Mathematical Economics

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 here.

We have completed the application involving the existence and uniqueness of a solution to the Bellman equation, and have been further occupied with another application of the Banach FPT, namely the Picard-Lindelof Theorem. You can find notes for the above here.

For purely illustrative purposes we have briefly stated Brouwer's FPT and subsequently examined one application involving the verification of the existence of Nash equilibria in a class of finite non-co-operative games. We have done so in a restricted setting that avoids the use of its generalization to correspendences, i.e. Kakutani's FPT. Our restrictive setting essentially ensures that the best response correspondence is actually a function. You can find notes on the above here. Notice that due to our brief encounter with Brouwer's FPT and the game theoretic application, those two subjects are not included in the exam material. However, Brouwer's FPT can be useful for the solution of Exercise 11 on the Perron-Frobenius Theorem. Should you decide to go for it (something that you are encouraged to do!) it would be useful to go through the material involving the derivation of the particular FPT.