APPENDIX I: Monetary Targets Subject to Nominal Bands
The evolution of the price level, p, and the exchange rates, s, determined by the equations (8)-(11) can be summarized in terms of two simultaneous stochastic differential equations (after elimination of y and i and normalizing by setting ȳ = 0) as
where Δ = κγ + λ. Alternatively, if we redefine variables s and p to denote deviations from long-run equilibrium, we may rewrite (A1) as
where A is the matrix of coefficients multiplying the vector of endogenous variables on the right hand side of (A1).
In order to obtain solutions to the system (A2), one begins by postulating a deterministic functional relationship s = f(p). Following the rules for stochastic differentiation one obtains
from which it follows that
Substituting for E(dp) and E(ds) from (A2) we obtain
where aij denotes the appropriate element of the matrix A. This second-order, nonlinear differential equation has no closed form solutions in general, but we have shown that it is possible to characterize completely the qualitative features of the relevant solutions (see Miller and Weller (1989)).
APPENDIX II: Cash Limits and Exchange Rate Overshooting
The effect of a (partial) cash limit on public spending is proxied by g0 adding the term g0-βp to the equation determining aggregate demand. (g0 is a measure of the demand effect of public expenditure without the limit, and β is an index of how hard the limit “bites”.)
It is easiest to calculate the effect of keeping such a limit permanently in operation before analyzing the “state contingent” use of such a device. Assuming that there is a fixed money supply target and that the exchange rate floats freely, one has the following equations to solve:
which, after substitution, can be written
Note that the locus of stationarity for the exchange rate (where Eds = 0), now has to satisfy the condition that
(1-κ(η + β))dp + ηλds = 0.
Hence, given cash limits, the condition for “no overshooting” becomes 1 - κη < κβ; instead of 1 - κη ≤ 0 as is true otherwise (i.e., the effect of β > 0 is to reduce overshooting).
Consider, for concreteness, the case where potential overshooting is just reduced to zero, so
κ(η + β) = 1, β > 0
In this special case, the stable manifold happens to coincide with the locus of stationarity, and in the absence of nonlinearities, it is appropriate to restrict the system to the stable manifold. So, with the exchange rate stationary at its equilibrium value, the solution is simply
In general, of course, the eigenvector with not have zero slope. But the slope will be less negative as a result of the cash limits.
To establish that cash limits enhance the stbility of the system, it is sufficient to observe that raising β above zero has no effect on the determinant of the matrix in B(5), while evidently increasing the trace in absolute value: so the size of the stable root must increase with β.
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University of Warwick and CEPR. This paper was written during our stay as Visiting Scholars in the European Department of the International Monetary Fund. We are grateful to Mr. Russo for suggesting the inclusion of fiscal policy; and to Robert Flood and Peter Garber for discussions on the monetary model. Responsibility for views expressed and for any errors remain with the authors.
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.