# Phd – Mathematics for Economists

#### MATHEMATICS FOR ECONOMISTS

Coordinator: Prof. Antonio PALESTRINI
eMail: a.palestrini@univpm.it
The course aims at providing the Ph.D students the basic math skills needed in order to follow proficiently the Econometric, Micro and Macro courses of the Ph.D program. The course is divided in two parts. In the first part, after a brief review of calculus, will be explained the basics of linear algebra, static optimization and discrete dynamic optimization. In the second part, the topics will be ordinary differential equations, calculus of variations and optimal control theory.
##### Mathematics for economists – Part I
Language: English/Italian Frequence: November Hours: 20
Professor: Antonio Palestrini eMail: a.palestrini@univpm.it Web: on DiSES
Objectives of the Course:
• Calculus Review
• Linear Algebra
• Systems of linear equations: Matrix operations
• Row reduction and echelon forms (LU factorization)
• Linear dependence, bases, subspaces
• Gram-Schmidt process (QR factorization)
• Cramer’s Rule
• Inplicit Function Theorem
• Eigenvalues and eigenvectors
• Symmetric matrices
• Positive definite matrices
• Singular Value Decomposition
• Matrix operators and differentiation of vectors and matrices
• Taylor Series Approximation
• Static Optimization
• Unconstrained optimization
• Constrained optimization with equality constraints: Lagrange’s method
• Kuhn-Tucker Theorem
• Value functions
• Introduction to Discrete Dynamic Optimization
• Dynamic programming
• The value function iteration
• Discrete optimal control theory
G. Tian, Mathematical Economics, http://econweb.tamu.edu/tian/ecmt660.pdf.
A. Quarteroni, F. Saleri, P. Gervasio, Scientific Computing with Matlab and Octave, Springer, 2010.
K. Sydsaeter, P. Hammond, A. Seierstad, A. Strom. Further Mathematics for Economic Analysis, Prentice Hal. 2005.
##### Mathematics for economists – Part II
Language: English/Italian Frequence: November Hours: 12
Professor: Maria Cristina Recchioni eMail: m.c.recchioni@univpm.it Web: on UnivPM
Objectives of the Course:
• Ordinary differential equations:
• Definition and examples
• Linear first order differential equations
• Separable equations
• Bernoulli differential equation
• Linear second order equations
• System of linear first order differential equations
• Calculus of Variations
• The Euler equation
• The general transversality conditions
• Second-Order conditions
• Introduction to Optimal Control Theory
• The simplest problem of optimal control
• The costate variable and the Hamiltonian function
• Maximum principle