Monetary Discretion, Pricing Complementarity and Dynamic Multiple Equilibria



state variable in this economy, the price set by firms in the previous period (P1,t = P0,t-1).
Define the normalized money supply as

(13)


(14)


mt = Mt/P1,t ,

and the normalized price set by adjusting firms in the current period as

p0,t = P0,t/P1,t .

We can then express all variables of interest as functions of these two normalized variables.
From (4), the normalized price level is a function of only
p0,t :

Pt    (п ʌ

=g = g(p0,t),
P
1,t

where

1     1 r1 1 —ε . 1-1 ι^-

g(po,t) ≡ [2p0,t +2]1-ε

Aggregate demand is a function of both p0,t and mt :

ct = c(p0,t,mt)


mt
g(po,t)


This follows from the money demand equation:

Mt = Mt Pι,t = mt
Pt Pι,t Pt g(po,t)

Further, since nt = [1no,t + 2nι,t] = [2co,t + 2cι,t], we can use the individual demands
together to show that total labor input is also pinned down by
p0,t and mt :

nt = n(po,t, mt) ≡ 1c(po,t, mt) ∙ [g (po,t)]ε (pd + l´

Leisure is the difference between the time endowment and labor input. Marginal cost is
∂ u(ct,lt)∕∂lt

ψ = wt = a u(e,l,)/ae = χct = ψ(m*' p°j)-

Another variable of interest is the gross inflation rate, Pt+1 /Pt It is determined by current
and future
p0 :

Pt+1                       g(p0,t+1)

— = π(p0f,p0f+1) ≡ ^pɪp0,t.                 <15)

This follows directly from writing the inflation rate as a a ratio of normalized variables:

Pt+1 = Pt+1/P0,t P0,t = g(P0,t+1)

Pt = Pt∕Pι,t Pι,t = g(po,t) p0,t

In a steady state, there is thus a simple relationship between inflation and the relative
price,
π = p0 .

11



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