# A Constructive Analysis of Convex-Valued Demand Correspondence for Weakly Uniformly Rotund and Monotonic Preference

Authors: Yasuhito Tanaka; Atsuhiro Satoh
Journal: Advances in Decision Sciences
 (2011)

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## Abstract

Bridges (1992) has constructively shown the existence of continuous demand function for consumers with continuous, uniformly rotund preference relations. We extend this result to the case of multivalued demand correspondence. We consider a weakly uniformly rotund and monotonic preference relation and will show the existence of convex-valued demand correspondence with closed graph for consumers with continuous, weakly uniformly rotund and monotonic preference relations. We follow the Bishop style constructive mathematics according to Bishop and Bridges (1985), Bridges and Richman (1987), and Bridges and Vîţă (2006).


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## Body

## 1. Introduction

Bridges ([1]) has constructively shown the existence of continuous demand function for consumers with continuous, uniformly rotund preference relations. We extend this result to the case of multivalued demand correspondence. We consider a weakly uniformly rotund and monotonic preference relation and will show the existence of convex-valued demand correspondence with closed graph for consumers with continuous, weakly uniformly rotund and monotonic preference relations.In the next section, we summarize some preliminary results most of which were proved in [1]. In Section 3, we will show the main result.We follow the Bishop style constructive mathematics according to [2–4].


## 2. Preliminary Results

Consider a consumer who consumesN goods. N is a finite natural number larger than 1. Let X⊂RN be his consumption set. It is a compact (totally bounded and complete) and convex set. Let Δ be an n-1-dimensional simplex and p∈Δ a normalized price vector of the goods. Let pi be the price of the ith good, then ∑i=1Npi=1 and pi≥0 for each i. For a given p, the budget set of the consumer is (2.1)β(p,w)≡{x∈X:p⋅x≤w},
where w>0 is the initial endowment. A preference relation of the consumer ≻ is a binary relation on X. Let x,y∈X. If he prefers x to y, we denote x≻y. A preference-indifference relation ≿ is defined as follows;(2.2)x≿y,iff¬(y≻x),
where x≻y entails x≿y, the relations ≻ and ≿ are transitive, and if either x≿y≻z or x≻y≿z, then x≻z. Also we have(2.3)x≿yiff∀z∈X(y≻z⟹x≻z).
A preference relation ≻ is continuous if it is open as a subset of X×X, and ≿ is a closed subset of X×X.A preference relation≻ on X is uniformly rotund if for each ε there exists a δ(ε) with the following property.Definition 2.1 (uniformly rotund preference).
Letε>0, x, and y points of X such that |x-y|≥ε, and z a point of RN such that |z|≤δ(ε), then either (1/2)(x+y)+z≻x or (1/2)(x+y)+z≻y.Strict convexity of preference is defined as follows.Definition 2.2 (strict convexity of preference).
Ifx,y∈X, x≠y, and 0<t<1, then either tx+(1-t)y≻x or tx+(1-t)y≻y.Bridges [5] has shown that if a preference relation is uniformly rotund, then it is strictly convex.On the other hand, convexity of preference is defined as follows.Definition 2.3 (convexity of preference).
Ifx,y∈X, x≠y, and 0<t<1, then either tx+(1-t)y≿x or tx+(1-t)y≿y.We define the following weaker version of uniform rotundity.Definition 2.4 (weakly uniformly rotund preference).
Letɛ>0, x and y points of X such that |x-y|≥ɛ. Let z be a point of RN such that |z|≤δ for δ>0 and z≫0 (every component of z is positive), then (1/2)(x+y)+z≻x or (1/2)(x+y)+z≻y.We assume also that consumers’ preferences are monotonic in the sense that ifx′>x (it means that each component of x′ is larger than or equal to the corresponding component of x, and at least one component of x′ is larger than the corresponding component of x), then x′≻x.Now, we show the following lemmas.Lemma 2.5.
Ifx,y∈X, x≠y, then weak uniform rotundity of preferences implies that (1/2)(x+y)≿x or (1/2)(x+y)≿y.Proof.
Consider a decreasing sequence(δm) of δ in Definition 2.4. Then, either  (1/2)(x+y)+zm≻x or (1/2)(x+y)+zm≻y for zm such that |zm|<δm and zm≫0 for each m. Assume that (δm) converges to zero. Then, (1/2)(x+y)+zm converges to (1/2)(x+y). Continuity of the preference (closedness of ≿) implies that (1/2)(x+y)≿x or (1/2)(x+y)≿y.Lemma 2.6.
If a consumer's preference is weakly uniformly rotund, then it is convex.This is a modified version of Proposition 2.2 in [5].Proof.
(1) Letx and y be points in X such that |x-y|≥ε. Consider a point (1/2)(x+y). Then, |x-(1/2)(x+y)|≥ε/2 and |(1/2)(x+y)-y|≥ε/2. Thus, using Lemma 2.5, we can show (1/4)(3x+y)≿x or (1/4)(3x+y)≿y, and (1/4)(x+3y)≿x or (1/4)(x+3y)≿y. Inductively, we can show that for k=1,2,…,2n-1, (k/2n)x+((2n-k)/2n)y≿x or (k/2n)x+((2n-k)/2n)y≿y, for each natural number n.
(2) Letz=tx+(1-t)y with a real number t such that 0<t<1. We can select a natural number k so that k/2n≤t≤(k+1)/2n for each natural number n. ((k+1)/2n-k/2n)=(1/2n) is a sequence. Since, for natural numbers m and n such that m>n, l/2m≤t≤(l+1)/2m and k/2n≤t≤(k+1)/2n with some natural number l, we have
(2.4)|(l+12m-l2m)-(k+12n-k2n)|=|2n-2m2m2n|<12n,((k+1)/2n-k/2n) is a Cauchy sequence, and converges to zero. Then, ((k+1)/2n) and (k/2n) converge to t. Closedness of ≿ implies that either z≿x or z≿y. Therefore, the preference is convex.Lemma 2.7.
Letx and y be points in X such that x≻y. Then, if a consumer's preference is weakly uniformly rotund and monotonic, tx+(1-t)y≻y for 0<t<1.Proof.
By continuity of the preference (openness of≻), there exists a point x′=x-λ such that λ≫0 and x′≻y. Then, since weak uniform rotundity implies convexity, we have tx′+(1-t)y≿y or tx′+(1-t)y≿x′. If tx′+(1-t)y≿x′, then by transitivity tx′+(1-t)y=tx+(1-t)y- tλ≿x′≻y. Monotonicity of the preference implies tx+(1-t)y≻y. Assume tx′+(1-t)y≿y. Then, again monotonicity of the preference implies tx+(1-t)y≻y.LetS be a subset of Δ×R such that for each (p,w)∈S,  (1)
p∈Δ,(2)
β(p,w) is nonempty,(3)
There existsξ∈X such that ξ≻x for all x∈β(p,w).In [1], the following lemmas were proved.Lemma 2.8 ([1, Lemma 2.1]).
Ifp∈Δ⊂RN, w∈R, and β(p,w) is nonempty, then β(p,w) is compact.Lemma2.8 with Proposition 4.4 in Chapter 4 of [2] or Proposition 2.2.9 of [4] implies that for each (p,w)∈Sβ(p,w) is located in the sense that the distance(2.5)ρ(x,β(p,w))≡inf⁡⁡{|x-y|:y∈β(p,w)}
exists for each x∈RN.Lemma 2.9 ([1, Lemma 2.2]).
If(p,w)∈S and ξ≻β(p,w) (it means ξ≻x, for all x∈β(p,w)), then ρ(ξ,β(p,w))>0 and p·ξ>w.Lemma 2.10 ([1, Lemma 2.3]).
Let(p,c)∈S, ξ∈X and ξ≻β(p,c). Let H be the hyperplane with equation p·x=c. Then, for each x∈β(p,c), there exists a unique point φ(x) in H∩[x,ξ]. The function φ so defined maps β(p,c) onto H∩β(p,c) and is uniformly continuous on β(p,c).Lemma 2.11 ([1, Lemma 2.4]).
Let(p,w)∈S, r>0, ξ∈X, and ξ≻β(p,w). Then, there exists ζ∈X such that ρ(ζ,β(p,w))<r and ζ≻β(p,w).Proof.
See the appendix.And the following lemma.Lemma 2.12 ([1, Lemma 2.8]).
LetR, c, and t be positive numbers. Then, there exists r>0 with the following property: if p, p′ are elements of RN such that |p|≥c and |p-p′|<r, w, w′ are real numbers such that |w-w′|<r, and y′ is an element of RN such that |y′|≤R and p′·y′=w′, then there exists ζ∈RN such that p·ζ=w and |y′-ζ|<t.It was proved by settingr=ct/(R+1).


## 3. Convex-Valued Demand Correspondence with Closed Graph

With the preliminary results in the previous section, we show the following our main result.Theorem 3.1.
Let≿ be a weakly uniformly rotund preference relation on a compact and convex subset X of RN, Δ a compact and convex set of normalized price vectors (an n-1-dimensional simplex), and S a subset of Δ×R such that for each (p,w)∈S(1)
p∈Δ,(2)
β(p,w) is nonempty,(3)
There existsξ∈X such that ξ≻x for all x∈β(p,w).Then, for each (p,w)∈S, there exists a subset F(p,w) of β(p,w) such that F(p,w)≿x (it means y≿x for all y∈F(p,w)) for all x∈β(p,w), p·F(p,w)=w (p·y=w for all y∈F(p,w)), and the multivalued correspondence F(p,w) is convex-valued and has a closed graph.A graph of a correspondenceF(p,w) is(3.1)G(F)=⋃(p,w)∈S(p,w)×F(p,w).IfG(F) is a closed set, we say that F has a closed graph.Proof.
(1) Let(p,w)∈S, and choose ξ∈X such that ξ≻β(p,w). By Lemma 2.11, construct a sequence (ζm) in X such that ζm≻β(p,w) and ρ(ζm,β(p,w))<(r/2m-1) with r>0 for each natural number m. By convexity and transitivity of the preference tζm+(1-t)ζm+1≻β(p,w) for 0<t<1 and each m. Thus, we can construct a sequence (ζn) such that |ζn-ζn+1|<ɛn, ρ(ζn,β(p,w))<δn and ζn≻β(p,w) for some 0<ɛ<1 and 0<δ<1, and so (ζn) is a Cauchy sequence in X. It converges to a limit ζ*∈X. By continuity of the preference (closedness of ≿) ζ*≿β(p,w), and ρ(ζ*,β(p,w))=0. Since β(p,w) is closed, ζ*∈β(p,w). By Lemma 2.9,p·ζn>w for all n. Thus, we have p·ζ*=w. Convexity of the preference implies that ζ* may not be unique, that is, there may be multiple elements ζ′ of β(p,w) such that p·ζ′=w and ζ′≿β(p,w). Therefore, F(p,w) is a set and we get a demand correspondence. Let ζ∈F(p,w) and ζ′∈F(p,w). Then, ζ≿β(p,w), ζ′≿β(p,w), and convexity of the preference implies tζ+(1-t)ζ′≿β(p,w). Thus, F(p,w) is convex.
(2) Next, we prove that the demand correspondence has a closed graph. Consider(p,w) and (p′,w′) such that |p-p′|<r and |w-w′|<r with r>0. Let F(p,w) and F(p′,w′) be demand sets. Let y′∈F(p′,w′), c=ρ(0,Δ)>0, and R>0 such that X⊂B¯(0,R). Given ε>0, t=δ>0 such that δ<ε, and choose r as in Lemma 2.12. By that lemma, we can choose ζ∈RN such that p·ζ=w and |y′-ζ|<δ. Similarly, we can choose ζ′(y)∈RN such that p′·ζ′(y)=w′ and |y-ζ′(y)|<δ for each y∈F(p,w). y′∈F(p′,w′) means y′≿ζ′(y). Either |y′-y|>ε/2 for all y∈F(p,w) or |y′-y|<ε for some y∈F(p,w). Assume that |y′-y|>ε/2 for all y∈F(p,w) and y≻ζ. If δ is sufficiently small, |y′-y|>ε/2 means |y-ζ|>ε/k and |y′-ζ′(y)|>ε/k for some finite natural number k. Then, by weak uniform rotundity, there exist zn and zn′ such that |zn|<τn, |zn′|<τn with τn>0, zn≫0 and zn′≫0, (1/2)(y+ζ)+zn≻ζ and (1/2)(y′+ζ′(y))+zn′≻ζ′(y) for n=1,2,…. Again if δ is sufficiently small, |y-ζ′(y)|<δ and |y′-ζ|<δ imply (1/2)(y+ζ)+zn≻y′ and (1/2)(y′+ζ′(y))+zn′≻y. And it follows that |(1/2)(y+ζ)-(1/2)(y′+ζ′(y))|<δ. By continuity of the preference (openness of ≻) (1/2)(y+ζ)+z′n≻y. Let y1=(1/2)(y+ζ). Consider a sequence (τn) converging to zero. By continuity of the preference (closedness of ≿) y1≿y′ and y1≿y. Note that p·y1=w. Thus, y1∈β(p,w). Since y∈F(p,w), we have y1∈F(p,w). Replacing y with y1, we can show that (y+3ζ)/4∈F(p,w). Inductively, we obtain (y+(2m-1)ζ)/2m∈F(p,w) for each natural number m. Then, we have |y-ζ|<η for some y∈F(p,w) for any η>0. It contradicts |y-ζ|>ε/k. Therfore, we have |y′-y|<ε or ζ≿y (it means |y-ζ|<δ+ε and ζ∈F(p,w)), and so F(p,w) has a closed graph.



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