Introduction

**Coordinator:** Prof. Antonio Palestrini

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.

Part I

**Language:** English/Italian**Frequence:** November; Hours: 20**Professor:** Antonio Palestrini**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

**Reading List:**

- 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.

Part II

**Language:**** English/Italian****Frequence:** November; Hours: 12**Professor:** Maria Cristina Recchioni**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

**Reading List:**

- Ordinary Differential Equations: Chapters 24, 25 (25.1-25.5).
- C.P. Simon, L. E. Blume, “Mathematics for Economics”, New York, Norton & Company Inc., 1994.
- Calculus of Variations and Optimal control
- A.C. Chiang, "Elements of Dynamic Optimization", Mc Graw Hill, Singapore, 1992.

**ADDRESS**

D i S E S

8, Piazzale R. Martelli

60121 - ANCONA ( I )

**CONTACTS**

Email: phdises@univpm.it

Phone: +39 71 220 7101

Fax: +39 71 220 7102