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.
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:
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.
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
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 (g – f’) 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.
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).
Bertola, Giuseppe, and Ricardo Caballero, “Target Zones and Realignments,” CEPR Discussion Paper 3986 (London: Centre for Economic Policy Research, March 1990).
Cutler, David M., James M. Poterba, and Lawrence H. Summers, “Speculative Dynamics,” NBER Working Paper 3242 (Cambridge, Massachusetts: National Bureau of Economic Research, January 1990).
Delgado, F., and B. Dumas, “Target Zones Big and Small” (unpublished; November 1990). Forthcoming in Currency Bands and Exchange Rate Targets, ed. by P. Krugman and M. Miller (Cambridge: Cambridge University Press).
Dixit, Avinash, “A Simplified Exposition of Some Results Concerning Regulated Brownian Motion” (unpublished; Princeton University, August 1988).
Flood, Robert P., and Peter M. Garber, “The Linkage between Speculative Attack and Target Zone Models of Exchange Rates,” NBER Working Paper 2918 (Cambridge, Massachusetts: National Bureau of Economic Research, April 1989).
Frankel, J., and K. Froot, “Using Survey Data to Test Standard Propositions Regarding Exchange Rate Expectations,” American Economic Review, Vol. 77 (March 1987), pp. 133–53.
Froot, Kenneth, and Maurice Obstfeld, “Exchange Rate Dynamics Under Stochastic Regime Shifts: A Unified Approach,” Harvard Institute of Economic Research Discussion Paper 1451 (September 1989).
Ichikawa, M., M. Miller, and A. Sutherland, “Entering a Preannounced Currency Band,” Economics Letters, Vol. 34 (December 1990), pp. 263–368.
Krugman, Paul R., “Target Zones and Exchange Rate Dynamics,” NBER Working Paper 2481 (Cambridge, Massachusetts: National Bureau of Economic Research, January 1988).
Krugman, Paul R., and Julio Rotemberg, “Target Zones with Limited Reserves,” NBER Working Paper 3418 (Cambridge, Massachusetts: National Bureau of Economic Research, July, 1990).
Miller, Marcus, and Paul Weller, “Solving Stochastic Saddlepoint Systems: A Qualitative Treatment with Economic Applications,” Warwick Economic Research Paper 309 (Coventry: University of Warwick (December 1988)).
Miller, Marcus, and Paul Weller, “Exchange Rate Bands and Realignments in a Stationary Stochastic Setting,” in Blueprints for Exchange Rate Management, ed. by M. Miller, B. Eichengreen, and R. Portes (London; San Diego, California: Academic Press, 1989).
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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.
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)).