Welcome to the ECOMATH Webpage!

Shortened course syllabus

What you see in this webpage is a shortened version of the course syllabus for Mathematical Economics (ECOMATH) available in Animospace.

Basic information

  1. You are enrolled in both ECOMATH V25.
  2. The prerequisites are ECOCAL2 and LBYCALC.
  3. ECOMATH and LBYMATH are co-requisites.
  4. ECOMATH is one of the prerequisites to MIC1ECO, MAC1ECO, ECONSTA.
  5. We meet for ECOMATH Mondays 1430-1600 face-to-face and Thursdays 1430-1600 synchronously (unless otherwise informed).
  6. We have two websites: one at Animospace and one here. I maintain a course diary here.
  7. Passing mark is 60%.

Course description

This course serves as an introduction to mathematics for economic analysis at the undergraduate level. The course discusses concepts on linear algebra, optimization and tools for comparative statics. Topics include determinants of matrices and matrix inversion, systems of linear equations, unconstrained and constrained optimization. The course also covers economic applications from consumer and production theory.

Main textbook

We will be using the following textbook (abbreviated as EMEA):

Sydsæter, K., Hammond, P., & Strøm, A. (2012). Essential Mathematics for Economic Analysis (4th ed.). Pearson Education Limited. Borrow from Internet Archive.

Course outline

Below is a tentative course outline and we may have to adjust, depending on the circumstances.

Weeks Topics Readings from EMEA
1-3 Matrix algebra Chapters 15 and 16
4-6 Single-variable optimization Chapter 8
7-9 Two-variable optimization Chapter 13
10-12 Constrained optimization Chapter 14

References

The following references have physical copies available at the DLSU Library Reserve Section, 8th Floor of the Henry Sy Hall and most are also available for digital borrowing through the Internet Archive.

Pemberton, M. & Rau, N. (2016). Mathematics for Economists: An Introductory Textbook (4th ed.). Manchester University Press. Borrow from Internet Archive.

Harrison, M. & Waldron, P. (2011). Mathematics for Economics and Finance. Routledge.

Klein, M. W. (2002). Mathematics for Economists: An Introductory Textbook (2nd ed.). Pearson Education Inc. Borrow from Internet Archive.

Hess, P. N. (2002). Using Mathematics in Economic Analysis (2nd ed.). Pearson Education Inc. Borrow from the Internet Archive.

Hoy, M., Livernois, J., McKenna, C., Rees, R., & Stengos, T. (2001). Mathematics for Economics (2nd ed.). The MIT Press. Borrow from Internet Archive.

Sydsæter, K. & Hammond, P. (1995). Mathematics for Economic Analysis. Pearson Education Inc. Borrow from Internet Archive.

The following references are more recent, meaning they come from the past five years. But we do not necessarily have physical or digital access to the newest editions, even if they may be available in the market. Of course, there are exceptions.

Margalit, D., Rabinoff, J. & Williams, B. (2024). Interactive Linear Algebra (UBC ed.). Visit online textbook draft.

Hoy, M., Livernois, J., McKenna, C., Rees, R., & Stengos, T. (2022). Mathematics for Economics (4th ed.). The MIT Press.

Sydsæter, K., Hammond, P., Strøm, A., & Carvajal, A. (2021). Essential Mathematics for Economic Analysis (6th ed.). Pearson Education Limited.

Because of accreditation requirements, I am compelled to brush aside my modesty and refer to the following materials which may still be relevant for the current course:

Pua, A. A. Y. (2024). Integral Calculus for Economic Analysis. Visit online textbook draft.

Pua, A. A. Y. (2023). Piecewise Interpolation, Numerical Differentiation, and Numerical Integration. Retrieved from https://apsacad.neocities.org/numerical-2023-10-05.html.

Course requirements

  1. Group quizzes (20%)
  2. Group homework assignments (10%)
  3. Long exams 1, 2, and 3 (weighted as 18%, 24%, and 28%, respectively): These requirements are timed and scheduled in advance. They are tentatively set on 2024-05-27, 2024-07-08, and 2024-07-29.

Class policies

Be guided by the student handbook regarding conduct and attendance. Additional policies may be given in class, if necessary. The biggest hurdle of any student is the inability to read, comprehend, and follow instructions. Make sure to keep yourself informed.

Refer to full syllabus for exam policy, late homework policy, and other policies related to online sessions.

Course diary

Long Exam 03, 2024-07-29, face-to-face

Lecture 19, 2024-07-25, online

Lecture 18, 2024-07-22, face-to-face

Lecture 17, 2024-07-18, online

  1. Worked on Section 13.7 Exercise 3: A simple one-parameter exercise to verify the envelope theorem
  2. Worked on Section 13.7 Exercise 4: About theoretical implications of the envelope theorem applied to profit maximization problems
  3. Motivated constrained optimization
  4. Introduced the Lagrange multiplier method
  5. Worked on Section 14.1 Examples 1 to 4: connections to economics
  6. Notes

Lecture 16, 2024-07-15, face-to-face

  1. Recap of Section 13.5 Exercise 3: you should complete the remaining parts of the exercise
  2. Introduced the meaning of comparative statics
  3. Motivated the idea behind the envelope theorem
  4. The computational convenience of having the envelope theorem

Lecture 15, 2024-07-11, online

  1. Finished Section 13.4 Exercise 5 on duopoly
  2. New language when applying the extreme value theorem: interior, boundary, open, closed
  3. Recipe for applying the extreme value theorem is the same, only the idea of boundary is different compared to Chapter 8
  4. Worked on Section 13.5 Example 1
  5. Started to work on Section 13.5 Exercise 3
  6. Notes

Long Exam 02, 2024-07-08, face-to-face

Coverage: Chapters 15, 16, and 8

Lecture 14, 2024-07-04, online

  1. Mainly about economic applications
  2. Notes

Lecture 13, 2024-06-18, face-to-face

NOTE: This is a Tuesday with a Monday schedule!!

  1. Group quiz 04

  2. Discussed Section 13.1

    • Pay attention to the notation of partial derivatives.
    • First order necessary conditions for a maximum or minimum
    • Used Desmos to plot bits and pieces of the function given in Section 13.1 Example 1: connections to shapes encountered in high school, e.g. parabolas, circles, ellipses
    • Spent time on traces and level curves of a surface
    • Section 13.1 Example 3: Interpreting first-order conditions

  3. Discussed first part of Section 13.2: Compare and contrast Theorems 13.1.1 and 13.2.1.

  4. To review functions of two variables, consider Sections 5.1 to 5.3 of the chapter Extensions to functions of two variables of my textbook draft in ECOCAL2 or Chapter 11 of EMEA.

  5. Reading assignment for next meeting: Sections 13.1 to 13.5

  6. Practice exercises: All examples and exercises of Sections 13.1 and 13.2

  7. Group HW02

Lecture 12, 2024-06-13, online

  1. Contrasting Section 8.6 with Sections 8.1 to 8.5
  2. Curve sketching and why it could be useful
  3. Worked on inflection points
  4. Worked on Review Problems for Chapter 8 Exercise 1
  5. Reading assignment for next meeting: Sections 8.6, 8.7 (Skip “More General Definitions of Concave and Convex Functions” and “Strictly Concave and Strictly Convex Functions”), 13.1, 13.2
  6. Notes

Lecture 11, 2024-06-10, face-to-face

  1. Group quiz 03
  2. Worked on the Extreme Value Theorem
  3. Worked on Section 8.5 Example 4

Lecture 10, 2024-06-06, online

  1. Finished concave and convex functions, Theorem 8.2.2 (magnitude of the first derivative is the focus), worked on Section 8.2 Exercises 5 and 10
  2. Worked on Examples 1, 2, and 3 of Section 8.3
  3. Reading assignment for next meeting: Sections 8.4 (Skip “The Mean Value Theorem”), 8.5, 8.6
  4. Practice exercises: All examples and exercises of Section 8.2 and 8.3
  5. Notes

Lecture 9, 2024-06-03, face-to-face

  1. Group quiz 02
  2. Worked on the idea behind the first derivative test: sign variation is the focus rather than magnitude
  3. Worked on Section 8.2 Examples 1 and 2

Lecture 8, 2024-05-30, online

  1. Moved intended discussion of Sections 16.9 and 12.11 to near the end of the course.

  2. Started motivation for Chapters 8, 13, and 14.

  3. Using non-calculus arguments to find maxima and minima, writing down arguments

  4. How to find maxima and minima more systematically

    • Pay attention to the vocabulary used. Stationary points and critical points seem to be used interchangeably. But they are not the same in other textbooks, see for example Paul’s Online Notes.
    • This distinction affects conclusions of Section 8.1, Figure 3.
    • Why is Theorem 8.1.1 called a necessary first-order condition?

  5. “Foreshadowing”: Local maximum, local minimum

  6. If first-order condition is only necessary, how do we check if we have a maximum or a minimum? Focused on the geometric picture first. Why is the first derivative involved here?

  7. Reading assignment: Sections 8.1, 8.2, 8.3

  8. Notes

Long Exam 01, 2024-05-27, face-to-face

What to bring for ECOMATH exam:

  1. Pens and extra pens
  2. Non-programmable scientific calculator
  3. University ID with picture

ECOMATH Coverage (updated as of 2024-05-23):

  1. Chapter 15, except 15.7 to 15.9
  2. Chapter 16, except 16.3, 16.5, Theorem 16.7.1, adjoints, The General Leontief model

Practice exercises: All examples and exercises in

  • Sections 15.1, 15.2, 15.3, 12.11
  • Sections 15.4 (skip Exercise 6c), 15.5 (skip generalization to \(n\) matrices for Exercise 6, skip Exercise 8c), 15.6
  • Sections 16.1, 16.4 (skip Exercise 14), 16.7 (skip Exercise 4 and modify instruction of Exercise 1 to find inverse using elementary row operations)
  • Only the exercises in Section 16.2 (Modify instruction of Exercise 1 to “Evaluate the following determinants using elementary row operations and rules from Theorem 16.4.1.”)
  • Only Exercises 1 (Modify instruction to “Evaluate the following determinants using elementary row operations and rules from Theorem 16.4.1.”) and 5 in Section 16.3
  • Only the exercises in Section 16.5: Modify the instruction to “Evaluate the following determinants using elementary row operations and rules from Theorem 16.4.1.”
  • Section 16.6
  • Only the examples of Section 16.8
  • For Section 16.9, focus on Example 1, Exercises 1, 2, 5
  • Review Problems for Chapter 15 (skip Exercises 10 to 12)
  • Review Problems for Chapter 16 (skip Exercises 15 and 16)

Lecture 7, 2024-05-23, online

  1. Wrap up properties of the inverse.
  2. Worked on some exercises.
  3. Notes

Lecture 6, 2024-05-20, face to face

  1. Group quiz 01: Covers reading assignment indicated in Lecture 5
  2. Emphasize the correct application of Rule F in Theorem 16.4.1 and contrast this with Section 15.6.
  3. Worked on the broad idea for Cramer’s rule and when it is ideal to use it.
  4. How to calculate an inverse using elementary row operations and how it is connected to solving systems of linear equations
  5. Reading assignment for next meeting: Section 16.6, 16.9, 12.11

Lecture 5, 2024-05-16, online

  1. Geometric interpretation of determinants: Why use the absolute value symbols as notation?

    1. Some neat connections to area and volume
    2. Connection to existence of solutions to systems of linear equations: When will the area of the parallelogram be equal to zero in terms of the row vectors which form the coefficient matrix?

  2. How to calculate determinants without introducing another new method?

    1. We avoid expansion via cofactors but exploit the structure of upper triangular matrices for calculating determinants faster.
    2. Use row operations and rules in Theorem 16.4.1 to calculate determinants.

  3. Get to see Cramer’s rule. Under what conditions will it be a good idea to use it?

    1. Use only when solving systems of \(n\) linear equations in \(n\) unknowns.
    2. Not very advisable when \(n\geq 3\).
    3. But interesting from a theoretical point of view: How did Cramer prove in general that the rule works? Answers may be found if you read the skipped sections of Chapter 16.

  4. Reading assignment for next meeting: Sections 16.4, 16.6, 16.7 (focus on the subsection “Finding Inverses by Elementary Row Operations), 16.9 (focus on the subsection”Homogeneous Systems of Equations”)

  5. Notes

    1. WARNING: I made a terrible mistake in my application of Rule F in Theorem 16.4.1. I will repeat this portion on May 20.
    2. I fix this here.

Lecture 4, 2024-05-13, face-to-face ONLINE

  1. Discuss parts of the group HW.

  2. Wrap up Gaussian elimination.

    1. Pay attention to free variables and the geometry of solutions of systems of linear equations.
    2. Pay attention to how Gaussian elimination can be used to determine whether a system of linear equations is consistent (under some conditions) or inconsistent.

  3. Wrap up other matrix operations.

    1. Properties of matrix multiplication
    2. Where powers of matrices come in
    3. Transposes
    4. Symmetric matrices and their connection to quadratic polynomials

  4. Move on to determinants of \(2\times 2\) matrices.

    1. Focus on systems of 2 linear equations in 2 unknowns.
    2. What are the connections to Cramer’s rule and determinants?
    3. Cramer’s rule in a systems of 2 linear equations in 2 unknowns is not best way to solve systems of linear equations but can be informative.

  5. Reading assignment for next meeting: Sections 16.1, 16.4, 16.6, 16.8

  6. Notes

Lecture 3, 2024-05-09, online

  1. Wrap up matrix multiplication: Why is matrix multiplication important?

    1. Connect to Section 15.3 Example 3 and systems of linear equations
    2. Matrices as linear transformations, matrix multiplication as composition of linear transformations

  2. Working through a simple version of a Leontief model

    1. A real example of input-output tables for the Philippines
    2. Some related information from the Asian Development bank
    3. Research by DLSU faculty on correcting input-output tables

  3. How are systems of nonlinear equations related to systems of linear equations?

    • Concentrate on functions of two variables first.
    • Review linearization here and differentials of functions of more than one variable here.
    • Continue with Sections 12.10 and 12.11.

  4. Started Gaussian elimination and row operations

  5. Reading assignment for next meeting: Sections 15.6, 15.4, 15.5, 16.1, 16.4

  6. Notes

  7. Group HW01

Lecture 2, 2024-05-06, face-to-face ONLINE

  1. Casting systems of equations into the form (1) found in page 546

    1. Important to know how to setup systems of linear equations properly
    2. Section 15.1, Exercise 6
    3. How do you solve systems of linear systems in a more systematic way?

  2. More vocabulary and operations for matrices: Section 15.2, Examples 1 to 6

  3. Matrix multiplication looks strange.

    1. Understand the meaning of summation notation. You will use this even outside of matrix multiplication.
    2. How do you interpret the components of \(c_{ij}\)?

  4. Reading assignment for next meeting: Sections 15.1, 15.2, 15.3, 16.9 (Example 1), 12.10, 12.11, 15.4

  5. Notes

Lecture 1, 2024-05-02, online

  1. Course administration:

    • Must NOT enroll in the lab course with section code V24!
    • Deadline for dropping course: May 8 with 90% refund, May 15 with 50% refund, May 29 with no refund
    • Deadline for withdrawal from course: May 30 to June 29 with no refund

  2. Advice:

    1. Forget what you know about study habits.
    2. How to write an email

  3. Learn to distinguish subscripts and superscripts.

  4. Introduce terminology and motivation for the study systems of linear equations.

  5. Section 15.1, Example 1

  6. Reading assignment for next meeting: Sections 15.1, 16.9 (Example 1 only), 15.2, 15.3, 15.6

  7. Notes