What Is the Natural Weight of the Current Old?

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Theoretical Economics Letters, 2013, 3, 257-261

http://dx.doi.org/10.4236/tel.2013.35043 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

Email: #[email protected], da[email protected]

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.

Copyright © 2013 SciRes. TEL

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.

Copyright © 2013 SciRes. TEL

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-

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

are around 10%. When, and if, there is an exter-

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

, 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.

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