Notes on Brouwer and Kakutani Fixed Point Theorems, 2026 Marcelo Gallardo. Notes related to Lecture 13 of ECON204 (2026).
Summary | Note
These notes present the proofs of two papers.
Milnor (Analytic proofs of the hairy ball theorem and the Brouwer fixed point theorem, American Mathematical Monthly 85, 1978) deduces Brouwer's fixed point theorem from the hairy ball theorem, which he proves by a volume computation.
Cellina (Approximation of set valued functions and fixed point theorems, Annali di Matematica Pura ed Applicata 82, 1969; A theorem on the approximation of compact multi-valued mappings, Atti della Accademia Nazionale dei Lincei 47, 1969) approximates an upper hemicontinuous correspondence with convex values by continuous functions whose graphs lie in a prescribed neighborhood of its graph; Kakutani's theorem then follows together with Brouwer's.
These topics were mentioned in Lecture 13 of ECON204 (2026), taught by Professor Chris Shannon at the University of California, Berkeley.
On the Continuous Dependence of Fixed Points on Parameters, 2026 Marcelo Gallardo. The problem discussed in this note arose in Lecture 5 of ECON204 (2026), taught by Professor Chris Shannon at the University of California, Berkeley.
Summary | Note
In dynamic programming the value function is the fixed point of a Bellman operator Tα on a function space, where α collects preferences, technology and the remaining primitives (Stokey et al., 1989). Comparative statics, continuity of the policy correspondence and consistency of structural estimators all presuppose that α ↦ z(α), the fixed point of Tα, is continuous.
Theorem 7.18 of de la Fuente (2000) asserts this under hypotheses that do not suffice in infinite dimensions — precisely the setting of dynamic programming. Section 2 locates the gap, following Shannon (2026): the printed proof uses a modulus common to all parameters, which the hypotheses do not supply. Proposition 3.1 then constructs a family in ℓ² meeting every hypothesis whose solution function jumps, so the statement is false and not merely unproved.
Two conditions restore it. Theorem 4.1 replaces the uniform bound by one holding near the parameter of interest, limsupα→α₀ βα < 1; this is what the comparative-statics exercise in the discount factor requires and the uniform bound excludes. Theorem 5.3 shows that when closed balls are compact — in particular when dim X < ∞ — no hypothesis on the moduli is needed.
Game Theory - Notes for 1ECO43, PUCP, 2025
Professor César Martinelli, and written by Marcelo Gallardo.
Summary | Lecture Notes
These lecture notes offer a deep dive into econometrics, structured around key themes from matrix algebra basics to statistical models.
These lecture notes offer a comprehensive exploration of game theory, structured around both static and dynamic models under varying information environments. The material opens with static games—first with complete information (Nash equilibrium) and then with incomplete information (Bayesian Nash equilibrium)—before moving on to extensive-form games with perfect and imperfect information, examining subgame perfect equilibrium as well as perfect Bayesian and sequential equilibria. It also covers cooperative solution concepts, including the Nash bargaining solution and Rubinstein’s alternating-offers model, and delves into infinitely repeated games with automaton-based strategy representations. An appendix presents decision theory under uncertainty, a proof of the minimax theorem via convex-set separation, and concise overviews of seminal papers by Myerson (1978), Kreps & Scheinkman (1983), Reny (1999), and Echenique & Saito (2015). The core textbook references are Osborne & Rubinstein (1994), Mas-Colell, Whinston & Green (!995), and Fudenberg & Tirole (1991).
Optimal Transport Theory and its Applications in Economics and Finance, 2024 Marcelo Gallardo and Carlos Cosentino. Final project for the course Introduction to Optimal Transport, taught by Johel Beltrán.
Summary | Lecture Notes
This document discusses applications of optimal transport theory in economics and finance, with a focus on computational methods like entropic regularization and the Sinkhorn-Knopp algorithm.
It covers topics such as matching markets stability, cost structure estimation, Credit Value Adjustment, and risk measures, aiming to provide detailed explanations and translations of complex results for students with a strong mathematical background.
The document includes an appendix to support understanding and is intended for advanced students interested in economic and financial applications of optimal transport.
About Brouwer Fixed Point Theorem and its Application in General Equilibrium, 2023 Marcelo Gallardo, Carlos Cosentino, and Eduardo Llamoca.
Summary | Lecture Notes
We develop a path towards the proof of Brouwer's Fixed Point Theorem and present an application in economic theory: the existence of the Walrasian Equilibrium.
Our goal is to provide the simplest, or at least one of the simplest, proofs for Brouwer's Fixed Point Theorem.
The only requirements are real analysis and general topology. Besides one Lemma which is not proved in its most general case, we prove all the results building up to the main theorem.
It is important to emphasize that this work does not introduce any new results in the literature. Instead, we focus on developing a clear and understandable approach to Brouwer's Fixed Point Theorem and its applications in general equilibrium.