Mathematical Methods in Economics - ECN115

Module Information

• Module Code: ECN115
• Module Title: Mathematical Methods in Economics and Finance
• Lecturer: Evgenii Safonov
• Email: [email protected]
• Office: GC332
• Office Hours: TBA
• Lectures: Wednesdays 9:00-11:00 in GCG10 Peston LT, Thursdays 12:00-14:00 in GCG10 Peston LT
• Classes: See My Timetable on QMplus

Teaching Assistants

• Mahmoud Shahin: [email protected]
• Zhenghao Liu: [email protected]
• Tallon Trent Howie: [email protected]

Module Content

• Refreshes existing mathematical knowledge and introduces mathematical tools used in undergraduate economics and finance.
• Themes:
  • Introduction: sets, numbers, mathematical induction, binomial formula
  • Elements of one-variable mathematical analysis and calculus
  • Constrained optimisation (one variable)
  • Introduction to multivariate analysis and optimisation
  • (If time permits) other topics including linear algebra

Assessment

• Three types:
  • Online quizzes (15%): Individual work
  • Graded problem sets (15%): Individual solutions, group discussions allowed
  • Final exam (70%): Individual work, centrally administered
• Timing:
  • Quiz 1 (7.5%): One hour, October 14th, choose time between 00:00 to 23:59
  • Problem Set 1(7.5%): Available 1 week in advance or earlier, November 3rd
  • Quiz 2 (7.5%): One hour, December 2nd; choose time between 00:00 to 23:59
  • Problem Set 2 (7.5%): Available 1 week in advance or earlier, December 10th
  • Final exam (70%): Two hours, January (paper and pen only)

Important Component

• Weekly homeworks (not graded): Essential for practice.
• A mix of basic and advanced problems.
• Solutions are posted online

Diagnostic Quiz

• Friday, September 27th
• Does not count towards the grade
• Objective: to assess background level in mathematics
• Format: 20 questions (mostly multiple choice, some requiring calculations)
• Duration: 80 minutes

Last Year's Materials

• Previous exam topics and exams from the past 2 years are available on QM+: ECN115 -> Key Module Information -> Syllabus
• Previous year's exam materials might be slightly different from this year's
• Solutions will be posted closer to the exams

Outline

• Elementary logic
• Sets
• Number systems and properties of real numbers
• Basic inequalities
• Summation operator
• Next week's material: Mathematical Induction

Sets

• A set is a collection of objects
• Elements are objects within a set denoted by ∈
• Two sets (A and B) are equal if each element in A is in B and each element in B is in A (A=B).
• An empty set is a set with no objects denoted by Ø or {}.
• A subset is a set where every element is also contained within another set, indicating a relationship of inclusion; if set A is a subset of set B, we denote this as A ⊆ B.
  • Conversely, a superset is one that includes all elements of a particular subset; thus, if set B contains all elements of set A, we can say B ⊇ A.
  • For example, if we have set A = {1, 2} and set B = {1, 2, 3, 4}, then A is a subset of B while B is a superset of A.
  • Understanding these concepts is fundamental in set theory, as they help categorize and understand the relationships between different sets.
• Not all concepts can be defined as sets, as showcased by Russell's paradox. This highlights a critical flaw in naive set theory, revealing contradictions arising when a set includes itself.
• Set relations and operations are fundamental concepts in set theory. Subsets refer to a set where all its elements are contained within another set. The intersection (∩) of two sets includes only the elements common to both sets. The union (∪) combines all unique elements from both sets. The difference (∖) illustrates the elements that belong to one set but not another, providing insight into the relationships between sets.
• This notation denotes a set that comprises all elements, denoted by x, which are derived from set A and simultaneously satisfy a specific condition or property, referred to as property P, thereby creating a defined subset with distinct characteristics.
• Set subtraction( A∖B)

Number Systems

• Natural numbers (N): {1, 2, 3,...}
• Integers (Z): {... -3, -2, -1, 0, 1, 2, 3,...}
• Rational numbers (Q): All expressions p/q where p ∈ Z and q ∈ N (q≠0)
• Real numbers (R): The set of points on the number line

Properties of Real Numbers

• Order Relation: In mathematics, an order relation provides a way to compare elements within a set. This could include total orders, where every pair of elements can be compared, or partial orders, where some elements may not be directly comparable. Understanding these concepts is critical in various fields, including analysis and algebra, as they lay the groundwork for discussing concepts like convergence and limits.
• Addition and Multiplication operations: These fundamental operations in mathematics exhibit several key properties. The commutative property states that changing the order of the numbers involved does not change the sum or product (e.g., a + b = b + a and ab = ba). The associative property indicates that the grouping of numbers does not affect their sum or product (e.g., (a + b) + c = a + (b + c) and (ab)c = a(bc)). Furthermore, every number has an additive inverse (a number that, when added to the original, results in zero) and a multiplicative inverse (a number that, when multiplied by the original, results in one). Lastly, the identity elements of addition (0) and multiplication (1) are crucial, as they do not alter the original value of a number when used in their respective operations.
• Least Upper Bound property: Also known as the supremum property, this concept asserts that any non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers. This property is essential in real analysis, ensuring that certain limits and bounds exist, thus providing a foundation for discussing continuity, compactness, and convergence in calculus and advanced mathematical theories.

Intervals of Real Numbers

• Closed interval [a, b] = {x ∈ R | a ≤ x ≤ b}
• Open interval (a, b) = {x ∈ R | a < x < b}
• Half-open/half-closed intervals ((a, b], [a, b))

Basic Inequalities

• The properties of real numbers, such as the distributive, associative, and commutative properties, play a crucial role in solving inequalities. Understanding how to manipulate these properties helps in rearranging and simplifying expressions. Additionally, techniques such as graphing, interval testing, and using the number line can effectively identify the solution sets for various types of inequalities.
• Basic techniques for solving inequalities often include isolating the variable, applying inverse operations, and carefully considering the impact of multiplying or dividing by negative numbers, which reverses the inequality sign. These foundational approaches are essential for ensuring accurate solutions in mathematical contexts involving inequalities.
• Example problems

Summation Operator

• Summation notation (Σ)
• Properties of summation:
  • (P1)
  • (P2)
  • (P3)
• Examples: calculations with the summation operator

Mathematical Induction

• A technique for proving statements that depend on a natural number (n)
• Requires a proven induction base (first case ) and induction step

Additional Notes

• The lecture notes cover a range of mathematical topics relevant to economics and finance.
• The module emphasizes understanding the underlying principles behind calculations, rather than just mechanical application of formulas.