Macroeconomics/Math Review

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[edit] 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.

[edit] 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, $ρ(x, y) = ρ(y, x)$ , 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 $\rho_{max} (x,y) = \max_{1\leq i\leq l} x_i-y_i$ ,

[edit] 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, $(S,ρ)$ where $S$ is the set of all bounded rational functions, and $ρ$ is some distance function. Once we have a metric space we can discuss convergence and continuity.

[edit] convergence

A sequence, $\{x_i\} \subset S$ , converges to $x$ , $x_i \rightarrow x$ , if $\forall \epsilon > 0, \exists N_\epsilon$ s.t. $ρ(x n, y n) < ε$ for $n > N ε$ ,

[edit] 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 $n, m > N ε$ ,

Question: does every Cauchy sequence converge?

[edit] Completeness

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

[edit] examples of completeness

$(\mathbb{R}, \rho_E)$ is complete.
$((0,1),ρ E)$ 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 $(0,1)$ ,
$([0,1],ρ E)$ is complete. Are all closed sets complete? A closed subspace of a complete space is complete.
$({0,1,2},ρ E)$ is complete.

[edit] Contraction Mapping

A mappting $T:S \rightarrow S$ is a contraction mapping on a metric space, $(S,ρ)$ , 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 $T X$ ,

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

[edit] examples of contraction mapping

$T x = .9 x$ is a contraction mapping on $[(0,1],ρ E)$ ,

Now we state the contraction mapping theorem.

[edit] Contraction mapping theorem

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

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.

[edit] Contraction Mapping criteria

For $S \subset \mathbb{R}^l$ and $ρ = ρ E$ , 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 $\underline{\vec{a}}=(a, a, \ldots, a)$ , $T_i(x_1+a, x_2+a, \ldots, x_l + a) \leq T_i(x_1,x_2,\ldots,x_l)+\beta a \forall i \Leftrightarrow T(x+\underline{a}) \leq TX + \beta \underline{a}$ .