Download full, postscript version of this paper.

Overlapping Generations Models with Realistic Demography: Static and
Dynamics.

by Antoine Bommier and Ronald D. Lee

Conclusion

The extensive literature on overlapping generation models is rich and
productive, yet it suffers from its reliance on simplistic demographic
assumptions which are largely unnecessary. The past literature has
mainly assumed only two age groups and perfect survival until the end
of the second of these. Theoretical results for two age groups
sometimes do not generalize, and without mortality one cannot study
the implications of its change. Such a crude model cannot even
simultaneously accommodate dependent childhood, productive mid years,
and retirement. Any kind of empirical implementation of these models
is virtually impossible.

This paper adopted a continuous age distribution with an arbitrary age
schedule of mortality. The core economic model was standard. The new
concept of reallocation system is sufficiently broad to include the
real world variety of individual and institutional mechanisms, ranging
from real capital formation through credit transactions to non-market
transfers through the family or the government. A simple two-by-two
categorization of these according to whether they are competitive
and/or conservative captures their analytically critical features. A
notation based on this categorization facilitates compact expressions.

Life cycle age profiles of planned survival-weighted earning and
consumption, in conjunction with the age distribution of the
population, give rise to an aggregate demand for wealth. If this
demand is exactly satisfied by aggregate holdings of capital, the
economy is said to be balanced. More generally, however, the demand
for wealth differs from the size of the capital stock, and wealth is
additionally held in some other form--typically as transfer
wealth. The difference between wealth and capital per head is called
the balance of the economy.

We have derived a considerable number of theoretical results in this
paper. Some of these extended the findings of other studies to a more
general demographic context or to economies with capital; some provide
simpler proofs; and some established new results. We began with a very
general expression for the evolution of wealth in non-steady state
economies and non-stable populations. We then considered closed market
economies. The previous result, applied to a market economy with a
stable population, led to a simple expression for the evolution of the
balance of the economy. This in turn implied that all steady state
market economies must be either golden rule or balanced. We also
showed that with rational agents with additive homothetic preferences
and some conditions on the production function, there will always
exist one balanced equilibrium and one golden rule equilibrium. Next
we established some welfare results for closed market economies. A
golden rule steady state was shown to be Pareto Optimal. When the
balance is positive, which occurs when the average age of consuming
exceeds the average age of earning, then a steady state with a higher
rate of population growth will permit a higher level of welfare in a
comparative static sense, and when the balance is negative, then a
steady state with more rapid population growth would be beneficial. A
balanced steady state is Pareto Efficient if "r" exceeds "n", but
inefficient if "r" is less than "n". Finally we derive some dynamic
results for closed market economies, showing that a balanced economy
cannot become unbalanced, and that in an unbalanced economy, the sign
of the balance cannot change. A balanced economy with "r" greater than
"n" is not stable, despite being Pareto Efficient. This set of results
achieves a synthesis, generalization and extension of a substantial
literature in this area.

In these hypothetical closed market economies, it is not entirely
clear how it is possible for unbalanced economies to occur and be
sustained by market institutions. In the real world, non-market
interage transfers are pervasive, and lead to positive or negative
transfer wealth, and thereby readily create and support unbalanced
economies. Unfunded public sector pension programs, or familial
support by adults of their elderly parents, for example, create
enormous positive transfer wealth which sustains positive balance in
the economy. Intended or unintended bequests, and publicly funded
education, create very substantial negative transfer wealth, tending
to create a negative balance (see Lee, 1994a and b). We considered
economies with transfers of this general sort, and showed how earlier
results generalized.  A steady state economy with transfers must
either be golden rule or have a balance related in a specific way to
the age profile of transfers. Results on the optimality, efficiency
and stability of the different kinds of steady-state were extended
straightforwardly, as well as the results on the dynamic.  Further
analysis led to a perturbation expansion about the golden rule case
which provided a convenient basis for evaluating the balance in the
golden rule case as well as outside of it. In golden rule, the balance
is given by the average per capita inflow of transfers times the
average age of receiving a transfer, minus the corresponding product
for the outflow of transfers, a result which holds for open economies
as well as closed ones. In closed economies, the inflows and outflows
of transfers must be equal, so the sign of the balance of the economy
is positive when the population-weighted average age of receiving
transfers exceeds that of making them, as appears to be the case in
the U.S. and some other industrial nations (Lee, 1994a and b).

Economists should not be put off by the apparent complexity of
realistic demographic models, models which in principle should permit
a much greater degree of generality and relevance to real world
phenomena and policy problems. The same model that has been
empirically implemented elsewhere to study the consequences of
population aging has here been related to a deeper theoretical
literature. We have shown that such models remain tractable, and that
comparative static, dynamic and welfare theoretic results can be
obtained.