Theoretical Economics Letters, 2013, 3, 257-261
http://dx.doi.org/10.4236/tel.[phone] Published Online October 2013 (http://www.scirp.org/journal/tel)
What Is the Natural Weight of the Current Old?*
Damien Gaumont1#, Daniel Leonard2
1Pantheon-Assas, Sorbonne Universities, ERMES EAC 7181 CNRS, CRED, Institute for Labor Studies
and Public Policies, Paris, France
2Flinders Business School, Flinders University, Adelaide, Australia
Received December 13, 2012; revised February 3, 2013; accepted February 20, 2013
Copyright © 2013 Damien Gaumont, Daniel Leonard. This is an open access article distributed under the Creative Commons Attri-
bution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
ABSTRACT
We consider a simple overlapping generations model with an externality à la Arrow-Romer [1,2] and a government
with fiscal powers. If it wishes to maximize a criterion depending on the lifelong utility of agents, is there a natural
weight for the utility of the current old? We show in a simple example that this weight depends on the specific features
of the model, in particular the length of the horizon, and cannot be chosen arbitrarily. Our result has a neat economic
interpretation [2].
Keywords: Overlapping Generations Model; Learning-by-Doing; Social Welfare
1. Introduction where t and t refer to the consumption of young
and old in period t respectively;
c d

is their subjective
rate of time preference and γ is the planner’s social dis-
count rate.
The overlapping generations model of Allais [3], Sam-
uelson [4,5] and Diamond [6] is ideally suited to the ex-
ploration of inter-generational issues. For a more recent
and thorough presentation of these models, see the clas-
sic reference: De la Croix and Michel [7].
Economics is a moral science. Welfare economics
should be a central part of the discipline (Atkinson [8], p.
192). Since there is usually more than one adult genera-
tion at any on time, one may reasonably ask—whose
welfare function is it? Are we saying to 50-year-old that
their welfare is judged by their 75-year-old parents? Or
the reverse? If the reverse, when does the baton pass?
The uneasiness surrounding this construction is apparent
when we consider the issue of the rate at which future
utility is discounted1 (see Atkinson [7], p. 195). Bern-
heim [9] also mentions that many of the usual concepts
of welfare are difficult to implement, since the concerned
individuals do not necessarily have the opportunity to
vote for them.
Most models assume that agents live for two periods;
the training of the young may or may not be explicitly
modeled. Steady states, temporary equilibria and inter-
temporal equilibria are studied and a role for government
intervention appears naturally if market imperfections
such as externalities are present. Public finance issues
can also be considered.
As it is assumed that agents live for two periods, an
objective measure of time is inherent to the model. Thus
the length of one period can be taken to be around 30
years. This observation has deep consequences for the
interpretation of policy recommendations. Here, our argument is that if government intervention
is warranted, the proposed policies should be acceptable
to people who are alive at the time. Policies that optimize
over the very long run but lower the welfare of the cur-
rent generations—compared with the status quo of no
intervention at all—have little chance of being adopted.
Hu [10] (p. 283) was well aware of this point. To sharpen
The traditional approach of the literature on overlap-
ping generations models has been to consider the welfare
of all generations from the present onward (De La Croix
and Michel [7], pp. 91-93). In such models, and with a
separable utility function, the planner’s criterion is
 
0
,
t
tt
t
Wucud












1Note that Atkinson [8] is talking here about the discount rate applied
to utility, not to the rate at which future consumption is discounted,
which takes account of differences in how well-off future generations
will be.
*We are grateful to Russell Davidson and participants of the economics
seminar at McGill University.
#Corresponding author.
C
opyright © 2013 SciRes. TEL

D. GAUMONT, D. LEONARD 258
our argument, we use the simplest criterion that reflects
this notion, namely the lifelong utility of people alive in
period 02. (The welfare function can be made to include
more generations as we discuss later.) In the usual nota-
tion, it is
  
100
Wuducud



1
,
where

and

are exogenous parameters.
The purpose of this note is to show that, when a spe-
cific horizon is adopted, there is a natural endogenous
value for

 (which we denote by

for simplicity
of notation) that is consistent with efficient government
intervention and cannot be arbitrary, contrary to the tra-
ditional approach of letting both

and

be exoge-
nous. Other horizons produce different specific results
but the main conclusion remains the same:

cannot be
chosen arbitrarily.
We begin with a standard Diamond-like model with an
externality of the learning-by-doing type (Arrow [1],
Sheshinski [11] and Romer [2]). The external effect is
that the aggregate stock of capital has a beneficial effect
on the efficiency of production by each firm. It can be
internalized through government intervention in the form
of fiscal policies that subsidize capital investment and are
financed by a tax on labor.
2. The Individuals
Individuals live for two periods: in the first period they
consume, save and inelastically supply one unit of labor.
In the second period they live off the revenue from their
savings. There is no population growth, ,
t
LLt

The
consumptions of young and old in period t are, respec-
tively, t and ; ct
dt
s
is savings; t

is the rate of tax
on wage income t; t
is the net wage rate and
t is the rent of capital. The individual’s subjective dis-
count factor is
w

1t w  
R
1

. An individual born in period t
solves the following program:
 
1
,1
max tt
cd tt
uc ud 




1
tt t
cs w  t
t
s
11tt
dR


The optimality condition is



1. t
duc u 

 t
(1)
3. The Firm
The production function exhibits constant returns to scale
to the factors hired by the firm but there is an externality.
The firm hires units of labor and produces t units
of good with t units of capital;
t
l
k
q


t
B
K
is a produc-
tivity factor that represents the externality where t
K
is
the aggregate stock of capital; however the firm is un-
aware of the structure of this productivity factor.

 
,,
ttt
BFqKklt
where


.,.F is homogeneous of degree one; therefore


,1 .
tt ttt
BFql Kkl
Normalizing the size of the firm at , we have, in
the usual notation:
1
t
l





,
ttt
tt
Bf
B
qK
L
k
kfk

 (2)
The firm maximizes profit





π
,
ttt
ttt
kkk
w
BL
Rk
f


 t
where t

is a government subsidy designed to internal-
ize the externality. Therefore




,
ttt t
BLRkf 
  k (3)


.
ttt t
BLf kkwfkt k




 (4)
An informed government that wishes to internalize the
externality in each period t would choose the efficient
subsidy as




,
tt
LB Lkkf 

t
in order to account for the role capital plays in enhancing
overall productivity.
Capital depreciates entirely in one period, hence the
dynamics of the economy are given by
1. tt
ks

4. The Government
The government is responsible for implementing the
planner’s policies and has fiscal authority. The govern-
ment balances its budget in each period, thus the con-
straint on its fiscal policy


,
tt

 is,
,
tt tt
wk



(5)
where t

is the rate of payroll tax and t

is the sub-
sidy to capital. Therefore we shall be able to express one
of the tax/subsidy parameters in terms of the other. To-
gether the wage rate equation, the efficient subsidy and
the budget constraint in period t yield an expression for
the payroll tax t

.





,
tt tttt
t
ttt
tt
BLf kfkk
LBLf kkk
wk k



 




2The utility of the young born in period -1 (hence those who are old in
p
eriod 0) can be included, but treated as exogenous, therefore it is
irrelevant. See for instance De la Croix and Michel [7], p. 91.

D. GAUMONT, D. LEONARD 259

 

  
1.
1
ttt
t
tt tt
tt
ttt
LBLkf kk
BLkf kfkk
KB K
BKfkkf k



 



t

(6)
This expression for t

can be interpreted as follows.
It is the ratio of the elasticity of the externality factor
with respect to aggregate capital over 1 minus the elas-
ticity of the firm’s output with respect to its capital stock.
Clearly, with a general CRS production-function and an
arbitrary externality effect, this ratio depends on current
capital stock and varies over time. The dynamic path of
t

is nonetheless constrained.
At this stage we must emphasize the following point in
order to show the essential nature of our argument: The
logic consequence of three elements, a competitive wage,
an efficient capital subsidy and a balanced budget in
every period, by itself determines the dynamic structure
of fiscal policies. This is done without any reference to
the planner’s objective.
In this extensive literature (Samuelson, [4,5], Lerner,
[12,13], De La Croix and Michel, [7]) the criterion is
often an infinite horizon welfare function or sometimes,
more simply, the steady state outcome. These views have
the advantage of supplying clear answers to real prob-
lems in an abstract world. However, steady state criteria
and infinite horizon optimal paths suffer from an imple-
mentation problem, namely that policies designed to
maximize such criteria may entail losses of utility for
several generations, including those alive at the time of
planning (See Gaumont& Leonard, [14], for some com-
pelling evidence). We argue that such policies stand little
chance of being implemented and therefore we look for a
more feasible criterion.
Here we assume that the government objective is sim-
ply to maximize the lifelong utility of people who are
alive at the time (see footnote 2). Therefore, it takes into
account the utility of the old and young in period zero,
plus the utility of those who will be old in period one. In
the traditional formulation the exogenous weight of the
old is

 , the ratio of the subjective rate of time
preference of households and the planner’s social dis-
count factor. Our purpose is to show that, for the optimal
choice of t

to be consistent with an efficient fiscal
policy, the weight of the current old must take on a natu-
ral value that depends on the specific features of the
model and the length of the horizon.
In order to make our argument simple and concise we
show by counterexample that the value of

cannot be
exogenous, even in a very standard version of the model.
5. A Simple Case
We use a logarithmic utility and a Cobb-Douglas produc-
tion with


t
BK K
t

 where 01

 measures the
strength of the externality3. Therefore conditions (1), (2),
(3), and (4) become, respectively,
11
,
tt
dR
t
c



,
ttt
qLkk



1,
tt t
RLk






1.
tt
wLk





We use (5) to eliminate t

from our calculations and
obtain


00
1d0 ,Lk






 (7)

00
11,
1
cL
0
k








11
,dRc
0


 
1
10
11
1,
R
dL
0
k






with


 
11
1
00
1
11
1
R
LL







 






k

1
d
(8)
The planner’s objective is to maximize a welfare func-
tion that incorporates the utility of the current old and the
lifetime utility of the current young,
00
ln lnlnWdc


 (9)
where

is, for the time being, treated as exogenous.
For the chosen production function the equation that
dictates the dynamicstructure of fiscal policies (6) sim-
plifies to
.
1
t






, (10)
Therefore in this specific model an efficient t

is
constant over time. This is because the two elasticities
are constant due to the Cobb-Douglas production func-
tion and the power function form of the externality factor.
This is so, irrespective of the planner’s choice of welfare
function4.
Turning now to the planner’s choice of an optimal fis-
cal policy to characterize the

value that maximizes (9)
and using the solved consumption Equations (7) and (8),
the first-order condition is:
3This is a very simplified version of Gaumont and Leonard [14], which
addresses the question of knowledge transmission among generations.
4Note that this is a convenient result as it guarantees inter-temporal
consistency, were the planner to redo this exercise in the next period
and thereafter.

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
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