Factors shaping the zero-bequest Su function are, however, intellectually challenging but realistically rather irrelevant niceties within the basically stationary
framework.
3. Relaxing the stationary assumption
It is a fact of life that the life-paths of successive cohorts are different with respect to cohort size, longevity, booms or recessions at various periods of life, and
several other factors. Consequently it is impossible for the age-specific, cross-sec- tional shares in income and consumption to remain constant over time. The shares
w
k
and c
k
in the descriptive framework must be substituted by w
x,t
and c
x,t
, respectively, indicating the shares in year t of the cohort born in x. As opposed to
Eq. 1, the starting point of accounting thus becomes w
ˆ
x,t
= a
t
w
x,t
cˆ
x,t
= a
t
1 − sc
x,t
11 with a crucial difference that has far-reaching implications:
x
w
x,t
= 1
and
x
c
x,t
= 1
but
t
w
x,t
1 and
t
c
x,t
1 12
i.e. cross-sectionally the shares still add up to 1 that’s why they are shares but there is no reason why longitudinally, over the life-path the sum of shares
‘collected’ by a cohort should be exactly unity. The Golden Rule congruence has been impaired: even at u = 1 longitudinal proportions are not identical with
cross-sectional ones.
The first concept to disappear with the stationary assumption is that of the representative individual. If the life-paths of successive cohorts are different they
cannot be expected to have identical preferences. Even if preferences did not — miraculously — change with time, the actual parameters of earning and consump-
tion would still be varying from cohort to cohort. Thereby the unified longitudinal utility maximization becomes conceptually infeasible.
It still remains possible, within the framework built on cross-sectional shares, to assert the zero-bequest function of the cohort born in x:
S
x
u =
k
c
x,x + k
− w
x,x + k
u
− k
k
c
x,x + k
u
− k
13 Because of Eq. 12, however, the convenient assurance of a trivial root at u = 1
is gone since the Golden Rule congruence is lost. The only case, when the existence of a unique root is guaranteed — not necessarily at u = 1 — is the simplest, solid
two-period case, whether ‘young’ or ‘old’. In the three-period case S
x
u has at most two roots — very inconveniently, however, it may have no roots at all. In
other words, it may be impossible at any relative interest factor u to balance the life-path of the cohort born in x by zero aggregate saving.
What is worse, S
x
u is cohort-specific. It would balance the life-cycle of cohort born in x — whether by zero or non-zero aggregate saving — but it would
certainly not, simultaneously, do the same for other cohorts, born before or after x. Yet there cannot exist more than a single aggregate saving rate in any one period
of time. At this point, it is not only the formal analysis, it is the conceptual framework that collapses: the aggregate saving rate ceases to be an appropriate
device for achieving zero bequest, for balancing the longitudinal life-paths.
Conclusion: the concept of the balanced longitudinal path itself requires a fresh look. It may appear far-fetched, but the truth is that a seemingly simple, technical
modification, the dismissal of the stationary assumption, results in the necessity of profound, conceptual rethinking. Bequest and aggregate saving must be substituted
by appropriate concepts that reflect the case of the non-stationary population in reality.
4. Issues for pension economics
