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

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1 MPRA Muich Persoal RePEc Archive A costructive aalysis of covex-valued demad correspodece for weakly uiformly rotud ad mootoic preferece Yasuhito Taaka ad Atsuhiro Satoh. May 04 Olie at MPRA Paper No , posted 4. May 04 0:5 UTC

2 A costructive aalysis of covex-valued demad correspodece for weakly uiformly rotud ad mootoic preferece Yasuhito Taaka Faculty of Ecoomics, Doshisha Uiversity, Kamigyo-ku, Kyoto, , Japa ad Atsuhiro Satoh Faculty of Ecoomics, Doshisha Uiversity, Kamigyo-ku, Kyoto, , Japa Abstract Bridges([4]) has costructively show the existece of cotiuous demad fuctio for cosumers with cotiuous, uiformly rotud preferece relatios. We exted this result to the case of multi-valued demad correspodece. We cosider a weakly uiformly rotud ad mootoic preferece relatio, ad will show the existece of covex-valued demad correspodece with closed graph for cosumers with cotiuous, weakly uiformly rotud ad mootoic preferece relatios. We follow the Bishop style costructive mathematics accordig to [], [] ad [3]. Keywords: costructive aalysis, demad correspodece, weakly uiformly rotud ad mootoic preferece. Itroductio Bridges([4]) has costructively show the existece of cotiuous demad fuctio for cosumers with cotiuous, uiformly rotud preferece relatios. We exted this result to the case of multi-valued demad correspodece. We cosider a weakly uiformly rotud ad mootoic preferece relatio, ad will show the existece of covex-valued demad correspodece with closed graph Correspodig author, yasuhito@mail.doshisha.ac.jp atsato@mail.doshisha.ac.jp

3 for cosumers with cotiuous, weakly uiformly rotud ad mootoic preferece relatios I the ext sectio we summarize some prelimiary results most of which were proved i [4]. I Sectio 3 we will show the mai result. We follow the Bishop style costructive mathematics accordig to [], [] ad [3]. Prelimiary results Cosider a cosumer who cosumes N goods. N is a fiite atural umber larger tha. Let X R N be his cosumptio set. It is a compact (totally bouded ad complete) ad covex set. Let be a -dimesioal simplex, ad p be a ormalized price vector of the goods. Let p i be the price of the i-th good, the N i= p i = ad p i 0 for each i. For a give p the budget set of the cosumer is β(p, w) {x X : p x w} w > 0 is his iitial edowmet. A preferece relatio of the cosumer is a biary relatio o X. Let x, y X. If he prefers x to y, we deote x y. A preferece-idifferece relatio is defied as follows; x y if ad oly if (y x) x y etails x y, the relatios ad are trasitive, ad if either x y z or x y z, the x z. Also we have x y if ad oly if z X (y z x z). A preferece relatio is cotiuous if it is ope as a subset of X X, ad is a closed subset of X X. A preferece relatio o X is uiformly rotud if for each ε there exists a δ(ε) with the followig property. Defiitio (Uiformly rotud preferece). Let ε > 0, x ad y be poits of X such that x y ε, ad z be a poit of R N such that z δ(ε), the either (x + y) + z x or (x + y) + z y. Strict covexity of preferece is defied as follows; Defiitio (Strict covexity of preferece). If x, y X, x y, ad 0 < t <, the either tx + ( t)y x or tx + ( t)y y. Bridges [5] has show that if a preferece relatio is uiformly rotud, the it is strictly covex. O the other had covexity of preferece is defied as follows; Defiitio 3 (Covexity of preferece). If x, y X, x y, ad 0 < t <, the either tx + ( t)y x or tx + ( t)y y.

4 We defie the followig weaker versio of uiform rotudity. Defiitio 4 (Weakly uiformly rotud preferece). Let ε > 0, x ad y be poits of X such that x y ε. Let z be a poit of R N such that z δ for δ > 0 ad z 0(every compoet of z is positive), the (x + y) + z x or (x + y) + z y. We assume also that cosumers prefereces are mootoic i the sese that if x > x (it meas that each compoet of x is larger tha or equal to the correspodig compoet of x, ad at least oe compoet of x is larger tha the correspodig compoet of x), the x x. Now we show the followig lemmas. Lemma. If x, y X, x y, the weak uiform rotudity of prefereces implies that (x + y) x or (x + y) y. Proof. Cosider a decreasig sequece (δ m ) of δ i Defiitio 4. The, either (x + y) + z m x or (x + y) + z m y for z m such that z m < δ m ad z m 0 for each m. Assume that (δ m ) coverges to zero. The, (x + y) + z m coverges to (x + y). Cotiuity of the preferece (closedess of ) implies that (x + y) x or (x + y) y. Lemma. If a cosumer s preferece is weakly uiformly rotud, the it is covex. This is a modified versio of Propositio. i [5]. Proof.. Let x ad y be poits i X such that x y ε. Cosider a poit (x + y). The, x (x + y) ε ad (x + y) y ε. Thus, usig Lemma we ca show 4 (3x + y) x or 4 (3x + y) y, ad 4 (x + 3y) x or 4 (x + 3y) y. Iductively we ca show that for k =,,..., k x + k y x or k x + k y y for each atural umber.. Let z = tx + ( t)y with a real umber t such that 0 < t <. We ca select a atural umber k so that k t k+ for each atural umber. ( k+ k ) = ( ) is a sequece. Sice, for atural umbers m ad l such that m >, t l+ m ad k m t k+ with some atural umber l, we have ( l + m l ) ( k + m k ) = m m <, ( k+ k ) is a Cauchy sequece, ad coverges to zero. The, ( k+ ) ad ( k ) coverge to t. Closedess of implies that either z x or z y. Therefore, the preferece is covex. Lemma 3. Let x ad y be poits i X such that x y. The, if a cosumer s preferece is weakly uiformly rotud ad mootoic, tx + ( t)y y for 0 < t <. 3

5 Proof. By cotiuity of the preferece (opeess of ) there exists a poit x = x λ such that λ 0 ad x y. The, sice weak uiform rotudity implies covexity, we have tx + ( t)y y or tx + ( t)y x. If tx + ( t)y x, the by trasitivity tx + ( t)y = tx + ( t) tλ x y. Mootoicity of the preferece implies tx + ( t)y y. Assume tx + ( t)y y. The, agai mootoicity of the preferece implies tx + ( t)y y. Let S be a subset of R such that for each (p, w) S. p.. β(p, w) is oempty. 3. There exists ξ X such that ξ x for all x β(p, w). I [4] the followig lemmas were proved. Lemma 4 (Lemma. i [4]). If p R N, w R, ad β(p, w) is oempty, the β(p, w) is compact. Lemma 4 with Propositio (4.4) i Chapter 4 of [] or Propositio..9 of [3] implies that for each (p, w) S β(p, w) is located i the sese that the distace exists for each x R N. ρ(x, β(p, w)) if{ x y : y β(p, w)} Lemma 5 (Lemma. i [4]). If (p, w) S ad ξ β(p, w) (it meas ξ x for all x β(p, w)), the ρ(ξ, β(p, w)) > 0 ad p ξ > w. Lemma 6 (Lemma.3 i [4]). Let (p, c) S, ξ X ad ξ β(p, c). Let H be the hyperplae with equatio p x = c. The, for each x β(p, c), there exists a uique poit φ(x) i H [x, ξ]. The fuctio φ so defied maps β(p, c) oto H β(p, c) ad is uiformly cotiuous o β(p, c). Lemma 7 (Lemma.4 i [4]). Let (p, w) S, r > 0, ξ X, ad ξ β(p, w). The, there exists ζ X such that ρ(ζ, β(p, w)) < r ad ζ β(p, w). Proof. See Appedix. Ad the followig lemma. Lemma 8 (Lemma.8 i [4]). Let R,c, ad t be positive umbers. The there exists r > 0 with the followig property: if p, p are elemets of R N such that p c ad p p < r, w, w are real umbers such that w w < r, ad y is a elemet of R N such that y R ad p y = w, the there exists ζ R N such that p ζ = w ad y ζ < t. It was proved by settig r = ct R+. 4

6 3 Covex-valued demad correspodece with closed graph With the prelimiary results i the previous sectio we show the followig our mai result. Theorem. Let be a weakly uiformly rotud preferece relatio o a compact ad covex subset X of R N, be a compact ad covex set of ormalized price vectors (a -dimesioal simplex), ad S be a subset of R such that for each (p, w) S. p.. β(p, w) is oempty. 3. There exists ξ X such that ξ x for all x β(p, w). The, for each (p, w) S there exists a subset F (p, w) of β(p, w) such that F (p, w) x (it meas 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)), ad the multi-valued correspodece F (p, w) is covex-valued ad has a closed graph. Proof. A graph of a correspodece F (p, w) is G(F ) = (p,w) S (p, w) F (p, w). If G(F ) is a closed set, we say that F has a closed graph.. Let (p, w) S, ad choose ξ X such that ξ β(p, w). By Lemma 7 costruct a sequece (ζ m ) i X such that ζ m β(p, w) ad ρ(ζ m, β(p, w)) < r m with r > 0 for each atural umber m. By covexity ad trasitivity of the preferece tζ m + ( t)ζ m+ β(p, w) for 0 < t < ad each m. Thus, we ca costruct a sequece (ζ ) such that ζ ζ + < ε, ρ(ζ, β(p, w)) < δ ad ζ β(p, w) for some 0 < ε < ad 0 < δ <, ad so (ζ ) is a Cauchy sequece i X. It coverges to a limit ζ X. By cotiuity of the preferece (closedess of ) ζ β(p, w), ad ρ(ζ, β(p, w)) = 0. Sice β(p, w) is closed, ζ β(p, w). By Lemma 5 p ζ > w for all. Thus, we have p ζ = w. Covexity of the preferece implies that ζ may ot be uique, that is, there may be multiple elemets ζ of β(p, w) such that p ζ = w ad ζ β(p, w). Therefore, F (p, w) is a set ad we get a demad correspodece. Let ζ F (p, w) ad ζ F (p, w). The, ζ β(p, w), ζ β(p, w), ad covexity of the preferece implies tζ + ( t)ζ β(p, w). Thus, F (p, w) is covex.. Next we prove that the demad correspodece has a closed graph. Cosider (p, w) ad (p, w ) such that p p < r ad w w < r with r > 0. Let F (p, w) ad F (p, w ) be demad sets. Let y F (p, w ), 5

7 c = ρ(0, ) > 0 ad R > 0 such that X B(0, R). Give ε > 0, t = δ > 0 such that δ < ε, ad choose r as i Lemma 8. By that lemma we ca choose ζ R N such that p ζ = w ad y ζ < δ. Similarly, we ca choose ζ (y) R N such that p ζ (y) = w ad y ζ (y) < δ for each y F (p, w). y F (p, w ) meas y ζ (y). Either y y > ε for all y F (p, w) or y y < ε for some y F (p, w). Assume that y y > ε for all y F (p, w) ad y ζ. If δ is sufficietly small, y y > ε meas y ζ > ε k ad y ζ (y) > ε k for some fiite atural umber k. The, by weak uiform rotudity there exist z ad z such that z < τ, z < τ with τ > 0, z 0 ad z 0, (y + ζ) + z ζ ad (y + ζ (y)) + z ζ (y) for =,,.... Agai if δ is sufficietly small, y ζ (y) < δ ad y ζ < δ imply (y + ζ) + z y ad (y + ζ (y)) + z y. Ad it follows that (y + ζ) (y + ζ (y)) < δ. By cotiuity of the preferece (opeess of ) (y + ζ) + z y. Let y = (y + ζ). Cosider a sequece (τ ) covergig to zero. By cotiuity of the preferece (closedess of ) y y ad y y. Note that p y = w. Thus, y β(p, w). Sice y F (p, w), we have y F (p, w). Replacig y with y, we ca show that y+3ζ 4 F (p, w). Iductively we obtai y+(m )ζ F (p, w) for each atural umber m. The, we have m y ζ < η for some y F (p, w) for ay η > 0. It cotradicts y ζ > ε k. Therfore, we have y y < ε or ζ y (it meas y ζ < δ ad ζ F (p, w)), ad so F (p, w) has a closed graph. Appedix: Proof of Lemma 7 This proof is almost idetical to the proof of Lemma.4 i Bridges [4]. They are differet i a few poits. Let H be the hyperplae with equatio p x = w ad ξ the projectio of ξ o H. Assume ξ ξ > 3r. Choose R such that H β(p, w) is cotaied i the closed ball B(ξ, R) aroud ξ, ad let c = + ( ) R ξ ξ. Let H be the hyperplae parallel to H, betwee H ad ξ ad a distace r c from H; ad H the hyperplae parallel to H, betwee H ad ξ ad a distace r c from H. For each x β(p, w) let φ(x) be the uique elemet of H [x, ξ], φ (x) be the uique elemet of H [x, ξ], ad φ (x) be the uique elemet of H [x, ξ]. Sice ξ β(p, w), we have φ (x) φ(x) x by covexity ad cotiuity of the preferece. φ (x) is uiformly cotiuous, so T {φ (x) : x β(p, w)} is totally bouded by Lemma 4 ad Propositio (4.) i Chapter 4 of []. 6

8 x ϕ(x) ϕ (x) ξ ξ H H Figure : Calculatio of φ(x) φ (x) Sice φ (x) φ(x) ad φ (x) = φ (x) + φ(x) we have φ (x) x, ad so cotiuity of the preferece (opeess of ) meas that there exists δ > 0 such that φ (x i ) x whe φ (x i ) φ (x) < δ. Let (x,..., x ) be poits of β(p, w) such that (φ (x ),..., φ (x )) is a δ-approximatio to T. Give x i β(p, w) choose i such that φ (x i ) φ (x) < δ. The, φ (x i ) x. Now from our choice of c we have φ(x) φ (x) < r for each x β(p, w). It is proved as follows. Sice by the assumptio φ(x) ξ < R, φ(x) ξ < R + ξ ξ. Thus, we have φ(x) φ (x) < r c R + ξ ξ ξ ξ = r c + ( ) R ξ ξ = r. See Figure. Let ad t = r φ (x ) ξ, η = t φ (x ) + ( t )ξ. The, η φ (x ) = r, ρ(η, β(p, w)) < r(+) (because φ(x ) φ (x ) < r ad φ(x ) β(p, w)), ad by covexity of the preferece η ξ or η φ (x ). I the first case we complete the proof by takig ζ = η. I the secod, assume that, for some k ( k ), we have costructed η,..., η k i X such that η k φ (x i ) ( i k), ad r( + k) ρ(η k, β(p, w)) <. As ξ η k > r (because ξ ξ > 3r), we ca choose y [η k, ξ] such that y η k = r r(+k+). The ρ(y, β(p, w)) < ad either y ξ or y η k. I the 7

9 former case, the proof is completed by takig ζ = y. If y η k, y + λ η k λ for all λ such that λ 0. The, either y + λ φ (x k+ ) for all λ ad so y φ (x k+ ), i which case we set η k+ = y; or else φ (x k+ ) η k λ for all λ ad so φ (x k+ ) η k, the we set η k+ = φ (x k+ ). If this process proceeds as far as the costructio of η, the, settig ζ = η, we see that ρ(ζ, β(p, w)) < r ad that ζ φ (x i ) for each i; so ζ x for each x β(p.w). Ackowledgmet This research was partially supported by the Miistry of Educatio, Sciece, Sports ad Culture of Japa, Grat-i-Aid for Scietific Research (C), , ad the Special Costs for Graduate Schools of the Special Expeses for Hitech Promotio by the Miistry of Educatio, Sciece, Sports ad Culture of Japa i00. Refereces [] E. Bishop ad D. Bridges, Costructive Aalysis, Spriger, 985. [] D. Bridges ad F. Richma, Varieties of Costructive Mathematics, Cambridge Uiversity Press, 987. [3] D. Bridges ad L. Vîţă, Techiques of Costructive Mathematics, Spriger, 006. [4] D. Bridges, The costructio of a cotiuous demad fuctio for uiformly rotud prefereces, Joural of Mathematical Ecoomics, vol., pp. 7-7, 99. [5] D. Bridges, Costructive otios of strict covexity, Mathematical Logic Quarterly, vol. 39, pp ,