Introductory Analysis

Real Space

Definition (closedness and convexness).

A set is closed if every limit point of is a point of . or equivalently, there exists no limit point of which is not contained in .
A set is convex if
is convex.

Definition (compactness).

Let be a metric space and . is compact if every open cover of has a finite subcover. i.e. for any open cover of , there exists a finite set s.t. since is finite, by letting for some , we have

However, since we only deal with real numbers in economics, by using Heine-Borel Theorem, we can use the properties of compactness easily.

Theorem (Heine-Borel Theorem).

Let and . then the following three properties are equivalent.

is closed and bounded
is compact
every infinite subset of has a limit point in .

the proof of @^3e8c14 Theorem 3 (Heine-Borel Theorem) is omitted, but the reader can refer to any undergraduate analysis textbook.

Functions

Homogeneity

Definition (homogeneity).

For any scalar , a function is homogeneous of degree if for all and all .

Definition (monotone fransform).

Let be an interval on . Then is a monotone transformation of if is a strictly increasing function on .
for instance, a monotone transformation of is

Definition (homothetic).

A monotone transformation of a homogeneous function is a homothetic function

Theorem (Euler's theorem).

Let be a homogeneous function of degree on . Then for all , we have or equivalently,

Proof.Let be HOD-. then we have then differentiate them with respect to . then we have
and by letting , we have the equality of which completes the proof □

Continuity

Definition (continuity).

Let and . The function is continuous at if for all .
and is continuous if it is continuous at every point in its domain .

Theorem (properties of continuous function).

let be a function. then the followings are equivalent:

is continuous on .
for each convergent sequence , then .
for each open set , is an open set.
for each closed set , is a closed set.

Proof.
Firstly, it is clear that 1 equals to 2. and for 3 and 4, the proof is delivered in the proof of @^f23fa1 Theorem 10 (check for continuous). □

Theorem (check for continuous).

Let be metric spaces and . is continuous on if and only if is open for every open set .

Proof. () ASM is continuous on X, and be open. and let ISTS that is an interior point of .

obviously, . and since is open, is an interior point of , i.e.
since is continuous at , we have
WTS . let . then . and by the previous equations, we have . and therefore
thus , and where is an arbitrary point of , which proves that is an interior point of .

() ASM is open for every open set . and let and is given arbitrary. WTS is continuous at , i.e.

since every neighborhood is open, is open. then by the ASM, is open.
since , we have .
thus is an interior point of ( def of open set), we have
thus for s.t. , we have which means . leading to .

this completes the proof. □

Corollary (check for continuous 2).

Let be metric spaces and . is continuous on if and only if is open for every closed set .

the proof of @^7bbc1c Corollary 11 (check for continuous 2) follows trivially from @^f23fa1 Theorem 10 (check for continuous).

Weierstrass Theorem

Theorem (Weierstrass' Extreme Value Theorem).

if is continuous and is closed and bounded, then has at least one maximum and minimum in the given interval.

The Extreme Value Theorem can be trivially derived from the following theorem of proving the bounded image of the continuous function with a compact domain.

Theorem (compactness of the image).

Let be function, where and is compact. then is compact.

Proof. Let be an open cover of . To show that is compact, ISTS that has a finite subcover.

as is continuous and is open, by @^7bbc1c Corollary 11 (check for continuous 2), is open for every .
since is open cover of , i.e. we have which means, that is an open cover of .
since is compact by the ASM, there exists a finite subcover s.t. thus we have therefore is a finite subcover of . □

Corollary (Corollary of previous Theorem).

Let be function, where and is compact. then is bounded on .

Proof. by @^1b537b Theorem 13 (compactness of the image), is compact. thus by @Heine-Borel Theorem Heine-Borel Theorem, is closed and bounded. □

Differentiability

Definition (differentiablility).

let be a function. then is differentiable at if or equivalently, or, and if the function is , where is an open set, it is differentiable at if there exists the derivative s.t. i.e.

Proposition (Jacobian matrix).

let be differentiable on open set , then the partial derivatives exists, and Jacobian matrix is where .
and if , then is called gradient vector.

Proposition (Hessian matrix).

let be differentiable on open set . Then Hessian matrix is where for any .

Taylor's Theorem

Theorem (Taylor's Approximation).

let be of on an open set . then the remainder s.t. , where

Quadratic Forms

Definition (quadratic form).

let is of quadratic form if then is symmetric matrix, and called

is positive definite if for all .
is positive semidefinite if for all .
is negative definite if for all .
is negative semidefinite if for all .
is indefinite if for some and for other .

Definition (leading principal matrix).

Let be matrix where then -th order leading principal matrix is

Theorem (positive and negative (semi)definite).

let is symmetric matrix. then

is positive (semi)definite if all leading principal matrix of are .
is negative (semi)definite if all leading principal matrix of are alternating signs. i.e.

Theorem (check for local max(min)).

Let for , where is open. and suppose is a critical point. then the followings hold:

If is negative definite, then is a local max.
If is positive definite, then is a local min.
If is indefinite, then is neither a local max nor a local min.

Concave and Quasiconcave

Definition (concave and convex functions).

let function be defined on the convex set .

is concave if
is strictly concave if
is convex if
is strictly convex if

Definition (quasiconcave and quasiconvex function).

let function defined on the convex set .

is quasiconcave if its upper contour sets are convex. i.e.
is strictly quasiconcave if
is quasiconvex if its lower contour sets are convex. i.e.
is strictly quasiconvex if

Remark (intuitive understanding of convex and concave).

is concave if
is quasi-concave if
is convex if
is quasi-convex if .

Theorem (concavity and negative (semi)definite).

let a function be where is convex. is concave if and only if the Hessian matrix is negative semidefinite for every . Moreover, is strictly concave if is negative definite.

Theorem (quasiconcavity and negative (semi)definite).

let a function be where is convex. is quasiconcave if and only if the Hessian matrix is negative semidefinite for every , in the subspace . i.e. Moreover, is strictly quasiconcave if is negative definite for every , in the subspace .

Implicit Function Theorem

Theorem (Implicit Function Theorem).

Let be , where .
Suppose and the determinant at as Then, there are an open neighborhood of , and of . and a unique function s.t. and furthermore, .

Corollary.

by the implicit function theorem, is given by

Envelope Theorem

Theorem (Envelope Theorem).

Let a function of . for the maximization problem let be a solution problem. and let . now suppose . then we have Pasted image 20240317202812.png

Proof. Using Chain Rule, we have where the last equality holds since for all , we have by the FOCs. □

Kuhn-Tucker Conditions

Theorem (Kuhn-Tucker Conditions).

Suppose that is a local maximizer of problem where . now let the constraint set denoted by .
Then there are multipliers such that

For every , i.e.
For every , which means that for any constraint such that .

Correspondence

Continuous Correspondence

Definition (correspondences).

Given a set , a correspondence is a rule that assigns a set to every .

unlike a function, correspondence allows is a set of multiple elements, or even allows the case of .

Definition (upper hemicontinuous; uhc).

Given and the closed set , the correspondence is upper hemicontinuous(uhc) if it has a closed graph and the bounded images of compact sets.
which means, for every compact set , is bounded where or, if for every sequence with and every sequence with , there exists a convergent subsequence s.t. .

Proposition (continuous function is uhc).

Given and the closed set , suppose that is a function. Then is an upper hemicontinuous if and only if it is continuous.

Definition (lower heminontinuous; lhc).

Given and a compact set , the correspondence is lower hemicontinuous(lhc) if for every sequence with , we can find a sequence and an integer s.t. for .

Pasted image 20240316232720.png

Definition (continuous correspondence).

A correspondence is continuous, if it is both upper and lower hemicontinuous.
Pasted image 20240316232706.png

Proposition (continuity concept for correspondences).

Let and be a continuous functions satisfying for all . then is continuous correspondence if

Berges' Maximum Theorem

Theorem (Berges's Maximum Theorem).

Let and . and let be a continuous function and let be a compact valued continuous correspondence.
Then the maximum value function is well defined and continuous, and the optimal policy correspondence is nonempty, compact valued, and upper hemicontinuous.

Proof. first, fix .
WTS#1 is nonempty

by ASM, is nonempty and compact, and is continuous.
since is maximum value of a continuous function on compact set, by @^c0889d Theorem 12 (Weierstrass' Extreme Value Theorem), is well-defined
as is well-defined on , therefore, is nonempty.

WTS#2 is compact valued

by @^3e8c14 Theorem 3 (Heine-Borel Theorem), ISTS that is closed and bounded.
note that , and is compact. then by Heine-Borel thm, is closed and bounded. thus its subset is also bounded.
now we show is closed set.
let be a sequence s.t. .
since , and is closed set, and every limit point of the closed set is the point of the set, we have .
since for all , and is continuous, then , i.e. . thus is closed.

WTS#3 is upper hemicontinuous

note that is fixed.
let be any sequence with . and let be a sequence with for all .
note that is continuous correspondence, i.e. both uhc and lhc.
since is uhc, there exists a subsequence that converges to
let . since is lhc, there exists a sequence with , and .
as , and by the def of , we have for all .
since for all and is continuous, we have , for any .
therefore, is upper hemicontinuous.

WTS#4 is continuous

note that is fixed.
let be any sequence with . and choose for all .
let and ISTS , i.e. .
since , there exists a sequence s.t. as is uhc, there exists a subsequence that converges to .
thus we have
similarly, there exists a subsequence s.t. as is uhc, there exists a subsequence that converges to . thus we have
thus we have , proving that is continuous.

this completes the proof. □

Fixed Point Theorems

Theorem (Brouwer's Fixed Point Theorem).

Let be nonempty, compact and convex set. and let be a continuous function. Then there exists an s.t. .

Theorem (Kakutani's Fixed Point Theorem).

Let be nonempty, compact and convex set. and let be a continuous function. Then there exists an s.t. .

Introductory Analysis

Real Space

Functions

Homogeneity

Continuity

Weierstrass Theorem

Differentiability

Taylor's Theorem

Quadratic Forms

Concave and Quasiconcave

Implicit Function Theorem

Envelope Theorem

Kuhn-Tucker Conditions

Correspondence

Continuous Correspondence

Berges' Maximum Theorem

Fixed Point Theorems

Other Advanced Topics

Frobenius' Theorem