State-dependent pricing with both positive and negative inflation. (Chaplin and Leahy, 1991.) Consider an economy like that of the Chaplin–Spooler model. Suppose, however, that m can either rise or fall, and that firms therefore follow a simple two-sided S_s policy: if pi − p∗ i (t) reaches either S or −S, firm i changes pi so that pi − p∗ i (t) equals 0. As in the Chaplin–Spooler model, changes in m are continuous.

Assume for simplicity that p∗ i (t) = m(t). In addition, assume that pi − p∗ i (t) is initially distributed uniformly over some interval of width S; that is, pi − p∗ i (t) is distributed uniformly on [X,X + S] for some X between −S and 0.

(a) Explain why, given these assumptions, pi − p∗ i (t) continues to be distributed uniformly over some interval of width S.

(b) Are there any values of X for which an infinitesimal increase in m of dm raises average prices by less than dm? by more than dm? by exactly dm? Thus, what does this model imply about the real effects of monetary shocks?