Currency Bands, Target Zones, and Price Flexibility
  • 1 0000000404811396 Monetary Fund
  • | 2 0000000404811396 Monetary Fund

Exchange rate behavior is analyzed in the context of a stochastic rational expectations model in which there are random shocks to the price-setting mechanism and in which the authorities choose to impose either nominal or real exchange rate bands. The effects of rules for realignment of the band are also examined. Results are compared with those that emerge from a simple monetary model subject to velocity shocks.


Exchange rate behavior is analyzed in the context of a stochastic rational expectations model in which there are random shocks to the price-setting mechanism and in which the authorities choose to impose either nominal or real exchange rate bands. The effects of rules for realignment of the band are also examined. Results are compared with those that emerge from a simple monetary model subject to velocity shocks.

Since 1979 macroeconomic policy in many European countries has been conducted within the constraints imposed by the exchange rate commitments of the European Monetary System (EMS). For member countries like France and Italy these arrangements have provided a useful anti-inflationary anchor in the form of an exchange rate link against a hard currency—the German mark; and the surrender of national monetary autonomy has been more obvious as recourse to realignments has been reduced and the pace of capital market integration deliberately accelerated. (In place of the current asymmetric arrangement, the Delors Report1 recommended the creation of a Central Bank for Europe to conduct monetary policy for the EMS.) At the global level, the Group of Seven industrial countries have been trying since 1987, with more or less success, to keep their currencies within informal bands against the U.S. dollar—although the extent of policy coordination involved has (like the bands themselves) been much less explicit.

In this paper we discuss what defending a currency band implies for monetary policy (at the edge of the band) and for the exchange rate within the band, using a version of the popular Dornbusch model where there are stochastic inflation shocks. We look both at nominal bands, which may be fully credible or subject to known rules for realignment, and at real currency bands (or target zones) as advocated by John Williamson since 1983.

The techniques used for the purpose were first developed by Paul Krugman in the context of a full employment, flexible price model where the shocks were those affecting the velocity of money. The explicit closed-form solutions that he and others have obtained to analyze the consequences of exchange rate commitments do not, however, carry over to the case where there is price inertia. Qualitative techniques (or numerical solutions) have to be used instead. But the principles underlying these qualitative solutions are essentially the same as for the monetary model.

To provide both an accessible introduction to the method of analysis and a benchmark for comparison, we begin with an outline of the methodology used and key results obtained for currency bands in the flex-price model. (Real bands cannot sensibly be discussed in this model, which assumes the real rate is constant.) We also discuss the “regime switching” that arises when reserves are limited.

The stochastic model of price inertia is presented in Section II, and the solutions associated with it are described (with the relevant analytical background supplied in the Appendices). We look first at the stabilizing effect of currency bands on the exchange rate. Although the S-shaped curve obtained resembles that for the monetary model, we note that it is associated with discrete monetary intervention (and argue that it is attributable to a locally reversible regime shift rather than to the regulation of fundamentals). Then the effect of a known realignment rule in “undoing” the stabilizing influence of the currency band is discussed, as is its influence on the real exchange rate. Lastly, the model is transformed into real terms, so as to consider the consequences of implementing Williamson’s target zones.

I. The Flex-Price Monetary Model

Perhaps because of its analytical tractability, the monetary model has been widely used to study the behavior of exchange rates in a stochastic environment where there are state-dependent changes in the conduct of monetary policy; see, for example, Krugman (1988), Froot and Obstfeld (1989), Flood and Garber (1989), and Bertola and Caballero (1990). In what follows we describe both the general solution to the model and the way in which different regimes imply specific boundary conditions to identify the particular solution. Closed-form results are reported, but we concentrate on a qualitative account of solutions and boundary conditions, to facilitate comparison with our treatment of the Dornbusch model in the next section.

The essential equations are as follows:


Equation (1) states the condition for equilibrium in the money market. On the left is the domestic money supply measured in logs and denoted m; on the right are the determinants of demand, where p is the log of the price level, y is the log of full employment output, v measures cumulative shocks to velocity, and E(ds)/dt is the expected rate of change of the exchange rate, with s denoting the log of the domestic currency value of foreign currency. (This definition of the exchange rate is the inverse of standard usage, but it has the useful feature that, under a neutral monetary expansion, the exchange rate rises along with other prices.)

In Krugman’s model of currency substitution, the term E(ds)/dt, the expected capital gain on foreign exchange, is a direct measure of the opportunity cost of holding domestic currency. But models with liquidity preference, where the opportunity cost of holding money is given by the (short) rate of interest on bonds, lead to the same solution paths for the exchange rate, as Froot and Obstfeld (1989) have shown. Changes in the term E(ds)/dt will also measure changes in the incentive to hold money as long as international interest differentials are based on currency arbitrage, so that i = i* + E(ds)/dt, where i and i* are domestic and foreign interest rates, respectively (and, for convenience, i* is taken to be constant).

Equation (2) states that purchasing power parity (PPP) always holds, so (in logs) the domestic currency price of any good equals its foreign price plus the price of foreign currency. (The asterisk denotes the variable in the foreign country.) Equation (3) indicates that the velocity variable, v, follows a Wiener process with variance σ2. Thus, velocity follows the continuous time equivalent of a random walk.

Domestic output is exogenous, as are all the foreign variables, so they may be set to zero (or subsumed into the velocity variable) to yield


where k = m + v denotes “economic fundamentals.” Thus, s – k is positively correlated with E(ds)ldt in such monetary models.

Krugman (1988) and Froot and Obstfeld (1989) describe a two-step procedure, which may be followed to solve explicitly for exchange rate behavior. First, find the family of functions of the form s = f(k) that satisfies equation (4) when velocity evolves according to equation (3), with the money supply held constant. Second, select whichever one of these solutions satisfies the boundary conditions appropriate to the currency regime in force. The general solution is derived algebraically in Appendix I; and the qualitative nature of the family of solution paths can be studied with the aid of Figures 1 and 2.

For a currency band (s,s) symmetrically placed around zero, so that s=s, the solution can be written in the form


where ρ = √(2/λσ2), and the parameter A is determined by the boundary conditions; the variety of solutions obtained for a constant money supply is shown by the three paths drawn passing through the origin of Figure 1. (Initially we assume that m = 0, so k = v; the effects that subsequent intervention may have on the money stock are considered later.) The line with a slope of 45 degrees is the free-float solution (FF), where, in the absence of bubbles, s = k and E(ds)/dt = 0. Thus, if the money supply is fixed and no regime change is expected, the exchange rate follows the same continuous-time random walk as velocity.

Since PPP is always preserved, the general price level must move in line with the exchange rate, so it is evident that constancy of the money supply is no guarantee of price stability. Despite its name, the monetary model does not, in these applications, lead to “monetarist” conclusions. On the contrary, the assumed behavior of velocity implies that active intervention, not a fixed rule for money, is needed to secure price stability; and the other solutions reflect the expectations of such intervention.

Those solutions shown as R’OR and B’OB in Figure 1 are nonlinear and diverge progressively from the 45-degree line for values of fundamentals increasingly far from the origin. Formally, this divergence reflects the nonlinear terms in equation (5), but it is not difficult to see how the curvature of these two solutions follows from the behavioral assumptions already made.

Figure 1.
Figure 1.

A Free Float, a Currency Band, and Realignment Prospects

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

Consider, for example, points N and B, on and below the free float solution, respectively. Given that m = 0, at both points the velocity variable is ν=k¯>0. At N, on the 45-degree line, the total increase of velocity above zero is exactly matched by the increase in the exchange rate (and in prices); money market equilibrium is thus preserved at N without any change to E(ds). At point B, however, the exchange rate (and p) has risen by less than v, so some extra inducement is required to ensure that the demand for money matches the given supply. The concavity of the function / at B, together with the diffusion of the velocity variable v, implies that E(ds)/dt is negative (in fact, E(ds)/dt = σ2f”(k)/2), and this—the anticipated capital loss on holdings of foreign currency—supplies the necessary inducement. (In models without direct currency substitution but with financial arbitrage, E(ds)/dt<0 leads to a fall in the international interest differential, i – i*, which increases liquidity preference indirectly.) Conversely, the convexity of the function at A ensures that the incentive to hold domestic money will be reduced as necessary to preserve money market equilibrium when real balances are lower than at N.

Which of these solutions is relevant for the exchange rate in a currency band depends on what rules are being used to implement it. In what follows the boundary conditions associated with marginal and intramarginal intervention are both described with reference to Figure 1, as is the effect of anticipated realignments.

A Nominal Currency Band with Marginal Intervention

It is clear that the exchange rate could be perfectly stabilized in this model if the money stock were to be continuously adjusted so as to offset all velocity shocks (dm = –dv = –σdz). But what happens if there is no accommodation as long as the exchange rate is within the band (s,s), so that the money stock is adjusted only when the exchange rate hits the edges of the band?

For the case where the band is symmetrical around zero (so that s=s) and where the intervention is infinitesimal, Krugman (1988) showed that the relevant solution should be tangent to the chosen bands; that is, the S-shaped curve, B’OB in Figure 1.

The intuitive justification for this tangency condition is that any other solution is inconsistent with money market equilibrium at the edge of the band.2 Consider point R, for example, where the exchange rate has just reached its upper limit, but the solution cuts the upper edge, as shown. If v were to increase by a small amount, s would—in the absence of intervention—be expected to rise above s¯ and this expectation is the only one consistent with equation (4), as already discussed. Consequently, intervention to hold s at s¯ if v increases (but not to prevent a decline if it decreased) will reduce E(ds)/dt at R and violate the conditions for equilibrium. Only at point B where the exchange rate path is smoothly tangent to the edge of the band will intervention (to prevent k from rising above k) leave E(ds)/dt unchanged as is required to preserve money market equilibrium.

Note that the amount of time that the exchange rate is expected to remain on the edge of the band must be infinitesimally short. To suppose otherwise means that over some finite period of time E(ds)/dt = 0; but this is only true for paths along the 45-degree line FF (where E(ds)/dt = (σ2/2)f” = 0). It also follows that the adjustment to the money stock necessary to defend the band itself must be infinitesimal, so as not to disturb E(ds)/dt. As a consequence, the money stock will be reduced infinitesimally each time the currency reaches the top of the band (and increased when it reaches the bottom). Inside the band, however, the money stock will remain constant, so the solution given in equation (5) continues to apply. (The implications of these policies for the behavior of interest rates inside a currency band in the monetary model with liquidity preference are analyzed in Svensson (1989).)

A Nominal Currency Band with Intramarginal Intervention

Flood and Garber (1989) show how a different set of boundary conditions, not involving smooth pasting, can sustain the same currency band. They suppose that the monetary authorities announce a “discrete” intervention rule, which specifies both the upper and lower limits of the fundamental k (denoted U, L, respectively) at which intervention will occur and the magnitude of the intervention at each limit. If, for example, the points chosen are equidistant from the origin and the rule is to accommodate exactly the accumulated velocity shock, then these boundary conditions select the solution path LB’OBU, as shown in Figure 1.

Assuming, as before, that initially m = 0, so k = v, then when velocity variable v reaches its upper limit, U, the authorities must immediately offset it by reducing the money supply, setting m = –U. Since this intervention is fully anticipated, there must be no discontinuous jump in the exchange rate, and the solution paths for the system immediately after the intervention must be those consistent with a lower money supply; graphically, fundamentals must jump from U to O, and the exchange rate will continue to satisfy the same solution path as before (shown as LB’OBU) with fundamentals now measured by k = v – U until the next intervention.

Note that any given currency band can be defended by any one of an infinite number of (fully credible) intervention rules. The same path, for example, could be selected by smaller discrete interventions at points closer to the edges of the band. In the limit, as the size of interventions tends to zero, the solution reverts to the S-curve identified by Krugman (where fundamentals do not jump when intervention takes place, since they are simply checked or “regulated” at k and k).

It is evident that the discrete intervention described by Flood and Garber occurs at a point when the exchange rate lies strictly within the interior of the band it is designed to support; that is, it involves intramarginal intervention. Moreover, it is not locally reversible; thus, if an intervention is provoked by a positive velocity shock, then an intervention of the same size will not occur if there is an immediately subsequent negative shock.

Realignment Prospects and the Inverse S-curve

Another example of an adjustment rule that is not locally reversible is described by Bertola and Caballero (1990). They show how expectations of realignment can reverse Krugman’s S-shaped curve, selecting instead a curve such as R’OR.

Their argument runs as follows. Assume that the authorities only intervene when fundamentals reach preannounced points ct – b, ct + b, where ct denotes the central parity applying at time t. Assume further that, when fundamentals reach either of these limits, there will be discrete intervention of Δm = ±b either to take the rate back to its existing central parity, or to take it to the center of a realigned band of the same width, with a central parity




depending on whether the upper or lower boundary is reached; and that the probabilities of either defense or realignment are known.

Given an initial central parity at the origin, and an upper intervention point at b in Figure 1, then the curve shown as R’OR will arise, given a sufficiently high probability π of realignment, so that the expected capital gain on foreign currency in the case of realignment (a shift from R to point O’) is matched by the expected capital loss in the case of defense (a shift from R back to the origin O). Since the increase in s above s¯ on realignment is less than the fall below s¯ on defense, it is evident that the assumed realignment probability along R’OR must be greater than one half. It is important to emphasize that for a given value of b, s¯ which identifies the width of the band, changes as π is changed (alternatively, if s¯ is fixed while π changes, b must change).

If the probability of realignment were zero, then, as the authors point out, their rule would be one of the intramarginal intervention rules considered by Flood and Garber. But as π rises, the S-shaped curve becomes steeper, coinciding with the 45-degree line when π =12, and reversing its curvature when π >12. Note that for sufficiently large π and certainly for π ≥12, all interventions take place at the edges of the currency band. Bertola and Caballero say that their interest in such high realignment probabilities is motivated by “the qualitative consistency of an inverted S-shape with EMS evidence on exchange rate behaviour within bands” (p. 10). In particular, they note that the inverted S-curve produces a different asymptotic distribution for the exchange rate with more weight on the middle of the band.

Currency Bands with Limited Reserves

In a recent paper examining how limited reserves will affect the sustainability of a currency band, Krugman and Rotemberg (1990) show how a “smooth-pasting solution” at the edge of the band can arise from discrete adjustments to the money stock caused by a “locally reversible” switch from a currency band to (temporarily) floating exchange rates.

The setting is the model of currency substitution where increases in v represent increased demand for foreign currency and where any monetary intervention designed to meet this demand involves depleting the limited stock of official foreign currency reserves. If it is the case that the money supply only varies when reserves do (so m = ln(R + C), where R is the quantity of reserves and C is a constant) then one can draw as in Figure 2 the locus that would correspond to a free float with zero reserves, shown as F’F’. Note that in this figure it is the velocity variable itself that is measured on the axis, not the fundamental k = m + v, as in Figure 1. For m = 0 (so that R = 1 – C), the solution for the exchange rate is that already identified as B’OB; but as reserves are depleted by marginal intervention, the S-shaped curve will shift to the right. Krugman and Rotemberg ask what would happen if the authorities were forced to suspend sales of foreign currency at s¯ when reserves finally run out, but remained willing to purchase foreign currency at s¯ For such a locally reversible regime switch between a currency band and a float, they conclude that the smooth-pasting solution tangent to the end of the band is appropriate; so the solution to equation (4) for the exchange rate outside the band takes the form

Figure 2.
Figure 2.

A Currency Band, Limited Reserves, and Temporary Floating

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008


where k = ln(C) + v, ρ = √(2/λσ2), and A2 is determined by the boundary condition that f’(k) = 0 when s =s¯. This is shown in Figure 2 as the exponential curve AE, which asymptotically approaches the free-float locus F’F’ as v tends to infinity.

Note that at point A, reserves have already been depleted by interventions that have shifted the S-shaped curve from tangency at B to tangency at A. Even though reserves are still positive, any further increase in velocity will take the rate outside the band, since it will trigger a stock shift out of domestic money (a speculative attack), which will reduce reserves to zero. The stock shift arises because of the change of curvature from the concavity of the S-shaped curve A ‘A (where E(ds) 0) to the convexity of AE (where E(ds) > 0), which enhances the attractiveness of foreign currency and depletes reserves to zero. The willingness of the authorities to continue to purchase foreign currency at s¯ means that a subsequent decline in v, which drives the exchange rate back to point A, will now cause a sudden speculative inflow into the domestic currency—the exact reversal of the attack that caused the collapse.

What the various cases shown in Figures 1 and 2 have in common is that the appropriate (rational expectations) path for the exchange rate is tied down by the specific rules for financial policy believed to be in force. These supply the boundary conditions needed to identify the particular solution appropriate to the regime in question. Firm expectations of intervention to defend a band, be it marginal or discrete, bend the rate toward the center of the band, for example; whereas prospects of realignment tend to do the opposite. Of particular interest is the observation that the smooth-pasting condition arises not only when there is marginal intervention at the edge of the band but also when there is a locally reversible shift of regime, as instanced by a temporary float.

The model with price inertia we now discuss is somewhat more complicated, because the solution path for the exchange rate directly affects fundamentals even within the band. But the same principles can be applied to determine the impact on the exchange rate of expected defense or possible realignment where the source of disturbance lies not in the velocity of money but in supply-side shocks to inflation.

II. A Model with Price Inertia

Although financial markets are taken to be forward-looking throughout our analysis, we focus on the simple case where the process of price adjustment is not. Specifically, we use a stochastic version of the popular Dornbusch model (1976) where the process of price setting is a simple Phillips curve. (The procedures we use can be extended to include forward-looking labor contracts, but we do not pursue this here.)

The following equations (7) through (10), to be used in this section, are expressions of, respectively, equilibrium in the money market and the goods market, currency arbitrage, and price adjustment.


where the variables are as defined in the last section.

Two of these behavioral equations are much as before, namely equation (7), which is the LM curve defining equilibrium in the money market, and equation (9), the arbitrage condition for the foreign exchange market. Note that disturbances to the velocity of money are omitted here, however. Instead, the price adjustment equation—where prices rise if gross national product (GNP) is high, and vice versa—is disturbed by white noise shocks. With prices evolving in this way, there is no guarantee that output will remain at any given level, nor that the exchange rate will remain at PPP. The log of the level of output, y, is therefore taken to be demand determined, where the level of demand depends on the real exchange rate (s – p) and on the interest rate, i (equation (8)).3

The dynamics of the system can be summarized as




Alternatively, by redefining the variables s and p to denote deviations from the long-run equilibrium, this may be simplified to


where A is as identified above. So long as the boundary conditions are independent of time, these stochastic differential equations may be solved in two stages. First one determines a family of functions of the time-independent form s = f(p); and then one imposes the relevant boundary conditions, in the form of currency bands around the equilibrium exchange rate, for example.

Beginning at the first stage involves deriving a differential equation defining the required family of functions, and examining its solutions using a qualitative approach. The equation is obtained by applying Ito’s lemma to f(p) to yield


Taking expectations, conditional on information at time t, we obtain


and then substituting for the expected changes in s and p from equation (12), we obtain the desired result:


where aij denotes the appropriate element of the matrix A.

For specific parameter values, numerical methods may be used to solve this nonlinear differential equation; but there is no general closed-form solution. In this respect it differs from the analogous equation of the monetary model, which has solutions of exponential form. The absence of closed forms is not a serious drawback, however, because for stochastic saddlepoint systems there exists a reasonably straightforward qualitative description of the family of solutions. Not surprisingly, their properties are intimately related to key features of deterministic solutions obtained when there is no stochastic disturbance—that is, when σ = 0—which for the “overshooting” Dornbusch model take the familiar saddlepoint form shown in Figure 3 (see Appendix II for a more formal treatment).

Figure 3.
Figure 3.

Phase Diagram for Non-Stochastic Case

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

The nature of symmetric solutions for the system once it is disturbed by stochastic shocks is shown in Figure 4 (see Appendix III for an analysis of the relation between these stochastic solutions and the deterministic phase paths). All the solutions shown pass through the origin, 4 but there are only two linear solutions, which are labeled SS and UU, since they correspond exactly to the stable and unstable eigenvectors of the deterministic system. There are in addition an infinity of nonlinear solutions, all of which are technically antisymmetric; that is, f(p) = –f(–p), and have a point of inflection at the origin.

Figure 4.
Figure 4.

Stochastic Solutions Through the Origin

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

Solutions with a slope at the origin less than that of the stable eigenvector diverge further and further below SS for increasing values of p (see, for example, the path ZZ). Those, such as WW, whose slope at the origin lies between the two eigenvectors, initially exhibit the same smooth divergence, but go through a point of inflection, after which they approach UU asymptotically. Paths whose slope at the origin are steeper than that of the unstable eigenvector are strictly concave to the right until they go through a point of inflection and approach UU asymptotically from above.5

It may be useful at this point to give an intuitive idea of how the presence of “noise” in the inflation rate can generate so many more solution paths linked to the origin and why they bend away from the stable eigenvector, as shown. For concreteness, consider the three points shown as M, A, and B, and the solutions passing through them. The first point, M, lies on the stable manifold s = θsp, where, on differentiating and applying the expectations operator, we find that E(ds) = θsE(dp). Since θs < 0 in this case, and the price level is expected to fall to equilibrium at the origin, an appreciation of the price of foreign currency is implied; that is, E(ds) > 0; thus, given arbitrage, domestic interest rates exceed those on foreign currency by the extent of the expected appreciation of the latter.

The second point, A, lies on YY, which represents price stability for the deterministic model and expected price stability in the stochastic case. At point A the interest differential is wider than at M, so arbitrage requires a larger anticipated change in the exchange rate.6 Since A lies on the line of expected stationarity (where E(dp) = 0, and s =f(p)), there would be no expected change in s in the absence of inflation shocks. But with noise in the inflation rate, the price level will depart from its current value over time, with the upward curvature of the function f leading to expectations of a higher exchange rate. (Formally, the upward bias required to match the interest differential is E(ds)/dt = (σ2/2)f” = i – i*.)

At point B below the stable eigenvector, the opposite curvature is required to satisfy arbitrage. There the interest rate differential is less than at N, but both E(dp) and f’ have increased in absolute value;7 so E(ds) would rise were it not for the concavity of the solution (which together with the noise reduces E(ds), as required for arbitrage).

A Nominal Currency Band

Having characterized the various solutions to the differential equation, we can proceed to the second stage and ask which of these is relevant for any particular currency regime (that is, we need to impose the boundary conditions associated with that regime). We consider first the monetary policy adjustments required (and the boundary conditions implied) by a commitment to keeping the exchange rate within a nominal band (s,s¯). Then we turn to the (very different) boundary conditions implied by a realignment rule.8

As a preliminary, we note that to keep the exchange rate pegged does not, as in the monetary model, involve the “perfect accommodation” of the stochastic disturbances; what is needed instead is to offset, by monetary policy, any effects that shocks to the price level might have on interest rates. From equation (11), one finds that the money supply rule necessary to achieve this is


where Δ = κγ + λ, and s¯ is the level of the peg. Thus, in the overshooting case (where 1 > κη > 0), the money supply must increase with the price level but less than in proportion; whereas in the undershooting case the money supply has to be cut when the price level rises.

What, then, are the boundary conditions implied by adjusting monetary policy at the edges so as to maintain a currency band (s,s¯)? Once again, the answer is that the stochastic solutions will be tangent to the edges of the band. The result, shown as C’B’ OBC in Figure 5, looks much like the S-shaped curve of the monetary model; but this similarity conceals two important differences. First, the fundamental p is not “regulated” when the edge of the band is reached. Consequently, the edges of the band C’B’ and BC also constitute part of the solution path (and the exchange rate can spend finite periods of time at the top or bottom edge of the band). Second, this smooth-pasting solution calls for discrete changes to the money stock at the edge of the band.

Figure 5.
Figure 5.

A Nominal Currency Band with Discrete Monetary Changes

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

One could indeed think of the proposed solution as consisting of three segments or regimes (being pegged at s, being pegged at s, and floating in between), with the regime switches being induced by stochastic shocks to the price level. Therefore, as was argued in Miller and Weller (1990), the reason for smooth pasting is not that there is a Brownian motion process being regulated by marginal intervention, but that there are locally reversible regime switches9 (compare Krugman and Rotemberg’s argument for smooth pasting in the monetary model discussed above). As in their case, the money supply makes a discrete adjustment each time a switch is triggered.

A solution is implemented as follows. The monetary authorities announce that when s hits s (or s), the money supply will be instantly adjusted so as to set domestic interest rates equal to world levels. Let us suppose that this occurs when p is above its long-run equilibrium value, say at p1 > 0 in Figure 5. So long as price shocks hold p above p1 the money supply is adjusted according to the rule in equation (15), but once p hits the value p1 the money supply is again immediately returned to its original level, and is then held constant so long as s < s < s. The required monetary adjustments are shown in the lower panel of Figure 5.

It may not be immediately obvious why the smooth-pasting solution is singled out by the intervention rule described above. There are, as we have seen, many other solutions to the stochastic system that cut the edge of the band, and might be considered possible candidates for a solution. How can they be ruled out? Consider, for example, the stable manifold itself. Suppose OA in Figure 5 were part of a solution path, and that the exchange rate has just entered the support regime at A. This means that the money supply has been adjusted so that i = i*. But over the next short interval of time dt, p is more likely to move back toward the origin than it is to move further away. If p falls, there will be a change of order dt in the value of s (as the rate moves up OA), whereas if p rises, s stays fixed at s. This means that s is expected, on average, to rise. But this is inconsistent with arbitrage, since i = i*. The same applies to all other solutions that cut the edge of the band. Only if there is a tangency at the edge of the band will the arbitrage condition be satisfied.10

The behavior of interest rates is significantly different from that implied by the monetary model. In the case of exchange rate overshooting being considered here, the higher the price level is, the higher interest rates are—until the point at which the exchange rate enters the support regime at the edge of the band, when there is a discrete downward jump in interest rates to the world level i*. The length of time the rate spends at the edge is random but could be quite short, since the price level process is still mean reverting.

The effects described above depend crucially on the assumption that the band is fully credible, so a return inside the band is confidently expected. Reaching the edge of the band may, however, trigger prospects of realignment. We therefore consider next how various realignment rules can affect the rate within the band, leading, for example, to a reversal of the usual smooth-pasting result.11

Discrete Realignment of a Currency Band

Suppose that the authorities announce a rule of the following form: whenever the exchange rate hits the top of the band, the band will be shifted upwards by an amount equal to half the total width of the band; and this realignment will be validated by a change in money supply designed to shift the long-run equilibrium exchange rate to the center of the new band. In this case, the exchange rate will follow the path A ‘OA, snaking around the 45-degree line in Figure 6, since any other trajectories would be found to imply fully anticipated gains or losses at the edge of the band.

Figure 6.
Figure 6.

Realignment Prospects and the Inverse-S Solution

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

In the original band, the exchange rate will follow the solid line, but when it hits the top edge of the band at point A, an upward realignment is triggered. By construction, A becomes the long-run equilibrium for the realigned band, and the rate now moves along the dotted trajectory centered at A. If subsequent price shocks drive the system back to point O, the realignment is reversed and the money stock is returned to its original level. Note that there is no smooth pasting here in the transition from one regime to another because of the locally irreversible nature of the regime shift.12

Though these conclusions apply equally in the overshooting case discussed earlier, we have chosen to illustrate the impact of the realignment rule in the undershooting case where κη > 1 and the stable manifold has a positive slope. The contrast between the S-shaped curve B’OB defined by smooth-pasting conditions attached to a credible band, and the inverse S-shape induced by expectations of realignment is immediately apparent; in Miller and Weller (1989), we noted that the various intermediate cases lying between the smooth-pasting solution and the inverse S-shape curve—including the stable manifold itself—can be obtained when the realignment rule discussed above is applied probabilistically. Bertola and Caballero (1990), who obtained similar results for the monetary model, have stressed the need to incorporate the effects of realignments in order to reconcile stochastic theories of currency bands with observed reality (at least for EMS currencies up until 1987).

The realignment rules discussed here involve not just a shift in the currency band but a discrete adjustment of monetary policy associated with it. To the extent that they are triggered by price increases they imply that local nonaccommodation gives way to global price accommodation. Another proposal that shares these features is monetary control inside real exchange rate bands, which will be examined next, after the model has been recast in real terms. Reformulating the model in this way makes clear that while real bands involve marginal monetary intervention, realignments correspond to discrete monetary intervention along the lines discussed by Flood and Garber (1989).

Target Zones

Williamson (1985) has argued forcefully that a sharp distinction should be drawn between nominal and real currency bands; and his target zone proposal is explicitly couched in terms of real currency bands. The monetary model cannot be used here, since, under the assumption of PPP, the real exchange rate is effectively constant. This is not the case in a model with price inertia, however, and a number of important differences between a nominal currency band and a target zone emerge in that context.

It is convenient to redefine variables in the model spelled out in equations (7)–(10), introducing the real exchange rate, c = s – p, and real balances, l = m – p. The transformed model can then be written as


Where, again, the variables are expressed as deviations from long-run equilibrium. The arbitrage condition now imposes the requirement that the real interest rate differential be equal to the expected depreciation of the real exchange rate.

The system is formally identical to that in equation (12), except that the matrix A’ differs from A in an obvious fashion. The only significant qualitative difference is that the stable saddlepath now always has a positive slope, as shown in Figure 7.

Figure 7.
Figure 7.

Target Zone for the Real Exchange Rate

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

In order to enforce a target zone, the authorities must now concentrate on the relationship between real balances and the real exchange rate. We assume, as before, that m is held fixed in the interior of the zone. But a willingness to adjust the stance of monetary policy at the edge of the band (by offsetting any price shocks that would take real balances and the real exchange rate further from the center of the band) means that policy is essentially one of regulating the fundamental—here, real balances. (Compare this with the regulation of velocity-adjusted money in the monetary model.) What this means is that infinitesimal intervention at the limits to a target zone will lead to a solution path such as A ‘A for the real exchange rate as depicted in Figure 7. The real exchange rate will never remain for more than an instant at the edge of the target zone, but will be reflected at the limits of its range of variation.13

In order to show what effect regulating real balances has on nominal variables, we need to consider the target zone illustrated in Figure 8. Its limits run parallel to the PPP line, and the path for the nominal exchange rate within the zone will drift up and down as the money stock is adjusted periodically to defend the zone. It is immediately clear that the behavior of the nominal exchange rate is strikingly different within a target zone involving perfect accommodation of shocks to prices at the edges, than it is within a nominal currency band, where defense must involve reversible switches of regime. Figure 8 reveals that the impact of a target zone is exactly the same as a nominal currency band whose limits are indexed to changes in the money stock, when those occur each time the exchange rate hits the edge of the band.

Figure 8.
Figure 8.

Target Zone as an Indexed Currency Band

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

In the long run, both the money stock and the price level will follow a random walk under a target zone arrangement. However, both will display short-run stability. This is in contrast to what happens to money and prices in the case of a nominal currency band, where both variables display a global stability, in the sense that they are always expected to adjust in the direction of long-run equilibrium.

This discussion is subject to one important qualification. In the case of a target zone, depending upon parameter values, there may exist a second smooth-pasting solution, shown as B’B in Figure 7. This solution is unstable and raises a potentially serious problem. If the intervention rule is formed in terms of what will happen when the exchange rate, rather than the fundamental, reaches a particular level, then the market will have no means of distinguishing between the two possible trajectories. This suggests the need for policy to be directed toward deterring the market from embarking upon the unstable path.14

It has been shown that a target zone for the real exchange rate may be implemented by infinitesimal adjustments to real balances. We note in conclusion that the argument advanced by Flood and Garber (1989) that such regulated solutions can alternatively be sustained by discrete interventions applies here too. An illustration is provided by the realignment rule for a nominal band studied in the previous section, under which discrete adjustments to the nominal money supply (and so to real balances) were triggered for certain at the edges of the band. It will be found that the path followed by the real exchange rate under such a rule is precisely of the form F’A ‘OAF shown in Figure 7, and that the shift in real balances on realignment corresponds to the shift from F (or F’) to the origin O. This makes the important point that periodic realignment of a nominal band is, for certain realignment rules, equivalent to defending a fixed real band by means of discrete intervention. Other realignment rules which involve larger or smaller discrete changes to real balances are analyzed in Miller and Weller (1989). As Figure 8 suggests, the smooth-pasting solution emerges in the limit where the change to real balances on realignment shrinks to zero.

III. Summary and Conclusions

Policy intervention to stabilize exchange rates has led to the macro-economic application of techniques more commonly found in finance theory. In their study of the behavior of sterling before its widely anticipated return to the gold standard in 1925, for example, Flood and Garber (1983) used the techniques of stochastic process switching. Extensive further use of these techniques has recently been made to study the operation of currency bands, with a focus on the experience of the EMS. This paper is designed to show that, although the technical details may seem esoteric, the main results are reasonably accessible.

As is to be expected, the policy implications depend to a great extent on the model used and the way in which shocks are introduced. We have looked in some detail at both a monetary model with velocity shocks and a Dornbusch-style model with shocks to the process by which prices are set.

The monetary model has attracted a good deal of attention, not least because it can be solved explicitly—and hence, subjected more easily to empirical testing. Given cumulative shocks to velocity, a policy of controlling the money supply cannot ensure price stability; and it is only by intervening at some point to offset these velocity shocks that finite limits can be set to these fluctuations in nominal values. But the credible expectation of such intervention can induce a good deal of stability before that, as Krugman (1988) demonstrated with the S-shaped solution for the rate inside a currency band.

Evidence from the EMS, however, strongly suggests that expectations of exchange rate realignment have played an important offsetting role—at least up until 1987—which can reverse this stabilizing effect. How important is the limitation of official reserves needed to defend currency bands in the EMS is a debatable issue (given the arrangements for extending credit). But in any case, introducing reserve limits suggests the need for considering locally reversible regime switches, even in the monetary model.

Much of the policy concern with exchange rate fluctuations, however, has been over the consequences for real exchange rates and international competitiveness, issues that cannot be addressed in the monetary model. In the main body of the paper, therefore, we have shown how the same techniques can be used to study the operation of currency bands (and their realignment) in a model with price inertia. Though the model is different (fixing the money supply will, for example, ensure long-run price stability), and the results cannot be expressed in closed-form solutions, nevertheless one finds similar tendencies for currency bands to stabilize the rate, and for realignment rules to undo this stabilization. Nominal bands were found to require discrete intervention at the edge, but this was not true of Williamson’s target zones where marginal intervention is sufficient. (The possibility of multiple equilibria inside the target zone implied that intramarginal intervention might be required, if only as a threat.)

The analysis in this paper can be considerably enhanced by allowing for time dependence in the solutions. This is appropriate when the market is expecting a policy shift at a reasonably predictable date, as was true in the interval before the United Kingdom’s entry into the exchange rate mechanism of the EMS in October 1990. Fortunately, this is straightforward in principle (see for example, Ichikawa, Miller, and Sutherland (1990)). As Svensson (1989) has shown, it is also possible to include systematic realignment risk inside the band without much difficulty.

To assess the impact of currency stabilization measures in recent years, it would be desirable to go further than the current literature, so as to include multiple sources of disturbances and a more realistic account of wage price setting. Another important task for future research is to introduce explicitly elements of market inefficiency into a model of currency bands, since such inefficiencies are likely to be a major reason for imposing a band in the first place. (For recent evidence on this issue see Frankel and Froot (1987) and Cutler, Porterba, and Summers (1990).) Development of a convincing theoretical framework within which one can analyze the welfare implications of asset market inefficiencies and strategies to mitigate their effects is still at an early stage, but recent work on noise trading holds much promise.

APPENDIX I Stationary Solutions for the Monetary Model

To obtain the desired family of functions, one can express E(ds)ldt in terms of the function f(k) and its first two derivatives, since by Ito’s lemma


when the money stock is kept constant.

Substitution into equation (4) yields the equation


One of the great attractions of working with the monetary model is that this differential equation has an explicit solution in terms of exponential functions, namely:


where ρ = √(2/λσ2) and A1, A2 are constants of integration (see Krugman (1988) and Froot and Obstfeld (1989), who show the effect of including a trend in velocity).

APPENDIX II Properties of the Saddlepoint Phase Diagram

In the absence of noise in the inflation process, equation (12) can be rewritten as


where A has roots of opposite sign. The integral curves (phase curves) for such a deterministic system shown in Figure 3 form a saddlepoint phase diagram. Some of the properties of this phase diagram are useful for characterizing solutions for the stochastic differential equations that result when noise is present. For convenience, these are listed below.

Eigenvectors and Eigenvalues (Roots)

Each of the eigenvectors (shown as SS, UU in Figure 3) satisfies the condition that


where θ represents the slope of the eigenvector, and ρ denotes the eigenvalue, one of the roots of characteristic equation |A–ρI| = 0. It follows, by substitution, that these slope coefficients can also be obtained as the roots of the quadratic equation


Notice that, from the signs of the coefficients appearing in A, the slope of the unstable eigenvector must always be positive:


but the sign of 65 depends on specific parameter values:




When κη < 1, the exchange rate overshoots the long-run equilibrium in response to a change in the money stock; therefore, the stable eigenvector slopes down to the right as in Figure 3. (The undershooting case, θs > 0, is considered in the text.)

Integral or Phase Curves

The solutions to the differential equation (18) are represented by the so-called integral curves shown in Figure 3. The slope of these curves at any point, denoted g(p, s) is given by the ratio of the two equations of motion:


See, for example, Petrovskii (1969) for further discussion.

These integral curves have turning points along the lines of stationarity shown as YY (where dp = 0) and II (where ds = 0) in Figure 3. Notice that the slope of YY (given by –a11/a12 = (Φγ + Φλη)/Φλη) is greater than unity, but less than θu. The slope of II, the locus of stationary points for the exchange rate, has the same sign as θs, and is at least as large in absolute terms.

APPENDIX III The Curvature of the Stochastic Solutions

The pattern of convexity and concavity of f(p) near the origin can be examined by expressing the curvature as a quadratic function of its slope at the origin, as follows:






Note that q(θ) is the quadratic expression already encountered in Appendix I, with its roots θu and θs (the slopes of the eigenvectors); it is positive for θs < θ < θu, and negative otherwise.

The curvature of stochastic solutions elsewhere may be studied using algebraic methods (as in Miller and Weller (1988)), but a simple geometric argument follows directly from rewriting the differential equation (13) in the form


and substituting to obtain


Note that


where g(p, s) is the slope of the (deterministic) integral curve. Thus, there will be curvature in the function/whenever its slope f’ differs from that of the deterministic phase curve at the same point. Specifically sgn(f”) = sgn (gf’) for E(dp) > 0; while for E(dp) < 0, sgn (f”) = sgn(f’ – g).

To show this relationship, in Figure 9 we superimpose the integral curves from the deterministic case on the stochastic solutions. Because of symmetry one needs to consider only half the plane.

Figure 9.
Figure 9.

Curvature of Stochastic Solutions

Citation: IMF Staff Papers 1991, 004; 10.5089/9781451956917.024.A008

In the half plane on the right-hand side, consider first those paths where E(dp) < 0; that is, those lying below YY, the line of expected stationarity for prices. In region B (beneath the line SS), g > f’, so f” < 0, and the curves diverge from SS as p increases; whereas in region A (between SS and YY), where f’ > g, the curvature is reversed.

Above the line of stationarity, however, points of inflection appear near the unstable eigenvector. In region A’ (above UU), f’ falls as p increases, until eventually it is tangent to an integral curve at point Ta. But the solution for the tochastic system through Ta must always lie below the phase curve for the deterministic system through the same point. It must therefore approach UU asymptotically, as shown. A similar argument can be applied to the solution path through the tangency point Tb.

Although the qualitative properties of the stochastic solutions are readily apparent, obtaining exact solutions will typically involve numerical methods (of shooting or power-series expansion), since the required integrals are not tabulated. An exception is the case where there is simple mean reversion in the fundamentals, so that dp = a11pdt + σdz; that is, a12 = 0. In this case, where YY and UU coincide with the vertical axis, the solutions can be found by using tabulated values of the confluent hypergeometric function, as noted by Froot and Obstfeld (1989) and Delgado and Dumas (1990).


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The report presented to the European Council in April 1989 by the Committee for the Study of Economic and Monetary Union.


A more formal justification of the “smooth-pasting” boundary condition applied here is provided in the papers by Froot and Obstfeld (1989) and Flood and Garber (1989). They observe that the problem studied by Krugman is equivalent to that of regulating a Brownian motion process. They appeal to the results of Harrison (1985) and the simplified discrete-time argument of Dixit (1988). The interpretation in terms of regulated Brownian motion is of considerable theoretical interest but is conceptually quite distinct from the notion of “locally reversible” regime switches to which we appeal to justify the smooth pasting in the Dornbusch model. Our logic is much closer to that applied by Krugman and Rotemberg (1988) for analyzing “switches” to periods of temporary floating, see below.


The choice of nominal interest rate here as the influence on output is for simplicity only. Nothing of substance in our analysis changes if we work with the real interest rate.


The reason is that we intend to consider only currency bands that are symmetric around the equilibrium of the origin. For currency bands that are not so placed, one would need to consider solutions that did not pass through the origin.


The patterns of concavity and convexity shown by these solutions in a neighborhood of the origin may be readily confirmed by the formal argument given in Appendix III. Properties of the solutions in the large are less easy to prove, but their pattern can be deduced relative to the saddlepath solutions shown in Figure 3 by rearranging equation (13), as is shown in Appendix III.


It is evident from equation (11) that E(ds) is increasing in s, given p.


Since, from equation (11), E(dp) is increasing in s, given p, the rate of disinflation must increase between M and B.


In both cases, we consider rules that are symmetric around the equilibrium; otherwise, we would have to consider a wider set of solutions to the fundamental equations (namely, all those that do not pass through the origin). The same principles may, of course, be applied to asymmetric cases, such as one-sided currency bands.


See Whittle (1983) for an account of matching conditions for diffusion processes at prescribed boundaries.


Note that for this argument to work it is essential that the switch from a floating to a fixed rate regime be instantaneously reversible. If this is not the case, then it is certainly possible that OA in Figure 5 could be part of the solution path. For, once the switch occurred, price shocks in either direction would then produce monetary adjustments to hold s at s, and there would be no violation of the arbitrage condition.


It is possible, however, to have stochastic prospects for realignment at the edge of the band, which prevent interest rates from moving to world levels but do not change the smooth-pasting condition; see Miller and Weller (1990).


A case of both mathematical and practical interest arises when one considers the effect of a realignment rule where the top of the old band becomes the bottom of the new one. This means that there is a reversible switch of regime at the edge of the band, and so smooth pasting should apply. With the larger realignment, the dotted reverse-S solution shown in Figure 6 will not overlap the original solution, but will slide further up the 45-degree line and have its lower end point at A; thus the smooth-pasting condition for the two solutions will be satisfied. It is not smooth pasting on to the edge of the band, however, because under the realignment rule considered there is no longer a transition to a (temporarily) fixed rate regime at the edge of the band.


Compare this with the discrete monetary intervention under the realignment rules examined in the last section. Under those rules real variables can be shown to follow a path shaped like the trajectory F’OF in Figure 7, with discrete intervention shifting real balances directly from F to O (compare Flood and Garber’s (1989) analysis of discrete intervention).


The idea that this second solution could be ruled out by a threat to adjust monetary policy within the band, if ever the market were to move in the perverse fashion that this solution implies, is discussed in Miller and Weller (1990). A more ambitious scheme, to use fiscal policy to help stabilize nominal income, was outlined in a blueprint for policy coordination (Williamson and Miller (1987)).

Long-Run Money Demand in Large Industrial Countries: Volume 38 No. 1
Author: International Monetary Fund. Research Dept.