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 -σ + η 0 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 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.