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

The log-linearized budget equations for generation s = T + T^R are:

c^τ⁺^τ R ,j ^ T⁺^τ R^j — (1 — τ⁰ ) y^τ⁺^τ R ,j y^τ⁺^τ Rj — trtrt — ΩΩ t

= (1 - δ) k^τ⁺^τRj ^T⁺^τR^j + m^τ⁺^τRj m^τ⁺^τRj

( τ ⁰y^τ⁺^τ Rj + pens + m^τ⁺^τ Rj ) π t,

j = 1, . . . , ne.

(30)

From the factor market equilibrium conditions (15) and (16) we obtain:

w t + αNt = αKt + g t + Zt,

^rt ⁺ (^α — 1)Nt — (^α — 1)Ki't ⁺^gt ⁺ zt.

(31)

From aggregate profits Ωt = (1 — gt)ZtNt ^αK^α, we derive

— (1 — α ) Nt = αKt + (1 — ь ) g t + Zt.

(32)

The aggregate consistency conditions (23c), (23b), and (23d), imply

ne τ+τ^R

KK = X X

j =1 s=2
ne τ
NN = XX

j =1 s=1

^μ(^j) ks,ksj
t + t R ^{k k}t ^,

^μ⁽^j) _ns,j^ ^sjn b ,
T + T^{R t}^,

ne τ+τ^R

Σ∖ ^{μ μ}(^j) sjjsjj

(33)

T-+ T ₊ T R^m^j^m t ■

j =1 s=2

Finally, the log-linearized budget constraint of the government (21) is given by:

ne τ+τ^R

X X T+T R τ ⁰y^s,j y s, ^— t^rt^rt = (¹^—^θ ) ^m7m t
j =1 s=1

ne T+T R _μ ( j ) T R

ʌ
θm θ t,

^— Σ _τ , _τR^τ⁰y^s,j⁺ (^θ^—¹)^m⁺ (^m^—^m) ^— TMTRpens ^πt

T + T^R T + T^R

j=1 s=1

(34)

where m = m — P”= ₁ (μ(j)/(T + TR))m¹^,j.

Next we derive the set of equations (26b). We begin with the log-linearized budget
equations of generation s = 1. From (5) we derive:

k 2^,j^μ 2+1 + θm 2 j m 2+1 + ( τ 0y¹^,j + m¹^,j ) ∏ t = (1 — τ⁰ ) y¹^,j ∙y 1^,j + trtrt + ΩΩ t — c¹^,j c _t¹^,j,
j = 1, . . . , ne.