3.1 Complementarity under homogeneous monetary policy
There are two mechanisms for complementarity in (17) and (18) that will be operative
in our analysis of both point-in-time and steady-state equilibria. First, holding fixed
the weights, P0,t has a positive effect on the right-hand side in (17): it enters linearly
with a coefficient of (εε¾) θt,t+imt+1, which is positive because firms are forward-looking
and the monetary authority raises nominal Mt+1 proportionately with P0,t . Hence the
specification of monetary policy has introduced some complementarity into an economy
in which it was previously absent.
Second, the weights in these expressions vary with the current price P0,t (or its normal-
ized counter part p0,t). This additional channel plays an important role in our analysis.
A reference value for the weight θt,t+1 is one-half, since (12) implies that the weight is
β∕(1 + β) with β close to one if if Pt = Pt+1. An upper bound on this weight is one: this
is a situation where firms place full weight on the future. Increases in the weight raise the
extent of the effect discussed above, i.e., they raise the coefficient ^ε>¾) θt,t+1mt+1 that
measures the extent of complementarity. The second mechanism is then that an increase
in P0,t (or its normalized counterpart p0,t) raises the weight on the future. This occurs
because a firm’s profits are not symmetric around its optimal price. As the firm’s relative
price rises, its profits decline gradually, asymptotically reaching zero as the price goes to
infinity. By contrast, as the price falls, the firms profits decline sharply toward zero and
may even become highly negative if the firm is not allowed to shut down its operations.8
Thus, when P0,t increases for all other firms, future monetary accommodation — and the
associated higher nominal price set by firms in the future — automatically lower’s the
firm’s future relative price. The costliness of a low relative price leads the firm to put
increased weight on future marginal cost.
3.2 Equilibrium analysis of steady states
To characterize steady-state equilibria for arbitrary constant, homogeneous monetary
policies, we impose constant m and p0 on the right hand side of (18). Steady-state
equilibria are fixed points of the resulting steady-state best-response function for p0 :
εχ
Po =----τm[1 - θ(po,Po) + θ(po, Po)po], (20)
ε-1
8 At this point in the analysis, we do not explicitly take into account the shut-down possibility. But,
when we calculate discretionary equilibria, we do verify that the equilibria are robust to allowing firms to
shut down.
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