Business Cycle Dynamics of a New Keynesian Overlapping Generations Model with Progressive Income Taxation

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 z_t being gov-
erned by the following AR(1) process:

ln zt = ρz ln zt-1 + εzt,                                                                    (14)

where ε_zt is i.i.d., ε_zt ~ 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 = gtαztKt^α-1Nt^1-α,                                                              (15)

wt = gt(1 - α)ztKt^αNt^-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 + βE_t {πt+ι} ∙
with κ ≡ (1 — φ)(1 — βφ)/φ > 0 and π_t is the percent deviation of the gross inflation
rate from its non-stochastic steady state level π.⁴

⁴A detailed derivation of this relation can be found in Herr and Mauβner (2005), Section A.4.

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