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

1,1       T+T^R,1         1,ne         T+T^R,ne 0

market income yt := [y_t^, , . . .y_t      ^, , . . . ,y_t^, , . . . ,y_t      ^, ]⁰,

• the rental rate of capital r_t , the real wage w_t , the aggregate capital stock K_t ,
effective aggregate labor input Nt , the beginning-of-period stock of real money
balances m_t, aggregate transfers Tr_t, and aggregate profits Ω_t. Thus, u_t is a
vector of ne[2(T + T^R) + T] + 7 elements.

We seek a representation of our model in the form

where the hat denotes percentage deviations from the non-stochastic steady state value
of a variable.

We first derive the set of equations (26a). The log-linearized Euler equations (6) are²⁰

ʌ      ∙                                                                        ∙                                                                i-            ∙                 -I

ʌtj = (γ(¹ - σ) - ¹) cs^{,j +} (¹ - γ)(¹ - σ) £ms^{,j - π}t] ^,

s= 1,...,T+T^R, j = 1,...,ne.,

m 1^,j = 0 ∀j = 1,..., ne.                                                                 (27)

The log-linearized Euler equations (9) are:

s = 1,2,... ,T, j = 1,2,... ,ne,                   (28)

where τ⁰ and τ⁰⁰ denote the first and second derivative of the tax function evaluated at
y^s,j , respectively. The ne(T + T^R) definitions of market income yield:

0 = У 1^,jУt^,j - we(1, j)n 1 ^,jn^{1 ,}j - we(1, j)n^{1 ,j}w^_t,

j = 1,...,ne,

rk^s,jkS^,j = y^s,jyS^,j - we(s,j)n^s,jns, - we(s,j)n^s,jw^_t - rk^s,jr_t,

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

rks,^jкS^,j = ys,^jyS^,j - rks,^j^t,

s=T+1,...,T+T^R,

j = 1, . . . ,ne.                                                                               (29)

²⁰We will use the ne equations for generation T + TR later to eliminate ^{^}T⁺T ^,j, which is a control
rather than a costate variable.

28

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