Real exchange rate misalignment—the sustained departure of the actual real exchange rate from its equilibrium value—has been a recurring policy problem in many developing countries. Overvaluation of the real exchange rate has had undesirable effects on net exports and growth in some countries, while undervaluation has created problems for monetary control and inflation in others. Getting the real exchange rate “right” in a world in which the fundamental structural determinants of the equilibrium real exchange rate are constantly changing remains one of the most important goals of economic policy in developing countries.

The problems associated with real exchange rate misalignment have led policymakers in developing countries to try to reduce the degree of overvaluation or undervaluation of their currencies by appropriate policies for the nominal exchange rate. The idea behind such policies is that the transition from, for example, a situation of overvaluation to one of real exchange rate equilibrium can be long and drawn out, particularly if institutional factors—such as wage- and price-setting behavior—are unfavorable. Under these circumstances, a change in the nominal exchange rate (a devaluation) can help to reestablish equilibrium more quickly and thereby mitigate some of the costs associated with the transition.

A well-known problem associated with the implementation of such policies is that the extent of real exchange rate misalignment—and in some cases even its sign—may be difficult to gauge. This is because information about the extent of misalignment requires knowledge of the level of the equilibrium real exchange rate, which depends both on structural factors (including trade and industrial policies, the degree of capital mobility, and the terms of trade) and on macroeconomic factors, such as the level and composition of government spending and taxation as well as the international macroeconomic environment. The problem of real exchange rate misalignment thus lies at the heart of both macro-economic and structural policy in developing countries.

In practice, the problem of estimating the extent of real exchange rate misalignment has been addressed in a variety of ways. One approach has been to determine some base period in which the actual and equilibrium real exchange rates were equal and then attempt to determine the extent by which the equilibrium real rate has changed as a result of changes in its fundamental structural determinants, so that a comparison can be made with the path of the actual real exchange rate.^{1} This approach, however, has not been used as frequently as one might think, probably because the effects of exogenous and policy-induced shocks on the equilibrium real exchange rate depend on structural relationships in the economy about which policymakers are likely to have insufficient information.

An alternative approach, which is simpler and more direct, involves using information from the parallel market to gauge the extent of real exchange rate misalignment.^{2} The existence of a premium on foreign exchange in the free market is taken to indicate an excess demand for foreign exchange at the official exchange rate, which in turn is interpreted as arising from an overvaluation of the domestic currency at the prevailing official exchange rate.

This intuition is buttressed by models in the literature in which the premium and the degree of misalignment both respond endogenously to some shock, such as an increase in the stock of credit. For example, models in the currency-substitution tradition (Calvo and Rodríguez (1977)) have been used by Edwards (1989) and Kamin (1993) to analyze the effects of unsustainable financial policies on the parallel market premium and the divergence of the real exchange rate from its long-run equilibrium value. The robust finding from these models is that, along the adjustment path, overvalued real exchange rates are associated with high premia. One interpretation of this relationship views the equilibrium exchange rate as a weighted average of the free rate and the official rate. Consequently, in many developing countries, exchange rate policy is designed to reduce the gap between the two rates by depreciating the official exchange rate (see, for example, Aghevli, Khan, and Montiel (1991)). In effect, the misalignment of the official rate is contained by targeting the premium at a reduced level.^{3}

This view of the relationship between the premium and real exchange rate misalignment has some empirical support. For example, studies of a number of devaluation episodes (see Edwards (1989) and Kamin (1993)) find that the parallel market premium often rises very rapidly in the period immediately preceding a major devaluation, and then falls off just after the devaluation. A positive correlation between the premium and the extent of underlying real exchange rate overvaluation is suggested by such cases.

Nonetheless, from an analytical standpoint, the case for treating the size of the parallel market premium as an indicator of the magnitude of real exchange rate misalignment seems far from obvious. Both the premium on foreign exchange in the free market and the real exchange rate in the official market are endogenous variables with complex macroeconomic roles. As such, the correlation between them should depend on the sources of shocks impinging on the economy. Moreover, the parallel market premium is an asset price, which can be expected to exhibit much greater volatility than the official real exchange rate, in particular by responding to transitory shocks that leave the equilibrium real exchange rate unaffected. The very different time series properties of the two variables raise some doubts about the reliability of the premium as an indicator of real exchange rate misalignment.

Empirically, parallel market premia tend to be a good deal more variable than real exchange rates in developing countries and can easily reach levels of several hundred percent without discernible changes in the underlying extent of real exchange rate misalignment. Furthermore, the sign of the correlation between the parallel market premium and the official real exchange rate seems to vary from country to country and time period to time period.^{4} For example, in the study cited previously, Kamin (1993) found that in about a third of the devaluation episodes he studied, the premium actually fell before an exchange rate correction. This certainly calls into question the presumption that there is a robust correlation between real exchange rate misalignment and the parallel market premium that is independent of the nature of the underlying shocks and the structure of the economy.

The purpose of this paper is to explore these issues in the context of a fully optimizing model of a developing country that simultaneously determines the degree of misalignment of the real exchange rate and the premium in the parallel market. The paper’s objective is to assess the reliability of the premium as an indicator of exchange rate misalignment by answering the following questions: Letting *e* and * e* denote respectively the actual and equilibrium real exchange rates, and letting

*b*denote (one plus) the premium in the free exchange market, can we identify a structural relationship

*e –*e = f(b– 1) that would provide

*quantitative*information about the degree of real exchange rate overvaluation from observations of the premium? If not, can we at least establish that the sign of (

*b*– 1) is the same as the sign of

*(e –*e ), so that the premium serves as a

*qualitative*indicator of misalignment? If neither of these can be established in a standard model, then the former cannot be taken as a reliable indicator of misalignment.

This paper is organized as follows. First, we set up the model by describing the consumer’s optimization problem and then discussing the equilibrium and solution of the model. We then proceed to demonstrate the central point of the paper, that adjustment in an economy (modeled along fairly standard lines) to even a simple shock (consisting in this case of a permanent productivity shock) can be rather complex. We show, in particular, that while the adjustment of the real exchange rate to its new long-run equilibrium is monotonic, that of the parallel market premium is not, implying that neither the sign nor the magnitude of the premium is informative about the extent of misalignment in this case. We end by summarizing the main conclusions.

## I. The Household’s Problem

Consider a small open economy in which agents derive utility from the consumption of traded and nontraded goods, denoted respectively by *c _{T}* and

*c*. The instantaneous utility function

_{N}*u*is thus given by

Consumption is subject to a cash-in-advance constraint of the form

where *m = M/P* is the stock of money deflated by the true consumption-based price index *P; c* is the real value of aggregate consumption, which is related to the consumption of traded and nontraded goods by *c = e*^{β}*(c _{T} + c_{T}/e)*, where

*e*is the price of tradables relative to nontradables (the real exchange rate) and 0 < β< 1 is a parameter whose economic interpretation will be given below; and α > 0. Apart from domestic money, household financial wealth includes foreign currency denominated securities, which are denoted by

*F*.

A “dual” exchange rate regime is assumed, involving a fixed “official” exchange rate *s* and a floating “unofficial,” or “parallel,” market exchange rate *v*. All commercial transactions are assumed to take place at the official rate *s*, while all other (financial) transactions take place at the market-determined rate *v*. We let *b* denote the ratio of the financial to the commercial exchange rate, so that *b = v/s* denotes one plus the premium in the parallel market. Thus, the real value (in terms of traded goods) of household financial wealth, denoted by *a*, may be written as

It is assumed that agents can engage in cross-transactions between official and unofficial markets. These leakages, denoted by *L*, carry a cost *T(L)*, where *T(0) = 0, T’(0) = 0*, and *T’’(0)* > 0.^{5} Thus, the increase in resources available to consumers by engaging in “fraudulent” activity (leakages) must be traded off against the costs associated with such activity, which are captured by the *T(L)* function. Taking into account these leakages, the instantaneous budget constraint of the representative household may be written as

where a dot above a variable denotes a time derivative and a circumflex denotes a proportional rate of change—that is, *y* denotes the real value (in terms of traded goods) of output of both traded and nontraded goods;^{6}*r ^{*}* is the world interest rate; and the last two terms in equation (4) respectively denote the costs associated with fraudulent transactions and the benefits associated with such activity. Thus, the change in real household wealth (per unit of time) is simply equal to income (which consists of income from production (y) plus interest earnings on foreign securities

*(r*plus capital gains

^{*}bF)*net*benefits associated with fraudulent activity.

The problem of the representative household is to choose values for its consumption of traded and nontraded goods (*c _{T}* and

*c*), foreign securities (F), and the amount of leakage into the free market (

_{N}*L*), to maximize the discounted sum of utility given by

subject to the budget constraint given by equation (4) and a transversality condition that requires the present value of household net worth to converge to a nonnegative number. In addition to the budget constraint (4) and the transversality condition, the optimality conditions for this problem are given by

The interpretation of these conditions is standard. The first two equations require that the marginal utility of consumption of each of the two goods be equated with the product of the marginal utility of wealth, λ, and the price of the good. The price itself involves two components. First, there is the market price (unity in the case of traded goods and the reciprocal of the real exchange rate *e* in the case of home goods). Second, because of the cash-in-advance constraint, there is the opportunity cost of holding money, which is α (the factor of proportionality between consumption and money) times the real interest rate,

The third optimality condition (equation (8)) relates the amount of leakage to the level of the parallel market premium. Given that *T(L)* is a convex function, the condition states that the higher the premium *b* – 1, the greater the amount of leakage between the official and parallel markets. Finally, equation (9) is the standard condition relating the path of the co-state variable to the difference between the subjective rate of time preference and the domestic real interest rate.

We proceed under the simplifying assumption that the instantaneous utility function in equation (1) is logarithmic:

Denoting aggregate consumption in terms of traded goods by Z = *c _{T} + c_{N}/e*, we have from equation (1’) and the first two optimality conditions that the consumption of traded and nontraded goods is given by

Thus, consumption of the two goods depends on the level of total expenditure Z and the real exchange rate *e*.^{7} We can also solve for the marginal utility of wealth in terms of aggregate expenditure and the domestic interest rate:

## II. Equilibrium and Solution of the Model

We begin by discussing some issues that affect the dynamics of the aggregate economy but that are exogenous from the point of view of the individual household and hence were ignored in the previous section. First, it should be noted that for the economy as a whole, the legal stock of foreign securities is fixed and the actual stock can therefore be altered only through leakages. Recalling that interest earnings on foreign securities are repatriated through the parallel market (since the official market is, by assumption, only for commercial transactions), we thus have

where the second term represents interest earnings on the existing stock of foreign securities.

Second, the country is assumed to face an upward-sloping supply of foreign loans—that is, the cost of funds to domestic private residents depends on the actual stock of foreign securities they hold. Specifically, it is assumed that *r ^{*} = r^{*}(F)*, with

*r*being a decreasing function that satisfies r

^{*}(F)^{*}(0) = δ, where δ is the rate of time preference. Thus, when the country is neither a net creditor nor a net debtor (

*F*= 0), the rate of interest its residents face is equal to the rate of time preference δ, but the rate of interest faced by the domestic residents rises with –F—that is, with the extent of the country’s net debtor position. This makes

*r*a decreasing function of

^{*}*F*.

^{8}

Third, in the previous section, the household took the production side of the economy as given. To close the model, however, the conditions of production must be specified. We adopt a simple approach here, wherein there is no investment or other dynamic problem that firms need to solve and production decisions maximize period-by-period profits. It is also assumed that all prices are flexible so that the economy operates continually at full employment.

A convenient way of modeling the supply side is to assume that the economy possesses a concave transformation frontier between traded and nontraded goods.^{9} It operates at the point of tangency between the frontier and a straight line with slope equal to (minus) the reciprocal of the real exchange rate. This implies supply functions of the form:

where θ is a productivity parameter in the traded goods sector to be discussed below.^{10} Using the definition of output in terms of traded goods *y*, together with the envelope condition, also implies

Finally, we can use this simple specification of production together with the demand functions given in equation (10) to solve for the equilibrium real exchange rate in terms of the level of total expenditure *Z*. Specifically, since in equilibrium the nontraded goods market must clear, we have

Differentiating equation (15) we can solve for the equilibrium real exchange rate in terms of total expenditure:

Clearly, an increase in total expenditure creates excess demand for home goods, thereby requiring an appreciation of the real exchange rate (a decrease in *e*) to restore equilibrium in the home goods market.

With these three issues in hand, the specification of the model is complete. Notice that the analytical set-up is a fairly standard one based on familiar components, so that the co-movements between the premium and the real exchange rate in response to shocks should be of particular interest.

We shall derive four dynamic equations, two of which refer to the “jumping” variables *b* and the domestic real interest rate *R*, and two of which involve the predetermined variables Z and *F*. We begin by deriving the dynamics of private consumption expenditure Z. Equation (11) links Z to the marginal utility of wealth. Proceeding in standard fashion, replace *R* in equation (11), differentiate with respect to time, and substitute from equation (9). The result is

which is the familiar condition relating the slope of the time path of consumption to the difference between the consumption real interest rate and the discount rate.^{11}

Substituting the cash-in-advance constraint into the expression for household wealth (recalling that *c = e ^{β}Z*) gives the following alternative expression for wealth:

Differentiating equation (18) with respect to time, equating the resulting expression for *Ż*:

Inverting the optimality condition in equation (8), we can write

where *H’(b – 1) = 1/T’’ (L) >* 0. More intuitively, equation (20) states that the amount of leakage is an increasing function of the parallel market premium *b*–1. Substituting equation (20) into equation (19) then gives

Equating the right-hand sides of equations (17) and (21) provides a dynamic equation for the domestic real interest rate:

The dynamic system that characterizes the economy comprises the following four equations:

Equation (23) simply follows from the definition of the real interest rate; equations (24) and (25) were given previously as equations (21) and (22) and are rewritten here for convenience; equation (26) incorporates the leakage function (20) and the dependence of the external interest rate facing domestic residents on the stock of foreign assets into the equation for *Ḟ* given previously as equation (12). The state variables are *Z* and *F*, while the non-predetermined variables are *b* and *R*.

Imposing the steady-state conditions *ḃ = Ż = Ṙ = Ḟ =* 0, we can solve for the steady-state values (where an overbar denotes a steady state):

The logic of the steady state is seen by observing that if the net stock of foreign securities reaches a constant value that differs from zero, this implies that the constant value reached by the premium *b* – 1 must also differ from zero (equation (26)) and that the value of the world interest rate must differ from the rate of time preference (by definition of the relationship between *F* and *r ^{*}(F)*). This would make the last term in equation (25) nonzero, which in turn implies that the term in curly brackets in that equation must also be nonzero. Since this term also appears on the right-hand side of equation (24), it would not be possible for expenditure

*Z*to reach a constant value, which contradicts the definition of the steady state. Therefore, the steady-state values must be those given by equations (27)–(30).

Our interest is in analyzing the co-movements of the premium and the real exchange rate in this economy when the economy is hit by an exogenous shock. For that purpose, we consider a simple shock, in the form of a permanent improvement in productivity. In the remainder of this section, we provide a formal solution for the response dynamics, which we then analyze in more detail in the section that follows. To analyze the dynamics in the vicinity of the new steady state, we linearize the system of equations (23)–(26) around the steady state:

The partial derivatives in matrix system (31) are given by

*b _{4} = –r^{*}’(F) > 0*,

*R _{1} = [(1+αR)/(α*

^{2}

*Z)]H’>0*,

*R _{2} = [(1+αR)/α>0*,

*R _{3} = [(1+αR)/(α*

^{2}

*Z)](1–y*

_{1}

*e’)>0*,

*R _{5} = –[(1+αR)/(α*

^{2}

*Z)]y*

_{2}

*<0*,

*Z _{1} = –H’/α<0*,

*Z _{3} = –(1–y*

_{1}

*e’)α<0*,

*Z _{5} = y*

_{2}

*α>0*,

*F _{1} = H’>0*,

*F _{4} = δ>0*,

The trace of the matrix on the right-hand side of (31) is given by

Trace = 2δ + *(y _{1}e’)/α>0*,

while the determinant is given by

Determinant = –r^{*}’H(1 + αδ) (1 – y_{1}e’)/α^{2}>0.

The fact that the trace is positive implies that not all roots of the system can be negative, and the fact that the determinant is positive implies that the number of positive roots is either two or four. Since there are two predetermined variables and two non-predetermined variables, the former case (two positive roots) implies saddle-point stability of the equilibrium, whereas the case of four positive roots implies global instability.

To determine which of these two cases is relevant, we examine the characteristic polynomial in more detail. Defining this polynomial by *P(λ)*, we have by definition

Expanding the expression for the polynomial, one obtains

Upon substitution for the coefficients as defined below matrix system (31), one obtains

*γ*_{0} = *–r ^{*}’H’(1 + αδ)(1–y*

_{1}

*e’)/α*

^{2}

*>0*,

*γ*_{1} = *r ^{*}’H’[δ + (y*

_{1}

*e’)/α]*–(δ/α)(1 + αδ)[(1–y

_{1}e’)/α + H’/(αZ)]<0,

*γ*_{2} = δ[δ + (y_{1}e’)/α]–[(1+αδ)/(α^{2}Z)][H’ + (y_{1}e’–1)^{2}/α] + r^{*}’H’≶ 0,

*γ*_{3} = 2δ + (y_{1}e’)/α > 0,

γ_{4} = 1.

By definition, γ_{0} is equal to the product of all the roots (the determinant), γ_{1} is equal to the sum of all products of three of the roots, γ_{2} is equal to the sum of all products of two of the roots, and γ_{3} is equal to the sum of all products of one of the roots (or the trace). Because γ_{1} is negative, it must be the case that at least one of the roots of the system is negative. This rules out the case in which all roots are positive and implies that the system is saddle-point stable, possessing exactly two positive roots and two negative roots.

The general solution of the model may therefore be written as

where the elements of each eigenvector may be solved from

for each of the four eigenvalues *λ _{i}, i* = 1 – 4. If we let λ

_{3}and λ

_{4}be the two positive eigenvalues, then convergence of the dynamic system requires ω

_{3}= ω

_{4}= 0. To solve for the remaining two scalars, we use the initial conditions, which imply

We can now write the solution of the system as

where Δ = *h*_{31}*h*_{42} – *h*_{32}*h*_{41}.

## III. Adjustment to a Productivity Shock

In this section, we analyze the properties of this solution, focusing specifically on the behavior of the premium and the real exchange rate. To begin, it is worth noting that the discussion that follows would not be altered in any important way if the source of the shock were an improvement in the (exogenous) terms of trade faced by the country^{12} or simply some productivity-enhancing policy reform. Alternatively, one may prefer to substitute an endowment structure for the simple transformation function described in the previous section. In this case, the shock would simply be an increase in the endowment of the tradable good. Finally, as mentioned previously, we confine the shock to the tradables sector mainly to simplify the analysis of its effect on the equilibrium real exchange rate (equation (16)). It may also be noted that the model can be used to analyze a variety of other disturbances, but this is not necessary for the purpose at hand, which is simply to illustrate the potential problems that may arise in using the parallel market premium to draw inferences about real exchange rate misalignment.

Consider then the effect of a permanent favorable productivity shock. As far as the steady state is concerned, all variables *( b, R, and F)* remain constant at their original steady-state values (given by equations (27), (29), and (30), respectively), except for expenditure, for which equation (28) implies that a positive (negative) shock causes the steady-state level of expenditure to rise (fall). From equation (16), therefore, the long-run value of the equilibrium real exchange rate decreases for a positive shock: there is an equilibrium real appreciation in the long run. Notice that the steady-state premium does not change in this model. This feature permits policymakers to feel confident about the equilibrium value of the premium after a shock—and thus about whether the observed premium is above or below its long-run value—without having to solve the model.

The issue, then, is whether this conveys information about the position of the real exchange rate relative to its long-run value.

To determine the out-of-steady state dynamics, it is useful to turn to Figure 1, which plots the level of expenditure *Z* on the horizontal axis and the level of (one plus) the parallel market premium *b* on the vertical axis. Imposing the steady-state condition *Ż* = 0 in equation (31) yields a negatively sloped locus in *b-Z* space, with the slope given by –(1 – *y _{1} e’)/H’*. Since the steady-state premium must be zero, the intersection of this locus with the horizontal line at

*b*= 1 determines the steady-state value of expenditure. Z

_{0}denotes the initial steady-state level of expenditure while

*A*, while the new steady state is located at point

*B*. The parallel market premium is zero (

*b*= 1), in both steady states, while the position of the

*Ż*= 0 locus is shifted to the right as higher spending is induced by the higher (post-shock) level of productivity. The arrows of motion relative to the new

*Ż*= 0 locus in the figure follow from the fact that

*Z*< 0, as established below equation (31).

_{3}Figure 1. Dynamics of Consumption and the Premium

Consider now the dynamics of the adjustment of *b* and Z from *A* to *B*. Since expenditure is linked to the stock of money by the cash-in-advance constraint, and since the money stock can only change over time through the process of hoarding, *Z* cannot jump on impact. This means that the instantaneous equilibrium of the economy must be somewhere along the perpendicular rising from point Z_{0}. To determine the paths followed by *b* and *Z* from an initial point on this perpendicular to the new steady state at *B*, we proceed in several steps. First, we derive slopes of the dominant and nondominant eigenvectors (that is, the eigenvectors associated with the smallest and largest of the negative eigenvalues, λ_{1} and λ_{2}, respectively). These loci have the property that the paths of *b* and Z cannot cross them, since if the economy is ever on one of these vectors, it can be shown to remain on it as it converges to the new steady state. From equations (38) and (40), the slope of the dominant eigenvector is equal to *h _{11}/h_{31}* while the slope of the nondominant eigenvector is equal to

*h*

_{12}/

*h*

_{32}. To determine the signs of the slopes, we use equation (35), which implies

where the signs in equations (42) and (43) follow from evaluating the polynomial P(λ) in equation (32) at λ = Z_{3}.^{13} Unless the initial values of the state variables *Z* and *F* place the economy precisely on the dominant eigenvector, the economy will converge to the steady state at *B* along the nondominant eigenvector drawn in Figure 1, since the influence of the largest negative root must gradually vanish over time.

It remains only to determine the sign of the jump in *b* resulting from the productivity disturbance. The Appendix establishes that, in fact, the premium and therefore *b* must make a discrete jump downward on impact, to a point such as *C* in Figure 1. This means that the adjustment path must follow the dark arrowed path in the figure: expenditure rises throughout the adjustment, but the premium falls on impact and then rises temporarily before falling back toward zero in the final stages of the adjustment.^{14}

Intuitively, the favorable productivity shock increases income, which is consistent with a permanent increase in household expenditure. However, the adjustment in spending cannot be instantaneous, because an increase in spending requires an increase in the money stock, and the latter can only occur gradually through the process of hoarding. Initially, then, spending remains below permanent income, and the associated saving results in the gradual accumulation of money, permitting spending to rise over time. This accounts for the monotonic adjustment of *Z*. At the same time, the path of *b* cannot be monotonic because, if it were, the stock of foreign exchange could not remain unchanged from one steady state to the next, as required by equation (30). With monotonic adjustment in *b*, foreign exchange balances would either be drawn down or accumulated continuously.

The contrast between the behavior of *b* and *Z* along the adjustment path has important implications for the use of the premium as an indicator of real exchange rate misalignment. Figure 2 plots the time paths of the various variables resulting from the adjustment of the economy portrayed in the first figure. As can be seen, the monotonic adjustment of expenditure implies that the adjustment of the real exchange rate is also monotonic by equation (16), while the path of the parallel market premium necessarily overshoots. This means that, throughout the adjustment, the currency is undervalued in real terms (obviously if the shock had been negative, the problem would have been real exchange rate overvaluation). However, the parallel market premium is initially negative (foreign currency sells at a discount in the unofficial market), then rises to a positive value, before falling again toward zero. Thus, only the first part of the adjustment process coincides with the conventional view that undervaluation (overvaluation) should be associated with parallel market discounts (premia). Clearly, the parallel market premium cannot by itself provide reliable information about even the sign of real exchange rate misalignment at a given point in time, let alone its magnitude.^{15}

Figure 2. Adjustment to a Favorable Productivity Shock

## IV. Conclusion

Because most developing countries maintain a managed official exchange rate, policymakers in such countries commonly have to make judgments about the extent of real exchange rate misalignment. The degree of misalignment is very difficult to measure, since the equilibrium real exchange rate is unobservable and depends on a range of structural and macroeconomic factors. Thus, economists looking at this issue have frequently used the parallel market premium—about whose equilibrium value they have more confidence—to draw inferences regarding the extent of real exchange rate misalignment.

The conventional wisdom is that there is a robust correlation between the extent of overvaluation or undervaluation of a currency and the parallel market premium, a view that appeals to intuition, as well as having both some analytical and empirical support. Against this view, however, one must set the observation that, since the degree of misalignment and the parallel market premium are both endogenous real variables, both the nature of shocks and the economic structure are likely to play a role in determining the sign and magnitude of co-movements between these variables. Moreover, available empirical evidence suggests that the premium is far more variable than any likely deviations of actual exchange rates from their equilibrium values, reaching several hundred percent with no ex post evidence of comparable magnitudes of real exchange rate misalignment.

This paper investigated, using a fully optimizing model of a small open developing country, how the parallel market premium and the real exchange rate jointly respond to an illustrative shock, taken here to be a permanent productivity disturbance. The analysis suggested that the informational content of the premium may be limited, since in response to a shock the premium was both positive and negative at various times along the adjustment path, while the degree of overvaluation of the currency was always positive for a negative shock and always negative for a positive shock.

The obvious policy implication of the analysis is that by itself the premium is an unreliable indicator of the sign and magnitude of real exchange rate misalignment. The premium may provide useful information about the relationship of the real exchange rate to its equilibrium level—for example, the magnitude of the initial jump in the premium will be correlated with that of the initial extent of real exchange rate misalignment—but it is dangerous to draw inferences about deviations of the actual exchange rate from the equilibrium real exchange rate based on observations of the premium at a given moment in time. In the model in this paper, other variables, such as the behavior of the trade balance, would provide more reliable information on the extent of real exchange rate misalignment.

The purpose of this appendix is to show that in response to a positive productivity shock, the premium must jump down on impact and that the adjustment of the economy to such a shock must be as presented in Figure 1. Figure 3 plots the stock of foreign securities *F* on the horizontal axis and the parallel market premium (plus unity) *b* on the vertical axis. The slope of the *Ḟ* = 0 locus (which is drawn relative to the new post-shock steady-state equilibrium at *B*) is equal to –*F _{4}/F_{1}* which is negative (equation (31)).

^{16}From equations (38) and (41), we can derive the slopes of the dominant and nondominant eigenvectors:

Figure 3. Dynamics of Foreign Assets and the Premium

where, by the fact that λ_{1} < λ_{2} < 0, the dominant eigenvector is more steeply sloped than the nondominant eigenvector, and both are more steeply sloped than the *Ḟ* = 0 locus, as drawn in Figure 3. As mentioned previously, unless the initial steady-state values of the two state variables place the economy on the dominant eigenvector, the path followed by the economy in Figure 3 eventually converges to the slope of the nondominant eigenvector.

The question, as in Figure 1, is where the economy jumps on impact along the line perpendicular to point *F* = 0, which is the new (and old) steady-state value of *F*. There are two possibilities, labeled I and II in Figure 3. Under path I, the premium jumps up on impact, and, given the arrows of motion drawn relative to the *Ḟ* = 0 locus, the premium must approach its steady-state level from below. Under path II, the premium jumps down on impact and, given the arrows of motion, must approach the steady state at *B* from above.

It may be recalled from Figure 1 that whether *b* jumps up or down on impact it must approach the steady state from above. But this rules out path I in Figure 3 since, in this case, *b* approaches the steady state from below. Therefore, *b* must jump down on impact, and the economy must follow the path indicated in Figure 1 (with the dynamics of the individual variables as drawn in Figure 2) and likewise path II in Figure 3.

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KaminSteven B.“Devaluation, Exchange Controls, and Black Markets for Foreign Exchange in Developing Countries,”Journal of Development EconomicsVol. 40 (1993) pp. 151–69.

KhanMohsin S. and Jonathan D.Ostry“Response of the Equilibrium Real Exchange Rate to Real Disturbances in Developing Countries,”World DevelopmentVol. 20 (1992) pp. 1325–34.

KharasHomi and BrianPinto“Exchange Rate Rules, Black Market Premia, and Fiscal Deficits: The Bolivian Hyperinflation,”Review of Economic StudiesVol. 56 (1989) pp. 435–48.

LizondoJ. Saul and Peter J.Montiel“Fiscal Policy and the Dynamics of Devaluation for a Small Country with Optimizing Agents” (unpublished; Washington: International Monetary Fund1991).

QuirkPeter J. and othersFloating Exchange Rates in Developing Countries: Experience with Auction and Interbank Markets Occasional Paper No. 53 (Washington: International Monetary Fund1987).

For an attempt at such an exercise, see Khan and Ostry (1992).

On the use of the premium as an indicator of real exchange rate misalignment in developing countnes, see Quirk and others (1987).

For an analysis of the Bolivian case, see Kharas and Pinto (1989).

See, for example, the discussion in Edwards (1989), chapter 4.

The first optimizing model of dual rates with leakages is Bhandari and Végh (1990). The main differences between their paper and ours relate to the goods structure and the economy’s access to world capital markets.

The production side of the economy is described in the following section.

Clearly, β is the expenditure share of nontraded goods.

Essentially, this assumption pins down the stock of foreign securities in the steady state.

For a similar modeling of the supply side, see Lizondo and Montiel (1991).

To simplify the analysis, it is assumed that the output (productivity) shock affects only the tradables sector. This implies that the productivity shock does not enter into equation (16) below, which relates the equilibrium real exchange rate to the level of total expenditure.

In any period, the effective price of consumption is equal to 1 + *αR*, which is the sum of its market price (unity) and the opportunity cost *(αR)* of holding, the a units of money needed to purchase a unit of the good. The consumption rate of interest, which is the true intertemporal price of consumption, is equal to the interest rate *R* less the rate of change of the effective price of consumption,

In this case, we could think of the model as comprising an exportables and a nontradables sector on the supply side and importables and nontradables on the demand side. Also, it should be noted that the analysis of a *negative* shock would be completely symmetric to that presented below for a positive shock.

Specifically from equation (32), it can be shown that as λ goes to –∞ then lim P(λ) > 0 and *P(Z _{3})* < 0. From these two facts, it follows that λ

_{1}< Z

_{3}< λ

_{2}< 0. The signs of the slopes in equations (42) and (43) follow directly.

It is interesting to note that the qualitative path followed by the economy in response to a real shock (an improvement in productivity) is the same as that followed in response to a nominal shock (for example, an exogenous change in the stock of money). Figure 1 is again illustrative. Suppose the *initial* steady state is *B* in Figure 1 and that there is an exogenous decrease in the stock of money. Since the model exhibits neutrality in the steady state, the new long-run equilibrium will be identical to the initial equilibrium: the new steady state is also point *B* in Figure 1. On impact, though, a decrease in the nominal money stock implies a decrease in expenditure in terms of traded goods (that is, a decrease in *Z*), since the official exchange rate is fixed and the cash-in-advance constraint must hold. Thus, on impact, the economy must move to a point along the perpendicular rising from Z_{0} in Figure 1. But then, as shown in the Appendix, the dynamic path followed by the economy must be as drawn in Figure 1—in particular, the premium must jump down on impact. Thus, the economy begins at point *B*, jumps to point C, and then follows the arrowed curve back to point *B*. Clearly, therefore, a nominal shock is capable of generating the same type of nonmonotonicity in the adjustment of the premium as the productivity shock does.

An alternative to looking at the premium at a point in time is to look at its *average* behavior over some period of time. Whether this yields a more reliable indicator depends on how long the economy spends in each phase of the adjustment path and on how long after the shock an assessment is made.

Because the new and old steady-state values of *b* and *F* are identical, the new and old steady states in Figure 3 coincide and are given by point *B*.