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

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 Vîta (2006).

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Yasuhito Tanaka 1 and Atsuhiro Satoh 2

Recommended by Chenghu Ma

1, Faculty of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan
2, Graduate School of Economics, Doshisha University, Kamigyo-ku, Kyoto 602-8580, Japan

Received 23 December 2010; Accepted 14 February 2011

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 consumes N 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 i th good, then ^∑i=1N_pi =1 and _pi ≥0 for each i . For a given p , the budget set of the consumer is [figure omitted; refer to PDF] where w>0 is the initial endowment. A preference relation of the consumer [succeeds] is a binary relation on X . Let x,y∈X . If he prefers x to y , we denote x[succeeds]y . A preference-indifference relation [succeeds, ~] is defined as follows; [figure omitted; refer to PDF] where x[succeeds]y entails x[succeeds, ~]y , the relations [succeeds] and [succeeds, ~] are transitive, and if either x[succeeds, ~]y[succeeds]z or x[succeeds]y[succeeds, ~]z , then x[succeeds]z . Also we have [figure omitted; refer to PDF] A preference relation [succeeds] is continuous if it is open as a subset of X×X , and [succeeds, ~] is a closed subset of X×X .

A preference relation [succeeds] on X is uniformly rotund if for each [straight epsilon] there exists a δ([straight epsilon]) with the following property.

Definition 2.1 (uniformly rotund preference).

Let [straight epsilon]>0 , x , and y points of X such that |x-y|≥[straight epsilon] , and z a point of ^RN such that |z|≤δ([straight epsilon]) , then either (1/2)(x+y)+z[succeeds]x or (1/2)(x+y)+z[succeeds]y .

Strict convexity of preference is defined as follows.

Definition 2.2 (strict convexity of preference).

If x,y∈X , x≠y , and 0<t<1 , then either tx+(1-t)y[succeeds]x or tx+(1-t)y[succeeds]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).

If x,y∈X , x≠y , and 0<t<1 , then either tx+(1-t)y[succeeds, ~]x or tx+(1-t)y[succeeds, ~]y .

We define the following weaker version of uniform rotundity.

Definition 2.4 (weakly uniformly rotund preference).

Let [varepsilon]>0 , x and y points of X such that |x-y|≥[varepsilon] . 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[succeeds]x or (1/2)(x+y)+z[succeeds]y .

We assume also that consumers' preferences are monotonic in the sense that if ^{x[variant prime]} >x (it means that each component of ^{x[variant prime]} is larger than or equal to the corresponding component of x , and at least one component of ^{x[variant prime]} is larger than the corresponding component of x ), then ^{x[variant prime]} [succeeds]x .

Now, we show the following lemmas.

Lemma 2.5.

If x,y∈X , x≠y , then weak uniform rotundity of preferences implies that (1/2)(x+y)[succeeds, ~]x or (1/2)(x+y)[succeeds, ~]y .

Proof.

Consider a decreasing sequence (_δm ) of δ in Definition 2.4. Then, either (1/2)(x+y)+_zm [succeeds]x or (1/2)(x+y)+_zm [succeeds]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 [succeeds, ~] ) implies that (1/2)(x+y)[succeeds, ~]x or (1/2)(x+y)[succeeds, ~]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) Let x and y be points in X such that |x-y|≥[straight epsilon] . Consider a point (1/2)(x+y) . Then, |x-(1/2)(x+y)|≥[straight epsilon]/2 and |(1/2)(x+y)-y|≥[straight epsilon]/2 . Thus, using Lemma 2.5, we can show (1/4)(3x+y)[succeeds, ~]x or (1/4)(3x+y)[succeeds, ~]y , and (1/4)(x+3y)[succeeds, ~]x or (1/4)(x+3y)[succeeds, ~]y . Inductively, we can show that for k=1,2,...,²ⁿ -1 , (k/²ⁿ )x+((²ⁿ -k)/²ⁿ )y[succeeds, ~]x or (k/²ⁿ )x+((²ⁿ -k)/²ⁿ )y[succeeds, ~]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/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 [figure omitted; refer to PDF] ((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 [succeeds, ~] implies that either z[succeeds, ~]x or z[succeeds, ~]y . Therefore, the preference is convex.

Lemma 2.7.

Let x and y be points in X such that x[succeeds]y . Then, if a consumer's preference is weakly uniformly rotund and monotonic, tx+(1-t)y[succeeds]y for 0<t<1 .

Proof.

By continuity of the preference (openness of [succeeds] ), there exists a point ^{x[variant prime]} =x-λ such that λ>>0 and ^{x[variant prime]} [succeeds]y . Then, since weak uniform rotundity implies convexity, we have t^{x[variant prime]} +(1-t)y[succeeds, ~]y or t^{x[variant prime]} +(1-t)y[succeeds, ~]^{x[variant prime]} . If t^{x[variant prime]} +(1-t)y[succeeds, ~]^{x[variant prime]} , then by transitivity t^{x[variant prime]} +(1-t)y=tx+(1-t)y- tλ[succeeds, ~]^{x[variant prime]} [succeeds]y . Monotonicity of the preference implies tx+(1-t)y[succeeds]y . Assume t^{x[variant prime]} +(1-t)y[succeeds, ~]y . Then, again monotonicity of the preference implies tx+(1-t)y[succeeds]y .

Let S be a subset of Δ×R such that for each (p,w)∈S ,

(1) p∈Δ ,

(2) β(p,w) is nonempty,

(3) There exists ξ∈X such that ξ[succeeds]x for all x∈β(p,w) .

In [1], the following lemmas were proved.

Lemma 2.8 ([1, Lemma 2.1]).

If p∈Δ⊂^RN , w∈R , and β(p,w) is nonempty, then β(p,w) is compact.

Lemma 2.8 with Proposition 4.4 in Chapter 4 of [2] or Proposition 2.2.9 of [4] implies that for each (p,w)∈S β(p,w) is located in the sense that the distance [figure omitted; refer to PDF] exists for each x∈^RN .

Lemma 2.9 ([1, Lemma 2.2]).

If (p,w)∈S and ξ[succeeds]β(p,w) (it means ξ[succeeds]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[succeeds]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 [straight phi](x) in H∩[x,ξ] . The function [straight phi] 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[succeeds]β(p,w) . Then, there exists ζ∈X such that ρ(ζ,β(p,w))<r and ζ[succeeds]β(p,w) .

Proof.

See the appendix.

And the following lemma.

Lemma 2.12 ([1, Lemma 2.8]).

Let R , c , and t be positive numbers. Then, there exists r>0 with the following property: if p , ^{p[variant prime]} are elements of ^RN such that |p|≥c and |p-^{p[variant prime]} |<r , w , ^{w[variant prime]} are real numbers such that |w-^{w[variant prime]} |<r , and ^{y[variant prime]} is an element of ^RN such that |^{y[variant prime]} |≤R and ^{p[variant prime]} ·^{y[variant prime]} =^{w[variant prime]} , then there exists ζ∈^RN such that p·ζ=w and |^{y[variant prime]} -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 3.1.

Let [succeeds, ~] 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 exists ξ∈X such that ξ[succeeds]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)[succeeds, ~]x (it means y[succeeds, ~]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 correspondence F(p,w) is [figure omitted; refer to PDF]

If G(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[succeeds]β(p,w) . By Lemma 2.11, construct a sequence (_ζm ) in X such that _ζm [succeeds]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 t_ζm +(1-t)_ζm+1 [succeeds]β(p,w) for 0<t<1 and each m . Thus, we can construct a sequence (_ζn ) such that |_ζn -_ζn+1 |<^{[varepsilon]n} , ρ(_ζn ,β(p,w))<^δn and _ζn [succeeds]β(p,w) for some 0<[varepsilon]<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 [succeeds, ~] ) ^ζ* [succeeds, ~]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[variant prime]} of β(p,w) such that p·^{ζ[variant prime]} =w and ^{ζ[variant prime]} [succeeds, ~]β(p,w) . Therefore, F(p,w) is a set and we get a demand correspondence. Let ζ∈F(p,w) and ^{ζ[variant prime]} ∈F(p,w) . Then, ζ[succeeds, ~]β(p,w) , ^{ζ[variant prime]} [succeeds, ~]β(p,w) , and convexity of the preference implies tζ+(1-t)^{ζ[variant prime]} [succeeds, ~]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[variant prime]} ,^{w[variant prime]} ) such that |p-^{p[variant prime]} |<r and |w-^{w[variant prime]} |<r with r>0 . Let F(p,w) and F(^{p[variant prime]} ,^{w[variant prime]} ) be demand sets. Let ^{y[variant prime]} ∈F(^{p[variant prime]} ,^{w[variant prime]} ) , c=ρ(0,Δ)>0 , and R>0 such that X⊂B¯(0,R) . Given [straight epsilon]>0 , t=δ>0 such that δ<[straight epsilon] , and choose r as in Lemma 2.12. By that lemma, we can choose ζ∈^RN such that p·ζ=w and |^{y[variant prime]} -ζ|<δ . Similarly, we can choose ^{ζ[variant prime]} (y)∈^RN such that ^{p[variant prime]} ·^{ζ[variant prime]} (y)=^{w[variant prime]} and |y-^{ζ[variant prime]} (y)|<δ for each y∈F(p,w) . ^{y[variant prime]} ∈F(^{p[variant prime]} ,^{w[variant prime]} ) means ^{y[variant prime]} [succeeds, ~]^{ζ[variant prime]} (y) . Either |^{y[variant prime]} -y|>[straight epsilon]/2 for all y∈F(p,w) or |^{y[variant prime]} -y|<[straight epsilon] for some y∈F(p,w) . Assume that |^{y[variant prime]} -y|>[straight epsilon]/2 for all y∈F(p,w) and y[succeeds]ζ . If δ is sufficiently small, |^{y[variant prime]} -y|>[straight epsilon]/2 means |y-ζ|>[straight epsilon]/k and |^{y[variant prime]} -^{ζ[variant prime]} (y)|>[straight epsilon]/k for some finite natural number k . Then, by weak uniform rotundity, there exist _zn and ^{zn[variant prime]} such that |_zn |<_τn , |^{zn[variant prime]} |<_τn with _τn >0 , _zn >>0 and ^{zn[variant prime]} >>0 , (1/2)(y+ζ)+_zn [succeeds]ζ and (1/2)(^{y[variant prime]} +^{ζ[variant prime]} (y))+^{zn[variant prime]} [succeeds]^{ζ[variant prime]} (y) for n=1,2,... . Again if δ is sufficiently small, |y-^{ζ[variant prime]} (y)|<δ and |^{y[variant prime]} -ζ|<δ imply (1/2)(y+ζ)+_zn [succeeds]^{y[variant prime]} and (1/2)(^{y[variant prime]} +^{ζ[variant prime]} (y))+^{zn[variant prime]} [succeeds]y . And it follows that |(1/2)(y+ζ)-(1/2)(^{y[variant prime]} +^{ζ[variant prime]} (y))|<δ . By continuity of the preference (openness of [succeeds] ) (1/2)(y+ζ)+_{z[variant prime]n} [succeeds]y . Let _y1 =(1/2)(y+ζ) . Consider a sequence (_τn ) converging to zero. By continuity of the preference (closedness of [succeeds, ~] ) _y1 [succeeds, ~]^{y[variant prime]} and _y1 [succeeds, ~]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|>[straight epsilon]/k . Therfore, we have |^{y[variant prime]} -y|<[straight epsilon] or ζ[succeeds, ~]y (it means |y-ζ|<δ+[straight epsilon] and ζ∈F(p,w) ), and so F(p,w) has a closed graph.

Acknowledgment

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.

References

[1] 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.

[2] E. Bishop, D. Bridges Constructive Analysis , vol. 279, of Grundlehren der Mathematischen Wissenschaften, pp. xii+477, Springer, Berlin, Germany, 1985.

[3] D. Bridges, F. Richman Varieties of Constructive Mathematics , vol. 97, of London Mathematical Society Lecture Note Series, pp. x+149, Cambridge University Press, Cambridge, UK, 1987.

[4] D. Bridges, L. Vîta Techniques of Constructive Mathematics , Springer, 2006.

[5] D. S. Bridges, "Constructive notions of strict convexity," Mathematical Logic Quarterly , vol. 39, no. 3, pp. 295-300, 1993.

Appendix

Appendix

A. Proof of Lemma 2.11

This proof is almost identical to the proof of Lemma 2.4 in Bridges [1]. They are different in a few points.

Let H be the hyperplane with equation p·x=w and ^{ξ[variant prime]} the projection of ξ on H . Assume |ξ-^{ξ[variant prime]} |>3r . Choose R such that H∩β(p,w) is contained in the closed ball B¯(^{ξ[variant prime]} ,R) around ^{ξ[variant prime]} and let [figure omitted; refer to PDF] Let ^{H[variant prime]} be the hyperplane parallel to H , between H and ξ and a distance r/2c from H , and ^{H[variant prime][variant prime]} the hyperplane parallel to H , between H and ξ and a distance r/c from H . For each x∈β(p,w) let [straight phi](x) be the unique element of H∩[x,ξ] , ^{[straight phi][variant prime]} (x) the unique element of ^{H[variant prime]} ∩[x,ξ] , and ^{[straight phi][variant prime][variant prime]} (x) the unique element of ^{H[variant prime][variant prime]} ∩[x,ξ] . Since ξ[succeeds]β(p,w) , we have ^{[straight phi][variant prime][variant prime]} (x)[succeeds][straight phi](x)[succeeds, ~]x by convexity and continuity of the preference. ^{[straight phi][variant prime]} (x) is uniformly continuous, so [figure omitted; refer to PDF] is totally bounded by Lemma 2.8 and Proposition 4.2 in Chapter 4 of [2].

Since ^{[straight phi][variant prime][variant prime]} (x)[succeeds][straight phi](x) and ^{[straight phi][variant prime]} (x)=(1/2)^{[straight phi][variant prime][variant prime]} (x)+(1/2)[straight phi](x) , we have ^{[straight phi][variant prime]} (x)[succeeds]x , and so continuity of the preference (openness of [succeeds] ) means that there exists δ>0 such that ^{[straight phi][variant prime]} (_xi )[succeeds]x when |^{[straight phi][variant prime]} (_xi )-^{[straight phi][variant prime]} (x)|<δ . Let (_x1 ,...,_xn ) be points of β(p,w) such that (^{[straight phi][variant prime]} (_x1 ),...,^{[straight phi][variant prime]} (_xn )) is a δ -approximation to T . Given x in β(p,w) , choose i such that |^{[straight phi][variant prime]} (_xi )-^{[straight phi][variant prime]} (x)|<δ . Then, ^{[straight phi][variant prime]} (_xi )[succeeds]x .

Now, from our choice of c , we have |[straight phi](x)-^{[straight phi][variant prime]} (x)|<r/2 for each x∈β(p,w) . It is proved as follows. Since by the assumption |[straight phi](x)-^{ξ[variant prime]} |<R , |[straight phi](x)-ξ|<^R2 +^{|ξ-ξ[variant prime] |2} . Thus, we have [figure omitted; refer to PDF] See Figure 1.

Figure 1: Calculation of |[straight phi](x)-^{[straight phi][variant prime]} (x)| .

[figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] Then, |_η1 -^{[straight phi][variant prime]} (_x1 )|=r/2n , ρ(_η1 ,β(p,w))<r(n+1)/2n (because |[straight phi](_x1 )-^{[straight phi][variant prime]} (_x1 )|<r/2 and [straight phi](_x1 )∈β(p,w)) , and by convexity of the preference _η1 [succeeds, ~]ξ or _η1 [succeeds, ~][straight phi][variant prime](_x1 ) .

In the first case, we complete the proof by taking ζ=_η1 . In the second, assume that, for some k (1≤k≤n-1 ), we have constructed _η1 ,...,_ηk in X such that [figure omitted; refer to PDF] As |ξ-_ηk |>r (because |ξ-^{ξ[variant prime]} |>3r ), we can choose y∈[_ηk ,ξ] such that |y-_ηk |=r/2n . Then, ρ(y,β(p,w))<r(n+k+1)/2n and either y[succeeds, ~]ξ or y[succeeds, ~]_ηk . In the former case, the proof is completed by taking ζ=y . If y[succeeds, ~]_ηk , y+λ/2[succeeds]_ηk -λ/2 for all λ such that λ>>0 . Then, either y+λ/2[succeeds]^{[straight phi][variant prime]} (_xk+1 ) for all λ and so y[succeeds, ~]^{[straight phi][variant prime]} (_xk+1 ) , in which case we set _ηk+1 =y ; or else ^{[straight phi][variant prime]} (_xk+1 )[succeeds]_ηk -λ/2 for all λ and so ^{[straight phi][variant prime]} (_xk+1 )[succeeds, ~]_ηk , then we set _ηk+1 =^{[straight phi][variant prime]} (_xk+1 ) .

If this process proceeds as far as the construction of _ηn , then, setting ζ=_ηn , we see that ρ(ζ,β(p,w))<r and that ζ[succeeds, ~][straight phi][variant prime](_xi ) for each i ; so ζ[succeeds]x for each x∈β(p·w) .