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Mathematics (ECO-CO-MATHS)

ECO-CO-MATHS

Description

This foundational mathematics course provides essential analytical tools for advanced study in Microeconomics and basic Macroeconomics. It focuses on linear algebra, topology in metric spaces, differential calculus in Euclidean spaces, nonlinear programming. The course also tries to obey the following highly demanding statement: 'If you state something, you must either provide a precise and recognized reference or prove it.'"

Learning outcomes:
    By the end of the module, students will:
•    Master key concepts and operations in linear algebra.
•    Understand basic topology in metric spaces.
•    Apply differential calculus in multidimensional settings.
•    Solve problems involving nonlinear optimization.
•    Be able to organize a proof.

Assessment:    
•    Final Exam (90%): Open-book and open-notes exam assessing theoretical understanding and problem-solving.
•    Homework Assignments (10%): 3 or 4 problem sets distributed during the course.

Students must be familiar with topics typically covered in undergraduate courses in calculus and linear algebra. Those topics are covered in the following 3 sets of books or notes.

    1.  Either: Clark, C., (1982), Elementary Mathematical Analysis, 2nd edition, Wadsworth Publisher of Canada, Ltd., Belmont, CA. Appendix 1, or Villanacci, A., (2020), Mathematics for Economics 1, Class Notes, Chapter 1.
    2.  Either: Apostol, T. M., (1967), Calculus, Volume 1, 2nd edition, John Wiley & Sons, New York, NY: Chapters 1-7 included, 9-10, or Spivak, M., (1980), Calculus, 2nd ed., Publish or Perish, Inc., Houston,TX: all but chapters16, 20, from 23 to 29, included.
    3.  Villanacci, A., (2024), Basic Linear Algebra, Metric Spaces, Differential Calculus and Nonlinear Programming: Chapters 1, 2 and 3; Sections 5.1 and 5.2.





Module structure

WEEK 1:

Linear Algebra:
Topics:
•    Vector spaces and subspaces
•    Linear independence, basis, and dimension
•    Linear functions and matrix representations
•    The dimension theorem
•    Determinant, Rank, invertibility, and systems of linear equations

WEEK 2
Topology in Metric Spaces:

Topics:
•    Definitions and examples of metric spaces
•    Open and closed sets
•    Sequences
•    Compactness
•    Continuous functions and Extreme Value Theorem
•    Correspondences. Lower and Upper Hemicontinuity:
•    The Maximum Theorem; a Fixed Point Theorem.

WEEK 3
Differential Calculus in Euclidean Spaces:

Topics:
•    Functions of several variables
•    Partial derivatives, directional derivatives and differentiability
•    The chain rule and other basic results on differentiation
•    Higher-order derivatives and Hessian matrices
•    The implicit function theorem

WEEK 4
Nonlinear Programming:

Topics:
•    Convex sets and convex functions
•    Optimization without constraints
•    Constrained optimizations and Karush-Kuhn-Tucker (KKT) conditions
•    The implicit function theorem and comparative static analysis

Dynamic Programming:
Topics:
•    The basic problemù
•    Principle of optimality
•    The value function
•    Euler-Lagrange approach
•    Economic applications


Bibliography and further readings

Main Reference:
•    Villanacci, A. (2026), Basic Linear Algebra, Metric Spaces, Differential Calculus and Nonlinear Programming, Class Notes
•    Villanacci, A., (2025), Basic topics in Dynamic Programming, Parts I, III and V, Class Notes.
A large part of Villanacci (2026) will covered in class. A small part of Villanacci (2025) will be covered in class.


Supplementary Optional Reading:
 I. Linear algebra
Lang S. (1971), Linear Algebra, second edition, Addison Wesley, Reading.
Lipschutz, S., (1991), Linear Algebra, 2nd edition, McGraw-Hill, New York, NY.

 II Some topology in metric spaces.
Lipschutz, S., (1965), General Topology, McGraw-Hill, New York, NY.
McLean, R., (1985), Class notes for the course of Mathematical Economics (708), University of Pennsylvania, Philadelphia, PA, mimeo.
Ok. E. A., (2007), Real Analysis with Economic Applications, Princeton University Press, Princeton NJ.

 III Differential calculus in Euclidean spaces
Apostol, T. M., (1974), Mathematical Analysis, 2nd edition, Addison-Wesley Publishing Company, Reading, MA.

 IV Nonlinear programming.
Cass D., (1991), Nonlinear Programming for Economists, University of Pennsylvania, Class Notes.

    V. Basic Dynamic Programming
Cugno, F., and Montrucchio, L., (1998), Scelte intertemporali; Teoria e modelli, (in Italian), Carocci, Roma.
Stokey, N. L., and Lucas, R.E., (1989), Recursive Methods in Economic Dynamics, Harvard University Press, Cambridge, MA.