Surplus analysis for overlapping generations
qJoaquim 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:[email protected] (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 com-ments. The usual caveat applies.
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 non-numeraire goods. The set of vectors ythat 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 functionC:>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 personi,i"1,2,N, is denotedXM ]>, whereXM -R.
Her utility function is of the form:u
i:XM ]>PR:ui(xi,y)"xi#vi(y), whereviis
a real-valued function with domain>.1
Writingx"(x
1,2,xN), de"ne afeasible allocationas a vector (x;y)3XM N]>
that satis"es+Ni/1x
i#C(y)"u, where, for convenience, the resource constraint
is written as an equality instead of the more common weak inequality. De"ne the set:>K "My3>Dthere exists anx3RNsuch that (x;y) is a feasible allocationN, and the social surplus function S:>K PR:S(y)"+Ni
/1vi(y)!C(y). The surplus
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
+
i/1
u
i(xi,y)" N
+
i/1
x
i#
N
+
i/1
v
i(y)"u!C(y)# N
+
i/1
v
i(y)"S(y)#u.
Therefore, (x;y) is (Pareto) e$cient wheneverymaximizesSon>K . But, is the
converse also valid? i.e., does the maximization ofSimply e$ciency? The answer depends on the consumption setsXM .
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.
1Many of the components of the vectorywill typically be utility irrelevant for personi. But the present notation provides simplicity and generality.
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$cient allocations where social surplus is not at its maximum. Figs. 1 and 2 illustrate. Let there be two people (N"2). Suppose that there are only two possible vectors,y0andyH, of nonnumeraire goods, i.e., >K "My0,yHN. Each "gure o!ers two utility mini-frontiers, one labelledy0and the other one labelledyH, constructed as follows. The utility mini-frontier labelledy0is the utility locus that results from produ-cingy0and distributing the remaining amount of the numeraire in all conceiv-able manners between the two persons. Note that the same level of social surplus, namely S(y0), corresponds to any point in the mini-frontier. The one labelledyHis constructed in a similar manner. The overall utility frontier is the outer envelope of all such mini-frontiers (i.e., one for eachy3>) ).
Fig. 1 re#ects the absence of bounds on the numeraire. The utility mini-frontiers are then in"nite straight lines of slope!1, which intersect the axes at the magnitude of the corresponding social surplus, plus the constantu: indeed, feasibility implies that, for N"2 and y"yH,u
2"x2#v2(yH)"
u!C(yH)!x
1#v2(yH)"u!C(yH)!u1#v1(yH)#v2(yH). As drawn, the
3The economy of this introduction is a special case of the one in Section 2 below, namely, with ¹"1, and with small notational di!erences: for instance, in the IntroductionNreplacesN
0#N1of
Fig. 2. The atemporal case: Lower bounds on the numeraire.
vectoryHyields a higher surplus and, thus,y*has a utility mini-frontier above that of y0. It follows that an allocation is e$cient if and only if its vector of nonnumeraire goods isyH.
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 pointD,u
2"0#v2(y0):
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$cient. But the converse is no longer true. The utility pair of point Acorresponds to an e$cient allocation, but surplus is not maximized there. Thus, the&global equiva-lence principle'no longer holds.
However, a&local'version of it does hold. The problem with pointAis that Person 2's"nal holdings of numeraire are too small there (Ais close toD, where she has zero). The local principle states that e$ciency and large enough indi-vidual holdings of numeraire imply surplus maximization. Formally, we say that there are lower bounds on the numeraire ifXM "R
`. Then we have the following
result.
¸ocal equivalence principle for atemporal economies.Assume that there are lower bounds in the numeraire,and let(x;y)be a feasible allocation.
(a) If y maximizes S on>K ,then(x;y)is e.cient.
(b) ¹here exists a real number B such that,if(x;y)is e.cient and if x
i'B,for
Here I extend these well-understood ideas to an economy with overlapping generations. The paper has two parts: "nite time and number of generations (¹(R, Section 2 below), and in"nite time and number of generations (¹" R, Section 3 below). The results of the"nite case are similar to the ones in the atemporal case, except that the model has now one numeraire for each period: the global equivalence principle holds in the absence of bounds on the numeraires, and the local version holds if lower bounds are present (Theorems 1}3).
The in"nite case is more subtle. First, no e$cient allocations exist in the absence of bounds on the numeraires (Theorem 4).4The 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 pointAin Fig. 2). And a local claim of the type&e$cient and large enough individual holdings of the numeraires imply surplus maximi-zation'is then basically vacuous, because of the incompatibility between e$ -ciency and large individual holdings of the numeraires.
But the local equivalence principle is recovered in the¹"Rcase 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 byt) coexist at eacht,t"1,2,¹. By convention,
Generation 0 lives only in period 1 as &old,' and Generation ¹ lives only in period¹as&young.'Fort"1,2,¹!1, Generationtlives for two periods:
as a young generation in period t, and as an old generation in period t#1. There are N
t people in Generation t. A person is identi"ed as &Person i
of Generationt1(i"1,2,Ni,t"0,2,¹), and indexed by the double subscript
(i,t).
2.2. Numeraires,other goods,resources and technology
A nonproduced good speci"c to periodt, called t-numeraire, is available in
u
tunits,t"1,2,¹. We view thet-numeraire as a private, transferable,
divis-ible and nonstorable good. Denote byx
i,t~1,tthe consumption oft-numeraire by
Personiof Generation (t!1) (recall thatt!1 is the old generation in periodt), and byx
ittthe consumption oft-numeraire by Personiof Generationt(tis the
young generation in periodt).
Moreover, in each periodtthere areM
tadditional private goods (Mt50) and Q
tpublic goods (Qt50). Denote byyt an allocation of such goods, and write y"(y
1,2,yT). The set of vectorsythat could be made available, perhaps using
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"
nite-dimensional Euclidean space.5Society's technology is de"ned by>and the cost
function:
C:>PRT:yP(C
1(y),2,CT(y)),
understood as follows:&in order to make the vector yavailable, society must spendC
t(y) units oft-numeraire, fort"1,2,¹'.
This formulation is rather general. It includes, as a particular case, the situation where the only relevant arguments ofC
t are those inyt.
2.3. Consumption sets and utilities
The consumption set of Personi of Generationt, fort"1,2,¹!1, and
i"1,2,Ntis denotedX]>, whereX-R2. Her utility function is of the form:
u
it:X]>PR:uit(xitt,xi,t,t`1,y)"xitt#jt`1xi,t,t`1#vit(y),
where j
t`1 is a positive real number and vit is a real-valued function with
domain>.
5The dimension ofy
tis (Nt~1#Nt)Mt#Qt,t"1,2,¹. Thus,>is a subset of a (+Tt/1[(Nt~1#
N
Similarly, fort"0 andi"1,2,N0(resp.t"¹andi"1,2,NT), Person
(i,t)'s consumption set isXM ]>,XM -R, and her utility is:
u
i0:XM ]>PR:uit(xi01,y)"xi01#vi0(y)
(resp.u
iT:XM ]>PR:uiT(xiTT,y)"xiTT#viT(y)).
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 timetandt#1, respectively. Ifj
t`1"1, then the two numeraires are
perfect substitutes with MRS equal to one, in other words, individuals do not discount the future (except possibly for nonnumeraire goods). Ifj
t`1(1, then
j
t`1 can be interpreted as Generation t's time discount factor for the later
numeraire.
Remark 2. Note that noisubscript appears inj
t`1: in other words, all members
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$cient allocations exist (two members of the same generation with di!erent MRS's between the two numeraires could achieve in"nite utilities by exchanging one numeraire against the other). If, on the contrary, lower bounds on the consumption of numeraires are assumed, as in Section 2.9 below, then di!erentj's among members of the same generation would imply that no Pareto e$cient allocation is interior, and Theorem 3 below would be vacuous.
Remark 3. As noted in footnote 1, the only relevant arguments inv
itwill often be
the public goods made available in periodstandt#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
it(y)"witt(yt)#ki,t`1wi,t,t`1(yt`1),
wherek
i,t`1is a real number, andwittandwi,t,t`1are real-valued functions, with
the following interpretations:
f w
itt(y) denotes the bene"ts from vector yt, which accrue to Person i of
Generationtwhen young;
f w
i,t,t`1(yt`1) denotes the bene"ts from vectoryt`1, which accrue to Personiof
Generationtwhen old;
f k
i,t`1 is Person (i,t)'s time discount factor for the bene"ts from vector y
that accrue to her when old; possibly, but not necessarily,k
particular, one may well have k
i,t`1Okh,t`1 for two members i and h of
Generationt(compare with the previous Remark).
2.4. Feasible and ezcient allocations
Afeasible allocationis a vector (x;y), where
x"(x
101;2;xN0,0,1;x111,x112;2;xN1,1,1,
x
N1,1,2;2;xitt,xi,t,t`1;2;x1,T~1,T~1,
x
1,T~1,T;2;xNT~1,T~1,T~1,xNT~1,T~1,T,
x
1TT;2;xNT,T,T)3RH
whereH"N
0#2+Tt/1~1Nt#NT,xi013XM ,i"1,2,N0,xiTT3XM ,i"1,2,NT,
(x
itt,xi,t,t`1)3X,t"1,2,¹!1,y3>, and, fort"1,2,¹,+Ni/1t~1xi,t~1,t#+Ni/1t
x
ittThe following de#Ct(y)"ut. "nitions are standard.
A feasible allocation (x@;y@)Pareto dominatesanother 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 dominatesanother feasible alloca-tion (x;y) if every person is better o!at (x@;y@) than at (x;y).
A feasible allocation (x;y) isPareto ezcientif 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>Dthere exists anx3RHsuch that (x;y) is a feasible allocationN.
Adopt the notational conventions j0"j1": 1, and C
0(y) :"0, for all
y3>. Write, for t"0, 1,2,¹,K
t": j0j12jt. De"ne the social surplus
function:
S:>KPR:S(y)"+T
t/0
K t
C
Nt
+
i/1
v
The social surplus is a weighted sum of utilities, where the weight of Person
i of Generation t is K
t.6 Recall that these weights are simply (products of)
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 thej'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 withj's less than one. Let¹"3. Fort"0, 1, 2, 3, letN
t"1, delete the subscriptiin the notation and denote byutthe utility level
of the only person in Generationt. Fix a particular vectory6 (with associated surplus levelS(y6)), as well as utility levelsu60andu63. The utility mini-frontier for
y6 in the (u
1,u2) plane can be derived as follows. Becauseu63"x33#v3(y6),x33
must be a constant, and, hence, because, by feasibility,x
23#x33#C3(y6)"u3,
i.e., the slope of the utility mini-frontier is!(1/j2), with absolute value greater than one as long as Generation 2 discounts its future numeraire and, thus,
j2(1. See Fig. 3.
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. Let¹(R.If an allocation(xH;yH)is such thatyHmaximizes 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 maximizationNe$ciency.'It triv-ially implies the weaker statement:&surplus maximizationNweak e$ciency.'
2.7. A lemma
Write Zfor the set of positive integersM1, 2,2N, andH for the set of time indices, which is the"nite setMt3ZDt4¹Nin Section 2 and the in"nite setZin Section 3.
GivenyH3>K , fort3M0NXH,i"1,2,Nt, de"ne
Dv
it[yH]:>K PR:Dvit[yH](y)"vit(y)!vit(yH),
DC
t[yH]:>K PR:DCt[yH](y)"Ct(y)!Ct(yH), DS[yH]:>KPR:DS[yH](y)"S(y)!S(yH).
Denote byAthe set of strictly positive sequences of real numbers, one for each generation ("nite sequences of the form (a0,a1,2,aT) in Section 2, or in"nite
sequences of the form (a0,a1,2,at,2) in Section 3).
GivenyH3>K anda3A, de"ne the functions:
G
i,t,t`1[yH;a]:>KPR, t#13H,i"1,2,Nt, G
tt[yH;a]:>K PR, t3H,
by the following expressions (which leave implicit the symbols &[yH]' in the notation for theDfunctions):
G
i01[yH;a](y)"
K
0Dvi0(y)
K
1
!K0a0
K
1N0
DS(y),
G
11[yH;a](y)"
K
1DC1(y)!+Nh/10 K0Dvh0(y)#K0a0DS(y)
K
1N1
,
G
i12[yH;a](y)"
K
1Dvi1(y)
K
2
!K1DC1(y)!+Nh/10 K0Dvh0(y)#K0a0DS(y)
K
2N1
!K1a1
K
2N1
DS(y).
Note that
G i12"
K
1
K
2
C
Dv
i1!G11!
a1
N
1
DS(y)
D
;fort'1 de"ne
G
tt[yH;a](y)"
+tq/1K
qDCq(y)!+qt~1/0+Nh/1q KqDvhq(y)#DS(y)+tq~1/0Kqaq
K tNt
, (1)
G
i,t,t`1[yH;a](y)"
K t K
t`1
C
Dv
it(y)!Gtt[yH,a](y)!
a
t N t
2.8. The absence of bounds on the numeraires
Assumption 1. (Absence of bounds on the numeraires).X"R2,XM "R.
Theorem 2. Let¹(Rand postulate Assumption 1. If (xH;yH)is weakly Pareto ezcient, thenyHmaximizesSon>K .
Proof. Suppose, by way of contradiction, that there is ay@3>K withS(y@)'S(yH). Write:
a6" 1
+Tq/0K
q
, (3)
and de"nex@by:
x@
i01"xHi01!Gi01[yH;a6](y@), i"1,2,N0,
2
x@
itt"xHitt!Gitt[yH;a6](y@), x@
i,t,t`1"xHi,t,t`1!Gi,t,t`1[yH;a6](y@), t"1,2,¹!1, i"1,2,Nt,
2
x@
iTT"xHiTT!Dvit[yH](y@)#DS(y@)
a6
N T
, i"1,2,NT.
I claim that (x@;y@) strongly Pareto dominates (xH;yH). By Lemma 1(ii), all persons in generations 0 to¹!1 are better o!at (x@;y@) than at (xH;yH). So are all persons in Generation¹, because
u
iT(x@iTT,y@)"x@iTT#viT(y@)
"xHiTT!v
iT(y@)#viT(yH)#DS(y@)
a6
N T
#v
iT(y@)
"u
iT(xHiTT,yH)#DS(y@)
a6
N T
'u
iT(xHiTT,yH), i"1,2,NT,
becausea6'0 andDS(y@)'0. By Lemma 1(i), the equalities&+Nt~1
i/1x@i,t~1,t#+Ni/1t x@itt#Ct(y@)"ut'are
satis-"ed fort"1,2,¹!1. We are left with showing that this is also the case for
t"¹. Recalling that+NT~1
i/1xHi,T~1,T#+iN/1T xHiTT"uT!CT(yH), we compute NT~1
+
i/1
x@
i,T~1,T#
NT
+
i/1
x@
iTT#CT(y@)
"u
T#DCT(y@)!
NT~1 +
i/1
G
i,T~1,T[yH;a6](y@)! NT
+
i/1
Dv
"u
where (3) has been used. h
Remark 6. Theorem 2 states that&weak e$ciencyNsurplus maximization.'It trivially implies the weaker statement:&e$ciencyNsurplus 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 onuorC(y).
Assumption 2. (Lower bounds on the numeraires).X"R2
`andXM "R`.
Assumption 3. There exist positive numbersv6 andCM such that
Dv
it(y)D(v6 for ally3>K, i"1,2,Nt, t"0, 1,2,¹,
DC
t(y)D(CM for ally3>K, t"1,2,¹.
Remark 7. Assumption 3 is automatically satis"ed when the functions v
itand
C
tare continuous and the set>K is compact.
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 Pareto ezcient, and if xHi,t~1,t5B, and xHitt5B,i"1,2,Nt,t"1,2,¹, then
Proof. Because, by Assumption 3,v
itandCtare bounded from above and from
below on >K , there exists a real numberB such that, recalling (1) and (2) and
de"ningGMiT[yH;a6](y)"Dv
iT[yH](y)!DS(y)a6/NT, wherea6is de"ned in (3), for
allyH3>K andy3>K we have that
G
tt[yH;a6](y)(B, t"1,2,¹!1, G
i,t,t`1[yH;a6](y)(B, t"0, 1,2,¹!1,i"1,2,NT~1,
GMiT[yH;a6](y)(B, i"1,2,NT.
Let (xH;yH) be Weakly Pareto E$cient and, for t"1,2,¹, let
xHitt5B,i"1,2,Nt, andx*i,t~1,t5B,i"1,2,Nt~1. Assume, as contradiction
hypothesis, that there exists ay@3>K such that S(y@)'S(yH). De"ne:
x@
i,t~1,t"xHi,t~1,t!Gi,t~1,t(y@), t"1,2,¹, i"1,2,Nt~1,
x@
itt"xHitt!Gtt(y@), t"1,2,¹!1, i"1,2,Nt, x@
iTT"xHiTT!GM iT(y@), i"1,2,NT.
Because G
i,t~1,t(y@)(B,t"1,2,¹,i"1,2,Nt~1,Gitt(B,t"1,2,¹!1, i"1,2,Nt, and GMiT(y@)(B,i"1,2,NT, we have that x@i,t~1,t'0 and
x@
itt'0,∀i,∀t. Lemma 1 and the proof of Theorem 2 shows that (x@;y@) strongly
Pareto dominates (xH;yH). Contradiction. h
Remark 8. Theorem 3 states that &weak e$ciency and large enough xH'sN
surplus maximization.'It trivially implies the weaker statement:&e$ciency and large enoughxH'sNsurplus maximization'.
3. In5nite horizon(¹"R)
3.1. The model
Now I extend the analysis to the case¹"R. The set>as well as the range
ofCare now subsets ofR=. Now Generation 0 lives only in period 0 as old, but, fort3Z, Generationtlives for two periods, and the consumption set of any of its members isX]>-R2]>. In obvious extension, afeasible allocationis a
vec-tor (x;y), where:
x"(x
101;2;xN0,0,1;x111,x112;2;xN1,1,1,xN1,1,2;2;xitt,
x
i,t,t`1;2)3R=,y3>, and fort3Z,
Nt~1 +
i/1
x
i,t~1,t# Nt
+
i/1
x
itt#Ct(y)"ut.
As before, de"ne:
domination, in particular, means that all persons in all generations, i.e., an in"nite number of consumers, are better o!.
3.2. Absence of bounds on the numeraires
Theorem 4. Let ¹"R. Under Assumption 1 (absence of bounds on the numeraires),the set of weakly Pareto ezcient allocations is empty.
Proof. Given a feasible allocation (x;y), choose a positive numbereand de"ne the numeraire allocationx@by:
"+Nt
i/1
x
i,t,t`1#
Nt`1 +
i/1
x
i,t`1,t`1#Ct`1(y)
"u
t`1,
the last equality following from the fact that (x;y) is feasible.
Second, everybody prefers (x@;y) to (x;y), because fori"1,2,N0:
u
i0(x@i01,y)"xi01#
e
N
0j1
#v
i0(y)'xi01#vi0(y)"ui0(xi01,y),
and fori"1,2,Nt,t3Z:
u
it(x@itt,xi@,t,t`1,y)
"x
itt!
e
N
tj12jt
! e
N
tj22jt
!2! e
N tjt
#j
t`1xi,t,t`1
#j
t`1
C
e
N
tj12jt`1
# e
N
tj22jt`1
#2# e
N tjtjt`1
# e
N tjt`1
D
#v
it(y)
"x
itt#jt`1xi,t,t`1#
e
N t
#v
it(y)
'x
itt#jt`1xi,t,t`1#vit(y)
"u
it(xitt,xi,t,t`1,y).
Thus, (x@;y) strongly Pareto dominates (x;y). h
Theorem 4 motivates focussing on economies with lower bounds on the numeraires: Assumption 2 will be postulated (with explicit mention) for the remainder of the paper.
3.3. Zero discounting of future numeraires
Next I show that, if future numeraires are not discounted, then e$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 atK50such that j
t51for all t5tK. Let(x;y) be
a feasible allocation. If there exists an e('0such that+Nt
i/1xitt5e(, for allt3Z,
Proof. Step 1: Dexnition ofx@. This step constructs a vectorx@such that (x@;y)
n2converges to a real number, to be denoted
bya(a+1.645). Without loss of generality, lettK'2 and de"ne the consumption allocationx@ as follows.
Step 2: (x@;y) satisxes the feasibility constraints. For period t"1,2,tK, the
argument in the proof of Theorem 4 applies. Ift'tK, we compute
Nt~1
Because, by assumption,x
"x
by the same argument as in the previous case.
Corollary.Let¹"R,postulate Assumption 2 (lower bounds on the numeraires) and let there be a tK50 such thatj
t51for all t5tK. If(x;y) is weakly Pareto
ezcient, then, givene'0,there exists atI(e)such that+NtI(e)
i/1xi,tI(e),tI(e)(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
t:"j0j12jt and that, by convention, j0"j1:"1 and
C
0(y) :"0, de"ne
>H"
G
y3>K Dthe series +=t/0
K t
C
Nt
+
i/1
v
it(y)!Ct(y)
D
convergesH
,and thesocial surplus function:
S:>HPR:S(y)"+=
t/0
K t
C
Nt
+
i/1
v
it(y)!Ct(y)
D
.Assumption 4. There exist ajM(1 and atM'1 such thatj
t4jM, for allt5tM,t3Z.
Assumption 5. There exist positive numbersu6,NM ,v6 andCM such that
Du
tD(u6, for allt3Z, N
t(NM for allt3M0NXZ,
Dv
it(y)D(v6 for ally3>K , i"1,2,Nt,t3M0NXZ,
DC
t(y)D(CM for ally3>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
n) :"(k1,k2,2,kn,2)
"
A
K1
N0 +
i/1
x i01,K1
N1
+
i/1
x i11,K2
N1 +
i/1
x i12,K2
N2 +
i/1
x
i22,2,Kt
Nt~1 +
i/1
x i,t~1,t,
K t
Nt
+
i/1
x itt,Kt`1
Nt
+
i/1
x
i,t,t`1, . .
B
(ii) The series+=
t/1KtCt(y),+=t/1Ktutand+=t/0Kt+Ni/1t vit(y)converge.
(iii) The function S is well dexned on>K ,i.e., >H">K.
Proof. (i) The sequence (k
n) is nonnegative by the assumption of lower bounds
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~1
By the same argument
Nt
Thus, the sequence of partial sums is bounded and the series+=
(ii) By an argument similar to the one in (i), the series+=
t/1KtDCt(y)Dconverges.
Therefore,+=
t/1KtCt(y) converges absolutely and, thus, it converges. The same
argument proves the convergence of+=
t/1Ktutand+=t/0Kt+Ni/1t vit(y).
(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 andyHmaximizes S on>K ,then(xH;yH)
with at least one inequality strict. Multiplying both sides of each inequality in (7) byK
t we have that K
tx@itt#Ktjt`1x@i,t,t`1#Ktvit(y)7KtxHitt#Ktjt`1xHi,t,t`1#Ktvit(yH),
i"1,2,Nt, t3M0NXZ, (8)
with at least one of the inequalities in (8) strict. By Lemma 2(ii), the series+=t/0K
t+Ni/1t vit(y@) and+t=/0Kt+Ni/1t vit(yH) converge.
Denote their sums, respectively, by
v@andvH. (9)
Denote by (k@
n) (resp. (kHn)) the specialization to the feasible allocation (x@;y@)
(resp. (xH;yH)) of the sequence (k
n) de"ned in the statement of Lemma 2(i), and
denote its in"nite sum bym@(resp.mH). The sequence
A
Kn) by inserting parentheses (its"rst term isk@1, its second term
isk@2#k@3, itsmth term isk@2 sum of the left-hand side (resp. right-hand side) terms of (8) converges to
we obtain
is also obtained by inserting parentheses in (k@
n) (its"rst term equalsk@1#k@2, and
But feasibility implies that
K
which, using Lemma 2(ii), implies thatm@"+=
t/1Ktut!+t=/1KtCt(y@), or, using
(9) and the de"nition of social surplus:
m@#v@"+=
Pareto ezcient, and ifxHitt7BandxHi,t~1,t7B,i"1,2,Nt,t3Z,thenyH maxi-mizes S on>K .
Proof.
Step 1. Dexnition and properties of a(
0,2,a(t,2 Write b"1/2KtM~1; for
De"ne the sequence
(c1,c2,2,cn,2)
Similarly, the second term,c2, can be written:
Step 2. Boundedness of the functions G
tt and Gi,t,t`1 on >K. Consider the
functionsG
tt[yH;a(],t3Z, andGi,t,t`1[yH;a(],t3M0NXZ,i"1,2,Nt, as de"ned
by (1) and (2) above, foryHas 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
tt[yH;a(](y)(B, for all yH, y3>K, and all t3Z, and that Gi,t,t`1[yH;a(](y)(B,
for allyH,y3>K, allt3M0NXZand alli3M1,2,NtN. We leave implicit, in what
follows, the symbols [yH;a(]. Consider"rstG
tt,t3Z, and lety3>K . We write
I claim that the last expression is bounded from above and from below by bounds that are independent fromyH,yort. By (13), (1!+t~1
t/0Kqa(q)/KtNt4b.
By Assumption 5,DDC
q(y)D(2CM andD+hN/1q Dvhq(y)D(2NM v6. The argument
lead-ing to (6) shows that the positive sequences int (which are independent from
Consider now, fort3M0NXZandi3M1,2,N
tN,Gi,t,t`1[yH;a(](y) as de"ned by
(2). The previous argument together with Assumption 6 shows the existence of a real number bM such that, for yH,y3>K andt3M0NXZ,G
i,t,t`1[yH;a(](y)(bM.
WriteB"maxMb`,b~N.
Step 3. Contradiction argument.
Let (xH,yH) be Pareto e$cient, and let xH
i,t~1,t7BandxHitt7B,t3Z,
i"1,2,N
t. I claim thatyHmaximizesS on>K . Suppose not, i.e., let there be y@3>K withDS(y@) :"S(y@)!S(yH)'0. De"ne
x@
i01"xHi01!Gi01[yH;a(](y@),i"1,2,N0,
x@
itt"xHitt!Gtt[yH;a(](y@), i"1,2,Nt, t3Z. x@
i,t,t`1"xHi,t,t`1!Gi,t,t`1[yH;a(](y@).
BecausexHi,t~1,t(BandxHitt(B,t3Z, we have thatx@
i,t~1,t'0 andx@itt'0,
for all t3Z, i.e., x@
i013R` and (x@itt,x@i,t,t`1)3R2`, for all t3Z. Lemma 1 then
implies that (x@;y@) dominates, in the strong Pareto sense, (xH;yH), contradicting the fact that (xH;yH) is weakly Pareto E$cient. h
References
