A stronger criterion for the weak weak axiom of revealed preference

ĐẠI HỌC QUỐC GIA TP HỒ CHÍ MINH TRƯỜNG ĐẠI HỌC BÁCH KHOA TRẦN TRỌNG HOÀNG TUẤN MỘT TIÊU CHUẨN MẠNH HƠN CHO TIÊN ĐỀ YẾU YẾU VỀ LÝ THUYẾT SỞ THÍCH ĐƯỢC BỘC LỘ A STRONGER CRITERION FOR THE WEAK WEAK AXIOM OF REVEALED PREFERENCE CHUYÊN NGÀNH: TOÁN ỨNG DỤNG Mã ngành: 8460112 LUẬN VĂN THẠC SĨ TOÁN HỌC TP HỒ CHÍ MINH, tháng 07 năm 2022 THIS THESIS IS COMPLETED AT HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY Advisor: Assoc Prof Dr Phan Thành An Examiner 1: Dr Nguyễn Bá Thi Examiner 2: Assoc Prof Dr Nguyễn Huy Tuấn Master’s thesis is defended at Ho Chi Minh City University of Technology on 15 July 2022 The board of the Master’s Thesis Defense Council includes: Chairman: Assoc Prof Dr Nguyễn Đình Huy Secretary: Dr Huỳnh Thị Ngọc Diễm Reviewer 1: Dr Nguyễn Bá Thi Reviewer 2: Assoc Prof Dr Nguyễn Huy Tuấn Member: Dr Cao Thanh Tình Verification of the Chairman of the Master’s Thesis Defense Council and the Dean of the Faculty after the thesis being corrected (if any) CHAIRMAN DEAN OF FACULTY Assoc Prof Dr Nguyễn Đình Huy Assoc Prof Dr Trương Tích Thiện THE SOCIALIST REPUBLIC OF VIET NAM Independence - Freedom - Happiness TASK SHEET OF MASTER’S THESIS Name: Trần Trọng Hoàng Tuấn Date of birth: 31 March 1993 Student code: 2070243 Place of birth: Bạc Liêu Major: Applied Mathematics Major code: 8460112 I THESIS TOPIC: A STRONGER CRITERION FOR THE WEAK WEAK AXIOM OF REVEALED PREFERENCE MỘT TIÊU CHUẨN MẠNH HƠN CHO TIÊN ĐỀ YẾU YẾU VỀ LÍ THUYẾT SỞ THÍCH ĐƯỢC BỘC LỘ II TASKS: - Study the criteria for the weak weak axiom of revealed preference introduced by Brighi in "Brighi, Luigi, A stronger criterion for the Weak Weak Axiom, Journal of Mathematical Economics Vol 40, 93-103 (2004)" - Study the stability introduced by An in "An, Phan Thanh, Stability of generalized monotone maps with respect to their characterizations, Optimization, Vol 55, 289–299 (2006)" w.r.t the criteria introduced by Brighi III DATE OF TASK ASSIGNMENT: 14 February 2022 IV DATE OF TASK COMPLETION: June 2022 V ADVISOR: Assoc Prof Dr Phan Thành An Ho Chi Minh City, 15 July 2022 ADVISOR HEAD OF DIVISION Assoc Prof Dr Phan Thành An Dr Nguyễn Tiến Dũng DEAN OF FACULTY Assoc Prof Dr Trương Tích Thiện ACKNOWLEDGMENT First of all, I would like to sincerely thank Ho Chi Minh City University of Technology for sponsoring my research in this thesis, and thank my thesis advisor, Assoc Prof Dr Phan Thành An for his careful and clear guidance In addition, I want to show my grateful attitude to my family members for the sympathy as they are always stand by me during my master thesis period Last but not least, I would like to deliver my gratitude to Division of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh City University of Technology for giving me such valuable opportunity to apply all knowledge I have learned during my master of mathematics Ho Chi Minh City, 15 July 2022 Author Trần Trọng Hoàng Tuấn i ABSTRACT In this thesis, the following stronger criteria (A), (B), (C), (A’) for the weak weak axiom of revealed preference introduced by Brighi in 2004 are presented: Criterion (A): For all x ∈ D and v ∈ Rn v T F (x) = ⇒ v T ∂F (x)v ≤ Criterion (B): For all x ∈ D and v ∈ Rn , if F (x) = and v T ∂F (x)v = 0, then ∀t > 0, ∃t ∈ (0, t) : v T F (x + tv) ≤ Criterion (C): For all x ∈ D and v ∈ Rn , if F (x) = and v T ∂F (x) = 0, then ∀t > 0, ∃t ∈ (0, t) : v T F (x + tv) ≤ Criterion (A’): v T p = 0, v T Z(p) = ⇒ v T ∂Z(p)v ≤ Criterion (C’): v T p = 0, Z(p) = 0, v T ∂Z(p) = 0, ∀t > 0, ∃t ∈ (0, t) : v T Z(p + tv) ≤ where F : D ⊆ Rn → Rn , Z(p) : P ⊂ Rn>0 → Rn We will present the concept of stability introduced by An in 2006 We also present that pseudo-monotone maps are not stable w.r.t the criteria (A), (B), (C), (A’) ii TÓM TẮT LUẬN VĂN Trong luận văn, ta nghiên cứu điều kiện (A), (B), (C), (A’) giới thiệu Luigi Brighi năm 2004 cho tiên đề yếu yếu lí thuyết sở thích bộc lộ sau: Điều kiện (A): Với x ∈ D v ∈ Rn v T F (x) = ⇒ v T ∂F (x)v ≤ Điều kiện (B): Với x ∈ D v ∈ Rn , F (x) = v T ∂F (x)v = 0, ∀t > 0, ∃t ∈ (0, t) : v T F (x + tv) ≤ Điều kiện (C): Với x ∈ D v ∈ Rn , F (x) = v T ∂F (x) = 0, ∀t > 0, ∃t ∈ (0, t) : v T F (x + tv) ≤ Điều kiện (A’): v T p = 0, v T Z(p) = ⇒ v T ∂Z(p)v ≤ Điều kiện (C’): v T p = 0, Z(p) = 0, v T ∂Z(p) = 0, ∀t > 0, ∃t ∈ (0, t) : v T Z(p + tv) ≤ F : D ⊆ Rn → Rn , Z(p) : P ⊂ Rn>0 → Rn Ta trình bày khái niệm tính ổn định giới thiệu Phan Thành An năm 2006 Ta trình bày hàm giả đơn điệu khơng ổn định với điều kiện (A), (B), (C), (A’) iii DECLARATION OF AUTHORSHIP I hereby declare that this thesis was carried out by myself under the guidance and supervision of Assoc Prof Dr Phan Thành An, and that the work contained and the results in it are true by author and have not violated research ethics In addition, other comments, reviews and data used by other authors, and organizations have been acknowledged, and explicitly cited I will take full responsibility for any fraud detected in my thesis Ho Chi Minh City University of Technology is unrelated to any copyright infringement caused on my work (if any) Ho Chi Minh City, 15 July 2022 Author Trần Trọng Hoàng Tuấn iv Contents ACKNOWLEDGEMENT i ABSTRACT ii TÓM TẮT LUẬN VĂN iii DECLARATION OF AUTHORSHIP iv LIST OF ACRONYMS, NOTATIONS AND CRITERIA vii INTRODUCTION 1 THEORETICAL BASIS 1.1 Consumer theory 1.1.1 Preference relation 1.1.2 Rational preference 1.1.3 Local nonsatiation 1.1.4 Utility maps 1.1.5 Budget sets 1.2 Demand maps and excess demand maps THE CRITERIA FOR THE WEAK WEAK AXIOM OF REVEALED PREFERENCE 2.1 Pseudo-monotone maps 2.2 Quasi-monotone maps 11 2.3 Weak weak axiom of revealed preference 13 v 2.4 Weak axiom of revealed preference 17 THE STABILITY OF GENERALIZED MONOTONE MAPS WITH RESPECT TO SOME CRITERIA 21 3.1 S-quasimonotone maps 21 3.2 The stronger version of Wald’s Axiom 23 3.3 Stability with respect to criterion (A) 25 3.4 Stability with respect to criterion (A’) 27 3.5 Stability with respect to criterion (B) and (C) 28 CONCLUSION 30 BIBLIOGRAPHY 32 vi LIST OF ACRONYMS AND NOTATIONS Acronym/Notation Meaning SWA Stronger version of Wald’s Axiom WA Weak Axiom of Revealed Preference WWA Weak Weak Axiom of Revealed Preference R Set of real numbers Rn Vector space of real numbers in n-dimension AT Transpose of matrix A ∂ F(x) Jacobian matrix of F(x) kak Euclidean norm of vector a LIST OF CRITERIA Criterion Page Criterion A Criterion B Criterion C Criterion A’ 15 Criterion C’ 15 Criterion LZ 17 Criterion LZ’ 17 Criterion D vii 23 Applied Mathematics Master’s Thesis Thus, g is also pseudo-monotone By Property 2.1.1, the set of zeroes of a pseudo-monotone map is convex, moreover g(1) = g(0) = 0, the map g is constant on [0, 1] and criterion (LZ’) cannot hold This contradiction completes the proof See [3] and other theorem proved above, we study the characteristics of demand map and excess demand map as follows: For demand map, criteria (A) and (LZ) characterize the WA For excess demand map, criteria (A), (LZ) and (C) characterize the WA In other words, the difference between characteristics of WA for demand map and excess demand map is pseudo-monotonicity We consider the following example of a map satisfying criteria (A) and (LZ) but not satisfying pseudo-monotonicity: Example 2.4.2 ([3]) Let F (x) = x3 (2 + sin(1/x)) and F (0) = Assuming that x < < y , we have (y − x)x3 (2 + sin(1/x)) ≤ 0, (y − x)y (2 + sin(1/y)) ≥ It follows that F (x) is not pseudo-monotone Since F (0) = when x = and F (0) = 0, criterion (A) holds To show that (LZ) holds, since x 6= 0, (y − x)F (x) = iff y = x, we need to consider the case x = We have ∀y 6= 0, F (y) 6= F (0) and F (y) < when y = 1/2πk or y = −1/(2k + 1)π , where k is any positive integer Thus, in any neighbourhood of of 0, the derivative of F (x) takes on negative values, i.e criterion (LZ) holds Next, we will consider the excess demand map satisfying criteria (A’) and (LZ’), but not satisfying WA Example 2.4.3 ([3]) We consider the excess demand map as follows: Z(p) = (h(p1 /p2 ), −p1 h(p1 /p2 )/p2 ) Trần Trọng Hoàng Tuấn 19 Applied Mathematics Master’s Thesis where h(s) = F (s − 1), with F (s) = s3 /(2 + sin(1/s)), F (0) = At the unique equilibrium p∗ = (1, 1), the Jacobian matrix of Z(p) is null, therefore criterion (A’) holds We consider criterion (LZ’) If Z(p) 6= 0, then we have v = because v T Z(p) = 0, criterion (LZ’) holds Next, we consider v = q − p∗ such that v T p∗ Excluding the trivial case v = 0, we obtain v2 = −v1 with v1 6= because v T p∗ = We consider g(t) = v T Z(p∗ + tv) = v1 h(s(t))(1 + s(t)) where s(t) = + tv1 − tv1 with t ∈ [0, 1] We calculate the derivative of g(t) as follows: g (t) = v1 s0 (t)[h0 (s(t))(1 + s(t)) + h(s(t)] = v T ∂Z(p∗ + tv)v If v1 > we choose t = 1/(v1 (4πk + 1)) for k sufficiently large such that t ∈ [0, 1], we have g (t) < If v1 < we choose t such that 1/(s(t) − 1) = −(2k + 1)π , we have g (t) < It follows that criterion (LZ’) holds However, the map Z(p) does not satisfies WA We choose p∗ = (1, 1) and q = (2, 1), we have (q − p∗ )T Z(p∗ ) = 0, (q − p∗ )T Z(q) = > Trần Trọng Hoàng Tuấn 20 Applied Mathematics Master’s Thesis Chapter THE STABILITY OF GENERALIZED MONOTONE MAPS WITH RESPECT TO SOME CRITERIA Definition 3.0.1 (see [1]) ∀a ∈ Rn , Z : Rn>0 → Rn , let Z + a be a map to be defined as (Z + a)(p) = Z(p) + a A map Z is said to be stable w.r.t the property A if there exists  > such that Z + a also has the property A for all a ∈ Rn , kak <  3.1 S-quasimonotone maps In this section, we study the criteria related to s-quasimonotone maps (see [1]) Definition 3.1.1 ([1]) A map Z is said to be s-quasimonotone iff ∃σ > such that (q − p)T Z(q) (q − p)T Z(p) −δ ≥0⇒ − δ ≥ |δ| < σ, kq − pk kq − pk Theorem 3.1.1 ([1]) The followings are equivalent: (i): Z is s-quasimonotone (ii): Z is stable w.r.t s-quasimonotonicity property (iii): Z is stable w.r.t pseudo-monotonicity property Trần Trọng Hoàng Tuấn 21 Applied Mathematics Master’s Thesis Proof We will show that (i) ⇒ (iii) Assuming that Z is s-quasimonotone, and there exists σZ satisfies definition 3.1.1 of s-quasimonotone Set  := σ We assume that q 6= p and for all a ∈ Rn , kak <  such that (q − p)T (F (p) + q) ≥ Therefore, we have (q − p)T Z(p) (q − p)T a ≥− kq − pk kq − pk Clearly, we have (q − p)T a − < σZ kq − pk Since Z is s-quasimonotone, we have (q − p)T a (q − p)T Z(q) ≥− kq − pk kq − pk It follows that (q − p)T (Z(q) + a) ≥ , i.e Z + a is pseudo-monotone We will show that (iii) ⇒ (i) Assuming that there exists  such that Z + a is pseudo-monotone for all a ∈ Rn satisfies kak <  Set σZ = , and assume that: (q − p)T Z(p) ≥δ kq − pk for all |δ| < σZ Choose a ∈ Rn such that kak = |δ| and − (q − p)T a = δ kq − pk Thus (q − p)T a (q − p)T Z(p) ≥− kq − pk kq − pk Therefore (q − p)T (Z(p) + a) ≥ Since Z + a is pseudo-monotone, we have (q − p)T (Z(q) + a) ≥ It follows that (q − p)T Z(q) ≥ δ kq − pk Trần Trọng Hoàng Tuấn 22 Applied Mathematics Master’s Thesis Consequently, Z is s-quasimonotone We will show that (i) ⇒ (ii) As have been shown above, there exists  > such that Z + a is pseudomonotone for all a ∈ Rn satisfies kak <  Choose a ∈ Rn such that kak < /2 For each a1 ∈ Rn satisfies ka1 k < /2, we have ka1 + ak <  Therefore, Z +a+a1 is pseudo-monotone with ka1 | < /2 Thus, as proof above, Z + a is s-quasimonotone for all a ∈ Rn satisfies kak < /2 We will show that (ii) ⇒ (i) Choose a = we have the proof completed 3.2 The stronger version of Wald’s Axiom In this section, we study criteria related to the stronger version of Wald’s Axiom (see [2]) Definition 3.2.1 (see [2]) The excess demand map Z satisfies Wald’s Axiom iff:  p Z(q) ≤  ⇒ Z(p) = Z(q) q T Z(p) ≤  T Definition 3.2.2 ([2]) A stronger version of Wald’s Axiom (SWA) is defined as follows : ∃σ > such that ∀δ, |δ| < σ, Z(p) 6= Z(q), q T Z(p) − δ ≤ 0, implies pT Z(q) + δ > Clearly, SWA implies Wald’s Axiom The criterion is defined as follows: Criterion (D): Z(p) 6= Z(q) ⇒ (p − p0 )T (Z(p) − Z(p0 )) 6= 0, ∀p0 ∈]p, q[ Theorem 3.2.1 ([2]) Assuming that diamP < ∞ SWA of Z implies s-quasimonotone property of −Z Proof Assuming that Z satisfies SWA with σ > We set σ1 = σ/diamP Trần Trọng Hoàng Tuấn 23 Applied Mathematics Master’s Thesis and assuming that (q − p)T Z(p) |δ1 | < σ1 , − δ1 ≤ kq − pk By Walras’ Law, since pT Z(p) = 0, it follows that q T Z(p) − δ1 kq − pk ≤ Set δ = δ1 kq − pk, we have |δ| < σ By SWA, we conclude that pT Z(q)+ δ > By Walras’ Law, we have pT Z(q) + δ (q − p)T Z(p) − δ1 = − < kq − pk kq − pk which implies that −Z is s-quasimonotone Theorem 3.2.2 ([2]) Assuming that kq − pk ≥ and Z satisfies criterion (D) We have the property of s-quasimonotone of −Z implies SWA of Z Proof Assuming that −Z is s-quasimonotone with σ > and q T Z(p) − δ ≤ 0, |δ| < σ, Z(p) 6= Z(q) By Walras’ Law, we have (q − p)T Z(p) δ − ≤ kq − pk kq − pk Set δ1 = δ/ kq − pk By condition kq − pk ≥ 1, we have |δ1 | ≤ |δ| < σ By the property of s-quasimonotone of −Z , we have (q − p)T Z(q) δ − ≤ kq − pk kq − pk Applying Walras’ Law, we obtain pT Z(q) + δ ≥ We need to show that pT Z(q) + δ > We assuming that pT Z(q) + δ = Applying Walras’ Law, we obtain (p − q)T Z(q) δ + = kq − pk kq − pk Trần Trọng Hoàng Tuấn 24 Applied Mathematics Master’s Thesis Since −Z is s-quasimonotone, we have (p − q)T Z(p) δ + ≤ kq − pk kq − pk By Walras’ Law, we obtain −q T Z(p) + δ ≤ Combining with q T Z(p) − δ ≤ 0, it follows that q T Z(p) = δ Therefore, q T Z(p) = −pT Z(q)δ Moreover, by Walras’ Law, we have (q − p)T Z(p) = (q − p)T Z(q) = δ For any p0 ∈]p, q[, we have (p0 − p)T Z(p) (q − p)T Z(p) = = δ1 kp0 − pk kq − pk Since −Z is s-quasimonotone, we have (q − p)T Z(p0 ) (p0 − p)T Z(p0 ) = ≤ δ1 kq − pk kp0 − pk It follows that (q − p)T Z(p0 ) ≤ δ On the other hand, we have (p0 − q)T Z(q) (p − q)T Z(q) = = −δ1 kp0 − qk kp − qk Since −Z is s-quasimonotone, we have (p − q)T Z(p0 ) (p0 − q)T Z(p0 ) = ≤ −δ1 kp − qk kp0 − qk It follows that (q − p)T Z(p0 ) ≥ δ Therefore, we conclude that (q − p)T Z(p0 ) = δ Thus (q − p)T Z(p) = (q − p)T Z(p0 ) = δ It follows that (p − p0 )T (Z(p) − Z(p0 )) = 0, this contradicts the criterion (D) 3.3 Stability with respect to criterion (A) Property 3.3.1 (see [1]) Pseudo-monotone maps are not stable w.r.t the criterion (A) Trần Trọng Hoàng Tuấn 25 Applied Mathematics Master’s Thesis Example 3.3.1 (see [1]) We consider the map F (x) to be defined as follows:    x4 (2 + sin x1 ) if x ∈ [−1, 0)    F (x) = if x =     −x4 (2 + sin ) if x ∈ (0, 1] x We have F (x)   > if x <  < if x > Consider the case x < 0, we have (y − x)F (x) ≤ ⇒ y < x ⇒ y < ⇒ (y − x)F (y) ≤ Consider the case x = we have (y − x)F (y) = yF (y) ≤ Consider the case x > 0, we have (y − x)F (x) ≤ ⇒ y > x ⇒ y > ⇒ (y − x)F (y) ≤ Thus F (x) is pseudo-monotone For k is integer and k ≥ 1, we set , 2kπ ak = − 8(kπ)4 xk = − We have F (xk ) + ak = We calculate the derivative of F (x) as follows:    1   x 4x(2 + sin x ) − cos x if x <    F (x) = if x =       −x2 4x(2 + sin ) − cos if x > x x Trần Trọng Hoàng Tuấn 26 Applied Mathematics Master’s Thesis We have F (xk ) = x2k (1 − )>0 kπ Thus, the criterion (A) is not satisfied Since limk→+∞ ak = 0, there not exists  > such that F + a satisfies (A) with |a| <  It follows that (A) is not stable w.r.t pseudo-monotonicity 3.4 Stability with respect to criterion (A’) Consider the map Z(p) as follows:  p1 p1 p  Z(p) = − F ( ), F ( ) p2 p2 p Taking v such that v T p = 0, we choose v = (−p2 , p1 )T We have p1 p21 v Z(p) = ⇔ F ( )(1 + ) = p2 p2 T and T v ∂Z(p)v = − (p21 + p22 )2 F ( pp21 ) + p1 p2 (p21 + p22 )F ( pp12 ) p32 We have T v ∂Z(p)v ≤ ⇔ (p21 + p22 )F ( pp12 ) + p1 p2 F ( pp21 ) p2 ≥ We consider cases: Case 1: F ( pp12 ) = We have F ( pp12 ) p2 ≥ Case 2: p2 = −p21 (−1 − p21 )F (− 1 ) + p1 F (− ) ≥ p1 p1 We choose F (x) as in example 3.3.1 and p2 ≤ 0, we have the map is not stable w.r.t (A’) for case Trần Trọng Hoàng Tuấn 27