Lecturer Carosi Laura – Sodini Mauro
The course aims to give a solid background in mathematics for economic
studies. Particular attention will be given to economic applications.
Part I. Topology, Fixed point theorem and separation
– The Euclidean spaces. Sequences in R and in Rn.
– Metric spaces: sequences, compactness, completeness. Fixed point
– Continuous functions on metric spaces. Continuous functions on
– Correspondence and fixed point theorems.
– Convex sets and separation theorems
Part II – Linear Algebra
– Vector spaces. Matrices. Determinant of a matrix.
– Eigenvector and eigenvalues.
– Diagonalization of a matrix. Canonical forms.
– Linear Functions. Linear Functions and Matrices.
Part III – Topics on Multivariable Calculus
– Gradients and Directional Derivatives.
– Differentiability and differential of a function.
– Taylor’s formula.
– Euler’s Theorem.
Part IV – Static optimization
– Implicit function theorem: applications.
– Unconstrained optimization.
– Optimization with equality constraints: Lagrange multipliers method.
– Optimization with inequality constraints: Kuhn-Tucker theorem.
– Generalized Convexity.
– Envelope theorems.
Part V- Dynamical systems
– System of difference equations.
– Systems of differential equations.
– Economic applications.
Part VI – Dynamic optimization
– Review of Reimann Integration.
– Optimality for continuous-time problems: Optimal Control by
Maximum Principle with several final conditions.
– Optimality for problems in discrete time: Maximum principle andoutline of dynamic programming: Bellman equation and Euler equation.