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

(35)

For generations s = 2, . . . , T the log-linearized budget equations (5) are:

k^s+1 ,j^_ts+11^,j + θm^s+1 ,^jmt+1^,j - (1 - δ) k^sj fcs^,j - ms,^jm^ss^j + (τ0y^sj + m^sj )πt

= (1 — τ⁰ ) y^s,j y s^,j + trtr_t + ΩΩ _t — c^s,j c ^s_t^,j,

j = 1, . . . , ne, s = 2, . . . ,T.                                                                          (36)

For generations s = T + 1, . . . , T + T ^R - 1 they are:

k^{s+1 ,j}k S .^l∣^{l ,j} + πm^s+1^,j m t+1^,j — (1 — δ ) k^s,jk s^,j — m^s,j m S^,j + ( τ ⁰y^s,j + m^s,j + pens ) π _t
= (1 — τ⁰ ) y^s,j y s^,j + trtr_t + ΩΩ _t — c^s,j ^ s^,j,

j = 1, . . . , ne, s = 2, . . . ,T.

(37)

The Euler equations (7) and (8) yield:

^s+1^,j     ^s^,j    β ^λ + ,j / 0      00 s+1 ,j∖

^λt +1 ^{— λ}t ^{— β}r λs,j ^{τ + τ y     π}t +1

_λs+1,j                         _λs+1,j

= ^β''_λsj τ ⁰0y^s+1 ,j^y t^{+ ,}j ^{— βl}' _λs,j^{(1 —} τ 0 )^r t+1^{,                                       (38)}

_{β λ}s+1,j                               _{β λ}s+1,j

∏-^λt+'1 ^j — ^λsj — I¹ + (¹ — ^β-P — Y)(¹ — σ) — ɪ]j ^πt+1

+ [(1 — γ)(1 — σ) — 1] (1 — β^λ+^j) rmS÷₁¹'^j
βλ^{s +1j}   s+_u

= ^—Y <¹ — ^σ ψ ^— ∏-C⁺^^j ■

j = 1, . . . , ne, s = 1, . . . ,T + T^R — 1.                                                     (39)

Note that in the equations for s = T + TR — 1 we must replace ΛT+T ^,j by the right
hand side of (27) for t + 1 and s = T + T^R. The remaining two equations are given by
the New Keynesian Phillips curve equation (17),

,ɔʌ ʌ I ^{(1 — φ )(1 — βφ} )

^βπ t+1 ^{— π} t +         g-------g t = 0^{,                                              (40)}

φ

and the log-linearized definition of the aggregate beginning-of-period real stock of
money m_t := M_t/P_t-1. Together with equation (18) this definition implies:

m t+1 — m t + ∏ t = θ t.                                                         (41)

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