Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol24.Issue4.Apr2000:

Journal of Economic Dynamics & Control
24 (2000) 623}650

Surplus analysis for overlapping generationsq
Joaquim Silvestre*
Department of Economics, University of California, Davis, CA 95616, USA
Received 1 February 1998; accepted 1 October 1998

Abstract
I extend surplus analysis to overlapping generations. In the atemporal quasilinear
model we have the &global equivalence principle': e$ciency is equivalent to surplus
maximization, provided that the numeraire is unbounded from below. If it is not, then
only a &local equivalence principle' obtains, requiring large holdings of numeraire. My
extension covers "nite and in"nite time. The "nite model has one numeraire per period,
but it otherwise parallels the atemporal model. The in"nite case is more subtle: the global
equivalence principle is lost, but the local principle is recovered if numeraires are
bounded and if consumers discount the old-age numeraire. ( 2000 Elsevier Science
B.V. All rights reserved.
JEL classixcation: D61; D62; D90; H41; H43; Q20; Q30
Keywords: Overlapping generations; E$ciency; Surplus maximization; Surplus analysis;
Cost}bene"t analysis; Time discount

1. Introduction
This paper provides an extension of traditional surplus analysis to a world of
multiple, overlapping generations.

* Corresponding author. Tel.: #1-530-752-1570; fax: #1-530-752-9382.
E-mail address: jbsilvestre@ucdavis.edu (J. Silvestre)
q
A previous version circulated with the title &Quasilinear, Overlapping-Generations Economies.'
I am indebted to Andreu Mas-Colell, Klaus Nehring and an anonymous referee for useful comments. The usual caveat applies.

0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 0 2 - 0

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Traditional surplus analysis is exact in the atemporal quasilinear model,
which can be summarized as follows. Let there be N consumers, indexed

1,2, N. A nonproduced, private, transferable and divisible good, called the
numeraire, is available in u units. Moreover, there are M additional private
goods and Q public goods: denote by y3RNM`Q an allocation of such nonnumeraire goods. The set of vectors y that could be made available, perhaps
using large amounts of the numeraire as input, and that respect any constraints
de"ning the consumption set of consumers, is denoted >. Society's technology is
de"ned by > and the cost function C : >PR, understood as follows: in order to
make the vector y available, society must spend C(y) units of the numeraire
good.
The consumption set of person i, i"1,2, N, is denoted XM ]>, where XM -R.
Her utility function is of the form: u : XM ]>PR: u (x , y)"x #v (y), where v is
i
i i
i
i
i
a real-valued function with domain >.1
Writing x"(x ,2, x ), de"ne a feasible allocation as a vector (x; y)3XM N]>
1
N
that satis"es +N x #C(y)"u, where, for convenience, the resource constraint

i/1 i
is written as an equality instead of the more common weak inequality. De"ne
the set: >K "My3>D there exists an x3RN such that (x; y) is a feasible allocationN,
and the social surplus function S: >K PR: S(y)"+N v (y)!C(y). The surplus
i/1 i
function is central in the surplus approach to welfare economics, where it
characterizes "rst and second best allocations, makes potential compensation
criteria operative and provides the foundations for cost}bene"t analysis. This
paper focusses on "rst best analysis: its formal structure can easily be adapted to
the other facets of welfare economics.2
A basic observation is that, at any feasible allocation (x; y), the sum of the
utilities equals social surplus plus a constant, because
N
N
N
N
+ u (x , y)" + x # + v (y)"u!C(y)# + v (y)"S(y)#u.
i i
i
i

i
i/1
i/1
i/1
i/1
Therefore, (x; y) is (Pareto) e$cient whenever y maximizes S on >K . But, is the
converse also valid? i.e., does the maximization of S imply e$ciency? The answer
depends on the consumption sets XM .
Consider "rst the case where consumption sets do not impose any bounds on
the consumption of the numeraire, and, in particular, negative amounts of the
numeraire are allowed: this is the quasilinear case in the strict sense, because
wealth e!ects are globally absent. Formally, let XM "R. Then we have the
following well-known fact.
1 Many of the components of the vector y will typically be utility irrelevant for person i. But the
present notation provides simplicity and generality.
2 Social surplus can often be interpreted as the sum of all consumers' and producer's surpluses
(minus any "xed costs). See Silvestre (1997) for an application of the model in Section 2 below to
environmental resources.

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

625

Fig. 1. The atemporal case: No lower bounds on the numeraire.

Global equivalence principle for atemporal economies. If there are no bounds on
the numeraire, a feasible allocation (x; y) is e.cient if and only if y maximizes S
on >K .3
But if such bounds are present, then there may be e$cient allocations where
social surplus is not at its maximum. Figs. 1 and 2 illustrate. Let there be two
people (N"2). Suppose that there are only two possible vectors, y0 and yH, of
nonnumeraire goods, i.e., >K "My0, yHN. Each "gure o!ers two utility minifrontiers, one labelled y0 and the other one labelled yH, constructed as follows.
The utility mini-frontier labelled y0 is the utility locus that results from producing y0 and distributing the remaining amount of the numeraire in all conceivable manners between the two persons. Note that the same level of social
surplus, namely S(y0), corresponds to any point in the mini-frontier. The one
labelled yH is constructed in a similar manner. The overall utility frontier is the
outer envelope of all such mini-frontiers (i.e., one for each y3>) ).
Fig. 1 re#ects the absence of bounds on the numeraire. The utility minifrontiers are then in"nite straight lines of slope !1, which intersect the axes at
the magnitude of the corresponding social surplus, plus the constant u: indeed,
feasibility implies that, for N"2 and y"yH, u "x #v (yH)"
2

2
2
u!C(yH)!x #v (yH)"u!C(yH)!u #v (yH)#v (yH). As drawn, the
1
2
1
1
2
3 The economy of this introduction is a special case of the one in Section 2 below, namely, with
¹"1, and with small notational di!erences: for instance, in the Introduction N replaces N #N of
0
1
Section 2, and a single subscript i replaces the double subscript (i, t). It follows that the equivalence
principles of this Introduction are special cases of the formal results of Section 2.

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Fig. 2. The atemporal case: Lower bounds on the numeraire.

vector yH yields a higher surplus and, thus, y* has a utility mini-frontier above
that of y0. It follows that an allocation is e$cient if and only if its vector of
nonnumeraire goods is yH.
But if there are lower bounds on the numeraire, say, if the "nal holdings of
numeraire are required to be nonnegative, then the utility mini-frontiers are no
longer in"nite lines: they are truncated as in Fig. 2. At point D, u "0#v (y0):
2
2
Person 2's "nal holdings of numeraire are there zero, hitting the nonnegativity
constraint. It is still true that if surplus is maximized (say, a utility pair in the
segment [B, C] is achieved), then the corresponding allocation is e$cient. But
the converse is no longer true. The utility pair of point A corresponds to an
e$cient allocation, but surplus is not maximized there. Thus, the &global equivalence principle' no longer holds.
However, a &local' version of it does hold. The problem with point A is that
Person 2's "nal holdings of numeraire are too small there (A is close to D, where
she has zero). The local principle states that e$ciency and large enough individual holdings of numeraire imply surplus maximization. Formally, we say that
there are lower bounds on the numeraire if XM "R . Then we have the following
`
result.

¸ocal equivalence principle for atemporal economies. Assume that there are
lower bounds in the numeraire, and let (x; y) be a feasible allocation.
(a) If y maximizes S on >K , then (x; y) is e.cient.
(b) ¹here exists a real number B such that, if (x; y) is e.cient and if x 'B, for
i
i"1,2, N, then y maximizes S on >K .

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

627

Here I extend these well-understood ideas to an economy with overlapping
generations. The paper has two parts: "nite time and number of generations
(¹(R, Section 2 below), and in"nite time and number of generations (¹"
R, Section 3 below). The results of the "nite case are similar to the ones in the
atemporal case, except that the model has now one numeraire for
each period: the global equivalence principle holds in the absence of bounds
on the numeraires, and the local version holds if lower bounds are present
(Theorems 1}3).
The in"nite case is more subtle. First, no e$cient allocations exist in the

absence of bounds on the numeraires (Theorem 4).4 The statement &e$ciency
implies surplus maximization' is then vacuous, and the converse statement is
actually false (because surplus may well attain a maximum on the set of feasible
allocations).
Second, and perhaps surprisingly, it matters whether individuals discount
future numeraires or not. If future numeraires are not discounted, then the
individual "nal holdings of the numeraires must be small at any e$cient
allocation (Theorem 5). This invalidates any &equivalence principle' for this case.
The statement &surplus maximization implies e$ciency' is here false, because one
can have surplus maximization and large individual holdings of the numeraires.
The statement &e$ciency implies surplus maximization' is also false (because of
the lower bounds: see point A in Fig. 2). And a local claim of the type &e$cient
and large enough individual holdings of the numeraires imply surplus maximization' is then basically vacuous, because of the incompatibility between e$ciency and large individual holdings of the numeraires.
But the local equivalence principle is recovered in the ¹"R case if lower
bounds on the numeraires are present and future numeraires are discounted.
Surplus maximization then implies e$ciency (Theorem 6) and e$ciency with
large individual "nal holdings of numeraire imply surplus maximization
(Theorem 7).

2. Finite horizon (¹(R)

2.1. Generations and persons
There are ¹#1 generations (¹(R), indexed 0, 1,2, ¹, and ¹ discrete
time periods indexed 1,2, ¹. An &old' generation (indexed by t!1) and
a &young' generation (indexed by t) coexist at each t, t"1,2, ¹. By convention,

4 Because it implies a constant MRS between two numeraires, my &quasilinear' OLG model
violates the curvature assumptions often imposed on OLG economies, which may be used to show
that the set of e$cient allocations is nonempty.

628

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Generation 0 lives only in period 1 as &old,' and Generation ¹ lives only
in period ¹ as &young.' For t"1,2, ¹!1, Generation t lives for two periods:
as a young generation in period t, and as an old generation in period t#1.
There are N people in Generation t. A person is identi"ed as &Person i
t
of Generation t1 (i"1,2, N , t"0,2, ¹), and indexed by the double subscript
i

(i, t).
2.2. Numeraires, other goods, resources and technology
A nonproduced good speci"c to period t, called t-numeraire, is available in
u units, t"1,2, ¹. We view the t-numeraire as a private, transferable, divist
ible and nonstorable good. Denote by x
the consumption of t-numeraire by
i,t~1,t
Person i of Generation (t!1) (recall that t!1 is the old generation in period t),
and by x the consumption of t-numeraire by Person i of Generation t (t is the
itt
young generation in period t).
Moreover, in each period t there are M additional private goods (M 50) and
t
t
Q public goods (Q 50). Denote by y an allocation of such goods, and write
t
t
t
y"(y ,2, y ). The set of vectors y that could be made available, perhaps using
1
T
large amounts of the numeraires as inputs, and that respect any constraints
de"ning the consumption set of consumers, is denoted >, a subset of a "nitedimensional Euclidean space.5 Society's technology is de"ned by > and the cost
function:
C: >PRT: yP(C (y),2, C (y)),
1
T
understood as follows: &in order to make the vector y available, society must
spend C (y) units of t-numeraire, for t"1,2, ¹'.
t
This formulation is rather general. It includes, as a particular case, the
situation where the only relevant arguments of C are those in y .
t
t
2.3. Consumption sets and utilities
The consumption set of Person i of Generation t, for t"1,2, ¹!1, and
i"1,2, N is denoted X]>, where X-R2. Her utility function is of the form:
t
u : X]>PR: u (x , x
, y)"x #j x
#v (y),
it
it itt i,t,t`1
itt
t`1 i,t,t`1
it
where j
is a positive real number and v is a real-valued function with
t`1
it
domain >.

5 The dimension of y is (N #N )M #Q , t"1,2,¹. Thus, > is a subset of a (+T [(N #
t
t~1
t t
t
t/1
t~1
N )M #Q ])-dimensional space.
t t
t

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

629

Similarly, for t"0 and i"1,2, N (resp. t"¹ and i"1,2, N ), Person
0
T
(i, t)'s consumption set is XM ]>, XM -R, and her utility is:
u : XM ]>PR: u (x , y)"x #v (y)
i0
it i01
i01
i0
(resp. u : XM ]>PR: u (x , y)"x #v (y)).
iT
iT iTT
iTT
iT
Remark 1. No particular interpretation of the numeraires has been imposed, but
it would be natural in many instances to visualize the t-numeraire and the
(t#1)-numeraire as being physically the same good located at the two di!erent
points in time t and t#1, respectively. If j "1, then the two numeraires are
t`1
perfect substitutes with MRS equal to one, in other words, individuals do not
discount the future (except possibly for nonnumeraire goods). If j (1, then
t`1
j
can be interpreted as Generation t's time discount factor for the later
t`1
numeraire.
Remark 2. Note that no i subscript appears in j : in other words, all members
t`1
of the same generation must have the same time discount rate for numeraire.
This is essential to the analysis. Otherwise, the absence of bounds on the
numeraires would imply that no Pareto e$cient allocations exist (two members
of the same generation with di!erent MRS's between the two numeraires could
achieve in"nite utilities by exchanging one numeraire against the other). If, on
the contrary, lower bounds on the consumption of numeraires are assumed, as
in Section 2.9 below, then di!erent j's among members of the same generation
would imply that no Pareto e$cient allocation is interior, and Theorem 3 below
would be vacuous.
Remark 3. As noted in footnote 1, the only relevant arguments in v will often be
it
the public goods made available in periods t and t#1, and the private goods
made available to Person i of Generation t in periods t and t#1 (barring
consumption externalities). For instance, one could have
v (y)"w (y )#k
w
(y ),
it
itt t
i,t`1 i,t,t`1 t`1
where k
is a real number, and w and w
are real-valued functions, with
i,t`1
itt
i,t,t`1
the following interpretations:
f w (y) denotes the bene"ts from vector y , which accrue to Person i of
itt
t
Generation t when young;
f w
(y ) denotes the bene"ts from vector y , which accrue to Person i of
i,t,t`1 t`1
t`1
Generation t when old;
f k
is Person (i, t)'s time discount factor for the bene"ts from vector y
i,t`1
that accrue to her when old; possibly, but not necessarily, k
"j : in
i,t`1
t`1

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

particular, one may well have k
Ok
for two members i and h of
i,t`1
h,t`1
Generation t (compare with the previous Remark).
2.4. Feasible and ezcient allocations
A feasible allocation is a vector (x; y), where
x"(x ;2; x 0 ; x , x ;2; x 1 ,
101
N ,0,1 111 112
N ,1,1
x 1 ;2; x , x
; ;x
,
N ,1,2
itt i,t,t`1 2 1,T~1,T~1
,x
,
x
; ;x
1,T~1,T 2 NT~1,T~1,T~1 NT~1,T~1,T
x ;2; x T )3RH
1TT
N ,T,T
where H"N #2+T~1N #N , x 3XM , i"1,2, N , x 3XM , i"1,2, N ,
0
t/1 t
T i01
0 iTT
T
(x ,x
)3X, t"1,2,¹!1, y3>, and, for t"1,2,¹, +Nt~1x
#+Nt
itt i,t,t`1
i/1 i,t~1,t
i/1
x #C (y)"u .
itt
t
t
The following de"nitions are standard.
A feasible allocation (x@; y@) Pareto dominates another feasible allocation (x; y)
if no person is better o! at (x; y) than at (x@; y@), and at least one person of is
better o! at (x@; y@) than at (x; y).
A feasible allocation (x@; y@) strongly Pareto dominates another feasible allocation (x; y) if every person is better o! at (x@; y@) than at (x; y).
A feasible allocation (x; y) is Pareto ezcient if there does not exist a feasible
allocation (x@; y@) that Pareto dominates (x; y).
A feasible allocation (x; y) is weakly Pareto ezcient if there does not exist
a feasible allocation (x@; y@) that strongly Pareto dominates (x; y).

2.5. The social surplus function
De"ne, as in the Introduction:
>K "My3> D there exists an x3RH such that (x; y) is a feasible allocationN.
Adopt the notational conventions j "j "
: 1, and C (y ) "
: 0, for all
0
1
0
y3>. Write, for t"0, 1,2, ¹, K "
: j j 2 j . De"ne the social surplus
t
0 1
t
function:

C

D

T
Nt
S: >K PR: S(y)" + K + v (y)!C (y) .
t
it
t
t/0
i/1

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

631

The social surplus is a weighted sum of utilities, where the weight of Person
i of Generation t is K .6 Recall that these weights are simply (products of)
t
parameters of the individual utility functions, and, hence, in no way do they
represent value judgements about the social relevance of a person's utility.
When the j's are less than one, these weights contrast with the unit weights of
the surplus function in atemporal economies, as seen in the Introduction. Some
intuition about their role can be gained by comparing the utility mini-frontiers
of the atemporal case, which, as in Fig. 1, have the slope of !1, with those in the
overlapping generations case with j's less than one. Let ¹"3. For t"0, 1, 2, 3,
let N "1, delete the subscript i in the notation and denote by u the utility level
t
t
of the only person in Generation t. Fix a particular vector y6 (with associated
surplus level S(y6 )), as well as utility levels u6 and u6 . The utility mini-frontier for
0
3
y6 in the (u , u ) plane can be derived as follows. Because u6 "x #v (y6 ), x
1 2
3
33
3
33
must be a constant, and, hence, because, by feasibility, x #x #C (y6 )"u ,
23
33
3
3
so must x be: call it x6 . By the same argument, x must also be a constant,
23
23
11
say x6 . Thus, u "x6 #j x #v (y6 ), and u "x #j x6 #v (y6 ). But,
11
1
11
2 12
1
2
22
3 23
2
again, x #x #C (y6 )"u , i.e.,
12
22
2
2
u "x #j x6 #v (y6 )
2
22
3 23
2
"u !C (y6 )!x #j x6 #v (y6 )
2
2
12
3 23
2
"u !C (y6 )!(1/j )(u !x6 !v (y6 ))#j x6 #v (y6 ),
2
2
2 1
11
1
3 23
2
i.e., the slope of the utility mini-frontier is !(1/j ), with absolute value greater
2
than one as long as Generation 2 discounts its future numeraire and, thus,
j (1. See Fig. 3.
2
6 Plus a constant term. Indeed, using the resource equality that de"nes a feasible allocation, one
can compute:
N0
N0
N1
N1
N1
T Nt
+ + K u "K + x #K + v #K + x #K j + x #K + v #2
t it
0
i01
0
i0
1
i11
1 2
i12
1
i1
i/1
i/1
i/1
i/1
i/1
t/0 i/1
Nt
Nt
NT
NT
Nt
+ x
#K + v #2#K + x #K + v
#K + x #K j
itt
t t`1
i,t,t`1
t
it
T
iTT
T
iT
t
i/1
i/1
i/1
i/1
i/1
"K

1

A

B A

B

N0
N1
N1
N2
+ x #+ x
#K
+ x #+ x
i01
i11
2
i12
i22
i/1
i/1
i/1
i/1

#2#K
T

A

B

NT
T
Nq
NT~1
+ x
#+ x
#+ K + v
i,T~1,T
iTT
q
iq
i/1
i/1
q/0 i/1

T
T
Nq
T
" + K (u !C )# + K + v " + K u #S(y).
q q
q
q
iq
q q
q/1
q/0 i/1
q/1

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Fig. 3. The overlapping-generations case with discounting of future numeraire.

2.6. The maximization of surplus implies ezciency
In order to facilitate the comparison among sections, the statement of any
theorem will make explicit all maintained assumptions.
Theorem 1. Let ¹(R. If an allocation (xH; yH) is such that yH maximizes S on >K ,
then (xH; yH) is Pareto ezcient.
Proof. Follows immediately from the observation that social surplus is
a weighted sum of utilities, with positive weights (see previous section). h
Remark 4. Note the theorem holds under any speci"cation of the consumption
sets, i.e., with or without nonnegativity constraints.
Remark 5. Theorem 1 states that &surplus maximization N e$ciency.' It trivially implies the weaker statement: &surplus maximizationNweak e$ciency.'
2.7. A lemma
We are assuming a "nite number of periods (¹(R) in Section 2, whereas
the in"nite case (¹"R) is analyzed in Section 3 below. It will be convenient to
adopt, in this subsection, an encompassing formulation that covers both cases.
(See Section 3 below for the de"nition of >K and S when ¹"R: Lemma 2(iii)
below shows that S is well de"ned under the assumptions of Section 3.)

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

633

Write Z for the set of positive integers M1, 2,2N, and H for the set of time
indices, which is the "nite set Mt3Z D t4¹N in Section 2 and the in"nite set Z in
Section 3.
Given yH3>K , for t3M0NXH, i"1,2, N , de"ne
t
Dv [ yH]: >K PR: Dv [ yH](y)"v (y)!v (yH),
it
it
it
it
DC [yH]: >K PR: DC [yH](y)"C (y)!C (yH),
t
t
t
t
DS[yH]: >K PR: DS[ yH](y)"S(y)!S(yH).
Denote by A the set of strictly positive sequences of real numbers, one for each
generation ("nite sequences of the form (a , a ,2, a ) in Section 2, or in"nite
0 1
T
sequences of the form (a , a ,2, a , 2) in Section 3).
0 1
t
Given yH3>K and a3A, de"ne the functions:
G
[yH; a]: >K PR, t#13H, i"1,2, N ,
i,t,t`1
t
G [yH; a]: >K PR, t3H,
tt
by the following expressions (which leave implicit the symbols &[yH]' in the
notation for the D functions):
K Dv (y) K a
G [ yH; a](y)" 0 i0 ! 0 0 DS(y),
i01
K N
K
1 0
1
K DC (y)!+N0 K Dv (y)#K a DS(y)
h/1 0 h0
0 0
G [ yH; a](y)" 1 1
,
11
K N
1 1
K Dv (y) K DC (y)!+N0 K Dv (y)#K a DS(y)
h/1 0 h0
0 0
G [ yH; a](y)" 1 i1 ! 1 1
i12
K
K N
2
2 1
K a
! 1 1 DS(y).
K N
2 1
Note that

C

D

a
K
G " 1 Dv !G ! 1 DS(y) ;
i1
11 N
i12 K
1
2
for t'1 de"ne
+t K DC (y)!+t~1 +Nq K Dv (y)#DS(y)+t~1 K a
q/0 h/1 q hq
q/0 q q, (1)
G [yH; a](y)" q/1 q q
tt
KN
t t
a
K
(2)
G
[yH; a](y)" t Dv (y)!G [yH, a](y)! t DS(y) .
it
tt
i,t,t`1
N
K
t
t`1

C

D

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Lemma 1. Let a3A, and let (xH; yH) be a feasible allocation, with yH3>K . Let y@3>K .
Dexne
x@ "xH !G [yH; a](y@), t3H, t(¹, i"1,2, N .
itt
tt
t
itt
!G
[yH; a](y{), t3HXM0N, t(¹.
"xH
x@
i,t,t`1
i,t,t`1
i,t,t`1
Nt~1
Nt
(i) + x@
#+ x@ #C (y@)"u ,
i,t~1,t
itt
t
t
i/1
i/1

for t3H, t(¹.

(ii) Assume that DS(y@) :"S(y@)!S(yH)'0. For i"1,2, N , if x@ 3X1 , then
i01
0
)3X, then
u (x@ , y@)'u (xH , yH); for t3H, t(¹, i"1,2, N , if (x@ , x@
itt i,t,t`1
t
i0 i01
i0 i01
, yH).
, y@)'u (xH , xH
u (x@ , x@
it itt i,t,t`1
it itt i,t,t`1
Proof. (i) Consider "rst the case t"1. Recall that, by convention, j "j "
0
1
K "K "1, and that, by feasibility, +N0 xH #+N1 xH "u !C (yH).
0
1
1
1
i/1 i01
i/1 i11
We compute
N0
N1
+ x@ # + x@ #C (y@)
i01
i11
1
i/1
i/1
N0
N1
N0
" + xH ! + G (y@)# + xH !N G (y@)#C (y@)
i01
i01
i11
1 11
1
i/1
i/1
i/1
K a
N0 K
"u !C (yH)! + 0 Dv (y@)# 0 0 DS(y@)
i0
1
1
K
K
1
i/1 1
K a
K N0
!DC (y@)# 0 + Dv (y@)! 0 0 DS(y@)#C (y@)
h0
1
1
K
K
1
1 h/1
"u .
1
For t'1, t(¹,
Nt~1
Nt
# + x@ #C (y@)
+ x@
i,t~1,t
itt
t
i/1
i/1
Nt
Nt~1
Nt~1
# + xH ! + G
" + xH
(y@)!N G (y@)#C (y@)
i,t~1,t
itt
i,t~1,t
t tt
t
i/1
i/1
i/1
K
Nt~1 K
(y@)# t~1 N G
(y@)
"u !C (yH)! + t~1Dv
i,t~1
t~1 t~1,t~1
t
t
K
K
t
t
i/1
K
# t~1 a DS(y@)!N G (y@)#C (y@)
t~1
t tt
t
K
t

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

635

Nt~1 K
"u #DC (y@)! + t~1Dv
(y@)
t
t
i,t~1
K
t
i/1
+t~1 K DC (y@)!+t~2 +Nq K Dv (y@)#DS(y@)+t~2 K a
K
q/0 h/1 q hq
q/0 q q
# t~1 q/1 q q
K
K
t~1
t
K
# t~1 a DS(y@)
t~1
K
t
+t K DC (y@)!+t~1 +Nq K Dv (y@)#DS(y@)+t~1 K a
q/0 h/1 q hq
q/0 q q
! q/1 q q
K
t
t~1K
t K
K
q DC (y@)!+
q DC (y@)
"u # t DC (y@)#+
t K
t
q
q
K
K
t
q/1 t
q/1 t
t~2 Nq K
t~1 Nq K
Nt~1 K
(y@)! + + qDv (y@)# + + q Dv (y@)
! + t~1Dv
i,t~1
hq
K
K hq
K
t
i/1
q/0 h/1 t
q/0 h/1 t
t~2 K
K
t~1 K
#DS(y@) + q a #DS(y@) t~1 a !DS(y@) + q a "u .
t~1
t
K q
K q
K
t
q/0 t
q/0 t
This proves (i).
(ii) For Person i of Generation 0, i"1,2, N :
0
u (x@ , y@)"x@ #v (y@)"xH !G (y@)#v (y@)
i01
i01
i0
i01
i0
i0 i01
a
"xH !v (y@)#v (yH)# 0 DS(y@)#v (y@)
i01
i0
i0
i0
N
0
a
"u (xH , yH)# 0 DS(y@)'u (xH , yH),
i0 i01
i0 i01
N
0
because both a and DS(y@) are positive.
0
For Person i of Generation t (t3H, t(¹, i"1,2, N ):
t
, y@)
u (x@ , x@
it itt i,t,t`1
"x@ #j x@
#v (y@)
itt
t`1 i,t,t`1
it
"xH #j xH
!G (y@)!j G
(y@)#v (y@)
itt
t`1 i,t,t`1
tt
t`1 i,t, t`1
it
j
a
"xH #j
xH
!G (y@)! t`1 Dv (y@)!G (y@)! t DS(y@) #v (y@)
itt
t`1 i,t,t`1
tt
it
it
tt
j
N
t`1
t
a
"xH #j xH
!G (y@ )!v (y@)#v (y*)#G (y@)# t DS(y@)#v (y@)
it
itt
t`1 i,t,t`1
tt
it
it
tt
N
t
a
, yH)# t DS(y@)
"u (xH , xH
it itt i,t,t`1
N
t
, yH),
'u (xH , xH
it itt i,t,t`1
because both a and DS(y@) are positive. h
t

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

2.8. The absence of bounds on the numeraires
Assumption 1. (Absence of bounds on the numeraires). X"R2, XM "R.
Theorem 2. Let ¹(R and postulate Assumption 1. If (xH; yH) is weakly Pareto
ezcient, then yH maximizes S on >K .
Proof. Suppose, by way of contradiction, that there is a y@3>K with S(y@)'S(yH).
Write:
1
a6"
,
+T K
q/0 q
and de"ne x@ by:
x@ "xH !G [yH; a6](y@),
i01
i01
i01
2

(3)

i"1,2, N ,
0

x@ "xH !G [yH; a6](y@),
itt
itt
itt
!G
[yH; a6](y@),
t"1,2,¹!1, i"1,2, N ,
"xH
x@
i,t,t`1
i,t,t`1
t
i,t,t`1
2
a6
x@ "xH !Dv [yH](y@)#DS(y@)
, i"1,2, N .
iTT
it
iTT
T
N
T
I claim that (x@; y@) strongly Pareto dominates (xH; yH). By Lemma 1(ii), all
persons in generations 0 to ¹!1 are better o! at (x@; y@) than at (xH; yH). So are
all persons in Generation ¹, because
u (x@ , y@)"x@ #v (y@)
iTT
iT
iT iTT

a6
"xH !v (y@)#v (yH)#DS(y@)
#v (y@)
iTT
iT
iT
iT
N
T
a6
"u (xH , yH)#DS(y@)
iT iTT
N
T
'u (xH , yH), i"1,2, N ,
T
iT iTT

because a6'0 and DS(y@)'0.
#+Nt x@ #C (y@)"u ' are satisBy Lemma 1(i), the equalities &+Nt~1x@
t
t
i/1 i,t~1,t
i/1 itt
"ed for t"1,2,¹!1. We are left with showing that this is also the case for
t"¹. Recalling that +NT~1xH
#+NT xH "u !C (yH), we compute
T
T
i/1 i,T~1,T
i/1 iTT
NT~1
NT
# + x@ #C (y@)
+ x@
i,T~1,T
iTT
T
i/1
i/1
NT~1
NT
"u #DC (y@)! + G
[yH; a6](y@)! + Dv (y@)#a6DS(y@)
T
T
i,T~1,T
iT
i/1
i/1

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

637

NT~1
N
K
K
K
T~1
"u # TDC (y@)! T~1 + Dv
(y@)# T~1
T K
T
i,T~1
N
K
K K
T~1 T~1
T
T i/1
T
T~1
T~2 Nq
T~2
] + K DC (y@)! + + K Dv (y@)#DS(y@)a6 + K
q q
q hq
q
q/1
q/0 h/1
q/0
a6
K NT
K
K
DS(y@)! T + Dv (y@)# T a6DS(y@)
# T~1N
T~1 N
iT
K
K
K
T~1
T i/1
T
T
1
NT~1
T~1
"u #
K DC (y@)! + K
Dv
(y@)# + K DC (y@)
T K
T T
T~1 i,T~1
q q
T
i/1
q/1
T~2 Nq
NT
! + + K Dv (y@)! + K Dv (y@)
q hq
T iT
i/1
q/0 h/1
T~2
1
#K
# DS(y@)a6 + K #K
q
T~1
T
K
T
q/0
1
1
"u # [!DS(y@)]# DS(y@)
T K
K
T
T
"u ,
T
where (3) has been used. h

C

D

C

C

D
D

Remark 6. Theorem 2 states that &weak e$ciency N surplus maximization.' It
trivially implies the weaker statement: &e$ciency N surplus maximization.'
2.9. Lower bounds on the consumption of the numeraires
Consider next the case of lower bounds on the numeraires. It is convenient to
choose zero as the lower bound. The choice is not particularly restrictive
because no sign restrictions are imposed on u or C(y).
Assumption 2. (Lower bounds on the numeraires). X"R2 and XM "R .
`
`
Assumption 3. There exist positive numbers v6 and CM such that
Dv (y)D(v6
for all y3>K , i"1,2, N , t"0, 1,2,¹,
it
t
DC (y)D(CM for all y3>K , t"1,2, ¹.
t
Remark 7. Assumption 3 is automatically satis"ed when the functions v and
it
C are continuous and the set >K is compact.
t
Theorem 3. Let ¹(R, and postulate Assumptions 2 (lower bounds on the
numeraires) and 3. There exists a real number B such that if (xH; yH) is weakly
5B, and xH 5B, i"1,2, N , t"1,2,¹, then
Pareto ezcient, and if xH
itt
t
i,t~1,t
yH maximizes S on >K .

638

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Proof. Because, by Assumption 3, v and C are bounded from above and from
it
t
below on >K , there exists a real number B such that, recalling (1) and (2) and
de"ning GM [yH; a6](y)"Dv [yH](y)!DS(y)a6/N , where a6 is de"ned in (3), for
iT
iT
T
all yH3>K and y3>K we have that
G [yH; a6](y)(B,
t"1, 2, ¹!1,
tt
G
[yH; a6](y)(B, t"0, 1,2, ¹!1, i"1,2, N
,
i,t,t`1
T~1
GM [yH; a6](y)(B,
i"1,2, N .
iT
T
Let (xH; yH) be Weakly Pareto E$cient and, for t"1,2, ¹, let
5B, i"1,2, N . Assume, as contradiction
xH 5B, i"1,2, N , and x*
i,t~1,t
t~1
itt
t
hypothesis, that there exists a y@3>K such that S(y@)'S(yH). De"ne:
x@
"xH
!G
(y@), t"1,2, ¹,
i"1,2, N ,
i,t~1,t
i,t~1,t
i,t~1,t
t~1
t"1,2, ¹!1, i"1,2, N ,
x@ "xH !G (y@),
itt
tt
t
itt
i"1,2, N .
x@ "xH !GM (y@),
iTT
iT
T
iTT
Because G
(y@)(B, t"1,2,¹, i"1,2, N , G (B, t"1,2,¹!1,
i,t~1,t
t~1 itt
'0 and
i"1,2, N , and GM (y@)(B, i"1,2, N , we have that x@
i,t~1,t
t
iT
T
x@ '0, ∀i, ∀t. Lemma 1 and the proof of Theorem 2 shows that (x@; y@) strongly
itt
Pareto dominates (xH; yH). Contradiction. h
Remark 8. Theorem 3 states that &weak e$ciency and large enough xH'sN
surplus maximization.' It trivially implies the weaker statement: &e$ciency and
large enough xH'sN surplus maximization'.

3. In5nite horizon (¹"R)
3.1. The model
Now I extend the analysis to the case ¹"R. The set > as well as the range
of C are now subsets of R=. Now Generation 0 lives only in period 0 as old, but,
for t3Z, Generation t lives for two periods, and the consumption set of any of its
members is X]>-R2]>. In obvious extension, a feasible allocation is a vector (x; y), where:
x"(x ;2; x 0 ; x , x ;2; x 1 , x 1 ;2; x ,
101
N ,0,1 111 112
N ,1,1 N ,1,2
itt
Nt
Nt~1
# + x #C (y)"u .
x
; )3R=, y3>, and for t3Z, + x
i,t~1,t
itt
t
t
i,t,t`1 2
i/1
i/1
As before, de"ne:
>K "My3> D there exists an x3R= such that (x; y) is a feasible allocationN.
The meaning of Pareto domination and e$ciency is the straightforward
extension of the previous de"nitions to the in"nite horizon. Strong Pareto

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

639

domination, in particular, means that all persons in all generations, i.e., an
in"nite number of consumers, are better o!.
3.2. Absence of bounds on the numeraires
Theorem 4. Let ¹"R. Under Assumption 1 (absence of bounds on the
numeraires), the set of weakly Pareto ezcient allocations is empty.
Proof. Given a feasible allocation (x; y), choose a positive number e and de"ne
the numeraire allocation x@ by:
e
x@ "x #
,
i"1,2, N ,
i01
i01 N j
0
0 1
e
,
x@ "x !
i11
i11 N j
1 1
e
e
#
,
i"1,2, N ,
x@ "x #
i12
i12 N j j
1
N j
1 1 2
1 2
e
e
x@ "x !
!
,
i22
i22 N j j
N j
2 1 2
2 2
e
e
#
,
i"1,2, N ,
x@ "x #
i23
i23 N j j j
2
N j
2 1 2 3
2 2
2
e
e
e
x@ "x !
!
!2!
,
itt
itt N j 2j
N j 2j
Nj
t 1
t
t 2
t
t t
e
e
"x
#
#
x@
i,t,t`1
i,t,t`1 N j 2j
N j 2j
t 1
t`1
t 2
t`1
e
e
#2#
#
,
i"1,2, N , t3Z, t'2.
t
Njj
Nj
t t t`1
t t`1
We "rst check that (x@; y) is a feasible allocation. For t3M0NXZ, we compute:
Nt`1
Nt
# + x@
#C (y)
+ x@
i,t,t`1
i,t`1,t`1
t`1
i/1
i/1
Nt
e
Nt`1
e
"+ x
#+ x
#N
#
i,t,t`1
i,t`1,t`1
t N j 2j
N j 2j
t 1
t`1
t 2
t`1
i/1
i/1
e
e
e
!N
#2#
#
t`1 N j 2j
Nj
N j 2j
t`1 1
t`1
t t`1
t`1 2
t`1
e
#2#
#C (y)
t`1
N j
t`1 t`1

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Nt
Nt`1
"+ x
#+ x
#C (y)
i,t,t`1
i,t`1,t`1
t`1
i/1
i/1
"u

t`1

,

the last equality following from the fact that (x; y) is feasible.
Second, everybody prefers (x@; y) to (x; y), because for i"1,2, N :
0
e
u (x@ ,y)"x #
#v (y)'x #v (y)"u (x , y),
i01 N j
i0 i01
i0
i01
i0
i0 i01
0 1
and for i"1,2, N , t3Z:
t
, y)
u (x@ , x@
it itt i,t,t`1
e
e
e
"x !
!
!2!
#j x
itt N j 2j
t`1 i,t,t`1
N j 2j
Nj
t 1
t
t 2
t
t t

C

D

e
e
e
e
#
#2#
#
#v (y)
#j
it
t`1 N j 2j
N j 2j
Njj
Nj
t 1
t`1
t 2
t`1
t t t`1
t t`1
e
"x #j
x
# #v (y)
itt
t`1 i,t,t`1 N
it
t
'x #j
x
#v (y)
itt
t`1 i,t,t`1
it
"u (x , x
, y).
it itt i,t,t`1
Thus, (x@; y) strongly Pareto dominates (x; y). h
Theorem 4 motivates focussing on economies with lower bounds on the
numeraires: Assumption 2 will be postulated (with explicit mention) for the
remainder of the paper.
3.3. Zero discounting of future numeraires
Next I show that, if future numeraires are not discounted, then e$ciency is
incompatible with large individual holdings of the numeraires.
Theorem 5. Let ¹"R, postulate Assumption 2 ( lower bounds on the
numeraires), and let there be a tK 50 such that j 51 for all t5tK . Let (x; y) be
t
a feasible allocation. If there exists an e( '0 such that +Nt x 5e( , for all t3Z,
i/1 itt
then (x; y) is not weakly Pareto ezcient.

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

641

Proof. Step 1: Dexnition of x@. This step constructs a vector x@ such that (x@; y)
strongly Pareto dominates (x; y). De"ne

GC D C

e9
e" min
2

C

D

D

1 ~1
1
1
1 ~1
1
1 ~1
#
#
#2#
,
,2,
,
j
j j
j 2j K j 2j K
j
jK
1
1 2
1
t
2
2
t
t

C

D H

~1
1
1
1
#
#2# #1
,
j 2j K j 2j K
jK
1
t
2
t
t
(clearly, e'0) and write:

D

C

1
1
1
.
#
#2#
j 2j K j 2j K
jK
1
t
2
t
t
1
As is well known, the series +=
converges to a real number, to be denoted
n/1 n2
by a (a+1.645). Without loss of generality, let tK '2 and de"ne the consumption
allocation x@ as follows.
e
,
i"1,2, N ,
x@ "x #
0
i01
i01 N j
0 1
e
,
x@ "x !
i11
i11 N j
1 1
e
e
x@ "x #
#
, i"1,2, N ,
i12
i12 N j j
1
N j
1 1 2
1 2
F
¸"e

C

D

1
1
1
e
#
#2#
x@ "x !
,
itt
itt N j 2j
j 2j
j
t
2
t
t
t 1
e
1
1
1
"x
#
x@
,
#
#2#
i,t,t`1
i,t,t`1 N j 2j
j 2j
j
t 1
t`1
2
t`1
t`1
i"1,2, N , t(tK ,
t
F

C

D

¸
x@ K K "x K K ! ,
itt
itt N K
t

¸
e
x@ K K "x K K # #
, i"1,2, N K ,
i,t,t`1
i,t,t`1 N K N K a
t
t
t
F
¸
e t~tK 1
x@ "x ! !
+
,
itt
itt N
q2
Na
t
t q/1
¸
e t~tK `1 1
+
x@
"x
# #
, i"1,2, N , t3Z, t'tK .
i,t,t`1
i,t,t`1 N
t
q2
Na
t
t q/1

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J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Step 2: (x@; y) satisxes the feasibility constraints. For period t"1,2, tK , the
argument in the proof of Theorem 4 applies. If t'tK , we compute
Nt
Nt~1
# + x@ #C (y)
+ x@
i,t~1,t
itt
t
i/1
i/1
Nt~1
Nt
¸
"+ x
# + x #C (y)#N
i,t~1,t
itt
t
t~1 N
t~1
i/1
i/1
K
e t~1~t`1 1
¸
e t~tK 1
+
#N
!N
!N
+ #C (y)
t~1N a
tN
tNa
t
q2
q2
t~1 q/1
t
t q/1
Nt~1
Nt~1
"+ x
# + x #C (y)"u ,
i,t~1,t
itt
t
t
i/1
i/1
because (x; y) is feasible.
'x
70, because
Step 3: x@ is nonnegative. Clearly, for t3M0NXZ, x@
i,t,t`1
i,t,t`1
(x; y) is feasible. Thus, we are left with proving that x@ 50, t3Z. Consider "rst
tt
the case t6tK . Then

C

D

e
1
1
1
x@ "x !
#
#2#
itt
itt N j 2j
j 2j
j
t 1
t
2
t
t
1
1
1
7x !e
#
#2#
itt
j 2j
j 2j
j
1
t
2
t
t
e(
7x ! ,
itt 2

C

D

because, by the de"nition of e,

C

D

1
e(
1
1 ~1
e6
#
#2#
2 j 2j
j 2j
j
1
t
2
t
t
i.e.,

C

D

1
e(
1
1
#
#2# 6 .
j 2j
2
j 2j
j
1
t
2
t
t
Because, by assumption, x 7e( , we have that x@ '0, t3Z. If t'tK , then, using
itt
itt
the de"nitions of a, ¸ and e:
e

¸
e t~tK 1
e t~tK 1
e = 1
x@ "x ! !
+ 7x !¸! + 'x !¸! +
itt
itt N
itt
itt
q2
q2
q2
Na
a
a
t
t q/1
q/1
q/1
1
1
1
"x !¸!e7x !e
#
#2# !e
itt
itt
j 2j K j 2j K
jK
1
t
2
t
t

C

D

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

643

1
1
1
e(
"x !e
#
#2# #1 7x !
itt
itt
j 2j K j 2j K
jK
2
1
t
2
t
t

C

D

'0,
by the same argument as in the previous case.
Step 4: Everybody is better ow at (x@; y). Clearly, for i"1,2, N , Person (i, 0) is
0
better o! at (x@; y) than at (x; y), because (x@; y) entails a utility gain of e/N '0.
0
For t"1,2, tK !1 and i"1,2, N , the argument in the proof of Theorem
t
, y)'u (x , x
, y).
4 shows that u (x@ , x@
it itt i,t,t`1
it itt i,t,t`1
For t"tK , i"1,2, N K , we compute, recalling that j K 51:
t
t`1
u K (x@ K K , x@ K K , y)"x@ K K #j K x@ K K #v K (y)
itt
it
it itt i,t,t`1
t`1 i,t, t`1

C

D

¸
¸
e
"x K K #j K x K K #v K (y)! #j K
#
itt
t`1 i,t,t`1
t`1 N K N K a
it
NK
t
t
t
¸
¸
e
7x K K #j K x K K #v K (y)! # #
it
t`1 i,t, t`1
itt
NK NK NK a
t
t
t
e
"u K (x K K , x K K , y)#
it itt i,t,t`1
NK a
t
'u K (x K K , x K K , y).
it itt i,t,t`1
Last, for t3Z, t'tK , we compute, recalling that j 71 for t7tK :
t`1
, y)"x@ #j x@
#v (y)
u (x@ , x@
it itt i,t,t`1
itt
t`1 i,t,t`1
it
e t~tK 1
¸
+
"x #j x
#v (y)! !
itt
t`1 i,t,t`1
it
q2
N a
N
t q/1
t
¸
e t~tK `1 1
+
#j
#
t`1 N
q2
Na
t
t q/1

C

D

e t~tK 1
¸
+
7x #j x
#v (y)! !
itt
t`1 i,t,t`1
it
q2
Na
N
t q/1
t

C

#

¸
e t~tK ~1 1
+
#
N
q2
Na
t
t q/1

D

1
e
"u (x , x
, y)#
it itt i,t,t`1
N a (t!tK #1)2
t
'u (x , x
, y). h
it itt i,t,t`1

644

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Corollary. Let ¹"R, postulate Assumption 2 (lower bounds on the numeraires)
and let there be a tK 50 such that j 51 for all t5tK . If (x; y) is weakly Pareto
t
ezcient, then, given e'0, there exists a tI (e) such that +NtI (e) x I I (e .
i/1 i,t(e), t(e)
Proof. Immediate from Theorem 5. h.
3.4. The local equivalence principle
The next two theorems make precise the &local equivalence principle' for
in"nite-horizon economies. Theorem 6 says that &surplus maximization implies
e$ciency (and, a fortiori, weak e$ciency),' whereas Theorem 7 says that &weak
e$ciency (and, a fortiori, e$ciency) plus large individual holdings of numeraire
imply surplus maximization'.
Recalling that K : "j j 2j and that, by convention, j "j "
: 1 and
t
0 1
t
0
1
C (y) "
: 0, de"ne
0
Nt
=
>H" y3>K D the series + K + v (y)!C (y) converges ,
it
t
t
i/1
t/0
and the social surplus function:

G

D

C

C

H

D

Nt
=
S: >HPR: S(y)" + K + v (y)!C (y) .
t
it
t
t/0
i/1
Assumption 4. There exist a jM (1 and a tM '1 such that j 4jM , for all t5tM , t3Z.
t
Assumption 5. There exist positive numbers u6, NM , v6 and CM such that
Du D(u6,
t
N (NM
t
Dv (y)D(v6
it
DC (y)D(CM
t

for all t3Z,
for all t3M0NXZ,
for all y3>K , i"1,2, N , t3M0NXZ,
t
for all y3>K , t3Z.

Lemma 2. Let ¹"R, and postulate Assumptions 2 ( lower bounds on the
numeraires), 4 and 5. Let (x; y) be feasible.
(i) The series obtained by summing the sequence
(k ) "
: (k , k ,2, k ,2)
n
1 2
n
N1
N1
N2
Nt~1
N0
" K + x , K + x , K + x , K + x ,2, K + x
,
i11 2
i01 1
i12 2
i22
t
i,t~1,t
1
i/1
i/1
i/1
i/1
i/1
Nt
Nt
+ x
,..
K + x ,K
itt t`1
i,t,t`1
t
i/1
i/1
converges.

A

B

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

645

(ii) The series += K C (y), += K u and += K +Nt v (y) converge.
t/0 t i/1 it
t/1 t t
t/1 t t
(iii) The function S is well dexned on >K , i.e., >H">K .
Proof. (i) The sequence (k ) is nonnegative by the assumption of lower bounds
n
on the numeraires. The convergence of the series will follow after showing that
the sequence of partial sums is bounded.
By feasibility, for t3Z, +Nt~1x
#+Nt x #C (y)"u , and because
i/1 i,t~1,t
i/1 itt
t
t
6!C (y)#u , i.e., using Assumption 5
+Nt x 70, we have that +Nt~1x
i/1 itt
i/1 i,t~1,t
t
t
Nt~1
+ x
6CM #u6, t3Z.
(4)
i,t~1,t
i/1
By the same argument
Nt
+ x 6CM #u6, t3Z.
(5)
itt
i/1
De"ne tH"2(tM !1). We can compute the partial sum of the "rst 2tH terms of
(k ), and apply (4) and (5) as follows:
n
2tH
N0
N1
N1
N2
+ k "K + x # + x
#K + x # + x
n
1
i01
i11
2
i12
i22
n/1
i/1
i/1
i/1
i/1
NtM ~1
NtM ~2
#+ x M
#2#K M
+ x M
t~1
i,t~2,tM ~1
i,t~1,tM ~1
i/1
i/1
NtM ~1
NtM
#K M j M + x M
M# + x M M
i,t~1, t
i,t,t
t~1 t
i/1
i/1
NtH~1
NHt
#2#K M j M 2j H + x H
H# + x H H
t~1 t
t
i,t ~1,t
i,t ,t
i/1
i/1
4(K #K #2#K M )(u6#CM )#K M jM 2(u6#CM )
t~1
1
2
t~1
#K M jM 2 2(u6#CM )#2#K M jM tH~tM `12(u6#CM )
t~1
t~1
"(K #K #2#K M )(u6#CM )
1
2
t~1
#K M 2(u6#CM )(jM #jM 2#2#jM tH~tM `1)
t~1
=
((K #K #2#K M )(u6#CM )#K M 2(u6#CM ) + jM q
t~1
1
2
t~1
q/1
jM
"(K #K #2#K M )(u6#CM )#K M 2(u6#CM )
.
(6)
t~1
1
2
t~1
1!jM

A

B A

B

A

B

A

B

A

B

Thus, the sequence of partial sums is bounded and the series += k converges.
n/1 n

646

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

(ii) By an argument similar to the one in (i), the series += K DC (y)D converges.
t/1 t t
Therefore, += K C (y) converges absolutely and, thus, it converges. The same
t/1 t t
argument proves the convergence of += K u and += K +Nt v (y).
t/0 t i/1 it
t/1 t t
(iii) Immediate from (ii). h
Theorem 6. Let ¹"R, postulate Assumptions 2 ( lower bounds on the
numeraires), 4 and 5. If (xH; yH) is feasible and yH maximizes S on >K , then (xH; yH)
is Pareto ezcient.
Proof. Suppose not, i.e., let (x@; y@) Pareto dominate (xH; yH). Then, writing
xH "x@ "0, we have that
i00
i00
#v (yH),
#v (y@)7xH #j xH
x@ #j x@
it
itt
t`1 i,t,t`1
it
itt
t`1 i,t,t`1
i"1,2, N , t3M0NXZ,
(7)
t
with at least one inequality strict. Multiplying both sides of each inequality in (7)
by K we have that
t
K x@ #K j x@
#K v (y)7K xH #K j xH
#K v (yH),
t itt
t t`1 i,t,t`1
t it
t itt
t t`1 i,t,t`1
t it
i"1,2, N , t3M0NXZ,
(8)
t
with at least one of the inequalities in (8) strict.
By Lemma 2(ii), the series += K +Nt v (y@) and += K +Nt v (yH) converge.
t/0 t i/1 it
t/0 t i/1 it
Denote their sums, respectively, by
v@ and vH.

(9)

Denote by (k@ ) (resp. (kH)) the specialization to the feasible allocation (x@; y@)
n
n
(resp. (xH; yH)) of the sequence (k ) de"ned in the statement of Lemma 2(i), and
n
denote its in"nite sum by m@ (resp. mH). The sequence

A

N1
N1
N2
N2
N0
K + x@ , K + x@ #K + x@ , K + x@ #K + x@ ,2,
i01 1
i11
2
i12 2
i22
3
i23
1
i/1
i/1
i/1
i/1
i/1
Nt
Nt
K + x@ #K
+ x@
,
t
itt
t`1
i,t,t`1 2
i/1
i/1
is obtained from (k@ ) by inserting parentheses (its "rst term is k@ , its second term
1
n
). Therefore, its in"nite sum is m@, i.e.,
#k@
is k@ #k@ , its mth term is k@
2m~1
2m~2
3
2
=
Nt
Nt
+ K + x@ #K
+ x@
"m@.
itt
t`1
i,t,t`1
t
t/0
i/1
i/1
Using (9) and repeating the argument for (xH; yH), we have that the in"nite
sum of the left-hand side (resp. right-hand side) terms of (8) converges to
m@#v@ (resp. mH#vH). Because at least one of the inequalities in (8) is strict,

B

A

B

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

647

we obtain

B

A

Nt
=
=
Nt
Nt
+ K + x@ #K
+ x@
# + K + v (y@)
itt
t`1
i,t,t`1
it
t
t
t/0
i/1
i/1
t/0 i/1
Nt
=
Nt
"m@#v@' + K + xH #K
+ xH
itt
t`1
i,t,t`1
t
t/0
i/1
i/1
=
Nt
# + K + v (yH)"mH#vH.
it
t
t/0 i/1
Now the sequence:

A

B

(10)

A

N1
N1
N2
N0
K + x@ #K + x@ , K + x@ #K + x@ ,2,
i01
1
i11 2
i12
2
i22
1
i/1
i/1
i/1
i/1
Nt`1
Nt
#K
K
+ x@
+ x
,
i,t,t`1
t`1
t`1
i,t`1,t`1,2
i/1
i/1
is also obtained by inserting parentheses in (k@ ) (its "rst term equals k@ #k@ , and
n
1
2
its mth term equals k@
#k@ ). Thus, its in"nite sum also converges to m@, i.e.,
2m~1
2m
Nt
=
Nt~1
# + x@ "m@.
+ K + x@
i,t~1,t
itt
t
t/1
i/1
i/1
But feasibility implies that

B

B

A

A

B

Nt
Nt~1
# + x@ "K (u !C (y@)), t3Z,
K + x@
i,t~1,t
itt
t
t t
t
i/1
i/1
which, using Lemma 2(ii), implies that m@"+= K u !+= K C (y@), or, using
t/1 t t
t/1 t t
(9) and the de"nition of social surplus:
=
=
=
m@#v@" + K u ! + K C (y@)#v@" + K u #S(y@).
t t
t t
t t
t/1
t/1
t/1
Similarly,
=
=
=
mH#vH" + K u ! + K C (yH)#vH" + K u #S(yH).
t t
t t
t t
t/1
t/1
t/1
But by (10), m@#v@'mH#vH, i.e., S(y@)'S(yH), contradicting the fact that
yH maximizes S on >K . h
Assumption 6. There exists a e6 '0 such that j 5e6 , for all t3Z, t'1.
t
Theorem 7. Let ¹"R, and postulate Assumptions 2 ( lower bounds on the
numeraires) 4, 5 and 6. There exists a real number B such that if (xH; yH) is weakly

648

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

7B, i"1,2, N , t3Z, then yH maxiPareto ezcient, and if xH 7B and xH
i,t~1,t
t
itt
mizes S on >K .
Proof.
Step 1. Dexnition and properties of a( ,2, a( ,2 Write b"1/2K M ; for
0
t
t~1
t3M0NXZ, de"ne

G

1!bK M
t~1 if t(tM !1,
M
~2 K
+t
a( "
(11)
q/0 q
t
b(1!j ) if t7tK !1,
t`1
and write a( for the in"nite sequence (a( ,2, a( ,2). We check that the a( 's are all
0
t
t
positive. If t(tM !1, then the sign of a( is that of 1!bK M "1'0. If t7tM !1,
t
t~1 2
then the sign of a( is that of 1!j 71!jM '0, because t#1 is then greater
t
t`1
than or equal to tM .
We can compute
1!+tM ~2 K a(
1!((1!bK M )+tM ~2 K / +tM ~2 K ) 1!1#bK M
q/0 q q"
t~1 q/0 q q/0 t "
t~1"b.
KM
KM
KM
t~1
t~1
t~1
(12)
De"ne the sequence
(c , c ,2, c ,2)
1 2
n
1!+tM ~1 K a( 1!+tM K a(
1!+tM `n~2K a(
q/0 q q,
q/0
q q, 2 .
q/0 q q,2,
"
:
KM
KM
KM
t
t`n~1
t`1
The "rst term, c , can be written, using (11) and (12):
1
1 b(1!j M )
1!+tM ~2 K a( 1 K M
q/0 q q ! t~1 a( M "b !
t
c "
~1
t
1
K M !1
KM
jM
jM
jM
t
t
t
t
t
1
1
"b !b #b"b.
jM
jM
t
t
Similarly, the second term, c , can be written:
2
b
K M a(
b
1
b
c
!
b(1!j M )"
!
#b,
c " 1 ! t t"
t`1
2 jM
jM
KM
jM
jM
jM
t`1
t`1
t`1
t`1
t`1
t`1
and recursively:

A

)
a( M
b
b(1!j M
c
t`n~1 "b,
!
c " n~1 ! t`n~2"
n jM
jM
jM
jM
t`n~1
t`n~1
t`n~1
t`n~1
i.e., (c ,2, c ,2) is the constant sequence (b,2, b,2).
1
n

B

(13)

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

649

Step 2. Boundedness of the functions G and G
on >K . Consider the
tt
i,t,t`1
functions G [yH; a( ], t3Z, and G
[yH; a( ], t3M0NXZ, i"1,2, N , as de"ned
tt
i,t,t`1
t
by (1) and (2) above, for yH as in the statement of the Theorem and for a( as
de"ned in Step 1. I claim that there is a positive number B such that
G [yH; a( ](y)(B, for all yH, y3>K , and all t3Z, and that G
[yH; a( ](y)(B,
tt
i,t,t`1
for all yH, y3>K , all t3M0NXZ and all i3M1,2, N N. We leave implicit, in what
t
follows, the symbols [yH; a( ].
Consider "rst G , t3Z, and let y3>K . We write
tt
t K
t~1 Nq K
q DC (y)! + +
q Dv (y)
G (y)" +
tt
q
KN
K t hq
q/1 t t
q/0 h/1 tN
= K
Nq
t~1
q
+ K a(
+ Dv (y)!DC (y)
# +
q q
hq
q
KN
q/0 t t h/1
q/0
t~1 Nq K
t K
q Dv (y)
q DC (y)! + +
"+
q
hq
KN
KN
t
t
t
t
q/0 h/1
q/1
t~1
t K
t~1
K
=
q DC (y) ! + K a(
q DC (y)
+
+
! + K a(
q q
q q
q
q
KN
KN
q/0
q/0 t t
q/0
q/t`1 t t
t~1
t~1 Nq K
q Dv (y)
+ +
# + K a(
q q
hq
KN
q/0
q/0 h/1 t t
t~1
= Nq K
q Dv (y)
+ +
# + K a(
q q
hq
KN
q/0
q/t h/1 t t
1!+t~1 K a( t~1 Nq
1!+t~1 K a( t
q/0 q q + + K Dv (y)
q/0 q q + K DC (y)!
"
q t
q hq
KN
KN
t
t
t t
q/1
q/0 h/1
=
+t~1 K a(
q/0 q q
+ j 2j DC (y)
!
t`1
q q
N
t
q/t`1
= Nq j 2j
+t~1 K a(
q/0 q q + + t
qDv (y) .
#
hq
N
j
t
t
q/t h/1
I claim that the last expression is bounded from above and from below by
bounds that are independent from yH, y or t. By (13), (1!+t~1K a( )/K N 4b.
t/0 q q t t
By Assumption 5, DDC (y) D (2CM and D+Nq Dv (y)D(2NM v6 . The argument leadq
h/1 hq
ing to (6) shows that the positive sequences in t (which are independent from
yH and y) (+t K ) and (+t K a( ) are bounded. Moreover, each term of the
q/0 q
q/0 q q
sequence in t (+= j 2j ) is positive and "nite, and the sequence is decreasing.
q/t t
q
It follows that there exist real numbers b` and b~ such that, for yH, y3>K and
t3Z,

A

C

A
A
A

BA
BA
BA

A
A

BA
BA

b~(G [yH; a( ](y)(b`.
tt

DBA

B A
B
B

B

BA

B
B

B

650

J. Silvestre / Journal of Economic Dynamics & Control 24 (2000) 623}650

Consider now, for t3M0NXZ and i3M1,2, N N, G
[yH; a( ](y) as de"ned by
t
i,t,t`1
(2). The previous argument together with Assumption 6 shows the existence of
a real number bM such that, for yH, y3>K and t3M0NXZ, G
[yH; a( ](y)(bM .
i,t,t`1
Write B"maxMb`, b~N.
Step 3. Contradiction argument.
7B and xH 7B, t3Z,
Let (xH, yH) be Pareto e$cient, and let xH
itt
i,t~1,t
i"1,2, N . I claim that yH maximizes S on >K . Suppose not, i.e., let there be
t
y@3>K with DS(y@) "
: S(y@)!S(yH)'0. De"ne
x@ "xH !G [yH; a( ](y@), i"1,2, N ,
i01
i01
0
i01
x@ "xH !G [yH; a( ](y@), i"1,2, N , t3Z.
itt
tt
t
itt
!G
[yH;a( ](y@).
"xH
x@
i,t,t`1
i,t,t`1
i,t,t`1
'0 and x@ '0,
(B and xH (B, t3Z, we have that x@
Because xH
itt
i,t~1,t
itt
i,t~1,t
) 3R2 , for all t3Z. Lemma 1 then
for all t3Z, i.e., x@ 3R and (x@ , x@
`
itt i,t,t`1
i01
`
implies that (x@; y@) dominates, in the strong Pareto sense, (xH; yH), contradicting
the fact that (xH; yH) is weakly Pareto E$cient. h

References
Silvestre, J., 1997. An e$ciency argument for sustainable use. In: Roemer, J.E. (Ed.) Property
Relations, Incentives, and Welfare. Macmillan, London, pp. 43}64.