75588 - Mathematical Tools and Methods

Learning outcomes

At the end of the course, students will acquire the basic knowledge of a range of mathematical concepts, tools and techniques that are of central importance in modern economic analysis. Students will be able, in particular, to apply fruitfully concepts and tools from calculus and optimization, geometric sums and series, linear algebra to environmental economics, resource valuation, and dynamic economic modelling.

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.

Basics of differentiation of one-variable functions. Derivative and linear approximation. Derivatives in economics. Rules of differentiation (including the "chain rule"). Monotonic functions. Inverse functions. Exponential and logarithmic functions. Rate of change and rate of growth.

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

 

Course contents - Mathematical Tools and Methods

 

Static optimisation of one-variable and two-variable functions

Two-variable functions. Linear functions, quadratic forms. Partial derivatives and linear approximation. Chain rules. An economic application: production functions. Homogeneous functions. Generalisation to multi-variable functions.

Implicit functions. Implicit differentiation. Applications to Economics.

Optimisation of one-variable and two-variable functions. Second derivatives, concavity and convexity, quadratic approximations. Critical points. Local and global maxima and minima. Applications to Economics.

Constrained optimisation. Equality constraints, Lagrange multipliers and their meaning. The case of inequality (and non-negativity) constraints: basic elements. Economic applications.

 

Basics of integration

Areas and integrals. Integration and differentiation: the fundamental theorem of Calculus. Rules of integration. Integration in economics. Methods of integration. Integrals over an infinite interval.

 

Basics of dynamic analysis in continuous time and discrete time

Continuous-time models. Solution to first-order differential equations. Separable equations. Linear equations with constant coefficients and with time-varying coefficients. Discrete-time models. Recursive models and solution to linear, first-order difference equations. Stationary solutions and their stability properties.

 

Dynamic analysis and recursive models in economics and financial mathematics

Exponential growth, growth towards an upper limit, logistic growth; cobweb dynamics; continuous compounding and discounting; stocks and flows; growth models in macroeconomics; recursive relations useful in financial calculus and asset valuation models; examples of intertemporal choice and intertemporal optimization problems arising in economics.

 

A glimpse at more advanced topics

Nonlinear dynamic equations, multiple equilibria and stability. Phase diagrams and qualitative analysis.

Higher-order dynamic systems (in particular, linear, two-dimensional continuous-time systems). Eigenvalues and eigenvectors, solution to linear systems of differential equations. Steady state and stability. Saddle-path stability.

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 questions, exercises and problems on the various topics of the course. Sample exams will be made available 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