Hu = (1 - λt)
t (1 + rt )Ω t
-ptu)Htu+j1+ptuHte+j1)+wtu.
(16)
The capitalised future pensions of workers Sw also enter the consumption rule. To simplify the
analysis we assume that pensions of workers and unemployed are identical. This does not seem to
be too strong an assumption since we assume that each member of the labour force has identical
characteristics they will spend equal proportions of their working life in unemployment. The
expression for the present value of future pensions is given by
Stwj =
1
λwε1σ
t εt+1
(1 + rt+1)Ω
_SJ + (1 - λw ) S
St+1 + Z1 ∖ z'ʌ St+1 .
t+1 (1 + rt+1 )Ωt+1
(17)
Notice, the marginal elasticity of substitution between consumption during working life and during
retirement enters this expression, because workers value consumption in the two states differently.
Aggregation across workers, unemployed and retirees is straightforward and yields the following
aggregate consumption rule
Ct =πt[(1-st)At+Ht+εt(stAt+St)] (18)
where s is the share of total assets held by retirees and H is the present discounted value of net
wages and unemployment benefits.
Notice also, like in the standard finite horizon model the dynamics of aggregate consumption
differs from the dynamics individual consumption. While the change of individual consumption is
only a function of the difference between the real rate of interest and the rate of time preference,
the aggregate rate of time preference becomes a positive function of financial wealth because
households belonging to different age cohorts have different financial wealth positions. This
implies that in the steady state the rate of interest will exceed the rate of time preference, because
high interest rates are needed in order to induce newly created worker households to save.
The life cycle feature also has consequences for the dynamics of assets and interest rates in open
economy models. With infinitely lived consumers the steady state requires the same rate of time
preference in all regions. In contrast with the life cycle model of consumption the effective rate of
time preference becomes a positive function of financial wealth, i. e. an endogenous distribution of
wealth will be generated in steady state equilibrium which equalises the effective rate of time
preference across regions. In other words those regions with above average rates of time
preference will, everything else equal, end up with a lower asset stock.
It is also instructive to compare consumption in this model to the standard infinite consumption
model. A first difference applies to permanent income. Permanent income consists of discounted
labour income throughout the expected working life as well as the expected present value of
pension income. Adjusting the discount factors of H and S with the respective probabilities of
staying in the two states takes care of the finite durations of work and retirement. Thus there is
saving for retirement.
There is another important feature of the utility function which can potentially work in the other
direction. Workers value the marginal utility of a unit of consumption differently between their
working life and retirement. This valuation depends crucially on the intertemporal elasticity of
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