75588 - Mathematical Tools and Methods

Learning outcomes

At the end of the course, the student will have an in-depth knowledge of the mathematical concepts and techniques that are of central importance in modern economic analysis. The student will be able, in particular, to apply fruitfully tools and concepts from static optimization, dynamical systems, and dynamic optimization to a wide range of topics and models in environmental economics.

Course contents

The following topics will be covered in the Crash Course in Mathematics

Linear equations. Linear inequalities. Sets, real numbers and functions. Quadratic functions and equations, power functions.

Sequences, series and limits. Geometric progressions in economics.

Differentiation of one-variable functions. Derivative, linear approximations and differentiability. Derivatives in economics. Rules of differentiation. Monotonic functions. Inverse functions. Exponential and logarithmic functions. Rate of change and rate of growth.

Maxima and minima of one-variable functions. Critical points. The second derivative. Optimisation. Convexity and concavity. Linear and quadratic approximations.

Basic elements of matrix algebra. Solution to systems of linear equations.

Course contents - Mathematical Tools and Methods

Multivariable calculus and static optimisation

Multivariable functions. Linear functions, quadratic forms. Partial derivatives. Linear approximations. The chain rule. An economic application: production functions. Homogeneous functions

Implicit functions. Implicit differentiation. Comparative statics.

Optimisation with several variables. Critical points. Global optima, concavity and convexity.

Constrained optimisation. Lagrange multipliers and their meaning. Economic applications. Quasi-concave functions. Envelope theorems. Inequality and non-negativity constraints.

Integration

Areas and integrals. Integration and differentiation: the fundamental theorem of calculus. Rules of integration. Integration in economics. Continuous compounding and discounting. Methods of integration. Infinite integrals. Differentiation under the integral sign.

Dynamic models: continuous-time and discrete-time dynamical systems

First-order differential equations. Linear equations with constant coefficients and with time-varying coefficients. Other kinds of first-order differential equations. Linear difference equations. Stationary solutions.

Higher-order systems. Eigenvalues and eigenvectors. Characteristic polynomial. Linear second-order differential equations. Linear second-order difference equations. Linear systems of differential equations and of difference equations.

Further dynamics. Nonlinear equations and nonlinear systems. Asymptotic behaviour, equilibria and stability for linear and nonlinear autonomous systems. Local analysis by linearization. Saddle equilibria. Phase diagrams and qualitative analysis. Cycles and chaos.

Intertemporal optimisation

Dynamic optimisation in continuous time. The basic problem and its variants. The Maximum Principle. Applications to resource economics. Problems with an infinite horizon. Optimal economic growth.

Dynamic optimisation in discrete time. The basic problem and its variants. Solution approaches: Hamiltonian, Dynamic Programming

Applications of dynamical systems and intertemporal optimisation to various areas of economics, in particular environmental economics.

Readings/Bibliography

Essential references

K. SYDSÆTER, P. HAMMOND, A. STRØM and ANDRÉS CARVAJAL. Essential Mathematics for Economic Analysis, 5th Edition. Pearson, 2016.

M. HOY, J. LIVERNOIS, C. McKENNA, R. REES, A. STENGOS, Mathematics for Economics, 3rd Edition, MIT Press, 2011.

An excellent alternative option is:

M. PEMBERTON, N. RAU. Mathematics for Economists: An Introductory Textbook, 4th Edition. Manchester University Press, 2016 (a student solutions manual is freely available from the publisher's website).

Teaching methods

Class lectures. During the class lectures, each topic will be illustrated by examples and worked-out exercises.

Assessment methods

Written examination.

According to the academic terms, the written exam of the first session can be taken in two steps: a midterm exam and a final exam, each weighted 50% in the final grade.

The written exam aims at testing the student's ability to understand and effectively apply the basic and advanced concepts and techniques learned in the course to specific mathematical (and economic) problems.

The written exam is based on exercises and problems on the various topics of the course. Sample exams will be made available on the VIRTUALE platform.

In case online exams will be envisaged by the University of Bologna, the exam will be run through Zoom and Exams Online - EOL (but the structure of the exam will remain the same). Detailed instructions on how to manage and hand in the online exam will be available on the course page on the VIRTUALE platform.

The maximum possible score is 30 cum laude, in case all anwers are correct, complete and formally rigorous.

The grade is graduated as follows:

<18 failed
18-23 sufficient
24-27 good
28-30 very good
30 e lode excellent

Teaching tools

Slides and graphics tablet. Class slides will be made available on the VIRTUALE platform. However, although class slides are generally very useful, they cannot replace textbooks.

Office hours

See the website of Roberto Dieci