A constructive analysis of convex-valued demand correspondence for weakly uniformly rotund and monotonic preference

Bridges([4]) 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 multi-valued 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 [1], [2] and [3].


Introduction
Bridges ([4]) 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 multi-valued 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 1 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 [4]. In Section 3 we will show the main result.

Preliminary results
Consider a consumer who consumes N goods. N is a finite natural number larger than 1. Let X ⊂ R N be his consumption set. It is a compact (totally bounded and complete) and convex set. Let ∆ be an n − 1-dimensional simplex, and p ∈ ∆ be a normalized price vector of the goods. Let p i be the price of the i-th good, then ∑ N i=1 p i = 1 and p i ≥ 0 for each i. For a given p the budget set of the consumer is β(p, w) ≡ {x ∈ X : p · x ≤ w} w > 0 is his 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; x ≿ y if and only if ¬(y ≻ x) 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 x ≿ y if and only if ∀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 1 (Uniformly rotund preference). Let ε > 0, x and y be points of X such that |x − y| ≥ ε, and z be a point of R N 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 (Strict convexity of preference). If x, y ∈ X, x ̸ = y, and 0 < t < 1, 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 3 (Convexity of preference). If x, y ∈ X, x ̸ = y, and 0 < t < 1, We define the following weaker version of uniform rotundity.
Definition 4 (Weakly uniformly rotund preference). Let ε > 0, x and y be points of X such that |x − y| ≥ ε. Let z be a point of R N such that |z| ≤ δ for δ > 0 and z ≫ 0(every component of z is positive), then 1 2 (x + y) + z ≻ x or We assume also that consumers' preferences are monotonic in the sense that if x ′ > 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. 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. Let x and y be points in X such that |x − y| ≥ ε. Consider a point Thus, using Lemma 1 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, . . . , 2 n − 1 k 2 n x + 2 n −k 2 n y ≿ x or k 2 n x + 2 n −k 2 n y ≿ y for each natural number n. 2. Let z = tx + (1 − t)y with a real number t such that 0 < t < 1. We can select a natural number k so that k 2 n ≤ t ≤ k+1 2 n for each natural number n. ( k+1 2 n − k 2 n ) = ( 1 2 n ) is a sequence. Since, for natural numbers m and n such that m > n, l is a Cauchy sequence, and converges to zero. Then, ( k+1 2 n ) and ( k 2 n ) converge to t. Closedness of ≿ implies that either z ≿ x or z ≿ y. Therefore, the preference is convex. Lemma 3. Let x 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 In [4] the following lemmas were proved. Lemma 4 (Lemma 2.1 in [4]). If p ∈ ∆ ⊂ R N , w ∈ R, and β(p, w) is nonempty, then β(p, w) is compact.
Proof. See Appendix.
And the following lemma.
Lemma 8 (Lemma 2.8 in [4]). Let R,c, and t be positive numbers. Then there exists r > 0 with the following property: if p, p ′ are elements of R N such that |p| ≥ c and |p − p ′ | < r, w, w ′ are real numbers such that |w − w ′ | < r, and y ′ is an element of R N such that |y ′ | ≤ R and p ′ · y ′ = w ′ , then there exists ζ ∈ R N such that p · ζ = w and |y ′ − ζ| < t.
It was proved by setting r = 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 1.
Let ≿ be a weakly uniformly rotund preference relation on a compact and convex subset X of R N , ∆ be a compact and convex set of normalized price vectors (an n − 1-dimensional simplex), and S be a subset of ∆ × R such that for each (p, w) ∈ S 1. p ∈ ∆.
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 multi-valued correspondence F (p, w) is convex-valued and has a closed graph.
A graph of a correspondence F (p, w) is If G(F ) is a closed set, we say that F has a closed graph.

Appendix: Proof of Lemma 7
This proof is almost identical to the proof of Lemma 2.4 in Bridges [4]. They are different in a few points.