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

2.1 Households

Households live T + T^R = 240 periods (corresponding to 60 years). Each generation is
of measure 1/(T +T^R). The first T =160 (=40 years) periods, they are working, the last
T^R = 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 M_t^s,j /P_t and capital k_t^s,j in period t. He maximizes expected life-time utility
at age 1 in period t with regard to consumption c_t^s,j , labor supply n_t^s,j , and next-period
s, s,

money balances M_t₊₁ , and real capital k_t₊₁ :

T+T^R

Et β^s-¹u ct^s+^,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 — n_t^ is assumed to be:

∣^^c,P^,¹^{— n})

γ^ln c ^{+ (1}^— γ)^ln M ⁺ η0⁽¹ 1-)η ^-C^γjΓ^ +_rl (1 -n)1 -η

1 -σ ⁺^η⁰ 1 -η

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 k_t¹^,j = 0, j ∈ {1, . . . , ne}, but receives an initial cash endowment from
the government M_t¹^j that is fixed in terms of the beginning of period price level Pt-1
and equal for the different productivity types, i.e., M_t¹^j /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 s}jʌ ( ^s⁺¹³ M^s3ʌ I P j

Pt k_t₊₁ — (1 — δ)k_t + M_t₊₁ — M_t + Ptc_t

Pt yt^,3

Pt-1 π

(3)

= Ptrtkt³ + Ptwte ( s,j ) n^s_t^,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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