Journal of Mathematical Economics 17 (1988) 29%308. North-Holland
VALUATION
EQUILIBRIUM
IN NON-CONVEX
Jean-Marc BONNISSEAU
AND PARETO
ECONOMIES*
OPTIMUM
and Bernard CORNET
CORE, VniversitP Catholique de Louvain, Louvain-le-Neuve, Belgium
CERMSEM,
VniversitP Paris 1 PanthPon-Sorbonne, Paris, France
Submitted November 1986, accepted January 1988
In this paper, we report an extension of the second welfare theorem when both convexity and
differentiability assumptions are violated. Our model allows various formalizations of the
marginal rule and considers the general setting of a topological vector space of commodities.
1. Introduction
The second welfare theorem for convex economies states that under
appropriate assumptions, at a Pareto optimal allocation, it can be associated
with a non-zero price ,vector such that each consumer minimizes his (her)
expenditure and each firm maximizes its profit or, equivalently, by a
redistribution of income, the optimal allocation can be obtained as a
Walrasian equilibrium. This result was firstly rigorously formulated and
proved by Arrow (1951) and Debreu (1951) in an economy with a finite
number of commodities and by Debreu (1954) to allow infinitely many
commodities. In the latter case, a topological vector space of commodities is
considered and Debreu’s result mainly rests on the assumption that the
aggregate production set of the economy has a non-empty interior.
When the convexity assumptions are violated, in particular, in the presence
of increasing returns, Guesnerie (1975) proposed to associate with a Pareto
optimal allocation, a non-zero price vector such that each consumer will
satisfy the ‘first-order necessary conditions’ for expenditure minimization and
each firm will satisfy the ‘first-order necessary conditions’ for profit maximization. Under convexity assumptions on the preferences of the consumers
and on the production sets ‘of the firms, the ‘necessary conditions’ become
also sufficient, so that the above statement coincides with the standard
(convex) formulation of the second welfare theorem. Several formalizations
and proofs of the above statement are now available for finite or infinite
*Support from the ‘Commissariat G&&al du Plan’ is gratefully acknowledged.
0304-4068/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)
294
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
dimensional commodity spaces. Apart from their generality, these results
differ in the way the ‘necessary conditions’ (or the marginal rule) are
mathematically formalized in the absence of convexity and differentiability
assumptions. For this formalization, Guesnerie uses the concept of tangent
(normal) cone of Dubovickii and Miljutin (1965), whereas Kahn and Vohra
(1984, 1985), Yun (1984), Quinzii (1986) and Cornet (1986) use the concept of
tangent (normal) cone in the sense of Clarke (1975) which was first
introduced by Cornet (1982) in the related problem of existence of equilibria
in non-convex economies. At this stage, it was worth pointing out that
Guesnerie’s result and the other ones (which use a different formalization of
the marginal rule) are not directly comparable, i.e., neither Guesnerie’s
implies one of the other results, nor the converse is true (see Remark 3.1).
We point out that, of all the above papers, Kahn and Vohra (1985) is the
only one to allow infinitely many commodities in the spirit of Debreu (1954)
and that furthermore they also consider an economy with public goods.
We consider here the general setting of a topological vector space of
commodities but we exclude the presence of public goods. Our first result
(Theorem 2.1) provides a (non-convex) version of the second welfare theorem
with the marginal rule formalized by Clarke’s normal cone. This theorem
generalizes the results of Debreu (1954) and Kahn and Vohra (1985) in the
absence of public goods. In an economy with a finite number of commod. ities, Theorem 2.1 also extends the results of Kahn and Vohra (1984),
Quinzii (1986) and Yun (1984). In our second result (Theorem 3.1), we
consider different ways to formalize the ‘first-order necessary conditions’ (or
the marginal rule), which include Clarke’s one and also the one of
Dubovickii and Miljutin used by Guesnerie. This will allow us to provide a
generalization of Guesnerie’s result (Corollary 3.1) in an infinite dimensional
setting and also to show the relationship between this result and the above
quoted ones.
The paper is organized as follows. In section 2, we recall definitions and
properties of Clarke’s tangent and normal cones which will be used later and
we state our first result, Theorem 2.1. In section 3, we state our second result
(Theorem 3.1) which implies the previous one and Guesnerie’s. Also some
additional definitions and properties of tangent and normal cones are
recalled. In section 4, the proofs of these results are given.
2. The model and a first result
We consider an economy with m consumers, n firms and we assume that
the commodity space L is a real topological vector space. We let Xi c L be
the consumption
set of the ith consumer and, for x=(x,, . . . , x,) in
.
.
.
x
x,,
we
let
P,(x) be the set of elements in Xi which are preferred to
X1 x
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
295
xi by the ith consumer. In other words, the preferences of the ith consumer
are described by a correspondence Pi, from X, x . . *x X, to Xi. We let Yrc L
be the production set of the jth firm and w in L be the vector of total initial
endowments.
Definition 2.1. An (m+n)-tuple ((xi’),($)) of elements of L is said to be a
feasible allocation if, (a) for all i, x~EX~, (b) for all j, J$’E Yj and (c)
CYl
xi*--&
yj=w = 0. An (m + n)-tuple ((XT), (yy)) of elements of L is said
to be a Pareto optimum (resp. a weak Pareto optimum) if it is a feasible
allocation and if there exists no feasible allocation ((xi),(yj)) such that, for all
i, XiE ClPi(Xr,. . . , x:), the closure of Pi(X:, . . . ,xz), and, for some i, E (1,. . . , m},
(resp., for all i,XiE P,(X:, . . . , Xz)).
Xi0 E Pi,(Xyy..
.) Xz)
In the following, we shall simply denote Pi(x:, . . . , A$) by Pff
Before stating our first result, we need to introduce some general
definitions together with the concept of tangent and normal cones in the
sense of Clarke. Firstly, we let L* be the set of all continuous real-valued
linear functions defined on L, and JV’be the set of open neighborhoods of 0 in L.
If x E L and p E L* we denote p(x) by p *x and, for Cc L, we let cl C, int C,
Co = {p E L* Ip *x 5 0, for all x E C} denote, respectively, the closure, the
interior and the negative polar cone of C. Let x be an element in cl C, we
then define the set H,(x) of hypertangent vectors to C at x, Clarke’s tangent
and normal cones, T,(x) and N,(x), to C at x, as follows:
~E>O,~NEA”,V~E(O,~),VX’E(X+N)~C:
H,(x) =
v E L(
x’+t(v+N)cC
Tc(x)=
1
VNNEN",~E~O,~N'E.N,
VtE(O,&),Vx'E(x+N')nC:
UELI
[x’+t(v+N)lnC#@
N,(x) = Tc(x)‘.
The above definitions of T,(x) and N,(x), due to Rockafellar (1979, 1980)
are extensions of the original ones of Clarke (1975) to the setting of
topological vector spaces. It is worth noticing that an element u in L belongs
to T,(x), if and only if, for all generalized sequences (nets) {x”} CC,
{tkj ~(0, + co) converging, respectively, to x and 0, there exists a generalized
sequence {u”}c_E converging to v such that, for all k, xk+ tkvkE C. Hence,
when L is a Banach space, the above definition coincides with the original
one, by Clarke (1983, Theorem 2.4.5). The definition of H,(x) is taken from
Clarke (1983) but it is different from the one in Rockafellar (1980) [who
considers a set, the interior of which contains H,(x)].
296
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
Definition 2.2. A subset C of L is said to be epi-Lipschitzian at x, an element
in cl C, if there exists an hypertangent vector to C at x, or equivalently, if
H,(x) is non-empty. The set C is said to be epi-Lipschitzian if it is
epi-Lipschitzian at every element x in cl C.
The following proposition gives two cases, of particular economic interest
under which a set is epi-Lipschitzian. It summarizes also the properties of the
above cones which will be used in the following. We recall that a subset C of
L is said to be a cone (with vertex the origin) if Ix belongs to C, for all ,I> 0
and XEC.
Proposition 2.1. Let C be a subset of L and let x be an element in cl C, then
the following assertions hold true.
(a) T,,,(x) = T,(x), NM(~) = N,(x) and H,(x) c H&x).
(b) T,(x) is a closed, convex cone, containing 0; H,(x) and H,,,(x) are open
convex cones (possibly empty) contained in T,(x).
is non-empty, then int T,(x) = H,(x)
[resp.
(c) 1. H,(x)
Cresp. Hd41
int T,(x) = H,,,(x)].
(d) Let us assume that C is convex. Then, Hc(x)={A(x’-x)II>O,
x’eint C},
x’EC} and Nc(x)={p~L*Ip~x~p*x’,
for all
Tc(x)=cl{A(x’-x+-O,
x’ E C}. Hence, C is epi-Lipschitzian if and only if C has a non-empty
interior.
(e) Let us assume that C+ Q c C, for some non-empty open cone Q. Then
Q c H,(x) and C is epi-Lipschitzian.
The proofs of (a), (b) and (e) are immediate, at the exception of the
fundamental properties, in (b), that T,(x), H,(x) and H,,,(x) are convex.
When L=R”, the convexity of T&x), as defined above, is proved by
Rockafellar (1979) who then noticed that his argument actually carries to an
arbitrary topological vector space L [Rockafellar (1980, p. 258)]. The
convexity of H,(x) and H,,,(x) is a consequence of the convexity of T,(x)
and of (c), which is exactly Corollary 2 of Rockafellar (1980). Assertion (d) is
Theorem 1 of Rockafellar (1980) when L is locally convex; a proof of it,
together with a more general assertion (Proposition 3.,l.d) will be given in
section 4 when L is an arbitrary topological vector space. Finally we point
out that all the assertions of the proposition are proved in the book of
Clarke (1983) when L is a Banach space.
We now state the main result of this section.
Theorem 2.1. Let ((xf),(yT)) be a weak Pareto optimum (resp. a Pareto
optimum and assume m > 1) such that, for all i, x: E cl Pr and, either, for some i,
J.M.
Bonnisseuu
and 8. Corner,
Equilibrium
and optimum
in non-convex
economies
297
Pi* (resp. cl Pr) is epi-Lipschitzian at xr, or, for some j, Yj is epi-Lipschitzian
at yj* .
Then, there exists a non-zero price vector p* in L* satisfying
(a) for all i, -p* E fVp;(xr);
(b) for all j, p* E Nyj(yf).
The proof of Theorem 2.1 will be given in section 4 as a consequence of a
more general result, Theorem 3.1, which will be stated in the next section.
We notice that, if Pr is epi-Lipschitzian at xf, then cl Pr is also epiLipschitzian at XT, since Hp;(x:)cH,,p:(x:)
[Proposition 2.1(a)], but the
converse is not true, in general, even in the convex case. We also point out
that, if m= 1, the notions of Pareto optimality and of weak Pareto optimality
are identical.
At this stage, it is worth discussing the link between Theorem 2.1 and
previous work in the literature on the same subject. The above theorem
generalizes a previous result of Kahn and Vohra (1985) who make the
stronger assumptions that, for all i, P: is epi-Lipschitzian and, for all j, Yj is
epi-Lipschitzian and that the commodity space L is ordered and locally
convex. When L is finite dimensional, Theorem 2.1 can be generalized in
several directions, for which we refer to Cornet (1986). However, Theorem
2.1 also generalizes the results of Kahn and Vohra (1984), Quinzii (1986) and
Yun (1984) who all assume that L= [w’, and that, for all i, Pi* + lfX$cP~
and, for all j, Yj- lQ’+c ri [which, by Proposition 2.1(e), implies that the sets
Pi* and q are epi-Lipschitzian].
It is worth also pointing out that our
definition of Pareto optimality is slightly different from the one used in Kahn
and Vohra (1984, 1985), Quinzii (1986) and Yun (1984), who assume
implicitly a free-disposal assumption in their definition, i.e., the condition (c)
in Definition 2.1 is replaced by (c’): X7= i XT-I;= i yj*--o 5 0, which requires
that the space is ordered [see the discussion in Cornet (1986), which is also
valid in the infinite dimensional setting]. The link with the result of
Guesnerie (1975) will be discussed in the next section.
Remark 2.1. We now assume that &=((Xi, Pi),(yi), w) is a convex economy,
in the sense that, for all i, all x in niXi, P,(x) is convex and, for all j, 5 is
convex. Then, by Proposition 2.1(d), the conditions (a) and (b) hold if and
only if
(a’) for all i, XT minimizes p* . xi over cl PI, and
(b’) for all j, yf maximizes p* .yj over the production
set 5.
Hence, in the terminology of Arrow and Hahn (1971), ((xr),(yr)) is a
compensated equilibrium, also called price quasi-equilibrium by Mas-Cole11
(1985). For the (standard) way to go, from price quasi-equilibria to price
298
J.M.
Bonnisseau
and B. Cornet,
Equilibrium
and optimum
in non-convex
economies
equilibria with respect to a price system [Debreu (1959)-J, i.e., from expenditure minimization to preference maximization by the consumers, we refer to
the three above books.
From the above remark, Propositon 2.1(d) and Theorem 2.1 (applied to
the economy &‘=((Xi, Pi), I;= I 5, w)) one deduces the infinite dimensional
version of the second welfare theorem for convex economies.
CoroZZary2.1. [Debreu (1954)]. The Pareto optimum ((xl),(yf))
is a price
quasi-equilibrium if, for all i, xf ECI PF, Pf’ is convex, c;= 1 5 is convex and,
either cj”=l Yj, or Pr, for some i, has a non-empty interior.
Remark 2.2. A second case of particular economic interest, under which
either Pr or Yj may be epi-Lipschitzian is given by Proposition 2.1(e) and is
pointed out by Kahn and Vohra (1985). Let us suppose that the commodity
space L is an ordered topological vector space whose positive cone Q has a
non-empty interior. Then rj is epi-Lipschitzian if q-Q c Yj, which is the
standard free-disposal assumption, and PT (resp. cl P”) is epi-Lipschitzian at
x* if PT + int Q c P: (resp. cl P* + int Q ccl P:), which is satisfied if the
preferences are monotonic and transitive. The assumption that the positive
cone Q has a non-empty interior is satisfied when (i) L is finite dimensional
or (ii) L= L, is the space of bounded functions on a measure space, endowed
with the supremum norm, a commodity space first considered by Bewley
(1972) in the related problem of existence of equilibria.
When the positive cone has an empty interior, as it is the case in several
examples of ordered commodity spaces of economic importance [see MasCole11 (1986)], the above discussion is no longer valid. One can expect,
however, to have, in the non-convex case, results analogue to the ones of
Mas-Cole11 (1986) in the convex case. In this paper we do not consider this
question and we only provide non-convex analogues of Debreu’s (1954)
result.
3. A further result and the link with Guesnerie’s
Before stating the main result of this section we first need to introduce
some additional definitions of tangent and normal cones (which will be,
respectively, larger and smaller than Clarke’s cones). If C is a subset of L and
x is an element of cl C we let
k,(X)={d+E>o,
t,(x)={27ELlVNEN,
3NEJv-,
VtE(O,E):X+t(U+N)cC},
3.5>0, VtE(O,&):[x+t(u+N)lnC#121},
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
n,(x) =
299
tc(x)O.
The first set kc(x) is the cone of interior displacement of Dubovickii and
Miljutin (1965), tc(x) is a related notion of tangent cone and n,-(x) is the
negative polar cone of t=(x). The above definitions are very much the
analogue of the ones of H,(x), T,(x) and N,(x) in section 2. There is,
however, a major drawback, since the cones kc(x) and tc(x) are no longer
convex, in general, and we have lost for these cones the fundamental
convexity properties of the cones H,(x) and T,-(x), which make them so easy
to work with.
The next proposition summarizes the properties of these cones, which will
be used in the following.
Let C be a subset of L and let x be an element of cl C, then
.assertions hold true.
Proposition 3.1.
the following
(a) t&x) is a closed
(possibly empty).
cone,
(b) H,(x) = k,(x) = k,,,(x)
containing
0; k,(x)
= int tc(x); H&x)
(c) T,(x)ct,(x)=t,,,(x)cc1{l(x’-X)(;1>O,X’EC},
N,(x) 3 ne(x) =n,,,(x) 3 {pe L*Ip.x2ppx’,
and k,,,(x)
= k&x)
are open
cones
= int tc(x).
for all x’E C>.
(d) Let us assume that C is convex. Then all the inclusions in (c) are equalities
C is epiand H,(x) = k,-(x) = {A(x’-x)11>
0, X’E int C}. Furthermore,
Lipschitzian tf and only tf C has a non-empty interior.
The proofs of (a), (b) and (c) are straightforward. The proof of (d) [which
implies assertion (d) of Proposition 2.11 will be given in section 4.
We can now state the main result of this section.
b e a weak Pareto optimum (resp. a Pareto
Theorem 3.1. Let* ((xt),(yj*))
optimum and assume that m> 1) such that, for all i, x: ~cl Pf. Let T
(i= l,..., m), Tj (j= l,..., n) be convex cones of L such that for all i,
Tct,;(xT),
for all j, Tjc tYj(yr), and
either,
for some i, @ # int T c 1+(x1) [resp. kc&x:)],
(I)
{ or,
for some j, 0 # int Tj c k,,(yT).
Then, there exists a non-zero price vector p* in L* satisfying
300
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum
for all i, -p* E Ty
and
in non-convexeconomies
for all j, p* E Ty.
We postpone to the end of this section the discussion on the assumptions
of the theorem and we now give some examples of cones q, Tj
The first example of convex cones T, Tj is given by Clarke’s tangent cones
IT;:=Tr;(xF) (i=l,...,
m), Tj=Tr,(yj*) (j=l,...,
n), which are subsets of tr;(x:)
and tr,(yj’), respectively. Theorem 2.1 is in fact a direct consequence of
Theorem 3.1 and the proof of Theorem 2.1 given in section 4, will consist in
checking that the epi-Lipschitzian assumption implies the above condition
(I).
The second example of cones IT;:,Tj is given by the cone of interior
displacements T = kr$xf) (resp. k,, r;(xT)), q= k,,(yf), in a situation where
they are convex. This is exactly the framework considered by Guesnerie
(1975). So, from Theorem 3.1, one deduces the following result, which for
finite dimensional commodity spaces, is exactly Theorem 1 of Guesnerie
(1975).
Corollary 3.1.
Let ((xr),(yj*)) be a weak Pareto optimum (resp. a Pareto
optimum and assume m > 1) such that, for all i, x: E cl Pi* and, for all i,j, the
sets kr;(xf) (resp. k,,r;(xt))
and kYj(yf) are convex. Then, there exists a
non-zero price vector p* in L* satisfying
(a’) for all i, -p* E [k&x:)]’
(b’) for all j, p* E [k,(yJ)]O.
(resp. [kclp;(xr)]o),
Remark 3.1.
We point out that the following inclusion int Trj(yj*) c kYj(yf)
may be strict, even when k,,(yj*) is non-empty, convex, and int T,,(yj*) is
non-empty. Hence, one may have [kyj(y~)]ofNYj(y~)
and the conclusion of
Corollary 3.1 may be stronger than the one of Theorem 2.1. To show this
assertion, consider the production set Yr in fig. 2, for which kr,(O)=
((x, y) 1y < min { -x, -(1/3)x}}, and int T,, (0) = int W”,.
Our third and last example of cones Ti, Tj is given by
Tj= trj(yj*) in a situation where they are convex. So we can state
Ti= tp;(x*),
Corollary 3.2. Let ((xF),(yr)) be a weak Pareto optimum (resp. a Pareto
optimum and assume m> 1) such that, for all i, XT ECU
P:, for all i, j, tr;(xr)
and tr,(yy) are convex and
(I bis)
either,
or,
for some i, @ #int tp;(xF) = kr;(xF) (resp. k,lP;(xi*)),
for some j, @ #int tr,(yT) = krj(yf).
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
301
Then, there exists a non-zero price vector p* in L* satisfying
(a”) for all i, - p* En&x:),
(b”) for all j, p* E nYj(yj*).
Remark 3.2. We also point out that the cones kYj(yT), kp:(xT) (resp.
tlj(yT), tp;(xt)) may not be convex and the above analysis is no longer valid.
This is the case for the following production set, of particular economic
importance,
Y={(yl,y,)~lRZIyl~O
and y,smax{O,-1
-yl}>
at y*=
(- 1,0) (where it has an ‘inward’ kink). However, it is easily shown to be epiLipschitzian and Theorem 2.1 can be applied; in other words, Pareto optima
can be ‘supported’ by prices in Clarke’s normal cone, but not in a smaller
cone, as in (b’) or (b”).
Remark 3.3. The following example shows that it is not possible, in
Corollary 3.2, to weaken assumption (I bis) by only assuming that, for some
j, int tYj(yT) #QI [or k,,(y,*) #a]
or, for some i, int tp;(x:) #fzr [or
kpf(xl)#$Z5]. In other words, the analogue of Theorem 2.1, with Clarke’s
cones replaced by the cones tp:(xr) and t,,(yT), does not hold, in general,
even under the assumptions that all these cones are convex. The same
example also shows that it is not possible, in general, to weaken assumption
(I) by only assuming that either, for some i, int T,cint t,:(xF), or, for some j,
int qcint tYi(yj*).
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Fig.
1
302
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
We consider the following economy with two goods, o = (2, f) as the vector
of initial endowments, one firm and one consumer, respectively, with a
production set Y and complete, transitive preferences, as represented on fig.
1.
Let XT=( 1, l), y: =( - 1, i). Then (x:, y:) is a Pareto optimum but one
easily sees that k,(y$ = l’@\{(O,y) 1y ZO}, hence tr(yr) = R2 and nr(y:) = (0).
Hence, the conclusion of Corollary 3.2 cannot hold. At this point, it is worth
stressing that, if L is finite dimensional, and if Y is closed, then at every
element y in aY, Clarke’s normal cone is not reduced to (01, i.e., N,(y)#(O}
[Clarke (1983, Corollary 2 of 2.5.5)].
The result of Theorem 3.1 is no longer true, in general, when
the tangent cone t=(x) is replaced by Bouligand’s tangent cone, T,-(x), which
we recall, is defined for a subset C of L and x E cl C as follows:
Remark
3.4
Fig. 2
J.M.
Bonnisseau
and B. Cornet,
Equilibrium
and optimum
in non-convex
economies
303
We
consider the following economy with two goods, o=(l, 1) as the
vector of initial endowments, two firms and one consumer. The production
sets of the firms, Y, and Y2 are represented in fig. 2. The consumption sets of
the consumer is I@+ and his preferences are represented by the utility
function ~(x~,x~)=rnin{x~,x~}.
Let x:=(1, l), y:=O and y:=O. Then
(x7, y:, y:) is a Pareto optimum but [Uy,(y:)]’ A [U,,(yt)]’ = (0).
4. Proofs
We begin with the proof of Theorem 3.1 from which we shall then deduce
the other results. We prepare its proof by a lemma.
Lemma 4.1. Let Ci, i=l,...,
k be arbitrary
elements in Cl Ci (i= 1,. . . , k) such that If= 1 X:
0 $ k&3
subsets
$
of L, and
let xf’ be
int C, + C:= 2 Ci. Then
+ i kh%+9.
i=2
Proof: By contraposition. Let us suppose that O=v,+u,+~~~+o,,
for some
u1 in k,-,(xf), and some vi in tc,(x:), i=2,. . . , k. Since v1 belongs to k,,(x:),
there exists .q >O and N1 EJV such that
x: + t(u, + N,) tint C,
for
all c E(O, Ed).
Since the set {(~,)EL~-~~~~=~x~E--N~}
exists N2~.M such that N2+...+N2c-N,.
t,-,(x7), there exists .q>O such that
[XF+t(Ui+N2)]nCi#0
for
is open and contains 0, there
Since, for i> 1, Ui belongs to
all tE(O,EJ,
i=2,...,k.
Let t,>O, to<min{eili=l,...,
k}; from above, we can choose, for i> 1, Xi
in the set [xi* + to(vi + N2)] n Ci and we let x1 =x7 +I!= 2 (xi*-Xi).
We claim that x1 belongs to int C,. Indeed, from above,
x,=x:+to
i
i=2
(x;-xi)/toEx:--o
i
(u i+N2)
i=2
cx~+t,(o,)+N,
tint C,.
304
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
Hence,
we
have
proved that xf=, x~=~~=i xi, x1 dint C,, xi~Ci
with the assumption of the lemma. This ends the
k), a contradiction
(i=2,...,
proof of the lemma.
3.1. Let ((XT), (yf)) b e a weak Pareto optimum (resp. a
Pareto optimum and assume m > 1) such that, for all i, XTECI P:. We first
assume that, for some i,E{l,. . . , m), say i,= 1,
Proof of Theorem
$3 #id
Tl c kp:(xf)
[resp. kclp;(xT)].
From the definitions of Pareto optimality one deduces that
o= f
xr-
i
i=l
resp.w= i
[
yr$intP:+
f
Pf-
i=2
j=l
XT- i
i=l
i
q;.,
j=l
yr $intclPT+
i
Pr-
i=2
j=l
f
j=l
q
1
.
But, for all i, x: E cl P: and, for all j, yr E 5, hence, by Lemma 4.1, noticing
that, for Cc L and x E cl C, one has trc( -x) = - tc(x),
0 $b,(XT)+ f t&T) - i
j=l
i=2
resp.O$k,,,;(x:)+
f
tYj(yT)
tp$xF)-
i=2
f
Recalling that, for all i, tp$xt) contains
contains Tj and 0, one deduces that
O$TcfintT,+
2
i=2
Tiu{O}-
1
tYj(yf) .
j=l
f
T and 0, and, for all j, tyj(yf)
Tju{O}.
j=l
But T is a non-empty open convex subset of L, since int Tl is non-empty
(and open), and T is the sum of non-empty convex subsets of L noticing that,
if T is a convex cone (possibly empty), then T u (0) is a non-empty convex
cone. Consequently, by Hahn-Banach’s
theorem [Bourbaki (1966, 11.5.2.,
Prop. l)], there exists a non-zero price vector p* in L* satisfying
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex
Ojp”(~~~i-~~~j)
for
all t,EintT,,
economies
tiEqU{O}
(i=2,...,m),
tjE?;U{O}
0’=1,...,n).
305
Clearly, the above inequality also holds for all t, E T1u(O), since p is
continuous and r, u (0) ccl T1 = cl int T1 [Bourbaki (1966, 11.2.b, Corollary
I)], recalling that r, is convex with a non-empty interior. Taking in the
above inequality, all the ti =O, tj=O, but one, we deduce that, for all i, j,
02 -_~*.t~
for
all CiE~,
and
O~p*.tj
for
all tj~ Tj,
or, equivalently, -p* E Ty (i= 1,. . . , m) and p* E Ty (j= 1,. . . , n). We leave to
the reader to adapt the above proof to the case where the assumptions of the
theorem are satisfied for some firm j. This ends the proof of Theorem 3.1.
At this stage it is worth pointing out the link between our proof and other
ones. The proof of Theorem 3.1 is similar to the ones of Guesnerie and
Quinzii, whereas Khan and Vohra use a product argument. We point out,
however, that Guesnerie (Quinzii) considers only the cone of Dubovickii and
Miljutin (resp. the interior of the cone of Clarke) to Pi* at xi* (i= 1,. .., m)
and to 5 at yj* (j=l,.,.,
n) since all these cones are assumed to be
non-empty. Here, the use of the closed cones tp;(x:) and tYj(yr) allows us to
make the interiority assumption only for one consumer, or one firm.
Proof of Theorem 2.1. It is deduced from Theorem 3.1 by taking T=
Tp;(xT), Tj= T,,(yj*) (Clarke’s tangent cones) and by checking that these cones
satisfy the assumptions of Theorem 3.1. Indeed, for all i,j, Tp;(xT) and T,,(yT)
are convex cones (Proposition
2.1) and Tp;(x:) c tp;(xF), T,,(yj*) c tYj(yj*)
(Proposition
3.1). Furthermore,
if, for some i, Pr [resp. cl Pr] is
epi-Lipschitzian
at xf, i.e., if HPf(xF) [resp. Hclpf(x$] is non-empty,
then,
by Proposition
2.1(c), @ # Hp;(x~) = int Tp$xr) = int T
[resp.
0 # H,,,;(xT) =int T,ip;(xF) =int Tp;(xr) = int Ti] and, by Proposition
3.1,
int Ti = Hp$xT) c k&x:) [resp. int Ti = int H,,&~) c kclp;(x:)]. Similarly, if,
for some j, Yj is epi-Lipschitzian at y;, then 0 =int Tjc k,:(yj*). This ends
the proof of Theorem 2.1.
Proof of Proposition 3.1, Part (d). Let C be a convex subset of L and x be
an element in cl C. To prove that the inclusions in (c) are equalities, since
306
J.M. Bonnisseau and B. Cornet, Equilibrium and optimum in non-convex economies
T,(x) is a closed subset of L, it suffices to show that the following inclusion
holds:
Indeed, let u= A(y-x), for some A>O and some YE C. We recall that u
belongs to T,(x) if for all NE JV there exist E> 0 and N’ E .M such that
[x’+t(u+N)]nC#@
for
all x’E(x+N’)nC,
all t~(O,.z),
and we check that the above statement holds for E = l/1 and N’ = -EN.
Indeed, for X’E(x + N’) n C and t ~(0, E), since C is convex and y belongs to
C,
Ex’+t(u-c-‘N’)cx’+t(u+N).
Hence UE T,(x), which ends the proof of the first part of(d).
We now prove the second part of (d). First, one easily checks from the
definitions that
and the proof of (d) will be completed by showing that
{A.(y-x)(A>O,yEintC}cH,(x).
Indeed, let u =A.(y-xx) for some 2 > 0 and some y E int C. Then, there exists
a neighborhood N E JV such that y + N c C. We now choose N’ E JV such
that N’ + N’ c N. Since HA(x) is a cone, to prove that u = A(y-x) E H,(x), it
suffices to show that y-x EH,(x) and, from the definition of H,-(x), that
x’+te’~C
for
all te(O, l),
Indeed, let t, x’ and e’ as above, then
x’ + te’ = t(x’ + e’) + (1 - t)x’
Et(x+N’+y-x+N’)+(l--)x’
Et(y+N)+(l--)x’EC,
x’~Cn(x+N’),
e’~(y-x)+N’.
J.M. Bonnisseau
and B. Cornet,
since y+ N c C, t l(0,l)
proposition.
Equilibrium
and optimum in non-convex
and C is convex.
economies
This ends the proof
307
of the
Remark 4.1. It is worth noticing that if C is a convex subset of L and,
either C has a non-empty interior, or L is finite dimensional, one further has
the equalities
H,(x) =k,(x) = {A(y-x)/I
>O, ycint C} =int T,(x).
The above equalities are consequences of Proposition 3.1(d) and the fact
that H,-(x) = int T,-(x), either when E is finite dimensional [Rockafellar (1979)]
or when H,(x) is non-empty [Rockafeller (1980)] and, by Proposition 3.1(d),
H,_-(x)is non-empty if and only if C has a non-empty interior.
It is also worth pointing out that (even when C is convex) the inclusion
H,(x) c int T,(x) may be strict [see Counterexample 1 of Rockafellar (1979)].
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