Lecture notes

These notes are my own write-up of the courses I took, not a transcript of them. The selection and order of the topics, the proofs, the examples and the translation into English are mine, so they are not a perfect match for what was covered in class, and any mistakes or typos are my own—please don’t hesitate to point them out!

• Econometrics, UC Berkeley, 2026 Econometrics
  Professor Michael Jansson, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Econometrics (ECON 240A, Part I), taught by Michael Jansson in Fall 2026. They cover probability theory, random variables and distributions, expectation, moments and generating functions, common families of distributions and exponential families, random vectors, conditioning and independence, random samples and sampling distributions, point estimation, sufficiency and UMVU estimation, and hypothesis testing, including optimal tests. The main reference is Casella and Berger (2024).
• Notes on Brouwer and Kakutani Fixed Point Theorems, 2026 Mathematical Economics
  Marcelo Gallardo. Notes related to Lecture 13 of ECON204 (2026), taught by Professor Chris Shannon (UC Berkeley).
  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.
• On the Continuous Dependence of Fixed Points on Parameters, 2026 Mathematical Economics
  Marcelo Gallardo. The problem discussed in this note arose in Lecture 5 of ECON204 (2026), taught by Professor Chris Shannon (UC Berkeley).
  Summary | Note

  In dynamic programming the value function is the fixed point of a Bellman operator T_α 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 T_α, 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, limsup_α→α₀ β_α < 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, PUCP, 2025 Microeconomic Theory
  Professor César Martinelli, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Game Theory (1ECO43), taught by César Martinelli in 2025-1. They cover static games with complete information (Nash equilibrium) and incomplete information (Bayesian Nash equilibrium); extensive-form games with perfect and imperfect information, with subgame perfect, perfect Bayesian and sequential equilibria; cooperative solutions, including the Nash bargaining solution and Rubinstein's alternating-offers model; and infinitely repeated games through the folk theorems. An appendix covers decision theory under uncertainty, a proof of the minimax theorem via convex-set separation, and overviews of Myerson (1978), Kreps and Scheinkman (1983), Reny (1999) and Echenique and Saito (2015).
• Real Business Cycles, PUCP, 2025 Macroeconomics
  Professor Paul Castillo, and written by Marcelo Gallardo.
  Summary | Lecture Notes (draft)

  Notes for the course Intermediate Macroeconomics (module 2), taught by Paul Castillo in 2025. They cover the real business cycle model as a stochastic general equilibrium economy with perfectly competitive, frictionless markets, its policy implications, permanent productivity shocks and extensions; real business cycles in small open economies, including the two-good case and Uribe's MX model; and credit cycles, following Bianchi (2011) on overborrowing and systemic externalities, and Kiyotaki and Moore.
• Partial Differential Equations, PUCP, 2024 Mathematics
  Professor Marcelo Flamarion, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Partial Differential Equations, taught by Marcelo Flamarion in 2024-2. They cover the theoretical, numerical and asymptotic approaches to the subject; the transport equation, the method of characteristics and shock formation; Burgers' equation and the Cole–Hopf transformation; the heat, wave and Laplace equations; separation of variables, Fourier series and transforms; and Duhamel's principle.
• Probability Theory, PUCP, 2024 Mathematics
  Professor Jonathan Farfán, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Probability and Statistics 1, taught by Jonathan Farfán in 2024-1. They cover probability theory from the measure-theoretic point of view: probability spaces, random variables and random vectors, integration, independence, the classical convergence theorems, L^p spaces, characteristic functions and conditional expectation.
• Stochastic Processes, PUCP, 2023 Mathematics
  Professor Johel Beltrán, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Stochastic Processes, taught by Johel Beltrán in 2023-2. They cover conditional expectation and its convergence theorems; discrete-time martingales, filtrations, adapted and predictable processes, and sub- and supermartingales; Poisson processes; Brownian motion; and stochastic calculus.
• Real Analysis 1, PUCP, 2023 Mathematics
  Professor Jesús Zapata, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Real Analysis 1 (MAT231), taught by Jesús Zapata in 2023-1. They cover differentiable maps; the Riemann integral on boxes and bounded sets of R^m, Fubini's theorem and sets of measure zero; homeomorphisms, diffeomorphisms and the change of variables theorem; surfaces, differentiable maps between them, tangent spaces and integration on surfaces; and, as complements, Sard's theorem, transversality and the index of a vector field.
• Functional Analysis, PUCP, 2023 Mathematics
  Professor Percy Fernández, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Functional Analysis, taught by Percy Fernández in 2023-1. They cover normed and Banach spaces, including L^p and sequence spaces; duality and Hilbert spaces; weak topologies; compact operators; and spectral theory.
• About Brouwer Fixed Point Theorem and its Application in General Equilibrium, 2023 Mathematical Economics
  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.
• Measure Theory, PUCP, 2022 Mathematics
  Professor Johel Beltrán, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course Measure Theory, taught by Johel Beltrán in 2022-2. They cover the extended real line, σ-algebras and the construction of measures, integration, the Lebesgue measure on R^d, change of variables and convolution.
• General Topology, PUCP, 2022 Mathematics
  Professor Rudy Rosas, and written by Marcelo Gallardo.
  Summary | Lecture Notes

  Notes for the course General Topology, taught by Rudy Rosas in 2022-1. They cover metric and topological spaces, compactness, connectedness, metrization, function spaces and the fundamental group.