WITHOUT THE LOWER SEMI-CONTINUITY OF THE CONSTRAINT MAPS.
Adib Bagh
Department of Mathematics.
Humboldt University-Berlin.
Abstract. We use graph convergence of set valued maps to show the existence of an equilibrium for an abstract economy without assuming the lower semi conti- nuity of the constraint maps.
Keywords: abstract economy, equilibrium, set convergence, set valued maps, L-majorized maps, Walrasian equilibrium.
J.E.L. classication :C62, D50.
A. Bagh, Humboldt University, Inst. of Mathematics, Unter den linden 6, D-1109 Berlin Germany email: bagh@mathematik.hu-berlin.de fax:49-30-20932232
1 Introduction
In 1975 Shafer and Sonnenschein 1975] proved the existence of equilibria for abstract economies without ordered preferences. Over the last twenty years, more general existence results appeared in the literature. To make a partial list of these results we mention Borglin and Keiding 1976], Yannelis and Prabhakar 1983], Toussaint 1983], Yannelis 1987], Tarafdar 1989], and Tan and Yuan 1994]. All of these results however, assume directly or indirectly the lower semi-continuity of the set valued map representing the constraints for each agent.
In this paper, we will use the concept of graphical convergence of set-valued maps to show that an equilibrium for an abstract economy exists without such an assumption.
Eliminating the requirement of the lower semi-continuity of the constraint map is not just technical improvement. It will actually make the setting of the abstract economy more realistic as we will demonstrate at the end of the paper.
We start by reviewing the mathematical and the economical concepts that we need. Then, we prove several results regarding the existence of equilibria for abstract economies without assuming the lower semi-continuity of the of the constraint map. Finally, we discuss some of the economical implications of the new results.
Preliminaries
In this section, we review some basic denitions and theorems regarding set convergence and set valued maps.
Set convergence
Let X be a topological vector space. Let 2X be the collection of all subsets of X. For a subset K in 2X we use the following notation:
clK denotes the closure of K in X. intK denotes the interior of K in X. conK denotes the convex hull of K.
Let C(X) be the collection of closed nonempty subsets of X. We are interested in set convergence in C(X). Recall that a set D is directed by a relation , if is reexive and transitive and for all 1 and 2 in D, 9 3 2D such that 3 1 and 3 2. We dene the collection of co nal subsets, N1(D), and the collection of residual subsets , N#(D), of a directed set D:
N
1(D) =fN DjN contains all the elements of D at or beyond some index 2Dg
N
#(D) =fN DjN contains some elements of D at or beyond each index 2Dg:
Let fC j 2Dgbe a net in C(X). We dene the Inner limit, Li, and the Upper limit, Ls:
LiC =fx2Xj9N0 2N1(D) such that x 0 2C 08 0 2N and x 0 ;!xg
2 LsC =fx2Xj9N0 2N#(D) such that x 0 2C 0 8 0 2N0 and x 0 ;!xg where in both denitions x 0 ;!x indicates the convergence of fx 0g to x in X. We say that C converges toC in C(X), and we write C ;!C, when LiC = LsC = C. This notion of set convergence is referred to in the literature as the Painleve-Kuratowski set convergence. In general the Painleve-Kuratowski set convergence is topological if and only if X is locally compact (cf. Klein and Thompson 1984]).
The following is a well known lemma
Lemma 1.1. LetfC j 2Dgbe a net inC(X) that is decreasing with respect to inclusion,
2 1 in D implies C 2 C 1. Then,
LsC \
2D
C
Proof. Let x 2 LsC , then 9N0 2 N#(D) such that 8 0 2N0 x 0 2C 0 and x 0 ;! x.
C are decreasing with respect to inclusion, hence for every 0 2D we have
x 0
2C
0 8
0
2N
0 such that 0 0:
However, C 0 is closed and hence x2C 0. Since this is true for any 0 2D, we get
x2
\
2D
C :
For more on set convergence, see Beer1993] and Klein and Thompson1984].
Set valued maps
Let X and Y be two topological vector spaces. We write F : X !! Y to indicate a set valued map from X to subsets of Y. We say F is convex-valued, closed-valued or open- valued, if 8x 2 X, F(x) is convex, closed, or open respectively. We dene the domain of
F, domF, to be
domF =fx2XjF(x)6=g:
If domF is all of X, then we say F is nonempty-valued. Let U be a subset in X. FjU denotes the restriction of F to U. The graph ofF, gphF, is a subset of X Y dened as follows
gphF =f(xy)2X Y jy2F(x)g:
We say F is open (resp. closed), if the graph of F is open (resp. closed) in X Y. Recall that F : X !! Y is lower semi-continuous(lsc.), if for any open set G in Y the set
3
fx2XjF(x)\G6=gis open in X. F is upper semi-continuous(usc.), if for any open set
G in Y the set fx 2 XjF(x) Gg is open in X. Furthermore, we say F has open lower sections(ols.), if for any y 2Y, the set fx 2Xjy 2 F(x)g is open in X. Similarly, we say
F has upper open section(uos.), if for all x 2X, the setF(x) is open in Y.
We write clF to indicate the map dened by clF(x) = cl(F(x)) 8x 2 X. Similarly, we write conF to indicate the map dened by conF(x) = con(F(x)) 8x 2 X. Let F and
G be two set valued maps from X to subsets in Y, then the map F \G is dened by
F \G(x) =F(x)\G(x) 8x2X.
We will need the following facts about maps from X into subsets of Y, Fact 1. F is open implies that F is ols., which in turn implies that F is lsc.
Fact 2. If F is lsc., then domF is open in X.
Fact 3. If Y is compact, then F is closed valued and usc. if and only if F is closed. In particular, if F is such that clF is usc., then clgphF = gphclF.
Fact 4. If F is lsc., then so are clF and conF. Fact 5. If F is open and G is ols., then F \G is ols.
Fact 6. If F is open and G is lsc., then F \G is lsc.
The proofs of Facts 1 and 2 are fairly obvious. The proofs of Facts 3 and 4 can be found in chapter 11 of Border1985] . The proofs of Facts 4, 5, and 6 can be found in Yannelis and Prabhakar 1983] and Yannelis 1987].
Finally we state a key lemma that we will use in proving our main results.
Lemma 1.2. Let fF j 2 Dg be a net of set valued maps form a topological space X to subsets of a topological spaceY. Suppose LsgphF gphF and suppose F is closed. Let
fx 0
g and fy 0g be such that y 0 2 F 0(x 0) 8 0 2 N0 2 N#(D): Furthermore, suppose that x 0 ;!x and y 0 ;!y. Then,y 2F(x).
Proof. Clearly, (x y ) 2 gphF and (xy) 2 LsgphF by the denition of the Upper limit of F . Hence, (xy)2gphF.
Selection theorems and abstract economies
Most equilibria results in abstract economies are based on selection theorems that lead to x point theorems. We recall that for a set valued map F, a selection is a continuous
4 function f such that 8x 2 X f(x) 2F(x). We say F has a xed point, if 9x 2 X such that x 2 F(x). We use ; = (XiAiPiI) to denote an abstract economy. I is the set of agents. Xi is the strategy set for the ith agent. Let X = Qi2I Xi, and for any x2 X, let xi be the ith component ofx. Pi :X !!Xi is a map that represents the preferences of the ith agent. Ai :X !!Xi is a map that represents the constraints for the ith agent. An equilibrium for this economy is a point x 2X such that for all i2I,Ai\Pi(x) =and
x
i 2 clAi(x): The main result
The idea that we use for all the proofs in this section is the same and it is fairly simple.
Instead of assuming the lower semi-continuity of the constraint maps, we \fatten" the graphs of theses maps. The \fattened" maps have open graphs, hence they are ols. and lsc. We apply the existing equilibrium results to the \fattened" maps and then we use Lemmas 1.1 and 1.2 to show that the equilibrium points of the \fattened" maps yield an equilibrium point for the original problem. Let X and Y be topological spaces. We index the local base of the zero element of the spaceX Y with an ordered setD. Thus for any two elements V 1 and V 2 of the local base at the zero element of X Y, we have
V 2
V 1
()
1
2
where 1 and 2 are elements in D. Let S : X !! Y. We can construct the maps
S :X !!Y, where
gphS = gphS+V : We also can construct the maps S :X !!Y, where
gph S = clgphS :
S is an open map for every 2D and clearly A is a closed map for every 2D. Lemma 1.3. Let S : X !!Y, where X and Y are compact topological vector spaces. Let
S :X !!Y be a collection of set valued maps de ned by gphS = gphS+V : Then,
LsgphS cl(gphS) where the upper limit is taking over all 2D.
Proof. In light of Lemma 1.1, we only need to show that \ 2DgphS clgphS. Let
x 62 clgphS. Since cl(gphS) is compact, 9 0 2 D such that x 62 gphS 0 and therefore
x62\
2DgphS .
5 We now prove a result that is based on a theorem of Tan and Yuan 1994] but with weaker conditions on the constraint maps. First, we recall the denition of locally L-majorized maps. We say that ~S : X !! Y is of class L, if ~S is ols. and 8x 2 X, x 62 con ~S(x). We say thatS :X !!Y is locallyL- majorized, if 8x2X, there exists open neighborhood Vx
of x and a set valued map ~Sx of class L such that S(x0)2 S~x(x0) 8x0 2Vx. We say that
S is L;majorized, if there exists an L-class map ~S such that S(x) S~(x) 8x 2X. We immediately have the following two lemmas
Lemma 1.4. LetX andY be topological spaces. Let S :X !!Y be a locallyL;majorized set valued map. Let T :X !!Y be an open map. Then, S\T is L-locally majorized.
Proof. Letx 2domS\T. Since S is locallyL-majorized, there exists a neighborhoodOx
of x and a class L map ~Sx such that S(x) S~x(x) 8x 2 Ox. ~Sx \T is ols. by Fact 5.
Furthermore, S \T(x) S~x\T(x) and x 62 con(~Sx\T)(x) 8x 2 Ox. Hence, S \T is locally majorized.
Lemma 1.5. corollary 5.1, Yannelis and Prabhakar1983]] Let X and Y be a linear topo- logical spaces. Let S :X !!Y be locally L-majorized. Then, S is L-majorized.
Furthermore, it is clear that the converse of Lemma 1.7. is true. Hence, from now on we will not distinguish between locally L-majorized maps and L-majorized maps.
We now state a simplied version of a result of Tan and Yuan 1994].
Theorem 1.6. theorem 3.2, Tan and Yan 1994]]. Let ; = (XiBiCiPiI) be an abstract economy with two constraint maps, Bi and Ci for every agent i 2 I. Suppose that for every i 2I, where I is possibly uncountable, the economy satisfy :
(a) Xi is a nonempty, compact, convex subset of a locally convex Hausdor space Ei. (b) for each x2X, Bi is lsc., nonempty, and convex valued map.
(c) Ci :X !!Xi is usc. andBi(x)Ci(x) 8x2X. (d) Bi\Pi is L-majorized.
(e) domBi\Pi is open in X.
Then, ; has an equilibrium 9x 2X such that
Bi\Pi(x) = and xi 2Ci(x) 8i2I:
The following theorem is our main result:
Theorem 1.7. Let ; = (XiAiPiI) be an abstract economy satisfying for all i2I : (a) Xi is a nonempty, compact and convex subset of a Hausdor locally convex space Ei. (b) Ai is nonempty, convex valued map such that clAi is usc.
6 (c) Pi is L-majorized.
(d) domAi\Pi is open in X. Then, ; has an equilibrium.
Proof. We index the local base at the zero element of X Xi with an ordered set Di. Let
D=Qi2IDi and let i denote the ith component of 2D. We direct D in the following manner: for any 2 and 1 in D, we have
2 1
()
i2 i1 8i2I:
Let i : D ;! Di be the projection map. We note that if N 2 N#(D), then i(N) 2
N
#(Di). We consider the relative topology on everyXi and the relative topology on X. Now for every 2D:
We construct Aii :X !!Xi :
gphAii = (gphAi+V i)\(X Xi): We also construct the maps Aii :X !!Xi, where
gph Aii = clgphAii:
For all i 2 I, Aii is open since it has an open graph in X Xi. Because of Fact 3, Aii is closed and usc. We also know that 8i 2 I Pi is L-majorized by some map ~Pi. Let
'
ii(x) = Aii \P~i(x), 8x 2 X. Since ~Pi is ols. and Aii is open, 'ii is L-majorized by Lemma 1.4. Furthermore, the setUii =fx2Xjdom'ii 6=gis open inX due to Fact 2.
Now for every 2 D, we apply Theorem 1.6. by letting Bi and Ci be the maps 'ii and
A
ii respectively. Thus, 9x 2X such thatx 62Uii andxi 2Aii(x ),8i2I. The reason
x 62U
ii is that otherwisexi 2con ~Pi(x ), which contradicts the fact that ~Pi is a class L map .
Since X is compact, 9x^2X and N 2N#(D) such that x ;!x^ and 8 2N,
x
62U
ii 8i2I:
Since
Ui Uii 8i2I
we get
x
62Ui 8i2I
and since for all i2I,Ui is open, we get
^
x62Ui 8i2I: (1)
7 Moreover, for all 2N,
x
i 2Aii(x ) 8i2I:
Hence,
(^xx^i)2Lsgph Aii cl(gphAi) = gphclAi 8i2I (2) where the upper limit is taking over all i 2 i(N) and the rst set inclusion holds because of Lemma 1.3. and the last equality holds because of Fact 3.
Finally, from (1) and (2) we conclude that for alli2I, ^x 62Ui and ^xi 2clAi(^x) and thus,
^
x is an equilibrium point for ;.
Note that in the proof of theorem 3.2 in Tan and Yan 1994], the idea was to \fatten" the values of the constraint maps, whereas the idea of our proof of theorem 1.7. is to \fatten"
the graphs of the set valued maps.
The nite dimensional case
Suppose I is a countable set and for all i2I, let Ei be IRn. We then can have a dierent set of conditions that guarantee the existence of an equilibrium. Furthermore, since IRn is rst countable, we will consider sequences instead of nets and for all i 2 I, we will take Di to be the set of natural numbers . Let X be IRn and for any C 2C(X) and for any x 2 X, let dC(x) = infy2Cd(xy), where d is the usual metric of IRn. For a xed
C 2 C(X), the function dC(:) is a continuous function from X to IR. The continuity of this function is immediate from the following observation: Let xn ;! x in X, then the triangle inequalities
dC(xn)dC(x) +d(xnx) and dC(x)dC(xn) +d(xnx) and the fact that d(xnx);!0 yield
limsupn dC(xn)dC(x)liminfn dC(xn): We dene the ";open fattening of C 2C(X) by
S"C] =fx2XjdC(x)<"g:
We also dene the ";closed fattening, S"C], by
S"C] = clS"C]:
Due to the continuity of dC(:), SC]" is an open set in X and clearly S"C] is a closed set of X.
The following lemmas are Lemma 1.1. and Lemma 1.3. in a metric setting.
8 Lemma 1.8. Let Cn be a sequence in C(X) that is decreasing with respect to inclusion,
then LsCn \
n1
Cn:
lemma 1.9. Let C be a compact set in C(X). Let "n be a sequence of reals that monoton- ically decrease to 0. Then,
LsS"nC]C:
Theorem 1.10. Let ; = (XiAiPiI) be an abstract economy. Assume that for every
i2I, where I is a countable set of agents, ; satis es : (a) Xi be compact and convex subset of IRn.
(b) Ai :X !!Xi is such that clAi is nonempty, convex valued, and usc.
(c) Pi :X !!Xi is lsc.
(d) Ui = domAi\Pi 6=is open in X. (e) 8x2X, x 62conPi(x).
Then, the abstract economy ; has an equilibrium.
Proof. The proof follows the outlines of the proofs of Theorem 1.7. and theorem 6.1.
of Yannelis and Prabhakar 1983]. For all i 2 I and 8n 2 IN, We construct the maps
Ani:X !!Xi and Ani:X !!Xi, where
gphAni=S1ngphAi] gph Ani= S1ngphAi]:
We know now that 8i 2 I and 8n 2 IN, Ani is open and Ani is usc. We also know from Lemma 1.9. that 8i2I,
Lsgph Ani gphAi: (3)
Now let'ni(x) =Ani\Pi(x),8x2X. Since Ani is open,'ni is lsc. by Fact 6. Furthermore, the set Uni =fx2Xjdom'ni6=g is open in X and 8n2IN,
Ui Uni 8i2I: (4)
X is a mertizable space and Uni is paracompct. By theorem 3.2" in Michael 1955], the restriction of 'ni to Uni has a selectionfni.
Let
Fni =
fni(x) if x2Uni
Ani(x) otherwise .
Then, Fni is usc.(lemma 6.1 in Yannelis and Prabhakar 1983]) with closed, convex, and nonempty values. Finally, 8n 2 IN, the map Fn : X !! X, where Fn = Qi2I Fni, is also
9 usc. by lemma 3 in Fan (1952, p. 124)]. Hence, 9xn such that xn 2 Fn(xn). Therefore,
8i 2I, we have xn 62 Uni (otherwise xni 2 conPi) and xni 2 Ani(xni). Since X is compact,
9x
2 X such that xn ;! x. >From (4) and from the fact that Ui is open, we have
x
62 Ui and
Ai\Pi(x) =8i 2I: (5)
>From (3), we have
(xxi)2Lsgph Ani gph Ai where the upper limit is taking over all n2IN. Hence,
x
i 2Ai(x)8i2I (6):
>From (5) and (6) we conclude that the economy has an equilibrium.
Note that above theorem is not a sepcial case of Theorem 1.7 since condition (c) of Theorem 1.7 does not imply condition (c) of Theorem 1.10.
In a subsequent paper, we hope to show that the method of \fattening" the constraint maps can be used in abstract economies where the set of agents is a measure space and thus we hope to be able to weaken some of the conditions of the existence results of Yannelis 1987].
Budget constraints and Walrasian equilibrium
In this section, we show that eliminating the requirement of the lower semi-continuity of the constraint map is signicant economically. One of the important ways to show the existence of a Walrasian equilibrium is to use the abstract economy approach where the agents are divided into consumers, producers, and an auctioneer.(see Debreu1959] and Gale and Mas-Colell1975]). Let us consider the case where the commodity space is IRm and the set of agents is nite, I = 1:::n. LetXi denote the ith consumer's consumption set, wi denote his initial endowment, and Ui denote his preference relation on Xi. For
j =i:::k,Yj denotes the jth producer's production set. ij(p) is a function that denotes the share of the ith consumer in the prots of jth supplier at the given price level p. The auctioneer is a player with a strategy set that consist of a simplex "IRm that represents the set of normalized prices. Our main concern in this model is the constraint map (the budget map) Ai for the ith consumer. We let X = Qni=1Xi, Y = Qkj=1Yj and for every
p2" and everyj, we letj(p) = supyj2Yj p:yj. Now the budget map for theith consumer is Ai:X Y "!!Xi, where
Ai(xyp) =fxi 2Xijp:xip:wi+Xk
j=1
ij(p)j(p)g:
10 In order to use the equilibrium results that currently exist in the literature, the budget map has to be lsc. However, to guarantee the lower semi continuity of such a map, one has to assume the continuity of each ij(:). More importantly, one has to assume that 8i and for all p2", 9xi 2Xi such that
p:xi <p:wi+Xk
j=1
ij(p)j(p): (y) The most common way to satisfy the above condition is to assume that ij(p) are non negative and 8i and 8p
p:wi >xinf
i
2Xip:xi: (z) The strict inequalities in (y) and (z) mean that consumers can not function when their consumption is compatible with their minimal income. The same problem occurs even when we use two sets of constraint maps and we assume that one of them is lsc. For example, Toussaint 1984] assumed that every agent i 2 I has two set valued constraint maps. The rst, Ai, is osc. The second, Bi, is nonempty and lsc. Furthermore, the two were related by the following
clBi(x) =Ai(x) 8x2X:
In the context of a Walrasian equilibrium,Ai and Bi are
Ai(xyp) =fxi 2Xijp:xip:wi+Xk
j=1
ij(p)j(p)g
Bi(xyp) =fxi 2Xijp:xi <p:wi+Xk
j=1
ij(p)j(p)g:
Clearly, assuming thatBi is nonempty and lsc. causes the same problem that we discussed earlier. For the results of this paper, no lower semi-continuity conditions are required for the constraint maps. Furthermore, we can allow for discontinuities in the functions ij(:) which might arise from taxes or welfare checks. In fact we can assume that
Ai(xyp) =fxi 2Xijp:xi i(p)g
where i(p) is the income function of the ith consumer. In order to satisfy condition (b) of Theorem 1.7.(or Theorem 1.8.), we only need i(p) to be an upper semi-continuous function of p.
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