APPENDIX: Bayesian Learning about the Realignment Probability
The realignment expectations discussed in the text can be derived from a model of Bayesian learning, as shown in Driffill and Miller (1991). The argument may be summarized briefly as follows.
After the exchange rate is pegged, the public still believes that realignments of size J may occur as a Poisson process--but they are not sure of the intensity, π. Assume specifically that their uncertainty is simply whether this intensity is high (πH) or low (πL) and suppose they start out immediately after the end of floating with initial probability PH(0) and PL(0)=1-PH(0) attached to each of these intensities, i.e., π(0)=pH(0) (πH-πL) + πL. If these probabilities are then updated in a Bayesian fashion, then as long as no realignments take place, PH declines exponentially towards zero. Specifically
where time t is measured from the date at which the rate was first pegged.
If a realignment occurs at time t, then this causes a discrete jump in PH(t) so
Thereafter PH will decline exponentially from its new high value just as in equation (A1), but measuring time from the date of the realignment.
Note that for large values of t,
and so, if πL = 0, then for large t
the result we use in the text.
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Marcus Miller is Professor of Economics at the University of Warwick and Alan Sutherland teacher economics at the University of York. We would like to thank Juan Carlos Di Tata for his comments. For financial support, we are indebted to the Economic and Social Research Council of the U.K. (grant no. R000231417, “The International Monetary System: An Analysis of Alternative Regimes.”) This paper was completed while Marcus Miller was a visiting scholar at the Research Department of the International Monetary Fund and he thanks the Research Department for its hospitality.