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:

ks+1 ,j^ts+11,j + θms+1 ,jmt+1,j - (1 - δ) ksj fcs,j - ms,jmssj + (τ0ysj + msj )πt

= (1 — τ0 ) ys,j y s,j + trtrt + ΩΩ t — cs,j c st,j,

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

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

ks+1 ,jk S .ll ,j + πms+1,j m t+1,j (1 — δ ) ks,jk s,j — ms,j m S,j + ( τ 0ys,j + ms,j + pens ) π t
= (1 — τ0 ) ys,j y s,j + trtrt + ΩΩ t — cs,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 λt βr λs,j τ + τ y     πt +1

λs+1,j                         λs+1,j

= β''λsj τ 00ys+1 ,jy t+ ,j βl' λs,j(1 τ 0 )r t+1,                                       (38)

β λs+1,j                               β λs+1,j

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

+ [(1 — γ)(1 — σ) 1] (1 — βλ+j) rmS÷11'j
βλs +1j   s+u

= Y <1 σ ψ ∏-C+^j

j = 1, . . . , ne, s = 1, . . . ,T + TR 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 + TR. 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
mt := Mt/Pt-1. Together with equation (18) this definition implies:

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

30



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