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₊⁺₁¹^,j , and next-period money m_t^s₊⁺₁¹^,j at age s =
1, . . . , T + T ^R - 1 in period t are as follows:

s,j
t

_M_ts,j

uc Γ , 1

nj = γ ( cS^j )^γ (1^σ) ) -¹ f

s_j (1-γ)(1-σ )

ms,j

πt

s,j
t

βEt

βE

ΛS+11^j (1 - ^δ + r,+1 (1 - τ0 y+1^j—±¹) ^ππt±ι))

si S+1+1 ,j ^m+ ^,j 1 n^s +1 ,jʌ

λ^s+1 ,j ^uM∕P ^ct +1 , P_t₊₁ , ¹^-ⁿt+1 J

^πt+1 ^πt+1

= βEt

∖++1 ,j ∩ √∣ ⁽_r^s+¹^,jA^γ⁽¹^σ) ⁽_ms+¹^,j⁾⁽¹^-^γ⁾⁽¹ ~^σ)^-¹^λt +1 + ⁽¹^-^γ) k^ct+1 ) m++1 )____________

^πt+1 ^πt+1

(6)

(7)

(8)

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

λ_t^s,jwte(s, j)

¹^-^τ0 (ys^,j—´ — i = un ^c⅛, M^-^,¹^-ⁿ^j = η0 (¹^-ⁿs^,j ) ^η ∙
π π P_t

(9)

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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