# Macroeconomics/Math Review

## Introduction

[edit | edit source]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, , 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

[edit | edit source]A **metric** is a function with the properties that it is non-negative, , symmetric, , and satisfies the triangle inequality,,

A common metric is euclidean distance, , Another is ,

### Space

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

### convergence

[edit | edit source]A sequence, , **converges** to , , if s.t. for ,

### Cauchy sequence

[edit | edit source]A sequence , , is called a **Cauchy sequence** if for ,

Question: does every Cauchy sequence converge?

### Completeness

[edit | edit source]The metric space, is **complete** if every Cauchy sequence converges.

### examples of completeness

[edit | edit source]- is complete.
- is not complete. Proof: let , So os Cauchy, but does not converge to a point in our set ,
- is complete. Are all closed sets complete? A closed subspace of a complete space is complete.
- is complete.

## Contraction Mapping

[edit | edit source]A mappting is a **contraction mapping** on a metric space, , if such that , Sometimes we write instead of ,

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

### examples of contraction mapping

[edit | edit source]- is a contraction mapping on ,

Now we state the **contraction mapping theorem**.

### Contraction mapping theorem

[edit | edit source]If is complete and is a contraction mapping, then with ,

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

[edit | edit source]For and , Let satisfy the following two conditions:

- (M, monotonic condition) and , and , if then ,
- (D, discout condition) , for , .

Then is a contraction mapping.