3 Equilibrium with homogeneous policy
We begin by studying the nature of equilibrium price-setting (p0,t) given an arbitrary ac-
tion by the monetary authority and given perfect foresight. We assume that the monetary
authority adopts a policy rule of the form
Mt = mtP0,t-1 , (16)
where mt is viewed as the policy variable. That is, the money supply is proportional to
the prices that adjusting firms set one period ago with a constant of proportionality mt .
We call this a homogenous monetary policy rule. This form of monetary accommodation
of past nominal variables is characteristic of optimal monetary policy under discretion, for
the following reason. The monetary authority is concerned about the real variables that
enter in private agents’ utility. It takes past prices as given, and there is no mechanism by
which the level of nominal predetermined prices necessarily constrains the behavior of a
discretionary policymaker.5 Thus, if we viewed M instead of m as the policy instrument,
we would find that the optimizing monetary authority adjusted Mt proportionally to
P1,t, just as is specified in (16). It will economize slightly on notation and computation
to view mt as the policy instrument, and there is no loss in generality. In a discretionary
equilibrium, mt will be chosen to maximize welfare; in this section mt is arbitrary.6
A homogenous money supply rule means that the future money supply depends on
the price set by adjusting firms today,
Mt+1 = mt+1P0,t.
Consequently, under homogeneous policy and using the preferences introduced above,
it follows from the efficient price-setting condition (11) that the nominal price set by
adjusting firms (P0,t) satisfies
Po,t = χ ((1 - θt,t+ι) mtPι,t + θt,t+1mt+1P0,t) (17)
ε - 1
in equilibrium. The derivation of (17) from (11) involves (i) using the fact that nominal
marginal cost is Ψt = Ptχct given the specific utility assumption; (ii) using the money de-
mand relationship (Mt = Ptct); and (iii) imposing the homogenous form of the monetary
5 The word “necessarily” appears because one could construct non-Markov equilibria in which all
agents agreed that P1,t did constrain the monetary authority. See previous footnote. We do not study
such equilibria.
6By contrast, under commitment, the monetary authority commits not to respond to P1,t, and the
choice is over sequences of Mt . King and Wolman (1999) study optimal policy with commitment in the
model used here.
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