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 zt 1),
the following equilibrium conditions hold:

1. π = θ

2. x = ɪ.
€— 1

3. r = 1 αKα-1N1 - δ
x

4. w = 1 (1 - α)KαN.

5. Ω = (1 - 1 ¢ KαN1 .

seigntSeignt Q     Pne P T+T R μ (j ) M Pne μ (j ) M1 j

. selgn     Pt     (     ) 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 + TR - 1)] variables with given initial conditions:
the capital and cash holdings of generations
s = 2, 3, . . . , T + TR.19 We summarize
these in the vectors
ktj := [kt2,j, kt3,j, . . . , ktT+TR,j]0, and mtj := [mt2,j, mt3,j, . . . , mtT +T R,j]0,
mts,j := Mts,j/Pt-1, j = 1, 2, . . . , ne. In addition, there are ne(T + TR - 1) + 2 variables
that are also predetermined at time
t. These variables are the ne(T +TR - 1) Lagrange
multipliers
λtj := [λt1,j, λt2,j, . . . , λtT+TR-1,j]0, j = 1, 2, . . . , ne, the inflation factor πt and
marginal costs
gt . The initial values of these variables must be chosen so that the
transversality conditions hold. For given vectors
xt := [kt2, mt2, kt3, mt3, . . . , ktne , mtne ]0
λt := [λt1 , λt2, . . . , λtne , 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 := [ct , . . . ct        , . . . , ct , . . . , ct          ]0,

1,1    T,1 1,ne T,ne 0

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

19Since we assume that the cash transfer to the newborn, Mt1j /Pt-1 remain unchanged, we can
ignore these additional
ne state variables.

27



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