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

(5)

where we define ms^,j ≡ M-^.

The necessary conditions for the households with respect to consumption c_t^s,j , s =
1, . . . , T + T ^R , next-period capital k_t^s₊⁺₁^1,j , and next-period money m_t^s₊⁺₁^1,j at age s =
1, . . . , T + T ^R - 1 in period t are as follows:

The optimal labor supply of the productivity j-type workers at age s = 1, . . . , T is
given by:

^{1 - τ}0 (ys^,j—´ — i = un ^c⅛, M^-^{, 1 - n j} = η0 (^{1 - n}s^,j ) ^η ∙
π π                    P_t

2.2 Production

The description of the production sector is similar to Bernanke, Gertler, and Gilchrist (1999).
A continuum of perfectly competitive firms produce the final output using differentiated
intermediate goods distributed on [0,1]. These goods are manufactured by monopolis-
tically competitive firms. Firms in the intermediate goods’ sector set prices according
to Calvo (1983).

Final Goods Firms. The firms in the final goods sector produce the final good with
a constant returns to scale technology using the intermediate goods Yt(j), j ∈ [0, 1] as
an input:

Yt = (ʃ Yt ('j )   dj) ‘ ¹ ∙                                                         (10)

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