Proceedings from the ECFIN Workshop "The budgetary implications of structural reforms" - Brussels, 2 December 2005

income risk. Therefore it seems reasonable to mitigate the impact of the stochastic income
variation of individuals between work and retirement by assuming risk neutrality. Children are
assumed not to make any economic decisions. Extending population dynamics to include children
is nevertheless useful since it allows to take care of different birthrates and initial youth
dependency ratios.

The decision problem of retiree and workers can now be formulated by postulating value functions
V ^r and V ^w over consumption for retirees (r) and workers (w) respectively as follows

V,¹   C + β-E, (V,+,∣x) } ^ρ                                              (3)

with z=(r, w). The value function is non-linear in the two arguments. However, instead of
complicating the analysis it is exactly this type of non-linearity of the value function which
generates risk neutrality with respect to stochastic income and which allows the derivation of
closed form decision rules which are functions of first moments of income only.

Consumption of retirees:

Each retiree j consumes out of pension income (w_t^r ) and financial wealth A_t^rj . Because of the life
insurance contract pensioners receive a premium on top of the interest rate which is equal to the
probability of death. Thus the budget constraint is given by

At^rj =(,+rt+λt^r)At^r-^j,+wt^r-Ct^rj                                                     (4)

Maximising (,0) subject to (20) yields the consumption Euler equation

Ct^r+^j, =(,+rt)βCt^rj.                                                                      (5)

From this first order condition together with the intertemporal budget constraint the following
decision rule for retiree consumption can be derived

Ct^rj = εtπt[At^rj +St]                                                                     (6)

where the marginal propensity to consume out of wealth is given by the following difference
equation

ε,π, = 1 -((1 + r )^σ-1 β (1 - χ )) ^П
εt+,πt+,

and the present value of retirement income is given by

Consumption of the population of working age:

Each member of the labour force J can either be employed or unemployed. Employed workers
receive net income w_t^e = (1 - t_t - ssc_t)w_t , the unemployed receive unemployment benefits w_t^u . An

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