Monetary Discretion, Pricing Complementarity and Dynamic Multiple Equilibria

In order for its maximization problem to be well-defined, the monetary authority
must have beliefs about the current and future distribution over private-sector equilibria.
Above, these beliefs were degenerate. Now that they are nondegenerate, the problem
is slightly more complicated. Letting α be the probability of the low-p0 outcome, the
monetary authority maximizes

{αu(c(m,po),l(m,po)) + (1 - α)u(c(m,p_o),l(m,po))} + βv⁰

where v⁰ denotes the future expected utility, which again cannot be influenced by the
current monetary authority. It is important to stress that the low and high p0 values are
influenced by the sunspot probabilities, since they satisfy the equations

p_o = ——  f----------------i^! m + f-------^βp0------г ! E Пп (p_o,po^ 1 m0 0 ,

0 m*[ V+ β E ∏ (Po,p0)^ε-1/     U+ βE ∏ (Po,P0)^ε-1/   ¹              ʃr

(24)

where expectations are taken over the distribution of the sunspot variable. For example,

= ^απ ζpo^,p_o´    ⁺(1 ^-a)π(po^,Po)^ε 1 ∙

Because the sunspot is i.i.d., this expression holds for both the low and high current value
of po. Note that uncertainty prevents us from writing (24) as the simple weighted average
that we used with perfect foresight.

5.1 Constructing Discretionary Equilibria

We can again apply the two computational approaches described in the previous section to
construct Nash equilibria. In implementing these, we assume that the monetary authority
and the private sector share the same probability beliefs.

5.2 Optimal discretionary policy

The relevant trade-offs for the discretionary monetary authority are illustrated in Figure
5. In panel A, there is a light solid line between the objective function for the low-po
private-sector equilibrium (the dark solid line) and the objective function for the high-po
private sector equilibrium (the dashed line): this is the monetary authority’s expected
utility objective, which is a weighted average of the two other objectives. The monetary
authority chooses an optimal action that is about 1.0202, which is more stimulative than

25

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