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ECON 213 ELEMENTS OF

MATHEMATICS FOR ECONOMISTS

ECON 213ELEMENTS OF

MATHEMATICS FOR ECONOMISTS

STUDY GUIDEFor Undergraduates

Level 2002015/2016 – 2016/2017 Academic Year

Dr. Monica Lambon-Quayefio , Economics Department, University of Ghana

STUDY GUIDEUGBS 302 RESEARCH METHODS

ECON 213 – ELEMENTS OF MATHS FOR ECONOMISTS - STUDY GUIDE 2015 Page 1 of 24

Acknowledgements

Many thanks to the Mr. Prince Senyo, the Teaching Assistant who played a critical role in the development and editing of this study guide and the development of the course slides and presentation of the course module on the University of Ghana Sakai Learning Management System.

STUDY GUIDEECON213–ELEMENTS OF MATHS

CONTENTS

COURSE DESCRIPTION...................................................................5

1. COURSE WEBSITE...................................................................5

2. INSTRUCTORS........................................................................5

3. Support Contact Information..................................................5

5. PROBLEM-BASED LEARNING APPROACH..................................6

7. LEARNING OUTCOMES............................................................7

7.1 KNOWLEDGE.................................................................................77.2 SKILLS...........................................................................................77.3 COMPETENCE................................................................................7

8. ASSESSMENT.........................................................................8

8.1 Individual Assignments.................................................................88.2 Group Term Paper – Research Proposal........................................8

9. RECOMMENDED TEXT.............................................................8

DETAILED CLASS SCHEDULE...........................................................9

10.SCHEDULE OF SESSIONS.........................................................9

11.SESSION 0 – INTRODUCTION TO ECON 213...............................9

11.1 Overview....................................................................................911.2 Goals and Objectives.................................................................911.3 Activities and Assignments........................................................9

12.SESSION 1 – REVIEW OF BASIC MATHEMATICS- PART ONE.......10

12.1 Overview..................................................................................1012.2 Goals and Objectives...............................................................1012.3 Activities and Assignments......................................................10

13.SESSION 2 – REVIEW OF BASIC MATHEMATICS- PART TWO......10




14.SESSION 3 – REVIEW OF BASIC MATHEMATICS – PART THREE

11


15.SESSION 4 – INTRODUCTION TO MATRIX ALGEBRA – PART ONE

11


16.SESSION 5 – INTRODUCTION TO MATRIX ALGEBRA – PART

TWO.........................................................................................12


17.SESSION 6 – INTRODUCTION TO MATRIX ALGEBRA – PART

THREE.......................................................................................12


18.SESSION 7 –DIFFERENTIAL CALCULUS- PART ONE ..................13


19.SESSION 8 – DIFFERENTIAL CALCULUS- PART TWO.................14

19.1 Overview..................................................................................1419.2 Goals and Objectives...............................................................14



19.3 Activities and Assignments......................................................14

20.SESSION 9 – DIFFERENTIAL CALCULUS – PART THREE.............14


21.SESSION 10 – DIFFERENTIAL CALCULUS – PART FOUR.............16


22.SESSION 11 – INTEGRAL CALCULUS -......................................16


APPENDIX A.................................................................................19



COURSE DESCRIPTION 1. COURSE WEBSITE

http://sakai.ug.edu.gh/XXXXXXXXXXX

2. INSTRUCTORSDr. Monica Lambon-QuayefioDept. of EconomicsUniversity of GhanaP. O. Box LG 57Legon, Accra

Email: [email protected]: RM 10),EPM Building (Econs Dept. Annex)

3. Support Contact InformationSchool of Continuing and Distance EducationUniversity of Ghana P. O. Box XXXXXXXLegon, Accra

Phone: XXXXXXXXXEmail: XXXXXXXXXXXX


mailto:[email protected]

mailto:[email protected]


4. OVERVIEWThe general objective of this course is to introduce the student to mathematical concepts, develop an understanding of how to formulate economic problems in mathematical terms and gain an understanding of how economic problems can be solved mathematically. Specifically, at the end of this course, the students should be able to

Solve calculus problems involving several variables, including partial derivatives of explicit and implicit functions

Solve both unconstrained and constrained optimization problems Apply calculus to different comparative static problems to find maximum and

minimum values of functions of several variables Apply the Lagrange multipliers approach to constrained optimization

problems Apply matrices in solving economic problem

5. PROBLEM-BASED LEARNING APPROACHProblem-based learning (PBL) is a student-centered pedagogy in which students learn about a subject through the experience of problem solving. The goals of PBL are to help the students develop flexible knowledge, effective problem solving skills, self-directed learning, effective collaboration skills and intrinsic motivation. This course will use a problem-based learning approach.

Working in groups, students identify what they already know, what they need to know, and how and where to access new information that may lead to resolution of the problem. The role of the instructor/lecturer/tutor is to facilitate learning by supporting, guiding, and monitoring the learning process. The tutor will help build students' confidence to take on the problem, and encourage the students, while also stretching their understanding.

6. COURSE FORMATThe course content will be delivered online through the SAKAI Learning Management System (Sakai LMS). The Sakai LMS will be used to deliver

• Video Lecture Sessions• Session Slides • Session Reading Materials• Assessments – Interim assessments and final exam



Announcements will be posted to the course website and/email accordingly. It is the responsibility of students to check on announcements made in class, on the Course Website, and through email.

7. LEARNING OUTCOMESThe learning outcomes for the course are outlined along three strands: knowledge, skills and outcomes. 7.1 KNOWLEDGEStudents must have knowledge on

1. The different types of functions –both linear and non -linear functions

2. How to solve simple algebraic equations 3. How to solve systems of equations algebraically4. What matrices are and how to transform systems of equations into

matrices5. Know how to apply matrices in solving systems of equations and its

application to solving economic problems6. Know what derivatives are and how to apply derivatives in

optimization problems7. Know the difference between definite and indefinite integrals

7.2 SKILLSStudents must be able to

1. Identify linear and non –linear functions 2. Find the solutions of systems of equations algebraically and using

matrices3. Apply techniques learnt from basic matrix algebra to determine the

determinants and inverses of square matrices4. Transform economic problems in matrix notation and determine the

solutions of economic problems5. Use derivatives to determine the maximum and minimum points of

functions6. Apply derivatives in solving both constrained and unconstrained

optimization problems in economics7. Apply the concept of integrals in solving economic problems such as

the determination of consumer and producer surplus.

7.3 COMPETENCEStudents should in the future be able to

1. Thoroughly understand mathematical methods and concepts employed in economic models, equilibrium analysis and comparative static analysis.



2. Employ mathematical techniques such as equations, functions, matrix algebra and differential calculus to model economic problems

3. Appropriately apply mathematical methods in problems and applications that aim to analyse economic problems.



8. ASSESSMENTThe assessment for this course has been designed to help all students to maximize their individual and group/team learning opportunities. A summary of the assessment tasks is provided below.

ITEM FORM OF ASSESSMENT

DELIVERED

MARKS

Problem Sets ( for some sessions)

Short answer ( worked out) problems

Alongside some SelectedSessions

5%

Interim Assessment

Middle of the Semester

25%

Written Examination

Semester Examination End of semester

70%

Total 100%

8.1 Problem SetsProblem sets will be provided at end of each session. These assignments will mostly be in the form of word problems or mathematical equations to be solved. Deadlines will be provided for each assignment with respect to scheduling of the sessions.

8.2 Interim Assessment An interim assessment will be held around the 8th week of the semester. The actual date will be announced on the announcement page of the course website.

8.3 Final ExamA final exam will be held at the end of the semester. The exact date will be announced on the announcement page of the course website.

9. RECOMMENDED TEXT 1) Chiang, A. C., “Fundamental Methods of Mathematical Economics”, McGraw

Hill Book Co., New York, 1984.2) Sydsaeter, K. and P. Hammond, Essential Mathematics for Economic Analysis,

2nd Edition, Prentice Hall, 20063) Dowling, E. T., “Introduction to Mathematical Economics”, 3rdEdition, Shaum’s

Outline Series, McGraw-Hill Inc., 2001.



DETAILED CLASS SCHEDULEThe course is organized into 11 SESSIONS along the following lines: (1) Overview; (2) Goals and Objectives; and (3) Activities and Assignments.

10. SCHEDULE OF SESSIONS

Week Session1 Sessions 02 Session 13 Session 24 Session 35 Session 46 Session 57 Session 68 Session 79 Session 810 Session 911 Session 1012 Session 11

11. SESSION 0 – INTRODUCTION TO ECON 21311.1 OverviewThis session is meant to introduce students to basic mathematical concepts and approaches in solving economic problems.

11.2 Goals and ObjectivesAt the end of the session, the student will

1. Understand the difference between mathematical and non-mathematical economics

2. Understand what economic models entail.

11.3 Activities and AssignmentsThis week, complete the following tasks:

1. Log onto the UG Sakai LMS course site: http://sakai.ug.edu.gh/XXXXXXXXX

2. Explore the online tools available in Sakai. 3. Watch the Video for Session 0 - Course Introduction4. Read the relevant chapters from the recommended texts5. Answer the questions in the problem set for submission ( if any)6. Visit the Chat Room and introduce yourself while discussing


http://sakai.ug.edu.gh/XXXXXXXXX


expectations for the course

12. SESSION 1 – REVIEW OF BASIC MATHEMATICS- PART ONE12.1 Overview This session serves as a refresher to students on basic mathematical concepts such as functions with more emphasis on one-variable functions, one-variable economic models, algebraic expressions involving variables as well as review of solving systems of equations using the elimination and the substitution methods.


1. Define what a function is and describe other related concepts2. Differentiate between linear functions and other types of functions3. Know what linear models are4. Know what systems of equations are5. Be able to find the algebraic solutions to systems of equations



2. Watch the Video for Session 1 – Review of basic Mathematics – Part One

3. Read the relevant chapters of the recommended texts as indicated on the session slides

4. Complete and submit the problem set provided and the end of the session if any.

13. SESSION 2 – REVIEW OF BASIC MATHEMATICS – PART TWO13.1 Overview This session goes beyond the simple linear function to non-linear functions. This session introduces students to quadratic and cubic functions. Students are also introduced to other non-linear functions such as the exponential and logarithmic functions.13.2 Goals and ObjectivesAt the end of the session, the student will

1. Know what non-linear functions are and differentiate between the various types

2. Be able to find the maximum and minimum of quadratic functionsECON 213 – ELEMENTS OF MATHS FOR ECONOMISTS - STUDY GUIDE 2015 Page

12 of 24



3. Know what exponential and logarithmic functions are4. Know the rules for working with logarithmic functions5. Be able to solve equations involving non-linear functions.



2. Watch the Video for Session 2 – Review of basic mathematics – part two


4. Complete and submit the problem set provided and the end of this session if any

14. SESSION 3 – REVIEW OF BASIC MATHEMATICS – PART THREE14.1 Overview In this session presents the application of the basic mathematical concepts in solving basic economic problems. Here, students are exposed to the art of formulating economic problems using functional notations and expressing word problems using algebraic expressions.14.2 Goals and ObjectivesAt the end of the session, the student will

1. Apply the concepts from the first two sessions in solving economic problems relating to demand and supply and national income determination.



2. Watch the Video for Session 3 – Review of basic Mathematics- part three







15. SESSION 4 – INTRODUCTION TO MATRIX ALGEBRA– PART ONE15.1 Overview Matrix algebra has several uses in economics as well as other fields of study. One important application of Matrices is that it enables us to handle a large system of equations. It also allows us to test for the existence of a solution to a system of equations even before we attempt solving them. This session explains the basic concepts and terminologies used in matrices on which the subsequent sessions on matrices will build on.

15.2 Goals and ObjectiveAt the end of the session, the student will

1. Describe what matrices are and understand the different terminology used in matrices

2. Know the different type of matrices3. Be able to express a system of linear equations in matrix form4. Know the basic matrix operations such as addition, subtraction and

multiplication5. Understand and prove the commutative, associative and distributive

laws of matrices15.3 Activities and AssignmentsThis week, complete the following tasks:


2. Watch the Video for Session 4 – Introduction to matrix algebra- part one



16. SESSION 5 – INTRODUCTION TO MATRIX ALGEBRA- PART TWO

16.1 Overview This session continues the study of matrix algebra, specifically focusing on how to find the solution to a system of equations using the inverse and determinants of matrices. In this session we consider the fundamentally important concept of inverse of a square matrix and its main properties in solving systems of linear equations. This session also discusses the application of the Cramer’s rule in solving a system of n linear equations and n unknowns.





1. Understand and be able to determine the determinant of 2 X 2 matrix

2. Understand and determine the matrix of co-factors of a 3X3 matrix3. Determine the inverse of square matrices4. Understand the properties of the inverse matrix5. Understand how the Cramer’s Rule works6. Apply the Cramer’s rule in solving systems of equations.



2. Watch the Video for Session 5 – Introduction to matrix algebra- part two



17. SESSION 6 – INTRODUCTION TO MATRIX ALGEBRA- PART THREE

17.1 Overview This session focuses mainly on the general applications of matrices in solving economic problems. Specifically the session will explain how matrix algebra can be used to determine the equilibrium levels of National Income, Consumption and Taxes in an economy. The Leontief input- output analysis will also be explored in this session to calculate industry demand for inputs and outputs17.2 Goals and ObjectivesAt the end of the session, the student will

1. Be able to determine the equilibrium levels of income , consumption and taxes using matrices

2. Interpret the Leontief input-output matrix3. Determine the total demand for industries’ inputs using matrices



2. Watch the Video for Session 6 – Introduction to matrix algebra -part three







18. SESSION 7 – DIFFERENTIAL CALCULUS- PART ONE18.1 Overview Many disciplines including economics are usually interested in how quickly quantities change over time. In order to predict or estimate the future demand of a commodity or predict the growth in population etc, we need information about the rates of change. This session introduces students to the concept of derivatives and presents some of the important rules for calculating it.


1. Be able to define the derivative of a function and determine the derivatives of various function types

2. Know the simple rules for differentiation: sum rule, the product rule, the quotient rule and the chain rule.



2. Watch the Video for Session 7 – Differential Calculus- part one3. Read the relevant chapters of the recommended texts as indicated

on the session slides4. Complete and submit the problem set provided and the end of this

session if any

19. SESSION 8 – DIFFERENTIAL CALCULUS- PART TWO19.1 Overview Economics basically studies the science of choice and the most common criterion for making a particular choice among alternatives involves the goal of maximizing something (such as a firm’s profit, consumer’s utility) or of minimizing something ( such as minimizing the cost of production for a firm). This session introduces students to the concept of optimizing economic functions by determining their maximum or minimum values using derivatives.





1. Be able to determine whether or not a given function is to have a maximum or a minimum point

2. Be able to determine the stationary point of a function3. Be able to classify stationary points as maximum or minimum

points.



2. Watch the Video for Session 8 – Differential Calculus- part two3. Read the relevant chapters of the recommended texts as indicated


session if any

20. SESSION 9 – DIFFERENTIAL CALCULUS – PART THREE20.1 Overview The previous two sessions on differential calculus focused on independent variable functions. Many economic activities however involve functions of more than one independent variable. To be able to measure the effect of a change in one independent variable on the dependent variable in a multivariable function, the partial derivative is needed. This session introduces students to the partial derivatives and also presents the rules of partial differentiation.20.2 Goals and ObjectivesAt the end of the session, the student will

1. Be able to determine the difference between the derivatives of a one variable function and a multivariate function.

2. Be able to determine the total differentiation of a multivariate function using partial differentiation.

3. Be able to perform unconstrained optimization







2. Watch the Video for Session 9 – Differential Calculus – part three3. Read the relevant chapters of the recommended texts as indicated


session if any

21. SESSION 10 – DIFFERENTIAL CALCULUS- PART FOUR21.1 Overview Solutions to economic problems frequently have to be found under constraints (eg. maximizing utility subject to a budget constraint or minimizing costs subject to some requirement of output such as production quota).This session introduces students to constrained optimization using the Lagrange multiplier/function which greatly facilitates this task. The session also presents some applications of constrained optimization in economics using the Lagrangian multiplier.21.2 Goals and ObjectivesAt the end of the session, the student will

1. Be able to set up the constrained optimization problem using the Lagrangian multiplier

2. Be able to optimize economic functions using constrained optimization approach.



2. Watch the Video for Session 10 – Differential Calculus- part four3. Read the relevant chapters of the recommended texts as indicated


session if any

22. SESSION 11 – INTEGRAL CALCULUS22.1 Overview The preceding 4 sessions have focused more on derivatives and its applications to solving economic problems. Some problems however require finding a function from information about its derivative. This is generally referred to as ‘inverse’ differentiation or integration. This session presents the concept of integration, introducing students to both definite and indefinite integrals as well as its application to determining consumer and producer surplus.





1. Be able to know how to determine the integral of a function2. Be able to know the basic rules of integration3. Be able to carry out indefinite integration4. Be able to determine the values of definite integrals5. Be able to apply the determination of integrals in solving economic

problems.



2. Watch the Video for Session 11 – Integral Calculus3. Read the relevant chapters of the recommended texts as indicated


session if any



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