Course : Mathematical Economics

Dear Students,

Below you can find a synopsis of our last three tutorials, as well as links for material that we covered in class.

In the third tutorial, we discussed in depth the notion of total boundedness and its connection with boundedness. We solved exercises 11 and 12 from Problem Set 2.  Regarding total boundedness, you are advised to study exercise 8 from the same problem set.

In the fourth tutorial, we proved that when a certain functional inequality between two metrics hold, then the convergence and the Cauchy property is inherited from one metric to the other. We also discussed how we should strengthen the functional inequality in order to get the preservation of completeness between the two metrics.

Finally, we discussed and proved an important result in mathematical analysis, according to which the set of bounded functions from a set X to a metric space Y is complete w.r.t. the unifrom metric if the codomain space Y is complete. Relevant notes on this result can be found here.   This material (the version that we followed in class) can also be found here.

Finally, in the fifth tutorial we discussed and proved some basic results about the principle of optimality. We motivated our analysis by connecting the ideas behind the principle of optimality with a standard example from growth theory. Afterwards, we stated a general dynamic programming problem, and its corresponding Bellman equation, and we provided brief proofs about the (conditional) equivalence of their solution sets. In the class we followed these notes, and we covered material until the end of Lemma 1, as well as Theorems 3 and 4. You are also advised to take a look at the notes by Dr. Zaverdas (they contain essentially the same proofs but with different notation), and to study the numerical example from page 9 onwards. 

Please do not hesitate to reach out for any questions.

Thanks

Pantelis