D. GAUMONT, D. LEONARD
260
11 .
11
This makes clear the meaning of our result in the sim-
plest terms: a short-term horizon, coupled with an effi-
cient tax/subsidy fiscal policy, cannot use an arbitrary
exogenous weight for the old generation.
Therefore the welfare-maximizing
value is
*11 1.
11
(11)
The expressions in (12) and (13) clearly depend on the
features of the model as well as on the length of the ho-
rizon selected by the government. Possible extensions
and alternatives have been explored. Additional calcula-
tions using (12), and in the simplest case(13), shows that
the natural value of
can vary enormously. Cases when
W also includes terms such as
1
ln c and
2
ln d yield
more complicated expressions. Another type of produc-
tion function such as tttt
qA lkB
, with tt
BLk
also yields simple results. A model with a different pro-
duction function might require more complex calcula-
tions but the constraint on the dynamic structure of t
given by (6) would still need to be accommodated with
the optimal choice of t
.
This welfare-maximizing tax depends on the values of
the parameters of the problem, including the weight of
the old generation, ,
which has so far been treated as
exogenous. Given *,
all the consumption variables can
be calculated and a complete solution obtained.
Finally, combining the dynamic fiscal structure (10)
and the planner’s choice of policy (11) we find that there
is only one value of
that is consistent with both. It is
therefore endogenous to the model and depends, among
other things, on the strength of the externality,
. We
denote it by
:
1.
1
(12) 6. Conclusion
In a simple two-period overlapping generations model
with an externality (à la Arrow-Romer [1,2]), when the
government has the power to tax the wage of the young,
we have shown that the “natural” value of the weight of
the current old—the value of the weight that reconciles
the maximization of the chosen welfare function with the
use of the efficient externality-correcting fiscal policy—
is endogenous to the model and depends on the strength
of the externality as well as on the government’s chosen
criterion. It is also true when the external effects are
non-existent. It is not possible to choose both the subjec-
tive rate of time preference of households and the plan-
ner’s social discount factor arbitrarily. The choice of the
value of the weight of the current old crucially depends
on the length of the social planner’s horizon.
We insist that our argument does not depend on the
existence of an externality, although it can accommodate
it. In order to make our point sharper, we now look at the
special case of no externality when there is no need for
government intervention as (10) makes clear, and the
natural
value is
0
1.
1
(13)
The expression in (13) is always less than 1 for sensi-
ble values of
0,1 2
. There is a similar result for
(12) but
0,1 2
cannot be assumed.
Proposition: The natural weight of the current old
cannot be exogenous but depends on the specific features
of the model (including the length of the planning hori-
zon). REFERENCES
With discount rates of 1% - 2% p.a. compounded over
a generation and a capital share of around 1/3, the values
of 0
are around 10%. When, and if, there is an exter-
nality, 0
, a government that wishes to optimize (9)
and impose an efficient tax/subsidy scheme in a competi-
tive economy must choose a weight for the old of the
current generation as given by (12). This weight, and
only this weight, will insure that welfare is maximized
and that the tax/subsidy scheme is efficient.
[1] K. Arrow, “The Economic Implications of Learning-by-
Doing,” Review of Economic Studies, Vol. 29, 1962, pp.
155-173. http://dx.doi.org/10.2307/2295952
[2] P. Romer, “Increasing Returns and Long Run Growth,”
Journal of Political Economy, Vol. 94, No. 5, 1986, pp.
1002-1037. http://dx.doi.org/10.1086/261420
[3] M. Allais, “EconomieetIntérêt,” Imprimerie Nationale,
Paris, 1947.
[4] P. A. Samuelson, “An Exact Consumption-Loan Model
of Interest with or without the Social Contrivance of
Money,” Journal of Political Economy, Vol. 66, No. 6,
1958, pp. 467-482. http://dx.doi.org/10.1086/258100
When there is no externality, there is no role for gov-
ernment, tax or subsidy. Therefore, 0
—which is the
only weight consistent with no taxes—is the weight
which the central planner must give to the current old if
its plans are to coincide with the competitive outcome.
The value of
(or, equivalently
) cannot be deter-
mined exogenously.
[5] P. A. Samuelson, “Reply,” Journal of Political Economy,
Vol. 67, No. 5, 1959, pp. 518-522.
http://dx.doi.org/10.1086/258223
[6] P. Diamond, “National Debt in Neoclassical Growth
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