Economic Theory 8, 155-166 (1996) Econom/c

Theory �9 Springer-Verlag 1996

Complete characterization of Yannelis-Zame and Chichilnisky-Kalman-Mas-Colell properness conditions on preferences for separable concave functions defined in LP+ and L"*

Cuong Le Van CNRS-CEPREMAP, 140 rue du Chevalerel, F-75013 Paris, FRANCE

Received: September 20, 1994; revised version April 20, 1995

Summary. Properness of preferences are useful for proving existences of an equilib- rium and of supporting prices in Banach Lattices. In this paper we characterize completely properness and uniform properness for separable concave functions defined in LP+. We prove also that every separable concave function which is well-defined in L p is automatically continuous.

1. Introduction

In 1980 (ref. 13), Chichilnisky and Kalman introduced the cone condition which is equivalent to the properness condition (see Chichilnisky, ref. 14) used by Mas-Colell in 1986 to prove an equilibrium existence theorem (ref. 15). Parallel to Mas-Colell's paper, Yannelis and Zame fief. 18) gave also an existence theorem if each agent has a commodity which is uniformly extremely desirable. Properness and extremely desirability, in general, are not comparable. But their uniform versions coincide (see e.g. Yannelis and Zame, ref. 18). In fact, the extreme desirability conditions are more general. Their point is to include preferences which are not necessarily transitive or complete, and could be interdependent. They coincide with the uniform properness or cone condition when preferences are transitive and complete. In Araujo and Monteiro (ref. 8), properness is called M-properness, and the extremely desirability, YZ-properness. In this paper, we will adopt these terminologies, but in order to correct the "mistake" mentioned by Chichilnisky, M-properness will be changed in CK-M-properness (Chichilnisky-Kalman-Mas-Colell properness).

Properness and uniform properness are very useful to prove the existence of equilibria in Banach Lattices or Riesz Spaces (see e.g. refs. 6,7,8,15,18). The Banach Lattices or Riesz spaces approach was introduced first by Aliprantis and Brown (ref. 1) and inspired many papers dealing with the existence of equilibrium in infinite dimensional economies with the positive orthant as consumption sets.

* The author would like to thank Rose-Anne DANA and three referees for their helpful remarks and suggestions.

156 C. Le V a n

The usefulness of properness was also demonstrated in ref. 2 where Aliprantis and Burkinshaw showed that, in a symmetric Riesz dual system economy, the fundamental welfare theorems are not valid without properness. Conversely (ref. 3), uniform YZ or C K - M properness is one sufficient condition to have non-zero prices supporting weakly Pareto optimal allocations (Second Welfare Theorem). In ref. 4, the same authors and Brown proved the identity between Edgeworth and Walrasian equilibria with the assumptions that the preferences are strongly monotone, uniformly proper and that the total endowment is strictly positive. In ref. 17, extreme desirability condition reveals to be again one sufficient condition to obtain the Core-Walras equivalence in separable Banach lattices whose positive cone has an empty interior and in an economy with an atomless measure space of agents.

But how one can characterize properness? It is well-known that it is a notion weaker than continuity, i.e. every continuous concave function is proper. A complete characterization for strictly increasing separable concave functions in L~_ is given in Araujo and Monteiro (ref. 8). In this paper, we characterize completely proper- ness for separable concave functions defined in LP+ without any monotonicity assumption.

In the LP-case, we obtain that a separable concave function which is well-defined in L p is automatically continuous.

Concerning the uniform properness, it has been pointed out (ref. 7) that a separable utility function which satisfies Inada condition can not be uniformly proper in LP+. More precisely, in the LP-case, uniform properness is equivalent to the boundeness (away from + oe and from 0) of the derivative of the criterion function (ref. 12). In this note we extend this result to L~ and the order interval. As a by-product, we obtain that every uniform proper separable concave function is Lipschitzian and, if it is defined in LP+, it has a continuous, concave extension in L p. This rejoins the result stated in ref. 16 by Richard and Zame: every continuous, concave, uniformly proper utility function defined in the positive orthant of a normed vector lattice has a continuous, concave extension on a convex set with non-empty interior.

The paper is organized as follows: Sections 2 and 3 deal with YZ and C K - M propernesses in LV+; in Section 4, we prove the continuity of every separable utility function of L p which is finite-valued; in Section 5, we study the uniform properness.

2. Characterization of Yannelis-Zame-properness in LP+

1 1 Let us consider the Riesz dual System (L p, L q) with - + - = 1 and p _ 1. We endow

P q L ~ with the z-Mackey topology, since (L ~, z) is locally convex-solid and (L ~ L 1) is a Riesz dual System.

A concave function u from LP+ into R is separable if there exists a concave function v from R + into R such that

1

(c) Vx~L~, u(x) = ~ v(x(t))dteR. 0

Character izat ion of properness 157

We denote by vr and v t the right and left derivatives of v. Without loss of assumption, we assume v continuous at the origin. We prove now that (c) implies YxeLP+, v (x )eL ~.

Lemma 2.1

Assume (c). Then Yx~LV+, v(x)~L 1.

P r o o f - L e t x~LP+ and let

11 = {tlv(x(t)) >_ 0},

12 = {t lv(x( t))< 0}.

If/1 or 12 is empty, then the statement is true. We can assume there exists a >_ 0 such that v(~) = O.

Define X l ( t ) = X(t) for teI1,

x l ( t ) = a for tCI2,

x2(t ) = ~ for t~ I 1,

Xz(t ) = x ( t ) for t e l 2.

Obviously, x 1 and x z belong to LP+. Then we have:

1 1 1 Iv(x(t))]dt = ~ v ( x l ( t ) ) d t - ~ v(x2(t))dt < + oo. []

0 0 0

Let us recall the definition ofYZ-properness. A concave function u from L~ into R is YZ-proper at xeLP+ if there exists (w, U), where w is a vector of L p, and U a neighbor- hood of 0 in L p such that:

Vz~ U, u(x + 2 w - 2z) > u(x), if 2 > 0 is sufficiently small and if x + 2w - 2zeLV+.

Throughout this paper we denote by A(x) the set:

A(x) = {hELPlx - 2[hI~LP+ for some 2 > 0, small}.

We have the following important result: for a Riesz dual System (LP, Lq), if x~LV++ then A(x) is dense in L p (see Aliprantis, Brown, and Burkinshaw, 1989, ref. 6).

We need the following Lemma:

Lemma 2.2 - Let x~LV+. If w~A(x) , then

i v~(x)w(t)dt < + oo and i vr(x)w(t)dt < + oo.

158 C. Le Van

P r o o f - I t follows the following inequalities, for 2 > 0

- - ~ < o 2 - -o

i v(x + 2 w ) - v ( X ) dt <_ i -- oe < vr(x)wdt o )" o

and the fact that: w~A(x) implies - w e A ( x ) . []

Theorem 2.1 - Let xeLP+ +. u is YZ-proper at x if, and only if, v , (x)eL ~, v l (x)sL q and there exists a e L p such that

1 1

vr(x)a + dt - ~ vt(x)a- dt > O. 0 0

(1)

O <

But

Proo f - i ) Assume u YZ-proper at x. Let (a, U) be the associated properness constants. One can assume U symmetric. Since A(x) is dense in L p, one can suppose aeA(x) (ref. 7). Let z~Uc~A(x) . Since a~A(x), one has a - z e A ( x ) . For 2 > 0 sufficiently small we have:

u(x + 2 ( a - z ) ) - u(x) ~ v(x + 2(a - z ) ) - v(x) dr.

2 - J o 2

i 1 v(x + 2(a-- z)) - v(x) dt < ~ vr (x ) (a- z) dt o 2~ o

i v(x + 2 (a -- z)) -- v(x) t dt < ~ v , (x ) (a -z )d t .

o )c o

Hence, Vze U c~ A(x),

1 1

v,(x)zdt < ~ v~(x)adt < + oo (Lemma 2.2) 0 0

1 1

vt(x)zdt < ~ vt(x)adt < + oo (Lemma 2.2) 0 0

and therefore: v,(x)EL q, v l(x)dY.

eque.ce * ' } inc easi.g is conca e 2 ~ 0 + , and is bounded above by ~o 1 v~(x) (a-z )d t < + o% converges (Beppo-Levi Theorem, see Brezis, ref. 11) to

1 1

v~(x)(a -- z) + dt - ~ vt(x)(a - z ) - dt > O. 0 0

Take z = 0 and we obtain (1).

C h a r a c t e r i z a t i o n o f p r o p e r n e s s 159

ii) Conversely, assume v , (x)eL q, vl(x)eL q and there exists a e L p sastisfying (1). Let

U = zeL '] v , ( x ) ( a - z ) + d t - ~ v t ( x ) ( a - z ) - d t > 0 . 0 0

U is a neighborhood of 0 in L p. Assume x+2(a - z ) eLP+ for z e U and 0 < 2 _< 2 o. One can easily prove, as in part i), that

lim u(x + 2 ( a - z ) ) - u(x) 1 1 = I v , ( x ) (a - z ) + dt - ~ v l ( x ) ( a - z ) - dt > O. 2 ~ 0 + 2 0 0

Hence u(x + 2 ( a - z ) ) > u(x) for 2 > 0 sufficiently small. In other words, u is YZ- proper at x with (a, U) as properness constants. []

Corollary 2.I - (Araujo and Monteiro, ref. 8)

Let xeL~++. Assume v concave and strictly increasing. Then u is YZ-proper at x if, and only if, vl(x)eL q.

P r o o f - W e have O<vr(x)<_vz(x ). Hence vl(x)eLq implies vr(x)eL q. Now, take a(t) = 1 for all t. Then ~ v~(x) a dt > O. []

Corollary 2.2 - Let xeLP++. Assume v differentiable in R++. Then u is YZ-proper at x iff v'(x)eLq\ {0}.

P r o o f - I f v '(x)eL~\ {O}, there exists a e L p such that

1

v'(x) adt < O. [] 0

3. Characterization of Chichilnisky-Kaiman-Mas-Coleil-properness in Lr+

The function u is CK-M-prope r at x if there exists a~L p and U, a neighborhood of 0 in L p, such that: V2> 0, VzsU , u ( x - 2 ( a - z ) ) < u(x).

We have the following important result:

Proposition 3.1 - Let E be a topological space, E' be its dual and f be a concave function from E into [ - ~ , + ~ [. Let x be a non-support point of dora(f), i.e., there exists no p~E' \{0} such that px <_ px', Vx'~dom(f).

Then: f is M-proper at x,c~Of(x) ~ ~ .

P r o o f - See Araujo and Monteiro (ref. 9) or Bonnisseau and Le Van (ref. 10).

Lemma 3.1 - Let xsL~+. Let f be a measurable function such that:

1

VheA(x) c~ LP+, ~ f (t)h(t)dt > 0. (2) o

Then, we have : f > 0.

160 C. Le V a n

P r o o f - Let heLP+. If x - heLP+ then x - hEA(x). Let f satisfy (2). Assume f ( t ) < 0 for tEI c [0, 1-1.

Define h:h(t) = 0 for tEI

h(t) = x(t) for t~I.

Obviously, hELP+ and x - hEL p. Hence x - heA(x) . From (2), one gets:

1

0 < S f ( t ) ( x - h ) d t = Sf( t )x( t)dt < 0: a contradiction. [] 0 I

Theorem 3 .1 - Let xeLP+. Then u is CK-M-proper at x if, and only if, 3gEL q such that:

vr(x ) <_ g < v,(x). (3)

P r o o f - First, observe that, for the Riesz dual System (L p, Lq), if xeLP+ then x is not a support point of dom(u) = L p +.

i) Assume u CK-M-proper at x. From proposition 3.1, there exists g E L q c~ au(x), i.e.:

1

V h e L p, u(x + h ) - u(x) <_ ~ g(t)h(t)dt < + oo 0

hELp+. The sequence f u ( x + 2 h ) - u ( x ) t is well defined, increasing for 2 ~ 0 + , Let 2 t_ )

bounded above by j~o g(t)h(t)dt, converges and its limit is: j~vr(x)h(t)dt. Hence vr(x) < g.

Le theLp+nA(x ) .Asprev ious l y the sequence{U(x -2h - - )2 -u (x ) } i swe l l -de f i ned ,

increasing, bounded above by -J~g(t)h(t)dt, and thus converges. Its limit is -~o vt(x)h(t)dt and verifies, VheLP+ c~ A(x), j~(v , (x ) - g)h(t)dt >_ O. From Lemma 3.1, vt(x) >_ o. ii) Conversely, assume there exists g e L q such that:

vr(x) <_ o <- v,(x).

Let h be such that x + heLP+. We have:

Vt, v(x(t) + h(t)) - v(x(t)) <<_ v,(x(t))h + (t) - v,(x(t))h- (t)

<_ g(t)h(t)

Hence u(x + h) - u(x) < 5~g(t)h(t)dt < + oo. In other words, ge~?u(x). []

Corollary 3.1 - (Araujo and Monteiro, ref. 8) Let xeLp++. Assume v concave and increasing. Then u is CK-M-proper at x if, and only if, v~(x)eL q.

P r o o f - If u is CK-M-proper at x and if v is increasing, one has: 0 < v~(x) < g, with g e L q. Hence vr(x)eL q.

Conversely, if v . (x)eL q, then (3) is fulfilled with g = vr(x). []

Charac te r i za t ion of p roperness 161

Corollary 3.2 - Assume v concave and decreasing. Then u is C K - M - p r o p e r at xeLV++ iff vt(x)eL ~. []

Corollary 3.3 - Assume v differentiable on R++. Then u is C K - M - p r o p e r at xeLV++ iff v ' (x)eL q. []

Remark 3.1 - If xeLV++ and v'(x) :/: O, then there is identi ty between YZ-properness and C K - M - p r o p e r n e s s at x.

4. The L p case w i t h p = 1, . . . . oo

In this section, we endow L ~ with the sup-norm. Assume now v defined on R and:

1

VxeLP, S v(x)dteR. 0

We have therefore: V x e L p, v (x )eL 1 ( L e m m a 2.1).

Lemma 4.1 - Let x e L p. Then, if W e L p,

v~(x)W(t)dt < + & and vr(x)W(t)dt < + oo.

In other words, v, and v l belong to L q.

P r o o f - See the p r o o f of L e m m a 2.2.

Theorem 4.1 - u is cont inuous.

P r o o f - Since the doma in o f u is L ~, we have just to p rove that u is cont inuous at 0. Since v is concave, we have:

V x e L p, Vt, i v ( x ( t ) ) - v(O)l <_ [iv,(x)[ + Ivr(0)[] Ix(t)].

Let x ' - ~ 0 in L p. F r o m Brezis, ref. 11, there exists a subsequence {x *} such tha t

xV(t) ~ 0, a.e

and IxV(t)l _< h(t), Vv, and a.e. with h e L p. We shall p rove that u(x ~) ~ u(O). Let I~ = {tIx*(t) <_ 0}. F o r

teI~, Ivr(x*(t))l <_ m a x {ivy(0)[, Iv,(-h(t)) l }, and for

t(i l i , [vr(x~(t))l <_ m a x {Iv~(0)J, I v,(h(t))J}. We have:

1 1

[v(xV(t))--v(O)] dt <- S ~l(h(t))lx~(t)] dt + ~ ~2(h(t))ix~(t)i dt o I~ i~

1

+ ~fvr(o)lix~(t)idt, 0

162 C. Le Van

with

and I ~ I ~ = [0, 1].

~1 (h(t)) = max { ] v~(0)[, ] vr( - h(t)) [ }

~2(h(t)) = max { [ v,(0)[, ]v,(h(t))[}

Obviously, 7J~(h), ~z(h) belong to L q, from Lemma 4.1. Hence:

i

]v(x*(t))-v(O)[ dt ~ [-[[ ~1(h) []Lq q- 1[ ~2(h) [[L~ + A] [Ix ~ ][L~" 0

Now, if l imvu(x ' )=l im, infu(x ") then, from the previous result, limvu(x ~) = u(0) since x ~ ~ 0 in L p. By the same argument one gets lira, sup u(x") = u(0) and the proof is complete. [ ]

Corollary 4.1 - If v is differentiable, then u is differentiable.

Proof -S ince u is cont inuous and since au(x) is reduced to the mapping h e L p---} ~v'(x)h(t)dt, u is differentiable. [ ]

5. Uniform properness in LP+

The function u is uniformly (YZ or C K - M ) - p r o p e r in LP++ if it is p roper for every xeLV++ and if the associated properness constants (a, U) are independent of x.

We recall that YZ and C K - M uniform propernesses are equivalent (Yannelis and Zame, ref. 18).

Proposition 5.1 - If u is uniformly proper then v is strictly monotonic .

P r o o f - A s s u m e the contrary: there exists s~R++ such that OeOv(s). Then we have: v,(s) <_ 0 <_ vt(s ). Let x be defined by x(t) = s for every t. Then u is not YZ-proper at x since condit ion (1) in Theorem 2.1 is not satisfied. [ ]

Theorem 5.1 - u is uniformly proper on LP++ if, and only if, there exist e and e' such that:

(4) Vs > 0, e' _< vr(s ) < vz(s) <_ e, (5) and ee' > 0.

Proof - i) By proposi t ion 5.1, one can assume, without loss of generality, that v is strictly increasing. Hence 0 < e' < e. Let a(t) be the function equal to 1 for every t. Let

U = xeLPle ' ( a - x ) + d t - ~ ( a - x ) - d t > O �9 0

U is a ne ighborhood of 0. Let xeLP++ and z ~ U such that x + 2 ( a - -z )EL v for 4 > 0

Characterization of properness 163

sufficiently small. Then:

u(x + ~(a-z))-u(x) lira

Since

1 1

= ~ vr(x)(a -- z) + dt - ~ vt(x)(a - z ) - dt. 0 0

1 1

vr(x)(a -- z) + dt >_ e' ~ (a - z) + dt 0 0

1 1

vl(x)(a -- z ) - dt >_ - e S (a - z ) - dt, 0 0

we have, for 2 > 0 sufficiently small:

u(x + 2(a - z)) - u(x) > O.

u is YZ-proper at x with (a, U) as properness constants. ii) Conversely, assume u uniformly proper on LP++. As previously, we can assume v strictly increasing. There exists a e L v and U(a), a ne ighborhood of a, such that, 'v'ze U(a), VxeLP++

u(x - 2z) < u(x) for 0 < 2.

Since u is strictly increasing, one can assume a strictly positive. Let e denote the constant function equal to 1. A(e) is dense in L p. Hence U(a) c~ A(e) r d?. Since L~ has no interior there exists z e U(a) c~ A(e) with z - r 0. Define for 2 > 0,

G(2) = u(e - 2z).

G is well-defined for 2 > 0 sufficiently small. Since u is uniformly proper, G is decreasing. Hence, z + r 0. Define x as follows:

x(t) = r o > 1 if z(t) >_ 0

x(t) = r I < 1 if z(t) < O,

and F(2) = u(x - 2z).

F is well-defined for 2 sufficiently small and decreasing.

Let F~ denote the left derivative of F when 2 is positive. We have:

1 1

FI(2) = - ~ v,(ro -- 2z+) z+dt + S vl(rl + 2 z - ) z - d t <_ O. 0 0

Let 2 go to O. Then:

o r

1 1

- - vt(ro) ~ z + dt + v~(rl) ~ z - dt <_ O, 0 0

1 1

Vr(r l ) I Z - d t ~_~ v t ( ro) I z + d t . (4) o o

164 C. Le Van

When r 1 ~ 0 , v~(rl), vl(r 1) increase and converge to vr(0 ). F rom (4), vr(0) < + oo and, Vs > 0, vr(s) < vl(s ) <_ v~(O) < + oo.

When r o ~ + oo, from (4), vl(ro) is bounded below away from 0. Since r~ < r o implies v,(ro) > vl(ro), one concludes that:

gs > O, vt(s) >_ vz(s) > 5' > O.

The proof is complete. [ ]

The same proof can be used to have the result stated by Cheng (ref. 12):

Theorem 5.2 - (Cheng, ref. 12)

Assume v is defined over R. Then u is uniformly proper if, and only if, there exist e and e' such that:

and~'>O. []

Finally, we have:

Vs, 5' < v , (s) < v~(s) < 5,

Theorem 5.3 - L e t co~LV++. Then u is uniformly proper on [0, co] if, and only if:

a) v,(co)eL q, vl(co)ffLq; b) 0 < Iv~(0)l < + oo.

Moreover , co can be used as a properness constant.

P r o o f - i) Assume v strictly increasing. Define

{ , } U = x ~ L p vr (co) (co-x)+dt -Sv , (O)(co-x) d t > O . 0

U is a ne ighbourhood of 0.

Let x e [0, co] ~ LP+ +. I f z e U and x + 2 ( c o - z ) e L p for 0 < 2 _< 2 o, then

lim u(x + 2 ( c o - z ) ) - u ( x ) 1 1 = ~ vr(x)(co - z ) + d t - 5 v , (x ) (co - z ) - d t .

a~O+ ~ 0 0

Since vr(x) >_ vr(~o ) and vl(x) <_ vr(0), u(x + 2(co- z)) > u(x) for ~ > 0, sufficiently smal l ii) I fu is p roper at co then v,(co)~L q and v~(co)~L ~ (Theorem 2.1). The technics in par t ii) of the proof of Theorem 5.1 can be used to obtain b).

Corollary 5.1 i) If u is uniformly proper then u is Liptschitzian. ii) If u is uniformly proper in L;+ then it has a continuous, concave extension in L p.

P r o o f - Assume u strictly increasing.

Characterization of properness 165

i) If u is uniformly proper, then, from Theorems 5.1, 5.2 and 5.3, there exists e > 0 such that, Vx 1, VX2, Vt, [V(xl(t)) - - v(x2(t))l < elxa(t) - x2(t)l. Hence [u(x 0 - u(xz) I <_ ~ II x l - x 2 [I. ii) Let u be uniformly proper in LP+. Define v 1 as follows:

vl(s ) = v ( s ) i f s eR+ v 1 (s) = v,(O)s if s e R _ (we assume v(0) = 0).

1

Obviously, Hi(x)= Sv~(x)dt is well-defined for x ~ L p. From Theorem 4.1, it is 0

continuous. []

Final remarks

1. YZ and CK-M-propernesses are defined as in Araujo and Monteiro (ref. 8). This slightly differs from the classical definitions of Yannelis and Zame (ref. 18) and of Mas-CoM1 (ref. 15). 2. Obviously, our results are true in the more general case with utility functions defined by

u(x) = S V(x(t)), t)l~(dt) Q

where xeLP(K2, d , #) with/ l o--finite, v(., t) is concave for every t, and v(x(t), t) is measurable for every x e L ' . [ ]

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