Course : Mathematical Economics

You can find here and here exercises corresponding to the notions that were examined in the first three lectures.

We have completed our basic vocabulary regarding balls in metric spaces by showing that those can be used in order to seperate points. This property (which does not generally hold for pseudo-metrics) is essential, since it among others provides with the uniqueness of limits in metric spaces.

We begun our study of metric properties with the finitary notion of boundness. The balls can be readily used in order to define it as a natural extension of the notion of boundness on the real line (w.r.t. t

We completed our general definitions with the notion of the metric subspace. This as well as  the notion of product metric spaces (with a finite number of factors) can be viewed as ways to construct further metric spaces from given one(s), with the resulting metrics carrying relevant information on the given one(s).

We begun studying properties of metric spaces via the definition of the open and the closed balls that the metric defines. We have shown that these cannot in any case be empty, and o

After some brief discussion of the course's scope and aims, we begun with the definition of a distance function (metric) w.r.t. a non empty set of reference as a real function defined on the product of this set with itself that satisfies positivity, separation, symmetry and triangle inequality. The example of the discrete metric showed that any such set bears at least one such function, and further examples implied that more than one metrics can exist, possibly inside structured families, some o

In the final lecture we initially revisited the issue of the uniform convergence of the self-composition of a contraction inside a complete and bounded metric space in order i) to provide with a more efficient description compared to the one provided in the previous lecture, and ii) to remark that the BFPT essentially implied the pointwise convergence (why?), while the further attribute of boundedness for the space strengthened the latter to uniform convergence. You can find notes for the above

In our final TA session we exerted effort into understanding how solutions to Bellman equations relate to solutions of a standard class of Stationary Dynamic Programming Problems. For this purpose, we showed two lemmas related to dynamic programming and used them along with what we have seen in class about bellman equations in continuous and bounded functional space. We saw under what conditions the solution in question exists and is unique. We also solved an example dynamic programming problem.

We have been occupied with issues involving the uniform convergence of uniformly Lipschitz functions, and the uniform convergence of the self-composition of a contraction inside a bounded and complete metric space (are those two issues somehow related?).

We have begun studying the framework for applications of the BFPT, derived the useful Blackwell's Lemma, and begun the application concerning the uniqueness of the solution to the Bellman equation.

You can find notes for the above here, here (th

We have been initially occupied with the non-topological notion of Lipschitz continuity and the subsequent notion of a contractive self-map.

We have then been occupied with issues on metric fixed point theory involving the derivation of the Banach FPT, some straightforward corrolaries, and some initial applications.

You can find notes on the above here and here.

During the third TA session we were mostly preoccupied with proving a lemma concerning the completeness property of general bounded functional spaces (general as in not necessarily mapping on the real numbers), as it is inherited by its image space. You can find the above lemma and its proof here and at the end of these notes here.

We also quickly went through and commented on an economic application of the notions of pointwise versus uniform convergence of sequences of functions and what the

Given the tutorial preparation of the notion of continuity of functions between metric spaces and a brief re-examination of this during the lecture, we have provided with a major application that establishes the continuity of the sup functional when properly restricted to possibly non-empty metric subspaces of the set of bounded real functions equiped with the uniform metric.

Using this and under further assumptions we were occupied with the issue of convergence of approximate maximizers under u