75588 - Mathematical Tools and Methods

Learning outcomes

At the end of the course, the student will have the 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 of the micro- and macroeconomics courses.

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.

Economic applications of dynamical systems. Price adjustment, Solow-Swan growth model, cobweb model, multiplier-accelerator model, Dornbusch overshooting model, dynamics of the stock of a renewable resource.

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

Readings/Bibliography

Essential references

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).

 

An excellent alternative option (with solutions manual) is:

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

M. HOY, J. LIVERNOIS, C. McKENNA, R. REES, A. STENGOS, Student Solutions Manual for Mathematics for Economics 3rd Edition, MIT Press, 2012.

Teaching methods

Classroom lessons

The exercises and problems presented and discussed in the classroom are important to properly understand all the parts of the program

Assessment methods

A (mandatory) written exam and an (optional) oral examination.

The written exam consists of a number of exercises of different levels of difficulty. The written exam aims at testing the student's ability to correctly and effectively apply the basic and advanced techniques learnt in the course to specific mathematical (and economic) problems.

The oral examination (requiring that the written part has been passed with a "sufficient" grade, at least 18/30) is optional. It must be regarded as a possible integration to the written exam, at the student's request. As such, it may affect the final grade both positively and negatively. The subject of the oral exam will be selected by the instructor among a number of theoretical topics and economic applications illustrated in the textbook(s). A detailed list of topics and further readings for the oral exam will be distributed in the class.

Office hours

See the website of Roberto Dieci