Monetary Discretion, Pricing Complementarity and Dynamic Multiple Equilibria

with a weight on the future of

_β ε-1

θ(P0,P0)= , βp0 ε-1 ∙                                (21)

1 + βp^ε₀^-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

r (po, m) = — ∙ [(1 — θ (po)) + θ (po) ∙ Po]
m^*
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