## Abstract

An optimizing model of dual exchange markets that are incompletely separated owing to the presence of fraudulent transactions is analyzed. The model is used to examine the implications of unanticipated and per¬manent changes in the commercial exchange rate and government spending. It is shown that these disturbances generate nonmonotonic responses in both the spread between the commercial and the financial rates and in capital flows. These results are contrasted with those obtained under complete market separation.

**T**here has been renewed interest recently in the operation of multiple exchange rate regimes, as represented by the number of recent contributions in this area (see Bhandari (1988) and Guidotti and Vegh (1988) for detailed bibliographies). Typically, dual exchange markets encompass a fixed rate for trade transactions (“commercial rate”) and a flexible rate for financial transactions (“financial rate”). An important feature of some of the new literature is the recognition that such multiple rate regimes often involve “leakage” between the two exchange markets. The presence of cross-transactions implies that despite the flextbility of the financial exchange rate, the capital account of the economy in question is nonzero, and net accumulation or decumulation of foreign assets may thus occur. This process was noted in the early literature (for example, Lanyi (1975)) and has been explicitly incorporated in non-optimizing models such as those found in Bhandari and Decaluwe (1987), Lizondo (1987), and Guidotti (1988).

The incorporation of the phenomenon of cross-transactions between the two markets has apparently not been extended to optimizing models. Thus, recent analyses of dual exchange markets incorporating optimizing behavior continue to assume that exchange markets can be, and in fact are, perfectly segmented (see, for example, Obstfeld (1986) and Guidotti and Vegh (1988)),^{1} In large part, this omission in optimizing models is due to the analytical difficulties encountered in attempting to deal with leakage in such a context.

The present paper is an attempt to incorporate leakages into an optimizing framework. Specifically, the model we construct is a fully optimizing one and similar in some respect to the complete separation model proposed by Guidotti and Vegh (1988). At the same time, intermarket leakage is explicitly introduced into the framework. Although such leakage is examined in nonoptimizing models, as in Bhandari and Decaluwe (1987), the present approach to this phenomenon is somewhat more realistic in its treatment of overinvoicing and underinvoicing associated with fraudulent transactions. Another contribution is the analytical characterization of the solution to a fourth-order differential equation model with two state variables using the technique originally suggested by Calvo (1987b).

The framework is used to examine the effects of certain domestic disturbances: specifically, devaluation and real expenditure changes. When the commercial exchange rate is devalued, a spread emerges, with the financial rate being relatively appreciated compared to the commercial rate. Thus, foreign currency is available at a discount in the financial market. With the passage of time, the discount diminishes progressively, turning into a premium in the financial market. The final phase of adjustment involves a declining premium until the steady state with a zero spread is ultimately restored. This nonmonotonic response is due to the presence of intermarket leakage, and other optimizing models that abstract away from leakage (for example, the models of Obstfeld (1986) and Guidotti and Vegh (1988)) are incapable of generating this type of response, which involves, first, a financial discount, and, subsequently, a financial premium.^{2}

In addition to predicting an oscillatory pattern of adjustment in the spread, the model also suggests a similar pattern for the capital account. Specifically, devaluation of the commercial exchange rate is followed by successive phases of capital inflows and outflows. These nonmonotonic movements in the spread and the capital account are also present when unanticipated and permanent changes in government spending take place.

The paper is organized as follows. Section I presents a simple optimizing model of a dual exchange rate economy in which no leakage is assumed to occur. This model is intended to serve as a benchmark case in order to facilitate comparison with the more substantive model incorporating leakage that is developed in Section II. The last section offers some concluding remarks.

### I. The Dual Exchange Rate Economy Without Cross-Transactions

This section discusses the case of a dual exchange rate economy in which the two exchange markets are perfectly segmented. This model is a particular case of the more general framework detailed in the next section. It enables one to evaluate the novel features introduced by the presence of leakage in the more general setting.

The framework is a small-country version of the two-country (complete separation) cash-in-advance model discussed by Guidotti and Vegh (1988). The exchange rate regime in place involves a pegged commercial rate, ** Ē**, and a freely flexible financial rate,

**,**

*Q*^{3}Because the financial rate is flexible and there is complete market separation, the real quantity of external assets (bonds) in the hands of the private sector is fixed at

^{4}

In contrast, as will be seen in the next section, the possibility of leakage permits domestic residents to purchase or sell foreign bonds. The ratio of the commercial rate to the financial rate is denoted by ** q;** namely,

*q ≡ Q / Ē*, which can be interpreted, as will be seen below, as the real (domestic) price of bonds. The world interest rate is a constant equal to

*r*. For the purposes of this section, it will be assumed that there is only one (nonstorable) good in the world whose domestic price is given via a purchasing-power-parity relationship)—that is,

*P = ĒP*. where

^{*}*P*and

*P*denote the domestic and foreign price level, respectively. In what follows the foreign price level

^{*}*P*is set equal to unity for simplicity.

^{*}The representative domestic consumer is endowed with a constant stream of the good, *y*. The consumer must use money to acquire the good; namely, *M* =αPc, where *M* stands for nominal money balances, *c* for consumption, and α>0.^{5}

The consumer faces the following optimization problem;^{6}

subject to

where a dot over a variable denotes its time derivative; the utility function, *U(·)*, is assumed to be increasing, twice-continuously differentiable, and strictly concave; δ denotes the constant rate of time preference; *m =M/P* denotes real money balances; and τ denotes real transfers from the government. Equation (1) is the asset-accumulation relationship; note that the real return on bonds equals (r + ^{7} As is well known, a dual exchange rate regime drives a wedge between world and domestic real interest rates. Equation (2) is the cash-in-advance constraint. Equation (3) defines real wealth.

The other actor in this economy is the government, which faces the following budget constraint:

where *f* denotes holdings of interest-bearing reserves, and *ḡ* indicates the (constant) stream of real government spending.^{8}

The money supply is given by

where *D*_{0} denotes the initial stock of domestic credit.

Solving the consumer’s problem, imposing the equilibrium condition b, =

where time subscripts have been dropped for convenience, and a subscript, unless otherwise indicated, indicates a partial derivative.

As can be observed, the system is block-recursive. In fact, equation (5), by itself, determines the behavior of *M*. Assuming that [(1/α) – r] > 0, equation (5) provides the only negative root of the system. Intuitively, this condition requires that consumption exceed interest gains on reserves when *M* increases to prevent an unbounded growth in *M*. Since there is only one state variable, *M*, the system exhibits saddle-path stability. As shown in Guidotti and Végh (1988), *M* and p, as well as p and *q*, move in opposite directions.

#### Devaluation

Consider an unexpected and permanent devaluation; that is, an increase in *Ē*^{9} The path of the different variables is illustrated in Figure 1. The increase in *Ē*, through the law of one price, implies a rise in the domestic price level that reduces real money balances. Owing to the assumptions of a freely fluctuating financial rate coupled with complete market separation, the public is prevented from exchanging bonds for money (which would have been possible under fixed exchange rates). Thus, the public cannot instantaneously restore its real money balances. It follows from the cash-in-advance constraint that consumption also falls on impact, thus generating a current account surplus. Given that the economy will be running current account surpluses while it is on the adjustment path, real money balances and, hence, consumption will also be increasing over time.^{10} The public foresees this increasing consumption path and bids down the real price of bonds, *q*, by attempting to sell bonds in order to smooth out consumption over time. The real domestic interest rate rises on impact and then decreases toward its steady-state level. This behavior of the real rate of interest induces an increasing consumption path for two reasons. First, as long as the real interest rate remains above the rate of time preference, consumers choose an increasing consumption path. Second, because the real interest rate (which is also the nominal interest rate) declines over time, future consumption is cheaper than present consumption.

The fact that there is only one state variable, M, ensures that the adjustment of all variables is monotonic. In particular, *q* rises toward its initial value, given by *r* /α.^{11} Both nominal money balances and consumption are higher in the new steady state, since the economy runs current account surpluses during the adjustment period.^{12}

#### Decrease in Government Spending

Consider an unanticipated and permanent decrease in government spending. The path of the different variables is the same as that illustrated in Figure 1 except for the behavior of consumption, as explained below. Since real money balances are given on impact, the decrease in government spending is translated on a one-to-one basis into a current account surplus. Consumption remains unchanged on impact but increases over time because money flows into the country during the adjustment process. Individuals foresee this increasing consumption path and try to smooth out consumption over time by selling bonds, thus bidding down their real price, *q*. Subsequently, *q* increases over time to its initial value. Naturally, in the aggregate, this increasing consumption path cannot be smoothed out, so that the domestic real interest rate rises on impact and declines throughout, while remaining above its steady-state level to induce such a path. Consumption is higher in the new steady state due to the current account surpluses run by the economy. Again, the adjustment of all variables, in particular that of *q*, is monotonic.

### II. The Dual Exchange Rate Economy with Leakage

This section modifies the benchmark model of the previous section in order to allow for the possibility of cross-transactions (leakage) between the two exchange markets. The economy is endowed with an exportable good, *y*, which it does not consume. Instead, the economy consumes an importable good, c. The relative price of the exportable good in terms of the importable good is given and, for simplicity, will be set equal to unity. It is convenient to begin the description with a specification of the budget constraint of the representative individual in nominal terms. Specifically, this is given by (only new variables are specified below; otherwise, the definitions of the previous section apply):

where *A* represents nominal wealth and *I* measures the real amount of leakage that is characterized by fraudulent overinvoicing *(I* ‹ 0) or underinvoicing of exports *(I* ›0). To the extent that the financial rate is relatively depreciated vis-á-vis the commercial rate, the term *I(Q — Ē)* measures the addition to the consumer’s resources resulting from the successful underinvoicing of exports. Similarly, if the financial rate is relatively appreciated with respect to the commercial rate, *I(Q — Ē)* measures the profits made from overinvoicing exports. It should be noted that, on an *aggregate* level, the stock of legal bonds is fixed. Therefore, in the aggregate the actual stock of bonds changes solely as a result of overinvoicing or underinvoicing of exports. The fraudulent activity is not without costs, however, which are captured in equation (8) by the term *h(I)*. This term is intended to capture the recurring transactional cost that would be incurred, for example, from the corrupt procurement of official approval or acquiescence in a de facto illegal transaction. In what follows, we assume a specific functional form for *h:*

where 0 ≤ ф < 1.^{13} We interpret ф as the indirect transactional cost associated with the maintenance of illicit activity, and a rise in ф indicates an increase in such costs. It will be assumed that the government levies a tax on bond holdings, given by the term *γ(qb)Qb*, where 0 < *γ* < l.^{14} This assumption causes the return on bonds to be dependent on the quantity of bonds held, thus leading to a well-defined steady state. This tax does not affect the choice of *l*, as the consumer’s first-order conditions make clear.^{15} The remaining terms in equation (8) are similar to those in equation (1) in the simplified model. Thus, *Ēbr* denotes the interest proceeds on holding foreign assets, *Qb* is the capital gain term, and Ē τand *Ēc* refer to nominal transfers and consumption, respectively.

For analytical purposes, it is preferable to recast equation (8) in real terms analogously to equation (1), taking into account the functional forms for *h(I)* and *γ(bq)*. Thus, after some rearrangement, the budget constraint may be expressed as

where

The representative consumer’s maximization problem may now be stated as

subject to equation (9) and

The optimality conditions for this control problem are given by (unless strictly necessary, time subscripts will be dropped for convenience)

where

Equation (12) is the familiar condition equating the marginal utility of consumption with the marginal utility of wealth, γ, times the effective price of consumption. The effective price of consumption is given by its market price (unity), plus the opportunity cost of holding the a units of money needed to purchase a unit of the good, ap. The expression for the domestic real interest rate, given by equation (16), is worth noting since it is affected by the presence of the tax on bond holdings. The parameter p affects the real return on bonds, because the consumer is deprived of a fraction *βqb* of the value of his bond holdings, *qb*. This implies that βq^{2}b^{2} is lost on taxes, which results in a negative marginal return of *2βq*^{2} *b*, or *2γqb* when expressed in terms of the financial rate. Thus, the real return on bonds now depends on the level of bond holdings. Equation (13) describes the path of the co-state variable. Equation (14) indicates the instantaneous level of leakage that results from falsifying export invoices. As one would expect, a positive (negative) spread induces underinvoicing (overinvoicing), which results in asset accumulation. Equation (15) indicates the path of *q*, which is influenced by the bond holding tax.

It will be assumed that the bonds that are taxed away from the consumer are returned in the form of lump-sum transfers. As before, government spending is exogenously given. The government budget constraint is thus

In equilibrium, the stock of bonds held by the private sector can only change as a result of fraudulent transactions; namely,

Combining the consumer’s optimality conditions (equations (12)—(15)) with equations (17), (18), and the definition of money supply (given by equation (4b)), the following four-equation dynamic system is obtained:

Equation (19) is the balance of payments equation. The presence of leakage implies that, in contrast to the no-leakage model (recall equation (5)), the level of *q* affects the change in money balances. This, in turn, implies that the balance of payments equation by itself no longer determines the behavior of money balances. Equation (20) describes the evolution of the real domestic interest rate. Equations (21) and (22) are familiar by now (note that (22) results from substituting (18) into (14)). The presence of leakage introduces some additional technical difficulties for the analytical solution to the dynamic system given by equations (19)—(22), since there are now two state variables in the economy. *At* and *b*.^{16}

Linearizing the system (19)-(22) around the steady state, we obtain

where

Equations (25a)-(25d) describe the steady-state values of the system. The steady-state value of *q* is unity, unlike the no-leakage case where it could differ from unity, depending on the ratio (r/δ). A value of *q* different from unity would be inconsistent with a steady state because there would be net variations in the private sector’s stock of bonds. The steady-state stock of bonds is constant (in the sense that it does not depend on either *Ē)*, because their real return depends on the amount that is held, and thus, in the steady state, there is a unique level of bond holdings that enables the consumer to equate the real return of bonds with the rate of time preference. The steady-state value of nominal money balances does not differ conceptually from the no-leakage case, in that it depends positively on *Ē* and negatively on

It follows from (24) that

The sign of equation (26b) follows from the assumption that [(1/α) - r]>0.^{17} From (26a) it follows that there is at least one positive characteristic root. Given (26a), equation (26b) indicates that there are either four positive roots or two positive and two negative roots. The former case can be ruled out since it would imply instability of the system. The remaining case, involving two positive and two negative roots, ensures that, given initial conditions for *M* and b, there exists a unique convergent continuous path that satisfies (23)—provided, of course, that in the general solution the arbitrary scalars that multiply the two positive roots are set to zero. Since it has been found that there are two negative roots (or. more precisely, two roots with negative real parts), the question of whether these roots can be either real or complex has to be addressed. The characteristic polynomial associated with matrix Γ is given by

where

Figure 2 depicts P(μ). Two pieces of information yield the conclusion that the two negative roots are real. It follows from equation (27) that for any parameter configuration, P(0)<0, and *P{A =* -[(1/α) - *r]}* < 0.

**Characteristic Polynomial**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

**Characteristic Polynomial**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

**Characteristic Polynomial**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

This implies, as Figure 2 makes clear, that the two negative roots are real roots, one being larger (in absolute value) than -[(1/α) - r], and the other being smaller. These roots will be denoted by μ_{1} and μ_{2}. respectively.

Let the characteristic roots of the matrix Γ be μ_{i}, *i =* 1, 2, 3, 4. Thus, if the μ_{i}s are pairwise distinct, the general solution to (23) is given by

where ω_{i}, *i* = 1, 2, 3, 4 are arbitrary scalars, and h_{1} = *(h _{i1}, h_{i2}, h_{i3}, h_{i4})* is the eigenvector associated with root μ

_{i},. As shown above, the matrix Γ has two negative real roots, denoted by μ

_{1}and μ

_{2}. Therefore, a necessary condition for convergence is that Μ

_{3}= Μ

_{4}= 0.

Recalling that Γh_{i} = μ_{i}h_{i} , it follows that

Since by definition an eigenvector cannot be a zero vector, equations (29a)-(29c) imply that all components of *h ^{i}, i* = 1,2, are different from zero.

^{18}Having established this fact, we can now characterize the behavior (outside the steady state) of the ratio of the paths of

*q*and

*b*. First note that along equilibrium paths that result from alternatively setting μ

_{i},

*i=*1,2, equal to zero, we obtain (recall that we have established that all

*h*s are different from zero)

_{i}It follows from (29c) that

These special paths are very useful in characterizing the dynamic behavior of the system. More specifically, the path (q_{t} - _{t} - _{i}. which is the greater (in absolute value) of the two negative roots, will be of particular interest. Such a path is called “the dominant eigenvector ray” (Calvo (1987b)). The path *(q _{t}-*

_{t}-

_{2}, is “the nondominant eigenvector ray.” If the system were to start on either of the eigenvector rays, it would travel along it toward the steady state, since both are solutions to system (23). For the latter reason, it is also the case that the convergent equilibrium path cannot cross any of the eigenvector rays.

The key fact in determining the behavior of the system can be derived from the following proposition (see Appendix for the proof).

Proposition. *For any initial values of* q *and* b *such that* (q_{0}-_{0} - _{13}/h_{14},

*Otherwise*,

The implications of this proposition are illustrated in Figure 3. From equation (22), we can determine the direction of the movement of *b* in the two zones divided by the line * $\dot{b}$ =* 0. The above proposition indicates that unless the initial values of

*q*and

*b*are such that they are on the nondominant eigenvector ray, the slope of the path

*(q*$\overline{q}$ ) / (b$\overline{b}$ )eventually converges to the slope of the dominant eigenvector ray. This means that depending on the value of

_{t}-_{t}-*q*after the jump following an unanticipated disturbance, the adjustment path will follow either the arrowed curve going from

*A*to

*O*or that going from S to

*O*. It can be seen that in either case, the adjustment of both variables is nonmonotonic.

It follows that it is necessary to derive the initial jump in *q* to determine which adjustment path the dynamic system will follow. Consider the convergent solution that follows from the general solution (28) at *t* = 0:

**Adjustment Paths of b and q**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

**Adjustment Paths of b and q**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

**Adjustment Paths of b and q**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

From (32a) and (32c)—and using equations (29a)-(29c)—the constants ω_{1} and ω_{2} can be computed. By substituting ω_{1} and ω_{2} into (32b), one obtains (recall that * $\overline{q}$) =* 1)

^{19}

where the coefficient of (M_{0} - _{1}, /ξ_{2}) > 1, *h*_{11} >0, and h_{21} < 0. Equation (33) indicates that if nominal steady-state money balances rise, then *q*_{0}*<* 1—that is, *q* jumps downward on impact. The opposite is true if steady-state nominal money balances decrease.

The behavior of *M* can be derived by considering the (A/, *b)* plane. Proceeding as before, it can be shown that

where

The implication of the last statement for the adjustment path will become clear when we consider the effects of a devaluation and of a decrease in government spending.

### Devaluation

Consider an unexpected and permanent devaluation; that is, an increase in *Ē*. Because nominal money balances increase across steady states, *q* declines on impact, as follows from equation (33). Therefore, the adjustment path of *q* in terms of Figure 3 is from *A* to *O*. The upper panel of Figure 4 illustrates the behavior of *M* and *b*. The initial steady state is *( $\overline{b}$,$\overline{M}$*

_{0}

*)*(point

*A*). As follows from equations (25a) and (25d),

*b*remains constant across steady_ states, whereas

*M*increases, so that the new steady state is given by

*(*$\overline{b}$ ,

_{1}) (point

*B)*. The arrowed curve describes the dynamic path of both variables from

*A*to

*B*. The arrowed curve cannot cross the dominant eigenvector ray (which slopes positively, as shown earlier), because if it touched it, it would stay on it indefinitely, since the dominant eigenvector ray is a solution of the differential system. It is clear that

*M*increases monotonically toward its new steady state. The stock of bonds, however, exhibits nonmonotonic behavior. It decreases initially (that is, there are capital inflows), then reaches a stationary point, after which it begins to increase (capital outflows) and reaches the same initial value. The lower panel of Figure 4 illustrates the path of

*q*that follows from Figure 3. On impact,

*q*falls as it also does in the no-leakage case (recall Figure 1). In the presence of leakage, however, as long as

*q*remains below unity, the private sector has an incentive to overinvoice exports, thus causing a decrease in the stock of bonds. The illegal reduction in the stock of bonds (that is, capital inflow) continues until

*q*reaches unity. Because the stock of bonds has to return to its original level (to equate the rate of return on bonds to the rate of time preference),

*q*has to overshoot its long-run value (unity) in order to induce underinvoicing of exports and accumulation of bonds (that is, capital outflows), which occurs until

*q*returns to unity.

The path of consumption mirrors the behavior of real money balances falling on impact and increasing over time to its new steady state. The domestic real interest rate, as was the case before, must rise on impact and decrease monotonically toward its steady-state level to induce this consumption path (see Appendix for details). The behavior of *M, c*, and p is therefore qualitatively the same as in the no-leakage case illustrated in Figure 1.

**Effects of a Devaluation Under Incomplete Market Separation**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

**Effects of a Devaluation Under Incomplete Market Separation**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

**Effects of a Devaluation Under Incomplete Market Separation**

Citation: IMF Staff Papers 1990, 001; 10.5089/9781451956863.024.A005

### Decrease in Government Spending

Consider an unexpected and permanent decrease in government spending. Figure 4 also applies in this case. The new level of steady-state nominal money balances is higher. Nominal money balances increase throughout the adjustment path (from *A* to *B* along the arrowed curve)— that is, the economy runs current account surpluses. Capital inflows in the first phase of the adjustment are followed by capital outflows, which return the stock of bonds to its initial level. For reasons analogous to the devaluation case, *q* jumps downward on impact, overshoots its long-run value, and then returns to unity. The behavior of consumption and the domestic real interest rate is the same as in the complete separation case.

Intuitively, with the decrease in government spending, more resources are available for the private sector to consume. This necessitates an increase in nominal (and, hence, real) money balances, which can only take place over time. Since the consumer foresees an increasing consumption path, he attempts to sell bonds to smooth out consumption over time, which produces an instantaneous fall in *q*. This in turn induces the public to overinvoice exports, which increases private asset holdings. This situation is reversed, however, when *q* increases above unity, thus inducing export under in voicing.

## III. Conclusion

This paper has constructed and analyzed a model of dual exchange rates characterized by the presence of fraudulent cross-transactions between the two exchange markets. Unlike previous analyses that have incorporated such leakage, the present model is a fully optimizing one.

The model is used to examine the effects of certain domestic disturbances, such as devaluation of the commercial exchange rate and real government expenditure disturbances. A new and interesting result is that following such disturbances, the behavior of at least some domestic variables, such as the exchange rate spread, is oscillatory. Specifically, when the commercial rate is devalued, for example, the financial rate is, in the first instance, relatively appreciated compared to the commercial rate. With the passage of time, the discount diminishes progressively and turns into a premium, with the financial rate being depreciated compared to the commercial rate. The final phase of adjustment involves a declining premium until the steady state with a zero spread is ultimately restored. Similar nonmonotonic behavior is predicted for the stock of bonds. Since, on impact, the financial rate becomes relatively appreciated compared to the commercial rate, a process of reduction of the stock of bonds (capital inflows) takes place until the spread reaches zero. Afterwards, however, the financial rate becomes relatively depreciated, which causes an illegal accumulation of bonds (capital outflows). These variations in the stock of bonds occur, in the aggregate, through under-invoicing or overinvoicing of exports. These patterns of adjustment appear to be in conformity with several actual experiences for the period following commercial devaluation.

The results of this paper may have important policy implications. For example, commercial devaluation is seen to result during the transitional phase in an increasing exchange rate spread, and account needs to be taken of this possibility when the performance of a devaluation strategy program is being assessed. Similarly, our results indicate that devaluation will involve periods of both capital outflows and inflows. The effect on the capital account of a devaluation will depend, therefore, on the time period during which the status of the capital account is ascertained. We recognize, of course, that the behavior of speculative capital movements is also substantially influenced by factors such as “credibility”; however, the effects of the latter factors would co-exist with those discussed in the text, and insofar as conflicting effects may be involved, the net effect of devaluation on the capital account may be difficult to predict.

## APPENDIX Proof of Proposition and Behavior of ρ

This Appendix presents the proof of the Proposition given in Section II and provides a description of the behavior of p.

### Proof of Proposition

Setting to, ω_{3} = ω_{4} = 0 in equation (28) yields

which implies,μ_{1} < μ_{2} given that that if ,ω≠_{2}

The second part of the proposition corresponds to the case in which ω_{2} = 0.

### Behavior of ρ

We now show how the initial jump in p can be established. Recall equation (21):

Consider the case of a devaluation. We have already established that, on the one hand, *q* falls on impact, which means that the last term on the right-hand-side of (21) falls on impact. On the other hand, * $\dot{q}$* becomes positive on impact, which implies that

*ρq*must rise on impact. Since

*q*falls, p must necessarily increase. Given the initial rise in ρ, proceeding as before, it can be established that the dominant ray eigenvector in the (b, ρ) plane slopes downward, which implies that the adjustment path of ρ is monotonic.

## References

Bhandari, Jagdeep S., “Exchange Rate Reform and Other Structural Disturbances in a Dual Exchange Rate Economy,”

*IMF Working Paper WP/88/99*(Washington: International Monetary Fund, 1988).Bhandari, Jagdeep S., and Bernard Decaluwe, “A Stochastic Model of Incomplete Separation Between Commercial and Financial Exchange Markets,”

Vol. 22 (February 1987).*Journal of International**Economics*,Calvo, Guillermo A., “Temporary Stabilization: Predetermined Exchange Rates,”

Vol. 94 (December 1986).*Journal of Political Economy*,Calvo, Guillermo A., (1987a), “Balance of Payments Crisis in a Cash-in-Advance Economy,”

Vol. 19 (February).*Journal of Money, Credit, and Banking*,Calvo, Guillermo A., (1987b), “Real Exchange Rate Dynamics with Nominal Parities: Structural Change and Overshooting,”

Vol. 22.*Journal of International Economics*,Feenstra, Robert C, “Anticipated Devaluations, Currency Flight, and Direct Trade Controls in a Monetary Economy,”

Vol. 75 (June 1985).*American Economic Review*,Flood, Robert P., and Nancy Marion, “Determinants of the Spread in a Two-Tier Foreign Exchange Market,”

*IMF Working Paper WP/88/67*(Washington: International Monetary Fund, 1988).Garza-Gomez, Jesus M., “Dual Exchange Markets and the Balance of Payments” (unpublished; Philadelphia: University of Pennsylvania, 1988).

Guidotti, Pablo E., “Insulation Properties under Dual Exchange Rates,”

Vol. 21 (November 1988).*Canadian Journal of Economics*,Guidotti, Pablo E., and Carlos A. Vegh, “Macroeconomic Interdependence under Capital Controls: A Two-Country Model of Dual Exchange Rates,”

*IMF Working Paper WP/88/74*(Washington: International Monetary Fund, 1988).Kamin, Steven B., “Devaluation, Exchange Controls, and Black Markets for Foreign Exchange in Developing Countries,”

*International Finance Discussion Paper*(Washington: Board of Governors of the Federal Reserve System, October 1988).Lanyi, Anthony, “Separate Exchange Markets for Capital and Current Transactions,”

(Washington), Vol. 22 (November 1975).*Staff Papers*, International Monetary FundLizondo, Jose S., “Exchange Rate Differential and Balance of Payments under Dual Exchange Markets,”

Vol. 26 (1987).*Journal of Development Economics*,Obstfeld, Maurice, “Capital Controls, the Dual Exchange Rate and Devaluation,”

Vol. 20 (February 1986).*Journal of International Economics*,Tornell, Aaron, “Insulating Properties of Dual Exchange Rates: A New Classical Model,”

*Discussion Paper No. 380*(New York: Columbia University, 1988).

^{}*

Jagdeep S. Bhandari is an economist in the European Department. He is currently on leave from West Virginia University where he is a Professor of Economics. He received his Ph.D. in economics from Southern Methodist University. He also holds a J.D. degree from Duquesne University and an IX.M. degree from Georgetown University.

Carlos A. Végh is an economist in the Research Department, He holds a doctorate in economics from the University of Chicago.

The authors wish to acknowledge helpful comments by Aaron Tornell and several colleagues in the Research Department.

^{}1

The possibility of intermarket transactions is briefly noted in Flood and Marion (1988). Tornell (1988) also recognizes the possibility of leakage; however, the model he uses abstracts from consumption-savings decisions, and illegal transactions are carried out by risk-neutral investors.

^{}2

Kamin (1988) reports empirical tests in which the behavior of the spread follows an oscillatory path as a result of a devaluation. On a theoretical level, the possibility of oscillations in the spread in the context of a portfolio-balance nonoptimizing model is also noted in Garza-Gomez (1988). In Tomell (1988) and Kamin (1988) such behavior occurs as a result of temporary disturbances. In contrast, this paper finds nonmonotonic behavior in a fully optimizing model for an unanticipated and permanent devaluation.

^{}3

Both exchange rates are defined as number of units of foreign currency per unit of domestic currency.

^{}4

The foreign price level is assumed to be constant, so that both the nominal and real quantities of foreign bonds in the hands of domestic residents are fixed.

^{}5

Feenstra (1985) derives cash-in-advance constraints in continuous-time models. For applications, see Calvo (1986, 1987a).

^{}6

It will be assumed that interest payments, being a current account item, are repatriated at the commercial exchange rate.

^{}7

Because the commercial exchange rate and foreign price level are fixed, real money balances do not depreciate.

^{}8

As pointed out by Obstfeld (1986), the fact that reserves earn interest implies that devaluation is neutral under fixed exchange rates.

^{}9

It is assumed that, as it is usually the case, the central bank does not monetize the nominal capital gains on reserves but instead creates a fictitious nonmonetary liability.

^{}10

Note that even if consumption rises over time, it is initially at a lower *level* than before the disturbance.

^{}11

The behavior of *q* will be dramatically altered when leakages between the commercial and financial markets are incorporated, as will be shown in the next section.

^{}12

As noted by Obstfeld (1986), the fact that steady-state consumption is higher does not mean that a devaluation improves welfare. On the contrary, the losses in utility attributable to diminished consumption in the short run outweigh the gains from a higher steady-state consumption. This follows from the fact that, for given resources, a constant path of consumption is preferable to a nonconstant path.

^{}13

This functional form permits considerable simplicity in that the resulting expressions are linear.

^{}14

It will also be posited that γ, the fraction of bonds taxed, takes the linear form *γ(qb) = βqb*, where β is a positive constant that is small enough to ensure that γ remains below unity. Thus, the larger the amount of bonds in the possession of the trader, the higher the proportion of bonds that is taxed.

^{}15

A more natural assumption might be that a fraction of the illegally acquired bonds are confiscated. However, this leads to analytical intractability. The introduction of the tax on bond holding enables us to solve an in finite-horizon optimizing model, with incompletely separated dual markets. Since, to the best of our knowledge, this has not been done previously, this model should be viewed as an attempt toward placing the theory of dual markets under incomplete separation on firmer microeconomic grounds.

^{}16

The solution procedure used is based on Calvo (1987b).

^{}17

Recall that this condition is also necessary in the no-leakage model to ensure the existence of the only negative characteristic root.

^{}18

For instance, if *h _{i1}* = 0. then (29a) implies that (1 -

*φ)h*

_{i3}

*= rh*

_{i4}. Substituting the latter into (29c) yields

*(r -*ω

_{i},)h

_{i4}= 0, whence it follows that

*h*= 0. This in turn implies, from (29a), that

_{i4}*h*

_{i3}

*=*0, which (from (29b)) implies that

*h*0. The other cases are straightforward.

_{i2}=