m, the monetary authority still has the same incentives to raise m, but it does so without
producing the curious behavior of the markup shown here.
In fact, it is not necessary to make a complicated set of fixed point computations in
this case. A tangency equilibrium is one in which dr(po,tmt∂p0,t+1 ,mt+1 ) = 1. Therefore, we
can simply solve the stationary version of the equation,
∂r(p0,t, mt, p0,t+1, mt+1)
P0,t----------«--------------= r(Pθ,t,mt,Pθ,t+1,mt+1h
∂p0,t
to calculate the equilibrium value of p0 (this is one equation in one unknown p0 because
the m = m0 drops out). We can then determine the relevant m from the equation
p0 = r(p0, p0, m, m).
In our numerical example, there is a consistent equilibrium with p0 = 1.17, so that
there is a 17% quarterly inflation rate in the pessimistic equilibrium with optimal discre-
tionary policy. The associated value of m/m* is 1.0295. This value is larger than the one
used to construct Figure 3, as it should be: a higher level of m is necessary to produce a
tangency equilibrium in the pessimistic case.
There are thus two steady-state equilibria with discretionary optimal monetary policy
in our quantitative example, one with low inflation and one with high inflation. The levels
of the inflation rates are quite different: about 2 percent (per quarter) in one case and
about 17 percent in the other.
5 Stochastic equilibria
The generic existence of two point-in-time equilibria and two steady-state equilibria for
arbitrary homogeneous policy suggests that it may be possible to construct discretionary
equilibria that involve stochastic fluctuations. We now provide an example of such an
equilibrium. We assume that there is an i.i.d. sunspot realized each period which selects
between the two private sector equilibria: in each period, the low-p0 outcome occurs with
probability 0.6, the high-p0 outcome occurs with probability of 0.4, and this is common
knowledge.9
9 Our model does not pin down the distribution of the sunspot variable. However, some restrictions on
that distribution are imposed by the requirement that every firm’s profits be nonnegative in each period.
For example, if α is 0.75 rather than 0.6, this condition is violated in the p0 state, and no discretionary
equilibrium exists. As in Ennis and Keister (forthcoming), it would be interesting to study whether
adaptive learning schemes would further restrict the distribution of the sunspot variable.
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