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COURSE INFORMATION
Course Title: MATH. FOR ECONOMICS AND BUSINESS II
Code Course Type Regular Semester Theory Practice Lab Credits ECTS
BUS 102 A 2 4 0 0 4 5
Academic staff member responsible for the design of the course syllabus (name, surname, academic title/scientific degree, email address and signature) Dr. Valmir Bame vbame@epoka.edu.al
Main Course Lecturer (name, surname, academic title/scientific degree, email address and signature) and Office Hours: M.Sc. Vasil Lino vlino@epoka.edu.al , Monday-Friday 8:30- 16:30
Second Course Lecturer(s) (name, surname, academic title/scientific degree, email address and signature) and Office Hours: NA
Language: English
Compulsory/Elective: Compulsory
Study program: (the study for which this course is offered) Bachelor in Economics (3 years)
Classroom and Meeting Time: E 406 Friday 13:30 - 15:30
Teaching Assistant(s) and Office Hours: NA
Code of Ethics: Code of Ethics of EPOKA University
Regulation of EPOKA University "On Student Discipline"
Attendance Requirement:
Course Description: Mathematics for Economics and Business II: Limits and Continuity. Average Rate of Change and Slope. Derivatives, Instantaneous Rate of Change, Higher Order Derivatives. Optimization, Concavity of Inflection Points. Maxima and Minima. Revenue, Cost and Profit Applications, Anti Derivatives, Rules of Integration, Differential Equations, Mathematics of Finance, Simple and Compound Interest, Present Value, Effective Interest, Future Value, Annuities.
Course Objectives: The aim of this course is to give the basic ingredients of mathematics for business and economics. Namely, functions, derivatives and differentials. Moreover, most of the models in economics appear in the form of linear models.
BASIC CONCEPTS OF THE COURSE
1 The concept of differentiation of functions of one variable and multivariable
2 Marginal economic functions
3 Optimization of functions using the derivatives and linear programming
4 Differentiation and integration, rules and methods
5 Matrices, operations with matrices. Solution of the systems using inverse matrix and Crammer's Rule
6 Differential equations as a concept which describes dynamic proceses.
7 Simple differential equations and their solutions for economics' problems.
COURSE OUTLINE
Week Topics
1 Differentiation & Marginal Functions. Derivative of the functions. Derivation rules. Marginal functions and elasticity of the economic functions. On the book “Mathematics for economics and business” Jan Jacque ninth edition, sections 4/1, 4/2, 4/3, 4/4, 4/5, page 268 – 328. During this week will explain the concept of the differentiation and rules, techniques for finding the derivatives of the mathematical or economical functions. The concept of the marginal economic functions and evaluation of the elasticity will explain as an application of the derivatives for the economic functions.
2 Optimization. Sections 4.6 and 4.7 (pages 329 – 371) are devoted to the topic of optimization, which is used to find the maximum and minimum values of economic functions. We will concentrate on the mathematical technique and to the applications these techniques to the real economic and business problems. Also, during this week will explain formulas for the derivatives of the logarithmic and exponential functions. The exercises will be real economic functions in exponential or logarithmic forms.
3 Elasticity. This week continuous the topic of calculus, multivariable functions and their differentiation (chapter five page 390-420). Knowing very well the differentiation, (finding the derivatives) of functions of a variable, students can find the partial derivatives, and elasticity of the economic functions. Mathematical techniques of differentiation of multivariable functions are very important and thy are placed at section 5.1; 5.1 page 390-420. The section 5.4 page 433-447 involves maximization and minimization of functions in which the variables are free to take any value, the so-called unconstraint optimization.
4 Revision Exercises. Quiz 1. Materials for week 4 are revision exercises. These exercises are about marginal economic functions, elasticity, of demand and supply function, rules of differentiation, and the optimization of economic functions of one variable. After revision is the Quiz 1.
5 Partial Elasticity. Week five is for constraint optimization, page 447-470. As a concept it is based on real life problems. For example, a company might wish to minimize the cost of its production, constraint by the need to satisfy the production quotas. A firm wants to realize the production with minimum work force, constraint by payment budget. There are two ways to solve such problems: the method of substitution (page 447-459), and the method of Lagrange multipliers.
6 Review. Exercises. Week six is for revision exercises and preparation for the midterm exam. During this week will be repeated the most important concepts like differentiation of function of one variable and the rules of derivations. The differentiation and rules of derivations of multivariable function. Also, will be repeated the methods of optimization of the economic problems.
7 Midterm. Midterm exam. Materials for midterm exam, the students must study at the end of the chapter four and five, multiple choice questions and examination questions, on page 376-388 and 474-482
8 Indefinite Integration. Definite Integration. Week eight deals with integration of functions of one variable, chapter six page 483-518. There are two sections on the book, the indefinite integration (page 483-497), and definite integration (page 499-510). The integration concept is entered as the opposite process of derivation. The rules and techniques of integration are applied for power function, natural logarithmic function and exponential function base “e”. Section 6.2 shows how integration can be used to find the area under the graph of a function. This process is called definite integration. We can apply the technique to supply and demand curves and so calculate producer’s and consumer’s surpluses. Definite integration can also be used to determine capital stock and to discount a continuous revenue stream.
9 Basic Matrix Operations. Matrix inversion. Week nine starts with chapter 7 page 523-581which introduces the concept of a matrix, which is convenient mathematical way of representing information displayed in a table. By defining the matrix operations of addition, subtraction and multiplication, it is possible to develop an algebra of matrices. In Section 7.2 page 545-561 you are shown how to calculate the inverse of a matrix. This is analogous to the reciprocal of a number and enables matrix equations to be solved. Inverse matrices provide an alternative way of solving systems of simultaneous linear equations. Section 7.3 page 564-573describes Cramer’s rule, an alternative way for solving systems of linear equations using the determinants and matrices.
10 Cramer`s rule. Week ten is for Section 7.3 page 564-573 and describes Cramer’s rule, an alternative way for solving systems of linear equations using the determinants and matrices. Also, during this week will be done exercises about the algebra of matrices and finding the inverse matrix of a square matrix.
11 Revision Exercises. Quiz 2. During the week eleven will be done revision exercises. Shall explain typical exercises of integration and matrices and applications on economic problems. These typical exercises and applications on economic problems will be used for the Quiz 2.
12 Graphical solution of linear programming problems. Materials for week twelve are from the chapter 8 (page 586-617) of the book. Linear programming is a constraint optimization problem where the constraints are inequalities. There are two sections. Section 8.1 (page 586- 602)describes the basic mathematical techniques and considers special cases when problems have either no solution or infinitely many solutions. Section 8.2 shows how an economic problem, initially given in words, can be expressed as a linear programming problem and how be solved using the graphical method.
13 Differential equations. Week thirteen is for differential equations, chapter 9 page 628-663. It is true that in economy situations are changed in discrete way (static) or continuous way (dynamic). To express the dynamic changes, we use the differential equations. Equations that involve the derivatives of an unknown function are called differential equations, and a method is described on the book for solving such equations (section 9.2 page 643-658). Exercise are simple economic situations which are solved using the simple cases of differential equations. Also, we shall show you how to solve dynamic systems in both macroeconomics and microeconomics.
14 Review. Exercises. Week fourteen is for revision exercises and preparation for the final exam. Students must study exercises at the end of the chapter 6 (page 513-522), chapter 7 (page576-584), chapter 8 (page 617-626), chapter 9 (page 658-663). Also, we shall explain the typical economic problems and their optimization using all methods of optimizations we know.
Prerequisite(s):
Textbook(s): Ian Jacques, Mathematics for Economics and Business, the ninth edition, Pearson 2018
Additional Literature: Knud Sydsaeter, Peter Hammond. Essential Mathematics for Economic Analysis, the fifth edition. Pearson 2016 Vasil Lino Matematika per Ekonomine dhe Biznesin 2 Tirane 2022
Laboratory Work:
Computer Usage:
Others: No
COURSE LEARNING OUTCOMES
1 Estimate the derivative of a function by measuring the slope of a tangent.
2 Derive the relationship between marginal and average revenue
3 Differentiate complicated functions using a combination of rules.
4 Determine the price elasticity for general linear demand functions.
5 Use the first and the second-order derivative to find maximum and minimum points of a function with one variable.
6 Perform implicit differentiation.
7 Use the first and the second-order derivative to find maximum and minimum points of a function with two variables.
8 Use the method of Lagrange multipliers to solve constrained optimisation problems.
9 Use methods of integration to calculate the consumer’s surplus, producer’s surplus, capital stock formation
10 Use matrix inverses to solve systems of linear equations arising in economics.
COURSE CONTRIBUTION TO... PROGRAM COMPETENCIES
(Blank : no contribution, 1: least contribution ... 5: highest contribution)
No Program Competencies Cont.
Bachelor in Economics (3 years) Program
1 Students define the fundamental problems of economics 5
2 Students describe key economic theories 4
3 Students critically discuss current developments in economics 3
4 Students appropriately use software for data analysis 2
5 Students critically contextualize the selection of an economic problem for research within scholarly literature and theory on the topic 5
6 Students apply appropriate analytical methods to address economic problems 5
7 Students use effective communication skills in a variety of academic and professional contexts 4
8 Students effectively contribute to group work 2
9 Students conduct independent research under academic supervision 4
10 Students uphold ethical values in data collection, interpretation, and dissemination 4
11 Students critically engage with interdisciplinary innovations in social sciences 3
12 Student explain how their research has a broader social benefit 1
COURSE EVALUATION METHOD
Method Quantity Percentage
Midterm Exam(s)
1
35
Quiz
2
5
Final Exam
1
45
Attendance
10
Total Percent: 100%
ECTS (ALLOCATED BASED ON STUDENT WORKLOAD)
Activities Quantity Duration(Hours) Total Workload(Hours)
Course Duration (Including the exam week: 16x Total course hours) 16 4 64
Hours for off-the-classroom study (Pre-study, practice) 16 2 32
Mid-terms 1 10 10
Assignments 0
Final examination 1 15 15
Other 4 1 4
Total Work Load:
125
Total Work Load/25(h):
5
ECTS Credit of the Course:
5
CONCLUDING REMARKS BY THE COURSE LECTURER

At the application of this program is important the usage of mathematical rules in the modeling and analyzing of the economic functions. On the focus will be the concept of the mathematical function and its application to economic functions. Important elements of these concepts are right implementation of their properties and features to the economic functions.

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