Monetary Discretion, Pricing Complementarity and Dynamic Multiple Equilibria

for the discussion of proposition 1.

Figure 1 provides the basis for a heuristic discussion of Proposition 1, based on the
best-response function r (). The dashed line in Figure 1 is the 45^o line; when r () crosses
this line the action of a representative adjusting firm (the horizontal axis) coincides with
the optimal action of an individual firm as described by r (). The solid line is r () for
m > m*. When m = m* it is easy to see from (22) that there is one steady state, and
it occurs at p0 = 1. An increase in m shifts r () upwards. It is thus clear that p0 =1is
not an equilibrium point with m > m*, but that there is a prospect for an intersection
point somewhere to the right as in the case illustrated in Figure 1. At any such “low”
stationary equilibrium, it must be the case that the slope is less than one (if r(p0,.) crosses
the 45^o line) or the slope is exactly equal to one (if it is a tangency). Let us call this first
equilibrium p0 .

Suppose the slope at a “low” stationary equilibrium is less than one, so that it is
not a tangency and corresponds to the case illustrated in Figure 1. As p0 becomes
arbitrarily large, θ → 1.Forlargep0, then, it follows that r(p0,.) approaches the line
(m∕m*)po from below. For high enough po then, r(po,.) > po since we are considering
an inflationary monetary policy (m > m*). We have assumed there was a fixed point at
which ∂r∕∂po < 1, and we have shown that r () lies above the 450 line for high enough
po, so there must be some other “high” p0 for which there is an equilibrium r(po) = po.
If m is high enough, the first fixed point does not exist, and r () lies everywhere above
the 45^o line. We label the two equilibria with an asterisk (*) and carry them over to our
discussion below.

There are two mechanisms at work to produce multiple steady-state rates of inflation
for arbitrary constant, homogeneous monetary policy. The first is that monetary pol-
icy is accommodative: if higher prices are set by other firms today, the future nominal
money stock will be higher in proportion. The second is that if all other firms raise prices
today and in the future, then the future inflation rate will rise and a single firm today
places higher weight on future nominal marginal costs, so that future monetary endogene-
ity becomes more influential on current price-setting. Looking ahead, the discretionary
equilibria we will construct below will involve constant, homogeneous monetary policy.
Necessarily, then, there will be multiple steady-state equilibria under discretion. However,
in order to construct those equilibria we cannot rely on the steady-state best-response
function.

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