3. Optimal pricing by firms in the current period (p0). The monetary authority must
have beliefs about the selection rule used to determine p0 when a contemplated
value of m leads to multiple equilibrium values of p0 .
Two conditions define a stationary perfect foresight equilibrium with discretion: (i)
the current and future monetary authority each choose the same action; and (ii) the
selection rule specifies that only one equilibrium will prevail in every period. It is common
knowledge which equilibrium will prevail.
As we noted above, it is the essence of discretion in monetary policy that certain
predetermined nominal variables are taken as given by the monetary authority. Here, the
current money supply is set proportionally to the previously set price, P1,t = P0,t-1. This
leads us to view m as the monetary authority’s choice variable. Our analysis of equilibrium
under arbitrary choice of m revealed that in general there were either two point-in-time
equilibria or no point-in-time equilibria, as long as future policy was expected to be
inflationary. This leads us to expect multiple discretionary equilibria. In this section we
analyze discretionary equilibria where there is a constant probability of 1.0 on one of the
two private sector equilibria.
4.1 Constructing Discretionary Equilibria
We look for a stationary, discretionary equilibrium, which is a value of m that maximizes
u(c, l) subject to the constraints above when m0 = m. We have used two computational
approaches to find this fixed point. A comparison of the two approaches is revealing
about the nature of the multiple equilibria we encounter.
The first computational method involves iterating on steady states. We assume that
all future monetary authorities follow some fixed rule m0 . Next, we determine the steady
state that prevails including the value of p00 . Then, we confront the current monetary
policy authority with these beliefs and ask her to optimize, given the constraints including
the selection rule. If she chooses an m such that |m - m0 | is sufficiently small, then we
have an approximate fixed point. If not, then we adjust the future monetary policy rule
in the direction of her choice and go through the process again until we have achieved an
approximate fixed point. This approach conceptually matches our discussion throughout
the text, but leaves open an important economic question: are the equilibria that we
construct critically dependent on the infinite horizon nature of the problem?
The second computational method involves backward induction on finite horizon
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