with a weight on the future of
β ε-1
θ(P0,P0)= , βp0 ε-1 ∙ (21)
1 + βpε0-1
Fixed points of the steady-state best-response function are constructed by simultaneously
varying current and future p0 on the right hand side. This is in contrast to fixed points of
the basic point-in-time best-response function, which are constructed holding fixed p0,t+1∙
3.2.1 Uniqueness occurs at zero inflation
A zero inflation steady state involves p0 =1∙ Such a steady state exists when the nor-
malized quantity of money is m* ≡ (^-ɪχ)-1∙ In this case, the weight on the future is
θ = β∕(1 + β^), which is roughly one-half. The zero-inflation steady state is asymptotically
optimal under full commitment in this model (see King and Wolman [1999]) and provides
an important benchmark. Furthermore, if m = m*, zero inflation is the unique steady
state; that is, po = 1 is the unique solution to (20) when m = m*.
3.2.2 Multiplicity or nonexistence must occur with positive inflation
We refer to any m>m* as an inflationary monetary policy, because if inflation is positive
in a steady state, then m>m* , as we now show. From (20), given that π = p0 in steady
state, we have
1π 1 *
m = --TT---=--τr-τ =-----------m—m ∙
( ε⅛ x)[1 - θ + θ∏] [θ + (1 - θ)( ɪ )]
Thus, π > 1 if and only if m>m* ∙
Proposition 1 states that under an arbitrary inflationary monetary policy, for low
values of m there are two steady-state equilibrium values of p0 . For high values of m,
no steady-state equilibrium exists. In a knife edge case there is a unique steady-state
equilibrium.
Proposition 1 There exists an me >m* such that for m ∈ (m* , me ) there are two steady-
state equilibria, and for m>me there is no steady-state equilibrium.
Proof. see Appendix. ■
From (20), steady-state equilibria for a given m are fixed points of r (p0; m) , where
we write the best-response function as
m
(22)
r (po, m) = — ∙ [(1 — θ (po)) + θ (po) ∙ Po]
m*
16