A stronger criterion for the weak weak axiom of revealed preference

THE SOCIALIST REPUBLIC OF VIET NAM Independence - Freedom - HappinessTASK SHEET OF MASTER’S THESISName: Trần Trọng Hoàng Tuấn Student code: 2070243 Date of birth: 31 March 1993 Place of

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

THIS THESIS IS COMPLETED AT

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGYAdvisor: 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:

1 Chairman: Assoc Prof Dr Nguyễn Đình Huy

2 Secretary: Dr Huỳnh Thị Ngọc Diễm

3 Reviewer 1: Dr Nguyễn Bá Thi

4 Reviewer 2: Assoc Prof Dr Nguyễn Huy Tuấn

5 Member: Dr Cao Thanh Tình

Verification of the Chairman of the Master’s Thesis Defense Council and theDean of the Faculty after the thesis being corrected (if any)

Assoc Prof Dr Nguyễn Đình Huy Assoc Prof Dr Trương Tích Thiện

THE SOCIALIST REPUBLIC OF VIET NAM Independence - Freedom - HappinessTASK SHEET OF MASTER’S THESIS

Name: Trần Trọng Hoàng Tuấn Student code: 2070243 Date of birth: 31 March 1993 Place of birth: Bạc Liêu Major: Applied Mathematics Major code: 8460112

- 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: 6 June 2022

V ADVISOR: Assoc Prof Dr Phan Thành An

Ho Chi Minh City, 15 July 2022

Assoc Prof Dr Phan Thành An Dr Nguyễn Tiến Dũng

DEAN OF FACULTY

Assoc Prof Dr Trương Tích Thiện

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 mythesis advisor, Assoc Prof Dr Phan Thành An for his careful and clearguidance

In addition, I want to show my grateful attitude to my family membersfor the sympathy as they are always stand by me during my master thesisperiod

Last but not least, I would like to deliver my gratitude to Division

of Applied Mathematics, Faculty of Applied Science, Ho Chi Minh CityUniversity of Technology for giving me such valuable opportunity to applyall knowledge I have learned during my master of mathematics

Ho Chi Minh City, 15 July 2022

Author

Trần Trọng Hoàng Tuấn

i

In this thesis, the following stronger criteria (A), (B), (C), (A’) for theweak weak axiom of revealed preference introduced by Brighi in 2004 arepresented:

Criterion (A): For all x ∈ D and v ∈ Rn

ii

TÓM TẮT LUẬN VĂN

Trong bài luận văn, ta nghiên cứu các điều kiện (A), (B), (C), (A’) đượcgiới thiệu bởi Luigi Brighi trong năm 2004 cho tiên đề yếu yếu về lí thuyết

sở thích được bộc lộ như sau:

Điều kiện (A): Với mọi x ∈ D và v ∈ Rn

Ta sẽ trình bày khái niệm về tính ổn định được giới thiệu bởi Phan Thành

An trong năm 2006 Ta cũng trình bày hàm giả đơn điệu không ổn địnhvới các điều kiện (A), (B), (C), (A’)

iii

DECLARATION OF AUTHORSHIP

I hereby declare that this thesis was carried out by myself under theguidance and supervision of Assoc Prof Dr Phan Thành An, and thatthe work contained and the results in it are true by author and have notviolated 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 HoChi Minh City University of Technology is unrelated to any copyrightinfringement caused on my work (if any)

Ho Chi Minh City, 15 July 2022

Author

Trần Trọng Hoàng Tuấn

iv

DECLARATION OF AUTHORSHIP iv

LIST OF ACRONYMS, NOTATIONS AND CRITERIA vii

1.1 Consumer theory 2

1.1.1 Preference relation 2

1.1.2 Rational preference 2

1.1.3 Local nonsatiation 2

1.1.4 Utility maps 3

1.1.5 Budget sets 3

1.2 Demand maps and excess demand maps 4

2 THE CRITERIA FOR THE WEAK WEAK AXIOM OF REVEALED PREFERENCE 5 2.1 Pseudo-monotone maps 5

2.2 Quasi-monotone maps 11

2.3 Weak weak axiom of revealed preference 13

v

2.4 Weak axiom of revealed preference 17

3 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

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

vii

Research contents and scope:

Chapter 1: Theoretical basis

We present concepts of consumer theory, demand maps, excess demandmaps, homogeneity of degree zero, Walras’ Law

Chapter 2: The criteria for the weak weak axiom of revealed ence

prefer-We present concepts of pseudo-monotone maps, quasi-monotone maps,

WA, WWA and criteria introduced by Brighi

Chapter 3: The stability of generalized monotone maps w.r.t somecriteria

We present the concept of stability introduced by An We also presentthat pseudo-monotone maps are not stable w.r.t the criteria (A), (B),(C), (A’)

1

The consumer or decision maker’s choices based on his preferences.

A  B will mean that A is at least as good as B

A  B will mean that A is strictly preferred to B

A ∼ B will mean that the decision maker is indifferent between A andB

1.1.2 Rational preference

Rational preference relation satisfies:

Completeness: we have A  B or B  A or both

Transitivity: if A  B and B  C, then A  C

1.1.3 Local nonsatiation

Preferences are locally nonsatiated onX if∀x ∈ X and∀ > 0,∃y ∈ X

such that ky − xk ≤  and y  x

2

Applied Mathematics Master’s Thesis 1.1.4 Utility maps

A map u : X → R is a utility map representing preference relation if

∀x, y ∈ X, we have

x  y ⇔ u(x) ≥ u(y)

Proposition 1.1.1 Only rational preferences relations can be sented by a utility map Conversely, if X is finite, any rational preferencerelation can be represented by a utility map

Property 1.1.2 Walras’ Law

If the preference is locally non-satiated, for any pair (p, w), and x ∈x(p, w), where, x(p, w) is optimal solution’s set of the following utilityoptimization problem:

max u(x) such that pTx ≤ w, x ∈ B(p, w) (1.1)

We have

pTx = w

Applied Mathematics Master’s Thesis

Proof We will prove by the contradiction Assuming that x ∈ x(p, w)

for pTx < w We have ∃ > 0 such that ∀y for kx − yk < , pTy < w.However, by locally non-satiation, ∃y such that p · y < w and y  x.Therefore x 6∈ x(p, w) This leads to the contradiction

1.2 Demand maps and excess demand maps

Suppose that we have x∗ is the optimal solution of the optimizationproblem (1.1) We define the demand map at the price p is D(p) = x∗

We choose e such that pTe = w We have excess demand map is Z(p) =D(p)−e We have the following properties related to excess demand maps

Z(p), where p is the price vector:

Property 1.2.1 Homogeneity of degree zero

Z(λp) = Z(p), ∀λ > 0

Property 1.2.2 Walras’ Law

pTZ(p) = 0

Applied Mathematics Master’s Thesis

Chapter 2

THE CRITERIA FOR THE

WEAK WEAK AXIOM OF

REVEALED PREFERENCE

In this chapter, we study the concepts related to pseudo-monotonemaps, quasi-monotone maps, weak axiom, weak weak axiom, and the cri-teria for these concepts (see [3], [5] and [10])

Applied Mathematics Master’s Thesis

This implies that F (x) is pseudo-monotone However, F (x) is not tone

mono-Property 2.1.1 (see [3]) If pseudo-monotone maps are continuous with

an open convex domain, the set of zeroes of such maps is convex

Next, we will consider the following criteria for the equivalence ofpseudo-monotone maps

Criterion (A): For all x ∈ D and v ∈ Rn

satisfies criterion (A)

However, if we choose x = 0, y = 1, we have

(y − x)F (x) = 0, (y − x)F (y) = 1 > 0

Thus x2 is not a pseudo-monotone map

Example 2.1.3 For x = (x1, x2)T, let F (x) = (x2, −x1)T We calculateJacobian matrix of F (x) as follows:

Applied Mathematics Master’s Thesis

We have vT∂F (x)v = 0 Moreover, F (x) = 0 when x = 0 Thus, vTF (x+tv) = vTF (tv) = 0 This implies that F (x) satisfies criterion (B)

Example 2.1.4 Let F (x) = −x3 We have F (x) = 0 and F0(x) = 0

when x = 0 Hence,

vF (x + tv) = vF (tv) = −t3v4 ≤ 0 (2.1)This implies that F (x) satisfies criterion (C) Also, F (x) satisfies crite-rion (B)

We have the following theorem

Theorem 2.1.1 ([3]) Let F : D → R be a C1 map on the open convexset D ⊆ R Then, the followings are equivalent:

(i): F is pseudo-monotone

(ii): F satisfies the criteria (A) and (C)

(iii): F satisfies the criteria (A) and (B)

Proof

We will prove that (i) ⇒ (ii) (see [3])

Let vTF (x) = 0 and put y = x + tv for any t > 0 close to 0 Since

(y − x)TF (x) = tvTF (x) = 0, and F (x) is a pseudo-monotone maps, wehave:

(y − x)TF (x) = 0 ⇒ 0 ≥ (y − x)TF (y) = tvTF (x + tv)

⇒ (1/t)vT[F (x + tv) − F (x)] ≤ 0 (2.2)Taking the limit as t approaches 0, we obtain

Applied Mathematics Master’s Thesis

We will prove by the contradiction Let us assume that criteria (A) and(C) hold, but criterion (B) is violated, i.e F (x) = 0, uT = vT∂F (x) 6= 0

and ∃t > 0 such that vTF (x + tv) > 0 for all t ∈ (0, t] Since uTu > 0,

by continuity of the Jacobian, there exists an open ball Bx centered at x,such that vTF (z)Tu > 0,∀z ∈ Bx Let us define the map:

f (s, t) = vTF (x + su + vt)

in a sufficiently small neighborhood of the origin of R2, so that all thevectors x + su + vt are in Bx By assumption, f is C1, and its partialderivatives are, respectively,

fs(s, t) = vT∂F (x + su + vt)u, (2.3)

ft(s, t) = vT∂F (x + su + vt)v

Since x + su + vt ∈ Bx, fs(s, t) > 0 Since f (0, 0) = vTF (x) = 0

and fs(0, 0) = v∂F (x)u = uTu > 0, the equation f (s, t) = 0 define

an implicit map, i.e there exists  > 0 and a map s(t) defined in theopen interval (−, ), such that s(0) = 0 and f (s(t), t) = 0 for all t Theimplicit map is differentiable and:

Applied Mathematics Master’s Thesis

and by the definition of implicit map, we have

We will prove that (iii) ⇒ (i) (see [5])

We will prove by the contradiction, let us assume thatF is not a monotone map, i.e there exist x, y ∈ D such that

pseudo-(y − x)TF (x) ≤ 0, (y − x)TF (y) > 0

Let v = y − x and consider the map f : I → R, defined on, I = {λ ∈ R :

x + λv ∈ D} and

f (λ) = vTF (x + λv)

By definition of f, f (0) ≤ 0, F (1) > 0 Since f is continuous, ∃τ ∈ [0, 1[

such that f (τ ) = 0, and f (λ) > 0, ∀λ ∈]τ, 1] Without loss of generality,assume that τ = 0 By definition of f, we obtain vTF (x + λv) > 0, ∀λ ∈]0, 1] This implies that F (x) 6= 0 since otherwise it follows from

This contradicts criterion (A)

Let xλ = x + λv and defined vλ := v − α(λ)F (x), where

Applied Mathematics Master’s Thesis

Since f (λ) = vTF (xλ), we have

vλTF (xλ) = vTF (xλ) − α(λ)F (x)TF (xλ) = 0

this implies that vλ belongs to the space orthogonal to F (xλ)

By definition of vλ, for λ > 0 close to 0, we obtain

λ→0 +ln (λ) = −∞

The derivative of ln (f (λ)) with respect to λ là f0(λ)/f (λ) has no upperbound when λ approaches 0 We have

limλ→0 +F (x)TF (xλ) = F (x)TF (x) > 0

Thus, the first term in the square blackets of (2.4) is not upper-boundedwhen λ approaches 0 Moreover, since the other terms inside the squarebrackets of (2.4) are bounded when λ approaches 0, we obtain

vλT∂F (xλ)vλ > 0

Applied Mathematics Master’s Thesis

for some λ sufficiently close to 0

This contradicts criterion (A) since vλ belongs to the space orthogonal to

Therefore, F (x) is not a pseudo-monotone map

Next, we will consider the criteria of equivalence of quasi-monotonemaps The regularity of map F to be defined as follows:

Let F : X → Rn be continuously differentiable on the open convexdomain X ⊆Rn,∀x ∈ X:

F (x) = 0 ⇒ det ∂F (x) 6= 0

We have the following theorem

Theorem 2.2.1 ([3]) Let F be regular, then the followings are lent:

Applied Mathematics Master’s Thesis

(i): F is pseudo-monotone

(ii): F is quasi-monotone

(iii): F satisfies criterion (A)

Proof

We will prove that (i) ⇒ (ii)

Pseudo-monotonicity trivially implies quasi-monotonicity

We will prove that (ii) ⇒ (iii)

We will prove by the contradiction Assuming that criterion (A) does nothold, i.e., ∃x ∈ X and v ∈ Rn such that vTF (x) = 0 và vT∂F (x)v > 0

By definition, F is not quasi-monotone

We will prove that (iii) ⇒ (i)

By Theorem 2.1.1, we only need to show that criterion (B) holds Let

Applied Mathematics Master’s Thesis

x ∈ X, v ∈ Rn, λ > 0 such that F (x) = 0, vT∂F (x)v = 0 We have toprove the existence of some λ ∈]0, λ] with vTF (x + λv) ≤ 0 Since F isregular, F (x) = 0 implies the existence of an open convex neighborhood

U of x such that F (y) 6= 0, ∀y ∈ U \{x} Consider some λ ∈]0, λ] with

xλ = x + λv ∈ U We will prove that vTF (xλ) ≤ 0 Assume the contrary,i.e vTF (xλ) > 0 We will prove the following claim that: vT∂F (x)w ≥

0, ∀w ∈ Rn, w 6= 0 Since vT∂F (x)v = 0, the above claim is true for any

w that is collinear to v Consider w is not collinear to v From (xλ −x)TF (xλ) = λvTF (xλ) > 0 it follows that (xλ − yν)TF (xν) > 0, for

yν = x + νw ∈ U and ν > 0 sufficiently small The segment [xλ, yν] iscontained in an open convex subset of U on which F does not vanish Wehave

From vT∂F (x)w ≥ 0 for all w 6= 0, it follows that vT∂F (x) = 0 Since F

is regular, ∂F (x) is not regular Hence, v = 0, contradicts vTF (xλ) > 0

2.3 Weak weak axiom of revealed preference

In this section, we study the criteria related to the weak weak axiom

of revealed preference (see [3])

Definition 2.3.1 (see [3]) The excess demand map Z satisfies the Weakweak axiom (WWA) iff for any pair of price vectors p and q, the following

Applied Mathematics Master’s Thesis

Let the vector q =

qTZ(p) = q1(p3 − p2) + q2(p1 − p3) + q3(p2 − p1),

pTZ(q) = −[q1(p3 − p2) + q2(p1 − p3) + q3(p2 − p1)]

Therefore:

qTZ(p) ≤ 0 ⇒ pTZ(q) ≥ 0

It follows that Z(p) satisfies WWA

Example 2.3.2 Consider the map Z(p) : P ⊂ R2>0 → R2 to be defined

Applied Mathematics Master’s Thesis

where f (x) = 1 + x2 We have pTZ(p) = 0, i.e Z(p) satisfies Walras’Law Moreover Z(λp) = Z(p), i.e Z(p) satisfies homogeneous of degreezero Since f (x) > 0, ∀x, we get qTZ(p) ≤ 0 ⇒ pTZ(q) ≥ 0, i.e Z(p)

satisfies WWA

Next, we will consider the criteria of equivalence of WWA Let Z be a

C1 excess demand map,p ∈ Rn>0, v ∈ Rn We have the following property:Property 2.3.1 ([3]) The excess demand map Z satisfies WWA ⇔ Theexcess demand map Z satisfies Walras’ Law and pseudo-monotonicity.The following criteria are defined as follows (see [3]):

We study the following theorem

Theorem 2.3.1 ([3]) The excess demand map Z satisfies WWA ⇔ Theexcess demand map Z satisfies criteria (A’) and (C’)

Proof Given Theorem 2.1.1 and Property 2.3.1, we need to show criteria(A’) and (C’) imply criteria (A) and (C)

Applied Mathematics Master’s Thesis

We will show that (A’) ⇒ (A)

Let vTZ(p) = 0 and notice that v can be written as v = u + αp with

α = (vTp)/(pTp)

Clearly, the vector u is orthogonal to both p and Z(p)

By Walras’ Law, we have

Thus (A’) implies (A)

We will show that (C’) ⇒ (C)

Let Z(p) = 0 and vT∂Z(p) = 0 Vector v can be written as v = u + αp

withα = (vTp)/(pTp), where the vector uis orthogonal to p Since α can

be negative, we choose t > 0 be sufficiently small so that (1 + αt) > 0.Then, by homogeneity of degree zero, we have

Z(p + tv) = Z(p + t(u + αp)) = Z(p + su) (2.5)where s = t/(1 + tα),

and using twice Walras’ Law yields:

pTZ(p + tv) + tvTZ(p + tv) = pTZ(p + su) + suTZ(p + su) (2.6)First notice that, by (2.5), we have