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Advances in Decision Sciences
Volume 2011 (2011), Article ID 960819, 7 pages
A Constructive Analysis of Convex-Valued Demand Correspondence for Weakly Uniformly Rotund and Monotonic Preference
1Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan
2Graduate School of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan
Received 23 December 2010; Accepted 14 February 2011
Academic Editor: Chenghu Ma
Copyright © 2011 Yasuhito Tanaka and Atsuhiro Satoh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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).
Bridges () 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.
2. Preliminary Results
Consider a consumer who consumes goods. is a finite natural number larger than 1. Let be his consumption set. It is a compact (totally bounded and complete) and convex set. Let be an -dimensional simplex and a normalized price vector of the goods. Let be the price of the th good, then and for each . For a given , the budget set of the consumer is where is the initial endowment. A preference relation of the consumer is a binary relation on . Let . If he prefers to , we denote . A preference-indifference relation is defined as follows; where entails , the relations and are transitive, and if either or , then . Also we have A preference relation is continuous if it is open as a subset of , and is a closed subset of .
A preference relation on is uniformly rotund if for each there exists a with the following property.
Definition 2.1 (uniformly rotund preference). Let , , and points of such that , and a point of such that , then either or .
Strict convexity of preference is defined as follows.
Definition 2.2 (strict convexity of preference). If , , and , then either or .
Bridges  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). If , , and , then either or .
We define the following weaker version of uniform rotundity.
Definition 2.4 (weakly uniformly rotund preference). Let , and points of such that . Let be a point of such that for and (every component of is positive), then or .
We assume also that consumers’ preferences are monotonic in the sense that if (it means that each component of is larger than or equal to the corresponding component of , and at least one component of is larger than the corresponding component of ), then .
Now, we show the following lemmas.
Lemma 2.5. If , , then weak uniform rotundity of preferences implies that or .
Proof. Consider a decreasing sequence of in Definition 2.4. Then, either or for such that and for each . Assume that converges to zero. Then, converges to . Continuity of the preference (closedness of ) implies that or .
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 .
Proof. (1) Let and be points in such that . Consider a point . Then, and . Thus, using Lemma 2.5, we can show or , and or . Inductively, we can show that for , or , for each natural number .
(2) Let with a real number such that . We can select a natural number so that for each natural number . is a sequence. Since, for natural numbers and such that , and with some natural number , we have is a Cauchy sequence, and converges to zero. Then, and converge to . Closedness of implies that either or . Therefore, the preference is convex.
Lemma 2.7. Let and be points in such that . Then, if a consumer's preference is weakly uniformly rotund and monotonic, for .
Proof. By continuity of the preference (openness of ), there exists a point such that and . Then, since weak uniform rotundity implies convexity, we have or . If , then by transitivity . Monotonicity of the preference implies . Assume . Then, again monotonicity of the preference implies .
Let be a subset of such that for each , (1), (2) is nonempty, (3)There exists such that for all .
In , the following lemmas were proved.
Lemma 2.8 ([1, Lemma 2.1]). If , , and is nonempty, then is compact.
Lemma 2.9 ([1, Lemma 2.2]). If and (it means , for all ), then and .
Lemma 2.10 ([1, Lemma 2.3]). Let , and . Let be the hyperplane with equation . Then, for each , there exists a unique point in . The function so defined maps onto and is uniformly continuous on .
Lemma 2.11 ([1, Lemma 2.4]). Let , , , and . Then, there exists such that and .
Proof. See the appendix.
And the following lemma.
Lemma 2.12 ([1, Lemma 2.8]). Let , , and be positive numbers. Then, there exists with the following property: if , are elements of such that and , , are real numbers such that , and is an element of such that and , then there exists such that and .
It was proved by setting .
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 of , a compact and convex set of normalized price vectors (an -dimensional simplex), and a subset of such that for each (1), (2) is nonempty, (3)There exists such that for all . Then, for each , there exists a subset of such that (it means for all ) for all , ( for all ), and the multivalued correspondence is convex-valued and has a closed graph.
A graph of a correspondence is
If is a closed set, we say that has a closed graph.
Proof. (1) Let , and choose such that . By Lemma 2.11, construct a sequence in such that and with for each natural number . By convexity and transitivity of the preference for and each . Thus, we can construct a sequence such that , and for some and , and so is a Cauchy sequence in . It converges to a limit . By continuity of the preference (closedness of ) , and . Since is closed, . By Lemma 2.9, for all . Thus, we have . Convexity of the preference implies that may not be unique, that is, there may be multiple elements of such that and . Therefore, is a set and we get a demand correspondence. Let and . Then, , , and convexity of the preference implies . Thus, is convex.
(2) Next, we prove that the demand correspondence has a closed graph. Consider and such that and with . Let and be demand sets. Let , , and such that . Given , such that , and choose as in Lemma 2.12. By that lemma, we can choose such that and . Similarly, we can choose such that and for each . means . Either for all or for some . Assume that for all and . If is sufficiently small, means and for some finite natural number . Then, by weak uniform rotundity, there exist and such that , with , and , and for . Again if is sufficiently small, and imply and . And it follows that . By continuity of the preference (openness of ) . Let . Consider a sequence converging to zero. By continuity of the preference (closedness of ) and . Note that . Thus, . Since , we have . Replacing with , we can show that . Inductively, we obtain for each natural number . Then, we have for some for any . It contradicts . Therfore, we have or (it means and ), and so has a closed graph.
A. Proof of Lemma 2.11
This proof is almost identical to the proof of Lemma 2.4 in Bridges . They are different in a few points.
Let be the hyperplane with equation and the projection of on . Assume . Choose such that is contained in the closed ball around and let Let be the hyperplane parallel to , between and and a distance from , and the hyperplane parallel to , between and and a distance from . For each let be the unique element of , the unique element of , and the unique element of . Since , we have by convexity and continuity of the preference. is uniformly continuous, so is totally bounded by Lemma 2.8 and Proposition 4.2 in Chapter 4 of .
Since and , we have , and so continuity of the preference (openness of ) means that there exists such that when . Let be points of such that is a -approximation to . Given in , choose such that . Then, .
Now, from our choice of , we have for each . It is proved as follows. Since by the assumption , . Thus, we have See Figure 1.
Let Then, , (because and , and by convexity of the preference or .
In the first case, we complete the proof by taking . In the second, assume that, for some (), we have constructed in such that As (because ), we can choose such that . Then, and either or . In the former case, the proof is completed by taking . If , for all such that . Then, either for all and so , in which case we set ; or else for all and so , then we set .
If this process proceeds as far as the construction of , then, setting , we see that and that for each ; so for each .
This research was partially supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research (C), no. 20530165, and the Special Costs for Graduate Schools of the Special Expenses for Hitech Promotion by the Ministry of Education, Science, Sports and Culture of Japan in 2010.
- D. S. Bridges, “The construction of a continuous demand function for uniformly rotund preferences,” Journal of Mathematical Economics, vol. 21, no. 3, pp. 217–227, 1992.
- E. Bishop and D. Bridges, Constructive Analysis, vol. 279 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1985.
- D. Bridges and F. Richman, Varieties of Constructive Mathematics, vol. 97 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1987.
- D. Bridges and L. Vîţă, Techniques of Constructive Mathematics, Springer, 2006.
- D. S. Bridges, “Constructive notions of strict convexity,” Mathematical Logic Quarterly, vol. 39, no. 3, pp. 295–300, 1993.