Overview:

• Objective: The program's initial year concentrates on core courses to establish a strong foundation in economics and quantitative methods. This foundation serves as the springboard for students to pursue their chosen specializations in the subsequent year.

• Mobility: For the academic year 2023-2024, students will spend their first semester in Paris, and the second semester in one of the five Consortium Universities: Universitat Autonoma de Barcelona (Spain), Université Paris 1 Panthéon-Sorbonne (France), Università Ca' Foscari di Venezia (Italy), Université catholique de Louvain (Belgium) and Warsaw School of Economics (Poland).

• ECTS Credit Distribution: Explore how 60 ECTS credits are divided to fulfill program requirements, with an emphasis on CORE scientific courses and language studies.

  • 56 ECTS Credits of Scientific Courses: 7 CORE courses worth 7 credits each, alongside 7 elective credits. 4 CORE courses in the first and 3 in the second semesters. Elective courses vary by institution.

  • Four ECTS Credits of Foreign Language Courses and Cultural Enrichment: You take two CORE courses worth two credits each. The courses include language classes and cultural excursions. They are offered in French, Italian, Polish, or Spanish. These courses are graded on a pass/fail basis.

Courses and Organization

You can participate in Economics and Mathematics tutorials without the pressure of grading or earning ECTS credits each semester. Furthermore, you'll have access to a computer laboratory with available tutorship.

• Course Organisation: Links to academic calendars, timetables, and syllabi for each partner university are provided here in Barcelona, Paris, Venice, Louvain-la-Neuve, Warsaw, and Kansas.

• Prerequisites: Apart from the regular program courses, additional mathematics courses may be provided for Quantitative Economics Master's students who lack prior exposure to them. These extra courses, potentially mandatory for some, as determined by the Director of Studies, do not contribute to the required 60 ECTS credits for the first year. Possible offerings include:

  • Logic and Sets (First Semester)

  • Multivariable Calculus (First Semester)

  • Linear Algebra (Second Semester)

  • Euclidean Algebra (Second Semester)

Syllabi for all first-semester courses

Macroeconomics 1: Economic Growth and Policy

Number of Credits: 7 credits.

Hours: 45 hours of Lectures and Tutorials.

General Presentation: This course examines theoretical Macroeconomics. It includes the following topics:

1. Economics Growth: the Solow model, growth accounting.

2. Economic Growth with Endogenous Saving Behavior: the Ramsey-Cass-Koopmans model.

3. Endogenous Growth: AK models, learning by doing.  

4. Economic Growth with Uncertainty: the Brock-Mirman model and an introduction to Real Business Cycle Models.

Books: 

• Romer, D., Advanced Macroeconomics, Mac Graw-Hill.

• Barro, R. J., and Sala-i-Martin, X., Economic Growth, MIT Press.

• Acemoglu, D., Introduction to Modern Economic Growth, Princeton University Press.

Prerequisites: Logic and set theory, multivariable calculus, static and dynamic optimization.

Remark: Depending on student’s levels and available time, some extra topics as Overlapping Generation Models, Labor Market Models (efficiency wages, matching models, the equilibrium rate of unemployment), Market Imperfections (price and wage inertia, imperfect competition and macroeconomics, coordination failures, and neo-Keynesian theory). 

Microeconomics 1: Markets and Decisions

Number of Credits: 7 credits

Hours: 60 hours of Lectures and Tutorials.

General Presentation: The course deals with individual decision-making by consumers and producers, both under certainty and uncertainty, competitive equilibria, and Pareto optimal allocations in pure exchange economies and in production economies.

Course Content:

Individual Decision Making

• Rational Behavior, Choice, and Market Demand. Consumption set, preferences, properties of preferences, utility representation, properties of utility functions, examples. Choice rules and the weak axiom of revealed preference. Budget constraint, utility maximization problem, competitive demand, properties, and computation on some examples. Differential characterization of the demand for a differentiable utility function. Expected utility theory and risk aversion.

• Production and Firm Behavior. Production set, transformation function, production function, examples. Competitive behavior, profit maximization, profit function, supply function, properties. Differential characterization of the supply for a differentiable transformation function. Cost minimization, cost function, demand function, properties. Relationship between profit maximization and cost minimization.

Equilibria & Optimality

• Pure exchange: the Edgeworth box. Definitions of competitive equilibrium and Pareto optimality in the Edgeworth box. Computation and geometric characterization of equilibria and Pareto optima in the Edgeworth box.

• Production economies: Private ownership economies, the definition of competitive equilibrium. Basic properties: Walras’s Law and price normalization. Computation of competitive equilibria. Notions of feasible allocations and Pareto optimal allocations in a production economy. Pareto optimality conditions in terms of marginal rates of substitution and marginal rates of transformation. First and Second theorems of welfare economics.

Books: Mas-Colell, A., Whinston, M.D., Green, J., Microeconomic Theory, Oxford University Press, 1995. Chapters 1-6, 10, and Chapters 15-16.

Prerequisites: Multivariable calculus and static optimization.

Remark: Depending on student’s levels and available time, some extra topics could be covered.

Optimization

Number of Credits: 7 credits

Hours: 30 hours of Lectures, 30 hours of Tutorials, and 20 hours of Tutorships.

General Presentation: This course deals with optimization in static and dynamic settings. This is motivated by models in microeconomics, macroeconomics, finance, statistics, where these tools are very important. 

Particular attention will be set on the development of the analytical tools necessary for understanding and proving some of the basic results that play a central role in unconstrained and constrained optimization, both in a static and dynamic framework.

Course Content:

The course program includes presentation of definitions and theoretical results, discussion and solution of problems on the following topics.

Basic concepts 

Topology of Rn: distance and norm; open sets; closed sets; neighborhood; frontier, closure, interior of a set; bounded and compact sets; convex combination and convex sets. Upper and lower bounds, supremum, infimum; maximum and minimum. Continuity. Differentiability; partial, directional derivatives. Eigenvalues and eigenvectors. Diagonalization. Quadratic forms. Definiteness and semi-definiteness of quadratic forms. Optimization in Economics. Existence result: the Extreme Value (Weierstrass) Theorem. 

Concavity/convexity in Optimization

Convex sets. Separation theorems. Concave and convex functions. Closure Properties. Concavity/convexity and properties of derivatives. Convexity and definiteness. Concavity/Convexity in Optimization: Local/global Theorem. Necessary and sufficient conditions. 

Static Optimization

Looking for unconstrained optima: FOC; SOC. Constrained Optimization (equalities, inequalities): Optimization problem with equality constraints. Necessary conditions for optimality: Theorem of Lagrange. The Lagrangian function: interpretation of the Lagrange multipliers. Equality constraints: Second order conditions. Sufficient conditions for local optimality. Sufficient conditions for global optimality. Optimization problem with inequality constraints. Necessary conditions for optimality: Kuhn-Tucker Theorem. Comparative static. Envelope result. Remarks on nonnegativity assumptions.

Dynamic Optimization 

Discrete time optimization: Finite Horizon Dynamic Programming. Markovian Strategy. Existence result. Backwards Induction Method: analysis. Sufficient conditions for the existence of solutions. Bellman Principle. Discrete time optimization: Stationary Dynamic Programming. Optimal strategy, value function.

Metric spaces, normed spaces, and fixed points

Metric spaces and normed spaces. Sequences. Complete metric spaces. Contractions. Fixed points. Banach's Fixed point Theorem. 

Books:

Sydsaeter K., Hammmond P., Seierstadt A., Strom A. (2005): Further Mathematics for Economic Analysis, Prentice-Hall. [Chp. 1-3, 13-14, Appendix A]

Sundaram R.K. (1999): A First Course in Optimization Theory, Cambridge University Press, Cambridge. [Chp.1-7, 11-12, Appendix A, B, C]

Prerequisites: differential and integral calculus in one variable; Linear Algebra: matrix and vector algebra, determinants, systems of linear equations, vector spaces and subspaces. [Chapters 2-5, Appendix A4, Chapters 7--14, 23,26--28 of Simon C.P., Blume L.E. (1994): Mathematics for Economists, W.W. Norton & Company Press, Cambridge, (1994)].

Remark: Depending on student’s levels and available time, some extra topics could be covered, for example integration.

Probability and Statistics

Number of Credits: 7 credits 

Hours: 30 hours of Lectures, 30 hours of Tutorials, and 20 hours of Tutorships.

General Presentation: This course introduces the student to the fundamentals of rigorous probability theory with some classical applications in statistics. It provides a clear and intuitive approach while maintaining a good level of mathematical accuracy without using the Lebesgue integral. No previous course in probability or statistics is needed.

Course Content:

Part 1: In this first part we introduce the basic notions of elementary probability theory. We start by the definition of a probability measure and discuss some of its properties. After defining the independence of sets we introduce conditional probabilities to obtain the so-called law of total probabilities and Bayes formula.  Then, the notion of random variable is extensively studied starting from the univariate case. We pay attention on the discrete and the continuous cases to define independence, moments and to present in detail special parametric families of univariate distributions (Uniform, Bernoulli, Binomial, Poisson, Geometric, Gaussian, Exponential, Gamma, Chi-squared).  What is more, we study the effects of some regular transforms on these random variables. After that, we extend our approach in the multivariate case starting with pairs of discrete and jointly continuous random variables (change of variables theorem). We also introduce the notion of moment generating function and discuss some of its powerful properties to handle practical computations. Finally, we introduce the notion of conditional distribution in order to be able to define in our setting the conditional expectation for discrete and jointly continuous random variables.

Part 2: The second part of the course starts with some complementary notions on random vectors and related matrix notations. We also spend time on the very important case of Gaussian vectors. Then, several important topics in statistics are introduced to conclude this course. We study properties of samples of random variables, with emphasis on the Gaussian case, introducing, in particular, some classical estimators of expectation and variance. Then we present a primer on theoretical estimation theory studying in particular maximum likelihood estimators and exploring some optimality criteria in relation to the Fisher information matrix. Finally, the basic concepts of classical test theory are presented insisting on likelihood ratio tests.

Books:     

Mood, A. et. al. (1974) Introduction to the Theory of Statistics, McGraw-Hill, Inc., NY (Chapters 1 to 5)

Rice, J. (2007) Mathematical Statistics and Data Analysis, Thomson, Berkley, CA (Chapters 1, 2, 3, 4, 6)   

Casella, G. and Berger, R.L. (1990, 2002). Statistical Inference. Wadsworth Publishing Co., Belmont, CA (Chapters 5 to 8).

Prerequisites: Logic and set theory, multivariable calculus, computation of sums with summation notation (change of index), and computation of Riemann integrals (primitives, change of variables, integration by parts).

Remark: Depending on student’s levels and available time, some extra topics as the convergence of random variables (strong law of large numbers, central limit theorem) or complementary aspects in estimation theory (sufficiency, Bayes estimation) could be covered.