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

6 Appendix

6.1 Non-stochastic steady state of the OLG model

In the stationary state state of the OLG model (constant money growth θ and z_t ≡ 1),
the following equilibrium conditions hold:

1. π = θ

2. x = ɪ.
€— 1

3. r = ¹ αK^α-1N^{1 -α} - δ
x

4. w = ¹ (1 - α)K^αN^-α.

5. Ω = (1 - ^{1 ¢} K^αN^{1 -α}.

seignt — Seignt Q     P^ne P T+T R μ (^j ) M ∣ P^ne μ (^j ) M^{1 j}

. ^sel^gn     P_t     (     ) 2-^j=1 2-^s=2 T+TR P + j=j=1 T+TR P '

6.2 The log-linear OLG model

In our model there are ne[2(T + T^R - 1)] variables with given initial conditions:
the capital and cash holdings of generations s = 2, 3, . . . , T + T^R.¹⁹ We summarize
these in the vectors k_t^j := [k_t^2,j, k_t^3,j, . . . , k_t^T+TR,j]⁰, and m_t^j := [m_t^2,j, m_t^3,j, . . . , m_t^{T +T R,j}]⁰,
m_t^s,j := M_t^s,j/P_t-1, j = 1, 2, . . . , ne. In addition, there are ne(T + T^R - 1) + 2 variables
that are also predetermined at time t. These variables are the ne(T +T^R - 1) Lagrange
multipliers λ_t^j := [λ_t^1,j, λ_t^2,j, . . . , λ_t^T+TR-1,j]⁰, j = 1, 2, . . . , ne, the inflation factor πt and
marginal costs g_t . The initial values of these variables must be chosen so that the
transversality conditions hold. For given vectors x_t := [k_t², m_t², k_t³, m_t³, . . . , k_t^ne , m_t^ne ]⁰
λt := [λ_t¹ , λ_t², . . . , λ_t^ne , πt, gt]⁰ the model’s equations determine the vector ut. The ele-
ments of this vector are

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

• consumption ct := [c_t , . . . c_t        , . . . , c_t , . . . , c_t          ]⁰,

1,1    T,1 1,ne T,ne 0

• working hours nt := [n_t , . . . n_t , . . . , n_t , . . . , n_t ]⁰,

¹⁹Since we assume that the cash transfer to the newborn, M_t^1j /Pt-1 remain unchanged, we can
ignore these additional ne state variables.

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