Macroeconomics/Math Review

IntroductionEdit

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.

MetricEdit

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  ,

SpaceEdit

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.

convergenceEdit

A sequence,  , converges to  ,  , if   s.t.   for  ,


Cauchy sequenceEdit

A sequence ,  , is called a Cauchy sequence if   for  ,


Question: does every Cauchy sequence converge?

CompletenessEdit

The metric space,   is complete if every Cauchy sequence converges.

examples of completenessEdit

  •   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 MappingEdit

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 mappingEdit

  •   is a contraction mapping on  ,


Now we state the contraction mapping theorem.

Contraction mapping theoremEdit

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 criteriaEdit

For   and  , Let   satisfy the following two conditions:

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

Then   is a contraction mapping.