Review of Mathematics for Economics Ryo Suzuki

Review of Mathematics

Ryo Suzuki

June 1, 2011

Remark

This paper is written in informal style because I made it to review basic mathematical stuff for myself. And I sometimes omit obvious assumption from a definition or theorem. For example A is nonempty in definition of compact set or so. So I recommend you to check formal definition, theorem or proof by reading the following references.

Next, I mention only most simple case in some theorems, that is I omit univariate Taylor’s theorem or other stuff. However, I think it is easy to proove general version of such a theorem so please try to do it.

Moreover, I’m sure that one of hte best way to understamd mathematics fully is solving problems by yourself, so please don’t misunderstand that you master it only by reading this paper.

Anyway, I’m glad if it is helpful to master mathematics for economics or to prepare for lecture of graduate shcool. And if you have any questions or find some mistakes, please feel free to send me messages. :D

1 Definition

1.1 Metric

Metric d : X_{× X →}Rsatisfies the followings

(1) ∀, y ∈ X d(, y) ≥ 0 , d(, y) = 0 ⇔  = y (2) ∀, y ∈ X d(, y) = d(y, )

(3) ∀, y, z ∈ X d(, y) + d(y, z) ≥ d(, z) And (X, d) called metric space

1.2 Open Set

∀ ∈ A, ∃ε > 0 s.t. Bε() ⊂ A

∗ Graduate School of University of Tokyo, Department of Economics, [email protected]

1.3 Closed Set

A^c: open

Remark

Of course, there are another definition of open and closed set. In fact, by the another definition, open set is defined as A = A^, where A^ is a set of interior point of A and closed set is defined as A = ¯A, where ¯A is closure of A.

Moreover, topological space is defined by collection of open sets, that is

(S, D) : topological space is a set S together with D (a collection of subsets of S) satisfying the following

(1)S ∈ D, ∅ ∈ D

(2)O1^{, O}2^,· · · , On∈ D ⇒ O1∩ O2∩ · · · ∩ On∈ D (3)∀λ ∈ Λ, Oλ∈ D ⇒ ∪λ_∈Λ^Oλ∈ D

and D⊂ 2^S is called a topology on S

In topological space, continuity of function ƒ : X→ Y is also defined as

∀O ⊂ Y : open ⇒ ƒ⁻¹(O) ⊂ X : open

and metric space such as (Rⁿ, d) is considerd as one of a topological space. Therefore, we can rewrite all of definitions.

However, when you apply mathematics to economic study, you usually have only to consider metric space such as (Rⁿ, d). So we consider only definitions or theorems on metric space in the following. But Heine-Borel theorem, which is about compactness inRⁿ is important to connect topological space and Euclid space, then please check later.

1.4 Bounded

∃λ ∈^{R, ∀}∈ A,  ∈ Bλ(0)

1.5 Supremum

sp A = min UA

1.6 Convergence

∀ε > 0, ∃ ¯m_∈N, s.t. ∀m_{≥ ¯}m, d(^m, ^∗) < ε

1.7 Cauchy Sequence

∀ε > 0, ∃ ¯m_∈N, s.t. ∀m, m^′_{≥ ¯}m, d(^m, ^m′) < ε

1.8 Completeness of Metric Space

∀{^m}^∞

m=1: Cchy on (X, d) _{⇒ {}^m}^∞

m=1^conergens

1.9 Continuous

∀ ∈ X, ∀ε > 0, ∃δ > 0, s.t. ∀^′∈ X, d(, ^′) < δ ⇒ d(ƒ (), ƒ (^{′)) < ε}

1.10 Compactness

∀{Oλ^}λ_∈Λ^{, A}⊂ ^∪

λ_∈Λ

O_{λ⇒ A ⊂}

n

∪

=1

O_λ

1.11 Convex Set

∀, ^′∈ X, ∀λ ∈ [0, 1], s.t. λ + (1 − λ)^′∈ X

1.12 Concavity

∀, ^′∈ X, ∀λ ∈ [0, 1], s.t. ƒ (λ + (1 − λ)^′) ≥ λƒ () + (1 − λ)ƒ (^′)

1.13 Quasi-Convexity

∀, ^′∈ X, ∀λ ∈ [0, 1], s.t. ƒ (λ + (1 − λ)^′) ≥ mx{ƒ (), ƒ (^′)}

1.14 Differentiable

∀ ∈ X, lim

h_→0

ƒ ( + h)_{− ƒ ()}

h ^{: ests}

1.15 Contraction Mapping

∃α ∈ (0, 1)∀, ^′∈ X, d(ƒ (), ƒ ()^′) < α d(, ^′)

1.16 Fixed Point

ƒ : X_{→ X , ∃}^∗ s.t. ^∗= ƒ (^∗)

1.17 Upper-Hemi Continuous

∀O ⊂ Y : open s.t.F() ⊂ O , ∃O^′⊂ X : open ,  ∈ O^{′ s.t. ′}∈ O^′⇒ F(^′) ⊂ O

1.18 Lower-Hemi Continuous

∀O ⊂ Y : open s.t. F() ∩ O 6= ∅ , ∃O^′⊂ X,  ∈ O^{′ s.t. ′}∈ O^′⇒ F(^′) ∩ O 6= ∅

1.19 Continuous of Correspondence

F : X ⇒ Y, : pper hem contnos nd oer hem contnos

1.20 Positive Definite

∀z ∈^Rn (z 6= 0), z^′D2ƒ ()z > 0

1.21 Positive Semidifinite

∀z ∈^{Rn , z′D2ƒ ()z}≥ 0

1.22 Hessian Matrix

D²ƒ =











∂²ƒ

∂₁∂₁ ^{· · ·}

∂²ƒ

∂₁∂_n

... ^...

∂²ƒ

∂n∂1 ^{· · ·}

∂²ƒ

∂n∂n











1.23 Jacobian Matix

Df = D(ƒ₁,_{· · · , ƒm}) =











∂ƒ₁

∂1 ^{· · ·}

∂ƒ₁

∂n

... ^...

∂ƒ_m

∂₁ ^{· · ·}

∂ƒ_m

∂₁











1.24 Real Number

R= {min UA | A ⊂^Q, ∀, b s.t. < b , b∈ A ⇒  ∈ A , 6 ∃ mx A}

Remark

The following theorems about continuity ofRare equivalent

⊲ Dedekind’s axiom

⊲ Least Upper Bound Axiom (Weierstrass’ Theorem(

⊲ Monotone Convergence Theorem

⊲ Cantor’s Intersection Theorem

⊲ Heine-Borel Theorem

⊲ Bolzano-Weierstrass Theorem

⊲ Cauchy’s Criterion for Convergence

⊲ Archimedes’ Axiom

1.25 Local Maximizer

∃ε > 0 s.t. ∀ ∈ Bε(⁰) ⊂ X, ƒ (⁰) ≥ ƒ ()

1.26 Global Maximizer

∀ ∈ X, ƒ (⁰) ≥ ƒ ()

2 Theorem

2.1 Heine-Borel Theorem

A_⊂Rⁿ: compct_{⇔ A ⊂}Rⁿ: cosed nd bonded

2.2 Weierstrass’s Theorem

ƒ : X_→R: contnos, X : compct_{⇒ ∃}^M, ^m∈ X s.t. ∀ ∈ X , ƒ (^M) ≥ ƒ () , ƒ (^m) ≤ ƒ ()

2.3 Intermediate Value Theorem

ƒ : [, b]_→R: contnos⇒ ∀ ∈ [ƒ (), ƒ (b)], ∃c ∈ [, b] s.t. ƒ (c) =  Proof

⊲ Suppose ƒ () <  < ƒ (b) without loss of generality

⊲ F() = ƒ ()− 

⊲ ∃ξ s.t. ∀∈ [, ξ), F() < 0 nd sp ξ : est (∵ ξ < b)

⊲ Point: We want to show c = sp ξ⇒ F(c) = 0

⊲ F(c) < 0⇒ ∃ε, F(c + ε) < 0

⊲ F(c) > 0⇒ ∃ε, F(c − ε) > 0

2.4 Mean Value Theorem

ƒ : [, b]_→R; dƒ ƒ erentbe⇒ ∃c ∈ (, b) s.t. ƒ^{′(c) =ƒ (b)− ƒ ()} b_{− } Proof

⊲ Point: F() = ƒ ()−^{ƒ (b)}_b^{−ƒ ()}_− ^

⊲ F() = F(b) nd F^′() = ƒ^′_{() −} ^{ƒ (b)}_b^{−ƒ ()}_−

⊲ Point: We want to show F^′(c) = 0

⊲ There exists maximizer or minimizer, then suppose c : maximizer

⊲ ^F()_^−F(c)_−c ≤ 0 ( > c) , ^F()_^−F(c)_−c ≥ 0 ( < c)

⊲ lim_→c^F()_^−F(c)_−c = F^′(c) = 0

⊲ Remark: ƒ () = g() = 0, then _g(b)^{ƒ (b)−ƒ ()}_−g() ^{= ƒ}

′_(c)

g^′(c)

2.5 Taylor’s Theorem

ƒ :R_→R: n dƒ ƒ erentbe t (, )⇒ ∃ξ ∈ (, ) s.t.

ƒ () = ƒ () +^ƒ

′_()

1! ^{( − ) +} ƒ ”()

2! ^{( − )}

2_{+ · · · +}^{ƒ(n−1)()}

(n − 1)! ^{( − )}

n₋₁₊^ƒ (n)_(ξ)

n! ^{( − )}

n

Proof

⊲ Point: F() = ƒ ()− ƒ () −^ƒ′_1!^()( − ) − · · · −^ƒ(n−1)_(n−1)!^()( − )ⁿ⁻¹

⊲ F^{n() = ƒn()}

⊲ F() = F^′() = · · · F^{(n−1)= 0}

⊲ Point: We want to show F() = ^ƒ

(n)_(ξ)

n! ^{( − )n}

⊲ Apply Mean Value Theorem

⊲ Point: _(−)^F()n = ^F′(1)

n(1−)^{n−1 = · · · =} Fⁿ(n)

n! ^{( ≤ n≤ n}−1≤ · · · ≤ 1≤ )

⊲ F() = ^F

n_(ξ)

n! ^{( − )n=} ƒⁿ(ξ)

n! ^{( − )n ( ≤ ξ = n≤ )}

⊲ Remark: Taylor’s Theorem is extension of Mean Value Theorem (∵ n = 1 Mean Value Theorem)

2.6 Inverse Function Theorem

ƒ :R_→R: C¹ , ƒ^′(⁰) 6= 0 ⇒ ∃ƒ^{−1: C1 , (ƒ−1)′(y0) = 1} ƒ^′(⁰)

2.7 Bounded and Increasing Sequence

{^m}^∞_m=1: bonded nd ncresng_{⇒ lim}

m_→∞^ m_{= }

Proof

⊲ We want to show ∀ε > 0, ∃ ¯^m∈^{Ns.t. ∀m}≤ ¯^{m, d(m}, ) < ε

⊲ Point: ∃∈^Rs.t.  = sp{^m}^∞

m=1 (∵ bounded)

⊲ Point: ∀^′ (< ), ∃ ¯m_∈Ns.t. ^′< ^m¯ ≤  (∵ if not  6= sp{m^}∞_m=1⁾

⊲ ε = − ^{′> 0}⇒ ∀m ≥ ¯^{m, d(m}, ) < ε (∵ increase)

2.8 Closed Set and Convergence

A_⊂Rⁿ: cosed_{⇔ ∃{}^m}^∞

m=1^{s.t. ∀m, }

m_{∈ A, lim} m_→∞^

m→ , then  ∈ A Proof (_⇒)

⊲ Suppose 6∈ A ⇔  ∈ A^c

⊲ Point: ∃δ > 0, Bδ() ⊂ A^{c (∵ Ac:open)}

⊲ On the other hand, ∀ε > 0, ∃ ¯^m∈^{Ns.t. ∀m}≤ ¯^{m, d(m}, ) < ε

⊲ Point: We want to show ∀m, ^m6∈ A

⊲ ∃m^{∗ s.t. ∀m}≥ m^{∗, d(m}, ) < δ

⊲ ∃m^{∗ s.t. ∀m}≥ m^{∗, m}∈ Bδ() ⊂ A^c

⊲ Contradiction

Proof (_⇐)

⊲ Suppose A: not closed⇔ A^{c: not open}

⊲ Point: ∃^′∈ A^c s.t. ∀δ > 0, B_δ(^′_{) 6⊂ A}^c (∵ ¬(∀^′ _{∈ A}^c, ∃δ > 0 s.t. B_δ(^′_{) ⊂ A}^c))

⊲ ∀δ > 0, Bδ(^′_{) ∩ A 6= ∅}

⊲ Point: We want to take sequence which converges to ^′

⊲ ∃¹∈ Bδ(^′_{) ∩ A}

⊲ Let {^m}∞_m=1such that d(^m+1, ^′) =¹₂d(^m, ^′)

⊲ ∀δ > 0, ∃ ¯^m∈^{Ns.t. ∀m}≥ ¯^{m, d(m, ′) < δ}

⊲ limm_→∞^m= ^′_{∈ A}^c

⊲ Contradiction

2.9 Bolzano-Weierstrass Theorem

A_⊂Rⁿ: compct_{⇔ ∃{}^m}^∞_m=1 s.t. ∀m, ^m_{∈ A, lim}

k_→∞^

m(k)→ , then  ∈ A

2.10 Cauchy Criterion for Convergence

mlim_→∞^

m_{=  ⇔ {}m_}∞

m=1^{: Cchy}

Proof (_⇒)

⊲ ∀ε > 0, ∃ ¯m s.t. ∀m, n_{≥ ¯}m, d(^m, ) < ^ε₂, d(ⁿ, ) <^ε₂

⊲ d(^{m, n}) ≤ d(^{m, ) + d(n, ) < ε}₂^+ε₂ Proof (_⇒)

⊲ Point: We want to show d(^{m, )}≤ d(^{m, ƒ) + d(ƒ, )}

⊲ ∀m≥ ¯^{m, d(m, m¯}) < δ (∵ triangle inequality)

⊲ ∀m≥ ¯^{m, m}∈ ¯^Bδ(^m¯_{) ≡ { ∈}Rⁿ_|d(^m¯, )_{≤ δ}}

⊲ ¯^Bδ(^m¯) : compact

⊲ Point: ∃{^m(k)}∞_k=1: converges to  (∵ Bolzano-Weierstrass Theorem)

⊲ ∃ ¯^{k, ∀k}≥ ¯^{k, d(m(k), ) < ε}₂

⊲ ∃ ¯^{m, ∀m, m′}≥ ¯^{m, d(m, m′) <ε}₂

⊲ m^∗= mx{m( ¯k), ¯m}

⊲ ∀m(k), m≥ m^{∗, d(m, )}≤ d(^{m, m(k)) + d(m(k), ) < ε}₂^+ε₂

2.11 Implicit Function Theorem

ƒ :R²_→R: C¹ , ƒ_y⁰, y⁰) = 0 , ƒ(⁰, y⁰_{) 6= 0}

⇒ ∃¹y = ϕ() s.t. (1) ƒ (, ϕ()) = 0 , (2) y⁰= ϕ(⁰) , (3) ^∂y

∂ ^{= −} ƒ_ ƒ_y^

2.12 Condition for Convexity

2.13 First Order Necessary Condition for Local Maximization

⁰: oc mzmzer_{⇒ ∇ƒ (}⁰) = 0 Proof

⊲ ∀∈ Bε(⁰), ƒ () ≥ ƒ (⁰⁾

⊲ Let ƒ () = ƒ (⁰₁,_{· · · },_{· · · }⁰

n⁾

⊲ ƒ (⁰_) ≥ ƒ (⁾

⊲ Point: ^{ƒ ()−ƒ (}

0

⁾

_−⁰_ ^{≤ 0 (> } 0

^{) ,}

ƒ ()−ƒ (⁰_⁾

_−⁰_ ^{≥ 0 (< } 0

 ⁾

⊲ _∂^∂ƒ_^{(0) = lim}→^o_

ƒ (_{)−ƒ (}⁰_)

_−⁰_ ^{= 0}

⊲ ∇ƒ (⁰) = 0

2.14 Second Order Necessary Condition for Local Optimality

⁰: oc mzmzer_{⇒ D}²ƒ (⁰) : negte semdeƒ nte Proof

⊲

2.15 Second Order Sufficient Condition for Local Optimality

2.16 Lagrangean Method

2.17 Karush-Kuhn-Tucker Theorem

2.18 Envelope Theorem

2.19 Duality Theorem

3 Problems

3.1

Discuss if A = {_∈R| 0 ≤  ≤ 1.  ∈^R\^Q} are open and/or closed.

3.2

Discuss ifZare open and/or closed.

3.3

Discuss if { +^p2b _{| , b ∈}Q} are open and/or closed.

3.4

Proove ∀⁰∈ X, ∀ε > 0, Bε(⁰) = { ∈ X | d(, ⁰) < ε} is open.

3.5

Proove if {A_}^∞_=1 is a sequence of open sets, then^∪∞_=1A_ is open.

3.6

Proove that {_}^∞

=1 such that ₁=^p2 and _n+1=^p2 +^p_n converges.

3.7

Proove that^∑∞

n=1^|n| < ∞, then^∑∞_n=1^2_n^{< ∞}

3.8

Proove that^∑∞

n=1^{|n| · b} 2

n< ∞, then^∑∞

n=1^{|n| · |bn| < ∞}

3.9

Construct a continuous function ƒ : [0, 1)_→Rwhich has neither maximum and minimum value.

3.10

Proove that if ∀, y_∈R, |ƒ () − ƒ (y)| ≤ ( − y)², then ƒ is constant.

3.11

Suppose ƒ^′() > 0 in (, b), then proove that inverse function g is differentiable and g^′(ƒ ()) = _ƒ′¹_()

3.12

Proove thet ƒ :R_→Ris differentiable but not of C¹ ƒ () =^¨

2_{sn(−1) 6= 0}

0  = 0

3.13

Proove that {_n}^∞_n=1 such that ₁= 1, n+1= 1 +_1+¹

n is Cauchy sequence.

3.14

4 Useful Expression

4.1 Basic

⊲ Suppose that {^m}∞_m=1converges to  ...

⊲ Let O be an open set ...

⊲ Therefore, ...

⊲ Take any ε > 0, then ...

⊲ Metric d satisfies the following ...

⊲ If there exists δ such that ...

⊲ By the definition of an open set, there exists ...

⊲ On the other hand, ...

⊲ Suppose the contrary, then ...

⊲ Hence, that is a contradiction.

⊲ In the same way, ...

⊲ Conversely, ...

⊲ Thus A is closed set.

⊲ A neccesary and sufficient condition for ...

⊲ Moreover, ...

⊲ By the assumption, ...

⊲ It followes that ...

⊲ Note that ...

⊲ , that is ...

4.2 Application

⊲ If F :^R→^Ris continuous, then ...

⊲ We can take ε1≥ ε2 without loss of generality, then ...

⊲ First, we show that ...

⊲ Since A is a compact set, ...

⊲ The same argument applies to ...

⊲ Cleary, ƒ ()≥ g() hold.

⊲ We consider the case of ...

⊲ Let ƒ be of C^{2 function.}

⊲ Take any ∈ Bε(⁰), where Bε(⁰_{) = { ∈}Rⁿ_{|d(, }⁰) < ε}

⊲ Consider the following maximization problem.

⊲ One can always claim that ∀∈ ∅ ,  ∈ A

⊲ Fix ε = 1, then cleary Bε() ⊂^R

⊲ Assume ⁿ≤ b^{nfor all n.}

⊲ Therefore we have Vn(k) =_1+β¹ log k + const. for all n by mathematical induction.

⊲ THe last inequality is from Cauchy-Swartz’s theorem.

⊲ Note that (^)′= ^ if and only if log  = 1

⊲ We simply apply the same way to ...

4.3 Example

Any sequence contained in a compact subset ofRhas a convergent subsequence.

Proof. First, we well proove.

5 References

⊲ ^{岡田章 『経済学}·^{経営学のための数学』}

⊲ ^{高木貞治『解析概論』}

⊲ ^{神谷和也,浦井憲}『経済学のための数学入門』

⊲ ^{石井恵一『線形代数講義』}

⊲ ^{宮川雅巳,水野眞治,矢島安敏『経営工学の数理I』}

⊲ ^{稲垣宣夫『数理統計学』}

⊲ ^{志賀浩二『位相への30講』}

⊲ ^{志賀浩二『解析入門30講』}

⊲ Carl P. Simon and Lawrence Blume. "Mathematics for Economists."

⊲ Rangarajan K. Sundaram. "A First Course in Optimization Theory."

⊲ Andrew Mscollel, Michael D. Winstein and Jeffery R. Green. "Microeconomic Theory."

⊲ Avinash Dixit. "Optimization in Economic Theory."

⊲ David G. Luemberger. "Optimization by Vector Space Methods."

⊲ Akihiko Matsui. Lecutre Note in Mathematics for Economics, University of Tokyo, 2010,

Summer

⊲ Kazuya Kamiya. Lecutre Note in Mathematics II, University of Tokyo, 2010, Summer

⊲ Tetsuya Kaji and Akitada Kasahara. Lecutre Note in Math Camp, University of Tokyo, 2011, Summer