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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