Monetary Discretion, Pricing Complementarity and Dynamic Multiple Equilibria

3.3 Point-in-time equilibria

Solving the monetary authority’s problem under discretion means computing the point-
in-time equilibria that correspond to all possible current policy actions, and then picking
the best action. Before studying this topic in detail in section 4, we here begin by
characterizing point-in-time equilibria for an arbitrary policy action in the current period.
Point-in-time equilibrium refers to the values of p0_,t that solve (18) for given current
and future monetary actions, and a given future price p0,t+1.The mechanisms described
earlier lead to the potential for multiple point-in-time equilibria. We assume that the
future money supply is given by mt+ι ∈ (m*,fn), (i.e. steady-state equilibria do exist
for the assumed value of m_t+1 and are inflationary) and that the future relative price
is consistent with one of the two steady-state equilibria that may prevail if that level of
mt+1 is maintained forever. Under these assumptions, there are either two equilibria in
the current period or equilibrium does not exist. Again, in a knife edge case equilibrium
is unique.

Proposition 2 If mt+1 ∈ (m*,m) is fixed, then there exists m such that for m_t < m
there are two equilibria in period t, and for m_t > m equilibrium does not exist in period t.

Proof. see appendix for a sketch. ■

Point-in-time equilibria are fixed points of the best-response function for current pe-
riod price-setters, which we write without time subscripts, using superscript prime to
denote next period:

P0 = ^r (po^,m^,p0^,m9 = m* £(1 ^- θ (po∙∕'0)) ^m + θ (po,p0) m⁰po]

No expectation operator appears because we are assuming, for the purposes of this sec-
tion, that there is no uncertainty about future m and — more importantly — future p0 .
Multiplicity of point-in-time equilibria occurs for much the same reason as multiplicity
of steady states. Because the future nominal money supply is endogenous, the current
price of other firms has an effect of more than one-for-one on a single firm’s desired price
if agents weight the future heavily, as they do if m⁰ > m* and po is high enough. Note
that as long as the future money supply is inflationary, there will be multiple equilibria
even for noninflationary current values of the money supply.

Figure 2 illustrates the multiplicity of point-in-time equilibria for m = m⁰ >m^* ,
for two different beliefs about future p0 . As above, the dashed line is the 45⁰ line that
identifies fixed points: the two points marked with asterisks (‘*’) on the 45^o line are

18

<<   17   18   19   20   21   22   23   >>

More intriguing information