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.