between Commercial and Financial Exchange Markets,”

(Amsterdam), Vol. 22 (February 1987), pp. 22- 55.*Journal of International Economics*Cumby, Robert, “Monetary Policy Under Dual Exchange Rates,”

(Guildford, England), Vol. 3 (August 1984), pp. 195- 208.*Journal of**International Money and Finance*Dornbusch, Rudiger, “The Theory of Flexible Exchange Rate Regimes and Macroeconomic Policy,”

*Scandinavian Journal of Economics*(Stockholm), Vol. 78 (May 1976), pp. 255- 75.Dornbusch, Rudiger, “Special Exchange Rates for Capital Account Transactions,”

*NBER Working Paper 1659*(Cambridge, Massachusetts: National Bureau of Economic Research, July 1985).Flood, Robert P., and Nancy Peregrim Marion,

*“The Transmission of Disturbances Under Alternative Exchange Rate Regimes with Optimal Indexing”*(Cambridge, Massachusetts), Vol. 97 (February 1982), pp. 43- 68.*Quarterly Journal of Economics*Frenkel, Jacob A., and Assaf Razin, “The Limited Viability of Dual Exchange Rate Regimes,”

*NBER Working Paper 1902*(Cambridge, Massachusetts: National Bureau of Economic Research, April 1986).Lanyi, Anthony, “Separate Exchange Markets for Capital and Current Transactions,”

International Monetary Fund (Washington), Vol. 22 (November 1975), pp. 714- 49.*Staff Papers*,Lizondo, José Saúl,

*“Exchange Rate Differential and Balance of Payments Under Dual Exchange Markets”*(unpublished; Washington: International Monetary Fund, 1984).Lizondo, Jose Saul, “Unification of Dual Exchange Markets,”

(Amsterdam), Vol. 22 (February 1987), pp. 57- 77.*Journal of International Economics*Reding, Paul,

*“Interest Parity in a Two-Tier Exchange Rate Regime: The Case of Belgium 1975-84,” Cahiers de la Faculté Des Sciences Economiques et Sociales de Namur, Serie Recherche No. 65*(March 1985).

^{}*

Mr. Gros, an economist in the Research Department, is a graduate of the University of Chicago. This paper was revised while the author was a Research Fellow at the Centre for European Policy Studies in Brussels. He wishes to thank colleagues in the Fund for helpful discussions and comments.

^{}1

See, for example, Dornbusch (1976) and (1985). The insulating effects of dual exchange rate regimes are also discussed in Flood and Marion (1982). Lizondo (1987) discusses the balance of payments effects of switching from a dual exchange rate regime to a unified floating or crawling peg system.

^{}2

The only exception seems to be Bhandari and Decaluwe (1987). But many other authors acknowledge that the difference between the two exchange rates should not become large enough to create problems in enforcing the separation between the two markets.

^{}3

A series of shocks would tend to raise the differential each time a shock hit the system, but after each shock the differential should, on average, decline.

^{}4

It is assumed here that interest payments are also converted at the financial exchange rate; thus the commercial exchange rate does not apply to all current account transactions.

^{}5

See Lanyi (1975) for a description of the various possibilities for traders to use foreign-trade-related financial transactions to circumvent the controls.

^{}6

This assumption is in contrast to the model of capital controls developed elsewhere. With capital controls, arbitrageurs incur costs only once, when they shift funds to the international market. Keeping the funds on the international market, where they earn higher interest rates, does not involve any additional costs. With a dual exchange rate regime, however, arbitrageurs have to make the entire round-trip each time to earn the arbitrage profit.

^{}7

As mentioned above, only situations in which the financial rate is at a discount compared with the commercial rate are considered here.

^{}8

See Reding (1985) for some empirical evidence for Belgium that shows that for the financial franc covered interest rate parity holds.

^{}9

Formally this is implied by the steady-state conditions *p=**F* = exp — *(i** +λ

^{}10

This result depends on the assumption that interest receipts on foreign assets have to pass through the financial markets.

^{}11

Changing the assumption that interest receipts on foreign assets have to pass through the financial market would imply for this model that the general equilibrium consists of four differential equations, with the foreign assets of the private sector in addition to the three variables of the system of equations (5)-(7). It would then become impossible to find an analytical solution. The gain in terms of tractability of the analysis might explain why the assumption that interest receipts pass through the financial market is commonly used in the literature. (See, for example, Dornbusch (1976) and Frenkel and Razin (1986).)

^{}12

The dates and the rates of devaluation of the Belgian franc against the deutsche mark (as measured by changes in central bilateral parity between the deutsche mark and the Belgian franc) were: September 24, 1979 (—2 percent); October5,1981 (–5.5 percent); February 22,1982 (–8.5 percent); June 14,1982 (–4.25 percent); March 21,1983 (–4 percent); and April 7,1986(2 percent). No changes occurred in the central parity between the deutsche mark and the Belgian franc during the realignments of November 30, 1979, March 23, 1981, July 22, 1985, and August 4, 1986.

^{}13

In Belgium, a certain class of exporters has the option of using either the financial or the commercial exchange rate. This means that the financial rate can never be lower than the commercial rate because the arbitrage costs apply only to outflows. This restriction is at variance with the model developed above, which implicitly assumes that arbitrage costs apply symmetrically to outflows and to inflows. The consequences of the asymmetry in the Belgian system will be taken into account whenever they become relevant in the remainder of the paper.