Monetary Discretion, Pricing Complementarity and Dynamic Multiple Equilibria

A Appendix

A.1 Proofs

Proof of Proposition 1. (i) The conditions which characterize a steady-state equilib-
rium are (20) and (21). Multiplying (20) by the denominator of (21) reveals that these
conditions are equivalent to

p0 = h (p0) ,                                        (25)

where

h (Po) = m + β (m - l´ Po                     (26)

m*

Steady-state equilibria are thus fixed points of h () , and fixed points of h () are steady-
state equilibria.

(ii) For P0 > 0 the function h () is strictly positive, strictly increasing, and strictly
convex.

(iii) Define Pe0 implicitly as follows:

Pe0:h⁰(Pe0)=1.

At Pe0,h() is tangent to the 45^o line. By differentiating h () , we find that

and Pe0 is decreasing in m.

Convexity of h () implies that if h (Pe0) > Pe0 then h () does not have a fixed point,
and if h (Pe0) <P0 then h () has two fixed points. We now need to show that for low m,
h (Po) < Po, and for high m, h (po) > Po- From (27) and (26), h (po) ^ Po is equivalent to

It is straightforward to show from (28) that there is a unique value ofm, call it me such
that h (po) ^ po for m ^ m. ■

Proof of Proposition 2.   (Sketch) From (18) and (19), point-in-time equilibrium

values of po are solutions to

fm- - ^mP = β μ^g(^p°)PoY'"¹ (m⁰ - m-) ,                (29)

po         g (po)

for fixed m⁰ >m^- and p⁰_o .

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