Monetary Discretion, Pricing Complementarity and Dynamic Multiple Equilibria

In each case, the second equality indicates the implications of the specific utility function
introduced above.

2.2 Firms

Firms produce output according to a linear technology, where for convenience we set the
marginal product of labor to one. So, for each type of firm, the production function is

cj,t = nj,t .

This implies that real marginal cost is unrelated to the scale of the firm or its type and
is simply

ψ_t = wt

and that nominal marginal cost is Ψt = Ptψ_t = Ptwt

Much of our analysis will focus on the implications of efficient price-setting by the
monopolistically competitive firm. The adjusting firms in period t are assumed to set
prices so as to maximize the expected present discounted value of their revenues, using
the household’s marginal utility as a (possibly stochastic) discount factor. That is, they
choose P0_,t to maximize their market value,

[λt(Po,t - Ψt)co,t + βEtλt+ιP÷-(Po,t - Ψt+1)c1,t+1].

Pt+1

As monopolistic competitors, firms understand that co,_t = (PP^t)^-εc_t and that c₁,t+1 =
(7P+7)-^εct+1, but take ct, Pt, ct+1 and Pt+1 as not affected by their pricing decisions. The
efficient price must accordingly satisfy

ε Pt^ε-1Ψt + βEt (Pt^ε+⅜+ι)
ε ^{- 1}   Pt^ε-1 + βEt (¾^

where we again give the result under the specific momentary utility function. In fact, this
reveals one motivation for the form of the particular utility function chosen. In general,
both aggregate demand (c_t) and the discount factor (λ_t) would appear in (10), but our
choice of a utility function that is logarithmic in consumption means that these two effects
exactly cancel out. With perfect foresight, the pricing equation can be written compactly
as

ε

P0,t = _{ε -} ι^[(1 ^- θt,t+ι) ^ψt + θt,t+ι^ψt+ι];                       ⁽¹¹⁾

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