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 $\overline{b}$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 + $\dot{q}$)/q (hereafter denoted by p), which is the domestic real rate of interest.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 D₀ denotes the initial stock of domestic credit.

Solving the consumer’s problem, imposing the equilibrium condition b, = $\overline{b}$, and taking into account (4a) and (4b), the following differential equation system is obtained (recall that P = Ē):

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 Ē⁹ 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.

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Effects of a Devaluation Under Complete Market Separation

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

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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

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 $\dot{a}=\dot{A}/\overline{E}$ and $q\equiv Q/\overline{E}\mathrm{.}$

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 βq²b² is lost on taxes, which results in a negative marginal return of 2βq² 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 $\overline{g}$ or 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 $\overline{g}$. Furthermore, if r = δ, nominal (and real) money balances would be the same with or without leakage.

It follows from (24) that

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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.

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Characteristic Polynomial

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

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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 μ₁ and μ₂. respectively.

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

where ω_i, i = 1, 2, 3, 4 are arbitrary scalars, and h₁ = (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 μ₁ and μ₂. Therefore, a necessary condition for convergence is that Μ₃ = Μ₄ = 0.

Recalling that Γh_i = μ_ih_i , it follows that

Since by definition an eigenvector cannot be a zero vector, equations (29a)-(29c) imply that all components of hⁱ, 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_i s are different from zero)

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 - $\overline{q}$) /(b_t - $\overline{b}$) that uses Μ_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-$\overline{q}$)/(b_t-$\overline{b}$) that uses the smaller (in absolute value) of the negative roots, μ₂, 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 (q₀-$\overline{q}$)/ (b₀ - $\overline{b}$) ≠ h₁₃/h₁₄,

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

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Adjustment Paths of b and q

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

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From (32a) and (32c)—and using equations (29a)-(29c)—the constants ω₁ and ω₂ can be computed. By substituting ω₁ and ω₂ into (32b), one obtains (recall that $\overline{q}$) = 1)¹⁹

where the coefficient of (M₀ - $\overline{M}$)) is positive because (μ₁, /ξ₂) > 1, h₁₁ >0, and h₂₁ < 0. Equation (33) indicates that if nominal steady-state money balances rise, then q₀< 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 ($\overline{b}$, $\overline{M}$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.

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Effects of a Devaluation Under Incomplete Market Separation

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

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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.