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 http://mpra.ub.ui-mueche.de/55889/ MPRA Paper No. 55889, posted 4. May 04 0:5 UTC
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, 60-8580, Japa ad Atsuhiro Satoh Faculty of Ecoomics, Doshisha Uiversity, Kamigyo-ku, Kyoto, 60-8580, 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, E-mail: yasuhito@mail.doshisha.ac.jp E-mail: atsato@mail.doshisha.ac.jp
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.
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
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
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
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
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
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), 053065, 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. 95-300, 993. 8