The solution to the differential equation for μ1t–
The differential equation for μ1t is:
The solution to this linear deterministic differential equation is straightforward:
and the only value for μ10 which ensures that μ1t is always finite makes μ1t time-invariant, i.e.:
The solution to the differential equation for μ2t–
The differential equation for μ2t is as follows:
it is possible to see that:
It follows that the solution to equation (56) is:
where is an arbitrary constant. The choice of is made to ensure that μ2t is finite at all times. There is a unique constant which satisfies this requirement and this is the following:
It is possible to show, using De L’Hopital rule, that the limiting value for μ2t as time tends to infinity is:
Barro, R. and D. Gordon, 1983, “Rules, Discretion and Reputation in a model of Monetary Policy”, Journal of Monetary Economics, pp.101–21.
Giavazzi, F. and M. Pagano, 1988, “The Advantage of Tying One’s Hands. EMS Discipline and Central Bank Credibility”, European Economic Review, pp.1055–75.
Winckler, G., 1991, “Exchange Rate Appreciation as a Signal of a New Policy Stance”, International Monetary Fund working paper, WP/91/32.
I wish to thank especially Martin Cripps and Marcus Miller. This paper also benefitted from the comments of John Driffil, Jonathan Thomas, and the participants in the seminars held at the European I Department and at the Research Department. I am grateful to Mr. Emmanuel Zervoudakis for his support and encouragement.
Besides this separating equilibrium, there exists also a pooling equilibrium.
It is possible to show that, in the deterministic case, the solution obtained when each period between realignments is treated independently holds also when links between realignments are taken into account. However, this is no longer true when asymmetric information is introduced.
If the simplifying assumptions of having πt = πPt and πt deterministic were removed, the problem would be stochastic and have the same solution for the government’s policy. In the rest of this paragraph these assumptions are maintained because the solution for the optimal policy is the same, whilst the distinction between actual and planned inflation and the stochastic nature of πt will be reintroduced in the next paragraph.
Notice that in this model it is not necessarily the case that there is a positive inflationary bias; in fact it is possible to have 0 as a time consistent inflation rate, or even a negative rate.
As already pointed out in the introduction, in the floating regime assumed here, PPP holds at all times, so that the government cannot affect the real exchange rate. In the pegged exchange rate system instead, the government’s decision to create inflation reduces competitiveness.
The Bts also affect St in an inverse way; the larger the Bts are, the smaller St is; however, if one considers only one Bt at a time, keeping the others fixed, the effect on St is infinitesimal, whereas there is obviously a finite effect on Bt and therefore the latter effect dominates.
The reduction in planned inflation entails also a reduction in actual inflation in expected terms.