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

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),
P1,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 m_t :

This follows from the money demand equation:

Mt = Mt Pι,t = mt
Pt Pι,t Pt g(po,t) ∙

Further, since n_t = [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) ≡ ¹ ∙ c(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 = ^χc^t = ^ψ(^m*' ^p°^j)-

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

Pt+1                       g(p0,t+1)

— = ^π(p⁰_f,^p0_f+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) p^0,t∙

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

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