A86003 Mathematics for Business, Economics and Finance

Scuola di Economia e Management
Syllabus
Academic Year 2013/14 Annual

Learning Objectives

This course aims to provide a solid undergraduate-level foundation to the theory and application of mathematical techniques to the fields of Economics, Finance and Management. By the end of the course students will be able to apply a number of quantitative tools to solve problems encountered in a modern business context. 

Learning targets

By following this course the student is expected to learn how to identify the relevant quantitative aspects of problems in business, economics and finance, and translate them into a formal mathematical statement which can then be solved using the techniques of mathematical analysis. 

Course Content

The course is one academic year long. The main topics covered in the first semester are: the theory of functions in one variable, one-dimensional differential and integral calculus, single variable optimization, matrix algebra and a brief introduction to vectors.  The main topics covered in the second semester are: the mathematics of interest rates, discounting, mortgages and project appraisal, the theory of functions in multiple variables, partial derivatives, unconstrained optimization, constrained optimization and linear programming. 

Course Delivery

The course is composed of circa 24 lectures on the first semester and circa 20 lectures on the second. Lectures will typically be composed of two parts: a first part in which the instructor will explain the theorical aspects of the mathematical technique to be covered, and a second part in which the instructor will conduct a guided problem-solving session in order to give the students the opportunity to apply the techniques discussed.

Studens are strongly advised to read the relevant sections of the textbooks (as specified in the syllabus) before the lectures. Also, regarding the problem-solving sessions, the instructor will publish online a number of exercises via “my.liuc.it”. Students are invited to attempt to resolve these problems before the lectures in which they will be used.

Attendance to the lectures is strongly advised to all students.

The instructor might publish online additional support material for the benefit of the students. This material is to be understood as complementary, and not substitute, to the one contained in the official textbooks of the course.

The course will also take advantage of some online resources to support the learning of mathematics, such as Wolfram Alpha (http://www.wolframalpha.com) and Pearson’s Education MyMathLab Global (http://www.mymathlabglobal.com).

 

Course Evaluation

This course is assessed via two written partial exams (40% each), to be taken at the end of the first and second semesters respectively, plus laboratory coursework (20%).

Students must register via the University administration in order to be able to take the exams. 

Syllabus

Session 0
Hours of lesson: 0
Instructor: A. Peņa

Topics:

 

FIRST SEMESTER

                                                                                                                                                           

Lecture 1

 

Why Learning Mathematics for Economics, Finance and Management?  (Part1)

·         Applications in Economics

·         Applications in Finance

·         Applications in Management

Reference:

·         tbd

Lecture 2

 

Introductory Topics

·         Integer numbers and real numbers

·         Powers

·         Intervals and Inequalities

·         Algebra rules

·         Summation notation

·         Set theory

Reference:

·         Sydsaeter et al. 1.1, 1.2, 1.3, 1.6, 2.1, 2.2, 3.1, 3.6

Lecture 3

 

Functions of One Variable I

·         Definitions, Domain and Range

·         Linear functions

·         Quadratic Functions

·         Graphs of Functions

·         Case Study: Supply and demand analysis

References:

·         Sydsaeter et al. 4.1, 4.2, 4.3, 4.4, 4.6

·         Jacques 1.5

Lecture 4

 

Functions of One Variable II

·         Polynomials

·         Power functions

·         Exponential functions

·         Logartithmic functions

·         Case Study: Revenue, cost and profit

References:

·         Sydsaeter et al. 4.7, 4.8, 4.9,4.10

·         Jacques 2.2

Lecture 5

Properties of Functions

·         Shifting graphs

·         Inverse functions

·         Composite functions

·         Distance in the plane, circles

References:

·         Sydsaeter et al. 5.1, 5.2, 5.3, 5.4, 5.5

Lecture 6

Differentiation I

·         Slopes of curves

·         Tangents and derivatives

·         Increasing and decreasing functions

·         Rates of change

·         Limits: an introduction

·         Case Study: Marginal functions

References:

·         Sydsaeter et al. 6.1, 6.2, 6.3, 6.4, 6.5

·         Jacques 4.3

Lecture 7

Differentiation II

·         Rules for differentiation

·         Sum, Products and Quotients

·         Chain Rule

References:

·         Sydsaeter et al. 6.6, 6.7, 6.8

Lecture 8

Differentiation III

·         Higher order derivatives

·         Derivatives of Exponential functions

·         Derivatives of Logarithmic functions

References:

·         Sydsaeter et al. 6.9, 6.10, 6.11

Lecture 9

 

Derivatives in Use I

·         Implicit differentiation

·         Differentiating the inverse

·         Case study: Supply and demand

References:

·         Sydsaeter et al. 7.1, 7.2, 7.3

Lecture 10

 

Derivatives in Use II

·         Linear Approximation

·         Polynomial Approximation

·         Taylor’s Formula

·         Case Study: Elasticity

References:

·         Sydsaeter et al. 7.4, 7.5, 7.6

·         Jacques 4.5

Lecture 11

 

Derivatives in Use III

·         Continuity

·         Limits

·         Intermediate value theorem

·         Newton’s method

·         Infinite sequences

·         L’Hôpital’s rule

References:

·         Sydsaeter et al. 7.8, 7.9, 7.10, 7.11, 7.12

Lecture 12

Single Variable Optimization I

·         Simple Tests for Extrema

·         Convex and concave functions

·         Case study: Harvesting

References:

·         Sydsaeter et al. 8.1, 8.2, 8.3

Lecture 13

Single Variable Optimization II

·         Extreme value theorem

·         Mean value theorem

References:

·         Sydsaeter et al. 8.4, 8.5

Lecture 14

Single Variable Optimization III

·         Local extreme points

·         Inflection points

·         Case Study: Optimisation in economic functions

References:

·         Sydsaeter et al. 8.6, 8.7

·         Jacques 4.6

Lecture 15

 

Integration I

·         Indefinite integrals

·         Definitions

·         General rules of integration

References:

·         Sydsaeter et al. 9.1

·         Jacques 6.1

Lecture 16

 

Integration II

·         Definite integrals

·         Area of definite integrals

·         Properties of definite integrals

·         The Riemann integral

·         Case Study: Consumer/Producer Surplus

References:

·         Sydsaeter et al. 9.2, 9.3, 9.4

·         Jacques 6.2.1, 6.2.2

Lecture 17

Integration III

·         Integration by parts

·         Integration by substitution

References:

·         Sydsaeter et al. 9.5, 9.6

·         Dowling 12.8, 12.9

Lecture 18

Integraion IV

·         Infinte intervals

·         A glimpse at differential equations

·         Case study: Logistic growth

References:

·         Sydsaeter et al. 9.7, 9.8

·         Jacques 9.2

Lecture 19

Matrix Algebra I

·         Description of matrices

·         Row vectors, Columns vectors

·         Basic Matrix operations

References:

·         Sydsaeter et al. 15.1, 15.2

·         Jacques 7.1.1, 7.1.2, 7.1.3

Lecture 20

Matrix Algebra II

·         Matrix multiplication

·         Rules matrix multiplication

·         Transpose

References:

·         Sydsaeter et al. 15.3, 15.4, 15.5

·         Jacques 7.1.4

Lecture 21

Matrix Algebra III

·         Gaussian elimination

References:

·         Sydsaeter et al. 15.6

·         Dowling 6.5

Lecture 22

Matrix Algebra IV

·         Determinants

·         Expansion by cofactors

·         Cramer’s Rule

·         Case study: The Leontief model

References:

·         Sydsaeter et al. 16.1, 16.2, 16.3, 16.4, 16.8

·         Jacques 7.3

Lecture 23

Matrix Algebra V

·         Inverse of a matrix

·         Properties of the Inverse

·         Solving Equations by matrix inversion

References:

·         Sydsaeter et al. 16.6, 16.7

·         Jacques 7.2

Lecture 24

Vector Algebra

·         Operations, Inner Product

·         Geometric interpretation

·         Cauchy-Schwartz inequality

·         Orthogonality

References:

·         Sydsaeter et al. 15.7, 15.8, 15.9

·         Jacques 4.6

 

Readings:

Session 0
Hours of lesson: 0
Instructor: A. Peņa

Topics:

 

SECOND SEMESTER

                                                                                                                                                           

Lecture 1

 

Why Learning Mathematics for Economics, Finance and Management?  (Part 2)

·         Applications in Economics

·         Applications in Finance

·         Applications in Management

Reference:

·         tbd

Lecture 2

 

Interest Rates I: Simple Compounding

·         Interest rate periods

·         Effective rates

·         Simple discounting

Reference:

·         Sydsaeter et al. 10.1

·         Zima and Brown 3.1,  3.2, 3.5

Lecture 3

 

Interest Rates I: Continuous Compounding

·         Accumulated value

·         Equivalent rates

·         Discounted value

·         Compound discounting

References:

·         Sydsaeter et al. 10.2

·         Zima and Brown 4.1, 4.2, 4.3

Lecture 4

 

Present Values and Discounting

·         Present discounted value

·         Geometric series

·         Harmonic series

References:

·         Sydsaeter et al. 10.3, 10.4

Lecture 5

Annuities

·         Accumulated value

·         Total present value

·         Continuous income stream

References:

·         Sydsaeter et al. 10.5

·         Zima and Brown 5.1, 5.2, 5.3, 5.4

Lecture 6

Mortgages

·         Amortizing of a debt

·         Outstanding principal

·         Mortgages

References:

·         Sydsaeter et al. 10.6

·         Zima and Brown 7.1, 7.2. 7.3

Lecture 7

Investment Appraisal

·         Present value as appraisal tool

·         Internal rate of return as appraisal tool

References:

·         Jacques 3.4

Lecture 8

Functions of two variables I

·         Domain, Range

·         Partial derivatives

·         Higher order derivatives

References:

·         Sydsaeter et al. 11.1, 11.2

Lecture 9

Functions of two variables II

·         Level curves

·         Geometric representation of partial derivatives

·         Distance formula

References:

·         Sydsaeter et al. 11.3, 11.4

Lecture 10

 

Partial Derivatives

·         Young’s theorem

·         Hessian

·         Case study: partial elasticities

References:

·         Sydsaeter et al. 11.6, 11.8

Lecture 11

 

The Chain Rule

·         Chain rule in one dimension

·         Chain rule in n dimensions

References:

·         Sydsaeter et al. 12.1, 12.2

Lecture 12

 

Tools of functional analysis I

·         Homogeneous functions

·         Homothetic functions

·         Euler’s theorem

·         Geometric aspects

References:

·         Sydsaeter et al. 12.6, 12.7

Lecture 13

Tools of functional analysis II

·         Linear Approximations

·         Tangent Planes

·         Rules for differentials

·         Increments

References:

·         Sydsaeter et al. 12.8, 12.9

Lecture 14

Unconstrained Optimization I

·         Necessary conditions

·         Critical points

·         First order conditions

·         Concavity and convexity

References:

·         Sydsaeter et al. 13.1, 13.2, 13.3

Lecture 15

 

Unconstrained Optimization II

·         Extreme value theorem

·         Interior and boundary points

·         Minima and maxima

·         Saddle points

·         Second order derivative test

References:

·         Sydsaeter et al. 13.5

Lecture 16

 

Constrained Optimization I

·         Lagrange multipliers

·         The Langrangian

References:

·         Sydsaeter et al. 14.1

·         Jacques 5.5

Lecture 17

Constrained Optimization II

·         Lagrange Multipliers: interpretation

·         Several solution candidates

References:

·         Sydsaeter et al. 14.2, 14.3

·         Jacques 5.6

Lecture 18

Linear Programming I

·         Objective function

·         Inequality constraints

·         Nonnegativity constraints

References:

·         Sydsaeter et al. 17.1, 17.2, 17.3

·         Jacques 8.1

Lecture 19

Linear Programming II

·         Case study: firm with N outputs and M resources

References:

·         Sydsaeter et al. 17.4

·         Jacques 8.2

Lectures 20-24

EXPERIENTIAL LAB

·         Projects in Business Cases  on Finance, Economics and Management using Wolfram Alpha

 

Readings:


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