28895 - Mathematical Economics

Learning outcomes

At the end of the course the student has acquired knowledge and skills essential to the study of dynamic economic systems. In particular, he/she is able to: - calculate explicitly the solution to systems of linear differential and difference equations; - study systems of nonlinear differential and difference equations using the phase diagram and through linearization around the steady state; - solve deterministic dynamic optimization problems in discrete time (dynamic programming) and continuous time (optimal control).

Course contents

1.1 Some Basic Mathematical Models; Direction Fields
1.2 Solutions of Some Differential Equations
1.3 Classification of Differential Equations

First Order Differential Equations
2.1 Linear Equations; Method of Integrating Factors
2.2 Separable Equations
2.3 Modeling with First Order Equations
2.4 Differences Between Linear and Nonlinear Equations
2.5 Autonomous Equations and Population Dynamics
2.6 Exact Equations and Integrating Factors
2.7 Numerical Approximations: Euler’s Method
2.8 The Existence and Uniqueness Theorem
2.9 First Order Difference Equations
Second Order Linear Equations
3.1 Homogeneous Equations with Constant Coefficients
3.2 Solutions of Linear Homogeneous Equations; the Wronskian
3.3 Complex Roots of the Characteristic Equation
3.4 Repeated Roots; Reduction of Order
3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
3.6 Variation of Parameters
3.7 Mechanical and Electrical Vibrations
3.8 Forced Vibrations
Higher Order Linear Equations
4.1 General Theory of nth Order Linear Equations
4.2 Homogeneous Equations with Constant Coefficients
4.3 The Method of Undetermined Coefficients
4.4 The Method of Variation of Parameters
Series Solutions of Second Order Linear Equations
5.1 Review of Power Series
5.2 Series Solutions Near an Ordinary Point, Part I
5.3 Series Solutions Near an Ordinary Point, Part II
5.4 Euler Equations; Regular Singular Points
5.5 Series Solutions Near a Regular Singular Point, Part I
5.6 Series Solutions Near a Regular Singular Point, Part II
5.7 Bessel’s Equation
The Laplace Transform
6.1 Definition of the Laplace Transform
6.2 Solution of Initial Value Problems
6.3 Step Functions
6.4 Differential Equations with Discontinuous Forcing Functions
6.5 Impulse Functions
6.6 The Convolution Integral
Systems of First Order Linear Equations
7.1 Introduction
7.2 Review of Matrices
7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues,Eigenvectors
7.4 Basic Theory of Systems of First Order Linear Equations
7.5 Homogeneous Linear Systems with Constant Coefficients
7.6 Complex Eigenvalues
7.7 Fundamental Matrices
7.8 Repeated Eigenvalues
7.9 Nonhomogeneous Linear Systems
Numerical Methods
8.1 The Euler or Tangent Line Method
8.2 Improvements on the Euler Method
8.3 The Runge–Kutta Method
8.4 Multistep Methods
8.5 Systems of First Order Equations
8.6 More on Errors; Stability
Nonlinear Differential Equations and Stability
9.1 The Phase Plane: Linear Systems
9.2 Autonomous Systems and Stability
9.3 Locally Linear Systems
9.4 Competing Species
9.5 Predator–Prey Equations
9.6 Liapunov’s Second Method
9.7 Periodic Solutions and Limit Cycles
9.8 Chaos and Strange Attractors: The Lorenz Equations
Partial Differential Equations and Fourier Series
10.1 Two-Point Boundary Value Problems
10.2 Fourier Series
10.3 The Fourier Convergence Theorem
10.4 Even and Odd Functions
10.5 Separation of Variables; Heat Conduction in a Rod
10.6 Other Heat Conduction Problems
10.7 The Wave Equation: Vibrations of an Elastic String
10.8 Laplace’s Equation
Boundary Value Problems and Sturm–Liouville Theory
11.1 The Occurrence of Two-Point Boundary Value Problems
11.2 Sturm–Liouville Boundary Value Problems
11.3 Nonhomogeneous Boundary Value Problems
11.4 Singular Sturm–Liouville Problems
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion
11.6 Series of Orthogonal Functions: Mean Convergence

Readings/Bibliography

https://www.amazon.com/Elementary-Differential-Equations-Boundary-Problems/dp/0470458313/ref=pd_sbs_14_1?_encoding=UTF8&pd_rd_i=0470458313&pd_rd_r=2FFE4JQ2NAKTABTH9B4Y&pd_rd_w=h5VkL&pd_rd_wg=RcFcK&psc=1&refRID=2FFE4JQ2NAKTABTH9B4Y

Teaching methods

Blackboard lectures.

Assessment methods

Written exams.

Teaching tools

Some supplementary material might be handed over in class.

Office hours

See the website of Enrico Bernardi