A86003 Mathematics for Business, Economics and Finance

Scuola di Economia e Management
Syllabus
Academic Year 2016/17 Annual

Learning Objectives

This course introduces students to fundamental concepts of calculus, linear algebra and financial mathematics. The emphasis is on the application of mathematical methods to solving practical problems in business, economics and finance.

At the end of the course students will be able to:

  • Apply mathematical reasoning to analyse essential features of various mathematical problems;
  • Solve microeconomic problems in one or more decision variables;
  • Solve economic and business problems involving the optimization with respect to one or more decision variables;
  • Understand the principles of financial mathematics.

Learning targets

The student will gain knowledge and understanding of

  • Elements of calculus of one variable and their applications in economics and business;
  • Elements of calculus of several variables and their applications in economics and business;
  • Linear algebra;
  • Basic problems in financial mathematics.

Course Content

Topics covered in Mathematics for Business, Economics and Finance are linear functions, polynomial functions, exponential and logarithmic functions, inverse functions, composite functions, limits, continuity and derivatives of functions of one and several variables, rules for differentiation, interpretations of the derivative, second and higher order derivatives, concavity and convexity, implicit differentiation, Taylor’s Formula, constrained and unconstrained optimization, integration, introduction to linear algebra, series, including geometric and harmonic series, ordinary differential equations, interest rates, present value and discounting, mortgages and investment appraisal.

Course Delivery

The course is made up of theoretical lessons, in-class assignments, homework assignments, case studies and experiential laboratory.

Lectures are held throughout the entire year. Students are responsible for learning the material before coming to class.

Students are responsible for consulting on a regular basis the website of the course on “my.liuc.it” where updates, additional material, slides and exercises for the course are posted.

  There will be a test at the end of the first semester.

Course Evaluation

Final exam:

The assessment is based on a written exam at the end of the course. The exam consists of eight exercises aimed at assessing the fundamental knowledge of calculus and linear algebra and the ability to solve economic, financial and business problems by means of mathematical tools. To pass the exam the student needs to achieve a grade higher than 18/30.

 

Partial exams:

For attending students only, the exam is made up of two written tests, one at the end of each semester. Sufficiency in both tests is required to pass the exam (sufficiency = grade higher than 18/30). Homework is assigned throughout the year and a positive evaluation of homework entitles the student to a maximum of 3 extra points on the score of the first test and 3 extra points on the score of the second test.

Syllabus

Session 0
Hours of lesson: 0
Instructor: D. Radi

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: C. Rossignoli

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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