policy rule (Mt = mtP1,t). From (17), the normalized price set by adjusting firms (p0,t)
satisfies
p0,t
(ɪ) ((1 - θt,t+ι) mt + θt,t+ιmt+ιPo,t)
(18)
≡ r(p0,t, mt, p0,t+1 ,mt+1).
The weight on future nominal marginal cost, which was defined in (12), can be written
in terms of current and future normalized prices as
where we are now explicit about how θt,t+1 depends on the present and the future. Equa-
tion (18) is a nonlinear difference equation in p0 and m that must be satisfied in a perfect
foresight equilibrium with homogeneous policy.
θ(p0,t, p0,t+1) =
βπ(po,t,po,t+ι)ε 1
1 + βπ(po,t,po,t+ι)ε-1 ■
(19)
We view p0,t on the left-hand side of (18) as describing what an individual firm finds
optimal given the actions of other price-setters and the monetary authority. On the
right hand side, p0,t then represents all other adjusting firms’ pricing behavior, and the
function on the right hand side represents the implications of those firms’ behavior for the
marginal revenues and costs of an individual firm. In other words, r(.) is a best-response
function for the individual firm. We restrict attention to symmetric equilibria, so that
prices chosen by all adjusting firms are identical. We define complementarity in terms of a
positive partial derivative of the response function with respect to its first argument. That
is: with perfect foresight, there is complementarity if ∂r(po,t, mt, p0,t+1, mt+1)∕∂po,t > 0.
In sections 3.2 and 3.3 below, we will make intensive use of the perfect-foresight best-
response function (18). First, we will use it to describe point-in-time equilibria; this
involves characterizing the fixed points for p0,t, taking as given mt,mt+1 , and p0,t+1 .
Second, we will use it to determine the model’s steady-state equilibria under constant
arbitrary policy. That is, we will impose p0,t = p0,t+1 and mt = mt+1 = m and determine
the equilibrium value(s) (fixed points) for p0. Both of these exercises will then serve
as inputs to our analysis of discretionary equilibria. There, (18) will summarize private
sector equilibrium for any action that the monetary authority contemplates, under perfect
foresight.7 With uncertainty, (18) will not hold exactly, but the mechanisms discussed
here will still be relevant.
7If we impose mt = mt+1 but allow p0t to differ from p0,t+1 , then (18) describes the dynamics of p0,t
for constant homogeneous policy. Such analysis might reveal interesting dynamics. However, it is not an
input into our analysis of discretionary equilibrium.
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