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)	NA
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- 17:30
Second Course Lecturer(s) (name, surname, academic title/scientific degree, email address and signature) and Office Hours:	NA
Teaching Assistant(s) and Office Hours:	NA
Language:	English
Compulsory/Elective:	Compulsory
Study program: (the study for which this course is offered)	Bachelor in International Marketing and Logistics Management (3 years)
Classroom and Meeting Time:	E 406
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 the differentiation of functions of one variable and multivariable.
2	Marginal economic functions
3	Optimization of the economic functions using the derivatives and linear programing
4	Differentiation and integration, rules and methods. Their usage for solution of the economic problems.
5	Matrices, operations with matrices. Crammer's Rules
6	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
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 International Marketing and Logistics Management (3 years) Program
1	Identify activities, tasks, and skills in management, marketing, accounting, finance, and economics.	5
2	Apply key theories to practical problems within the global business context.	4
3	Demonstrate ethical, social, and legal responsibilities in organizations.	5
4	Develop an open minded-attitude through continuous learning and team-work.	4
5	Use technology to enable business growth and sustainability.	4
6	Synthesize creativity needed for marketing notion with scientific method and numerical skills, for achieving business sustainability.	4
7	Apply the concepts and structures of modern marketing in global context at private and public sectors.	5
8	Integrate the management of logistics, supply chain and in total operations with corporate goals and strategies.	4

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

This program will use mathematical methods for optimization of the economic functions. Primary will be modeling of the economic function as the mathematical function of one or many variables. As, the economic functions have a good approach with mathematical functions, the solution would find, using the mathematical methods and rules.