Macroeconomics/Math Review

Introduction

We have a Bellman equation and first we want to know if there exists a value function that satisfies the equation and second we want to know the properties of such a solution. In order to answer the question we will define a mapping which maps a function to another function, and a fixed point of the mapping is to be a solution. The mapping we discussed is a mapping on the set of functions, which is a bit abstract. So today we will look at the math review.

So first we consider a set, $S\subset \mathbb {R} ^{l}$, For us what it will be relevant to describe a sort of distance between any two points in a set. We will use the concept of a metric.

Metric

A metric is a function $\rho :S\times S\to \mathbb {R}$ with the properties that it is non-negative, $\rho (x,y)\geq 0$, symmetric, $\rho (x,y)=\rho (y,x)$, and satisfies the triangle inequality,$\rho (x,z)\leq \rho (x,y)+\rho (y,z)$,

A common metric is euclidean distance, $\rho _{E}(x,y)={\sqrt {\sum _{i=1}^{l}(x_{i}-y_{i})^{2}}}$, Another is $\rho _{max}(x,y)=\max _{1\leq i\leq l}x_{i}-y_{i}$,

Space

A space, is a set of objects equipped with some general properties and structure

We may be interested in a metric space, a space with a metric such as, $(S,\rho )$ where $S$ is the set of all bounded rational functions, and $\rho$ is some distance function. Once we have a metric space we can discuss convergence and continuity.

convergence

A sequence, $\{x_{i}\}\subset S$, converges to $x$, $x_{i}\rightarrow x$, if $\forall \epsilon >0,\exists N_{\epsilon }$ s.t. $\rho (x_{n},y_{n})<\epsilon$ for $n>N_{\epsilon }$,

Cauchy sequence

A sequence , $\{x_{i}\}\subset S$, is called a Cauchy sequence if $\forall \epsilon >0,\exists N_{\epsilon }s.t.\rho (x_{n},y_{m})<\epsilon$ for $n,m>N_{\epsilon }$,

Question: does every Cauchy sequence converge?

Completeness

The metric space, $(S,\rho )$ is complete if every Cauchy sequence converges.

examples of completeness

• $(\mathbb {R} ,\rho _{E})$ is complete.
• $((0,1),\rho _{E})$ is not complete. Proof: let $x_{n}={\frac {1}{n+2}}$, So $\{x_{n}\}$ os Cauchy, but does not converge to a point in our set $(0,1)$,
• $([0,1],\rho _{E})$ is complete. Are all closed sets complete? A closed subspace of a complete space is complete.
• $(\{0,1,2\},\rho _{E})$ is complete.

Contraction Mapping

A mappting $T:S\rightarrow S$ is a contraction mapping on a metric space, $(S,\rho )$, if $\exists o\geq \beta <1$ such that $\rho (tx,Ty)\leq \beta \rho (x,y)\forall x,y\in S$, Sometimes we write $T(x)$ instead of $TX$,

This means that any two points in our set, $S$, are mapped such that after the mapping the distance between the points shrinks.

examples of contraction mapping

• $Tx=.9x$ is a contraction mapping on $[(0,1],\rho _{E})$,

Now we state the contraction mapping theorem.

Contraction mapping theorem

If $(S,\rho )$ is complete and $T:s\rightarrow S$ is a contraction mapping, then $\exists !x^{*}$ with $Tx^{*}=x^{*}$,

We will prove this theorem for a general metric space later on. However, we must remember that it is necessary for this proof that the space be complete.

Let us now look at a criteria to verify that a mapping is a contraction mapping.

Contraction Mapping criteria

For $S\subset \mathbb {R} ^{l}$ and $\rho =\rho _{E}$, Let $T:S\rightarrow S$ satisfy the following two conditions:

• (M, monotonic condition)$\forall x=(x_{1},x_{2},\ldots ,x_{l})\in S$ and $y=(y_{1},y_{2},\ldots ,y_{l})\in S$, and $T=(T_{1},T_{2},\ldots ,T_{l}$, if $x_{i}\geq y_{i}\Leftrightarrow x\geq y$ then $T_{i}x\geq T_{i}y\Leftrightarrow Tx\geq Ty$,
• (D, discout condition) $\forall x=(x_{1},x_{2},\ldots ,x_{l})\in S$, for ${\underline {\vec {a}}}=(a,a,\ldots ,a)$, $T_{i}(x_{1}+a,x_{2}+a,\ldots ,x_{l}+a)\leq T_{i}(x_{1},x_{2},\ldots ,x_{l})+\beta a\forall i\Leftrightarrow T(x+{\underline {a}})\leq TX+\beta {\underline {a}}$.

Then $T$ is a contraction mapping.