Consider a continuous-time version of the Mankiw Reis model. Opportunities to review pricing policies follow a Poisson process with arrival rate α > 0. Thus the probability that a price path set at time t is still being followed at time t + i is e−αi

. The other assumptions of the model are the same as before.

(a) Show that the expression analogous to (7.81) is a(i) = φ(1 − e−αi

)

[1 − (1 − φ)(1 − e −αi

)]

.

(b) Consider the experiment of a permanent fall in the growth rate of aggregate demand discussed in Section 7.7.

That is, until t = 0, all firms expect m(t) = gt

(where g > 0); thereafter, they expect m(t) = 0.

(i) Find the expression analogous to (7.83).

(ii) Find an expression for inflation, p (t), for t ≥ 0. Is inflation ever negative during the transition to the new steady state?

(iii) Suppose φ = 1. When does output reach its lowest level? When does inflation reach its lowest level?