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



2.1 Households

Households live T + TR = 240 periods (corresponding to 60 years). Each generation is
of measure 1
/(T +TR). The first T =160 (=40 years) periods, they are working, the last
TR = 80 periods (=20 years), they are retired and receive pensions. Agents are also
heterogeneous with regard to their productivity level
e(s, j). The productivity e(s, j)
depends on the age
s and the idiosyncratic productivity type j {1, . . . , ne}. Individ-
ual productivity is non-stochastic, and an individual will not change his productivity
type
j over his life time. The mass of type-j agents in each generations is constant and
denoted by
μ (j ). The s-year old household with productivity type j holds real money
holdings
Mts,j /Pt and capital kts,j in period t. He maximizes expected life-time utility
at age 1 in period
t with regard to consumption cts,j , labor supply nts,j , and next-period
s,            s,

money balances Mt+1 , and real capital kt+1 :

T+TR

Et       βs-1u cts+,js-1

s=1


Mj 1 1

Pt+s-1


s,j
t
+s-1


(1)


where β is a discount factor and expectations are conditioned on the information set
of the household at time
t. Instantaneous utility u (ct, Mt, 1 — nt^ is assumed to be:

M

^c,P, 1 — n)


γln c + (1 γ)ln M + η0(1 1-)η -
CγjΓ^ +rl (1 -n)1 -η

1      + η 0  1


if σ = 1

if σ 6= 1,


(2)


where σ > 0 denotes the coefficient of relative risk aversion.3 The agent is born
without capital
kt1,j = 0, j {1, . . . , ne}, but receives an initial cash endowment from
the government
Mt1j that is fixed in terms of the beginning of period price level Pt-1
and equal for the different productivity types, i.e., Mt1j /Pt-1 =: m1 > 0 for all t and j .

The s-year old working agent with productivity type j faces the following nominal
budget constraint in period
t:

P (s++1 3           sjʌ   ( s +1 3   Ms3ʌ I P j

Pt kt+1   (1 — δ)kt   + Mt+1  — Mt   + Ptct

Pt yt,3

Pt-1 π


(3)


= Ptrtkt3 + Ptwte ( s,j ) nst,3 + Pttrt + Pt Ω t — PtT

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

3We follow Castaneda, Dlaz-Giminenez, and Rlos-Rull (2004) in our choice of the functional form for
the utility from leisure. In particular, this additive functional form implies a relatively low variability
of working hours across individuals that is in good accordance with empirical evidence.



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