Profit maximization implies the demand functions:
Yt ( j ) = PPj^ ) Yt, (11)
with the zero-profit condition
Pt =( ∕1 Pt ( j )1 -j * " ∙ (12)
Intermediate Goods Firms. The intermediate good j ∈ [0, 1] is produced with
capital Kt(j) and effective labor Nt(j) according to:
Yt(j)= ztKt(j)αNt(j)1-α∙ (13)
All intermediate producers are subject to an aggregate technology shock zt being gov-
erned by the following AR(1) process:
ln zt = ρz ln zt-1 + εzt, (14)
where εzt is i.i.d., εzt ~ N(0,σ%).
The firms choose Kt(j) and Nt(j) in order to maximize profits. In a symmetric equi-
librium profit maximization of the intermediate goods’ producers implies:
rt = gtαztKtα-1Nt1-α, (15)
wt = gt(1 - α)ztKtαNt-α, (16)
where gt denotes marginal costs.
Calvo price setting. Let φ denote the fraction of producers that keep their prices
unchanged. Profit maximization of symmetric firms leads to a condition that can be
expressed as a dynamic equation for the aggregate inflation rate:
πt = -κxt + βEt {πt+ι} ∙
with κ ≡ (1 — φ)(1 — βφ)/φ > 0 and πt is the percent deviation of the gross inflation
rate from its non-stochastic steady state level π.4
(17)
4A detailed derivation of this relation can be found in Herr and Mauβner (2005), Section A.4.
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