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CONTENTS
PREFACE..........iv
GENERAL GUIDE TO SYMBOLS AND SHORTHAND..........vi
CHAPTER 1: BASIC ARITHMETIC CONCEPTS..........1
Introduction..........1
Types of Numbers..........1
Basic Arithmetic Operations..........2
Fractions, Decimals, Percentages..........5
Mixed Calculations..........10
Exponents..........12
Roots..........14
Logarithms..........16
Inequalities..........18
CHAPTER 2: BASIC ALGEBRA AND ECONOMIC FUNCTIONS..........20
Key Terms in Algebra..........20
Variables..........20
Constants and Parameters..........21
Symbols..........21
Coefficients..........22
Subscripts and Superscripts..........22
Common Variable Symbols in Economics..........23
Equations: identity, behavioral equation, and equilibrium condition..........24
Functions and Dependencies in Economics..........26
Examples of Common Functions and Dependencies in Economics: Examples of Functions in Micro-economics
and Examples of Functions in Macro-economics..........28
Different Ways of Representing Functions Mathematically : Algebraic Equations and Types of Functions, Polynomial or Algebraic Functional Forms, and Non-Algebraic Functional Forms..........37
CHAPTER 3: BASIC COORDINATE GEOMETRY: GRAPHS OF FUNCTIONS..........42
Introduction..........42
Representing Functions by Graphs..........42
Two Dimensional Space and the Other Things Being Equal Assumption..........46
Graphs of Different Functional Forms..........46
Average Value (or Function)..........53
Slope of a Line, Relative Changes of Variables, and the Marginal Amount..........53
Applications: Zero Slope (Horizontal Line), Undefined Slope (Vertical Line), Positive Slope (Up-sloping Line), and Negative Slope (Down-sloping Line)..........54
CHAPTER 4: CALCULUS, SLOPES OF NONLINEAR CURVES AND MARGINAL ANALYSIS..........59
Slope of a Nonlinear Curve..........59
Use of Differential Calculus to Find the Slope of a Nonlinear Curve..........60
Limiting Process..........60
Basic Rules of Differential Calculus..........61
The Total Differential and Implicit Functions..........68
Common Applications of Differential Calculus in Economics: Maximization and Minimization..........71
Partial Derivatives..........79
Elasticity..........82
Price Elasticity of Demand..........83
Cross Price Elasticity of Demand..........84
Income Elasticity of Demand..........84
Price Elasticity of Supply..........84
Integral Calculus..........86
CHAPTER 5: FURTHER TOPICS IN ALGEBRA..........90
Set Theory: Definitions; Set Notation; Graphs of Sets; Examples of Common Sets in Economics; Set Operations..........90
Simple Simultaneous Equations and Equilibrium Analysis..........102
Theory of the Quadratic Function..........110
Direct and Inverse Variation and Equilibrium Analysis..........114
CHAPTER 6: OTHER FORMULAS FOR ECONOMICS AND BUSINESS..........128
Summation (or Sigma) Notation..........128
Simple Interest..........133
Compound Interest..........134
Present Value..........138
Net Present Value (NPV) and Internal Rate of Return (IRR)..........142
Arithmetic and Geometric Progressions..........145
Index Numbers..........151
Real and Nominal Economic Variables..........153
Depreciation Methods..........158
CHAPTER 7: BASIC ECONOMIC STATISTICS..........160
Introduction..........160
Measures of Central Tendency..........160
Measures of Variation..........163
Permutations and Combinations..........169
The Binomial Theorem..........172
Basic Probability..........173
Probability Distributions..........176
Normal (or Z) Distribution..........176
t-Distribution and Hypothesis Tests..........177
Confidence Intervals..........180
The Binomial Distribution..........182
Chi-Squared (X^{2} ) Distribution..........185
Correlation and Regression..........187
Correlation..........187
Linear Regression..........189
Example of Regression Output..........191
Partial Regression Coefficients..........192
CHAPTER 8: BASIC MATRIX ALGEBRA..........194
Basic Definitions..........194
Vectors and Scalars..........195
Matrix Operations..........196
Solution of a System of Linear Equations..........205
Cramer's Rule..........207
Other Matrix Properties..........210
APPENDIX A: UNIT ANALYSIS..........213
APPENDIX B: ELEMENTARY MULTIPLIER ANALYSIS AND EQUILIBRIUM NATIONAL INCOME DETERMINATION..........217
APPENDIX C: BASIC QUANTITATIVE CONCEPTS FOR MANAGEMENT DECISIONS..........229
Break Even Analysis..........229
Linear Programming..........232
Network (CPM/PERT) Models..........251
Inventory Models..........257
Game Theory and Decision Analysis..........261
Markov Analysis..........269
Queueing Models..........270
Simulation..........271
APPENDIX D: SOME USEFUL FORMULAS AND FACTS..........276
APPENDIX E: GENERAL REVIEW PROBLEM SET..........278
APPENDIX F: SELECTED TABLES..........287
INDEX..........292
PREFACE
This handbook is designed to correct identified weaknesses in the mathematical and analytical ability of both undergraduate and graduate students of economics and business. The author's goal is to equip students (particularly those with inadequate math preparation) with the basic math concepts and skills needed for modern economic analysis and business decision making in an increasingly quantitative and computer-analytic business environment.
Deficiencies in basic high school math concepts (in arithmetic, algebra, and geometry) hamper students' ability to understand key economic and business concepts like elasticity, graphs, slopes (or marginal values, or relative changes), financial analysis, and the optimization (maximization or minimization) of economic and business objective functions. Standard textbooks in economics, business and college math assume prior knowledge of most of these basic math concepts.
Aptitude tests measure verbal and math or quantitative skills needed for college that students are supposed to have internalized. But many colleges often admit students with weak math backgrounds. Remedial math courses for such students often focus on general math concepts and do not necessarily relate these concepts specifically to pertinent economic and business problems. Further, because some entering graduate students may have worked for long periods of time in fields making little or no use of math, they forget most of the basic concepts.
All the above reasons led to this handbook, which serves as a simplified quick reference source and self-study guide for students to supplement standard textbooks in economics and business which presuppose full grasp of basic math concepts and hence devote little or no time to explaining them. Further, researchers, analysts, businessmen, policymakers, and administrators may also gain from a handbook written in a very simplified and summary format for the average less-mathematically inclined reader.
Some characteristics of this handbook need explanation. First, the general approach is to do away with unnecessary mathematical proofs, formulas, terms and symbols that may confuse the reader. Important symbols and concepts are explained with highly simplified examples to facilitate understanding.
Second, the basic rules and operations of arithmetic, algebra, and geometry are concisely presented, followed by the relevant economic and business concepts and the associated math concepts used. The emphasis is on economic and business relevance of the math concepts.
Third, key formulas and graphs are given with worked examples and a few self-practice questions to help readers understand the concepts.
Finally, notes are provided after each formula or concept is introduced to stress extra important points.
There are eight chapters and six appendices. The first chapter deals with basic arithmetic concepts like the number system, fractions, decimals, percentages, exponents, roots, logarithms and inequalities.
Basic algebra concepts like variables, constants, parameters, symbols, equations, functions and examples of common economic / business functions are presented in chapter 2. Further, chapter 3 deals with graphs of economic / business functions and the slopes of lines (or relative changes of variables, or marginal analysis).
In chapter 4, the basic rules of differential calculus are used to find the slopes (or marginal values) of nonlinear economic and business functions which may have maximum and minimum turning points. The concept of elasticity and the basics of integral calculus are also presented.
Chapter 5 encompasses further topics in algebra like set theory, simultaneous equations, theory of the quadratic function, direct and inverse variation, and their common applications to economics and business.
Miscellaneous topics like math for finance, arithmetic and geometric progressions, index numbers, and real and nominal economic / business variables are highlighted in chapter 6. Basic probability and economic / business statistics are covered in chapter 7, while the fundamentals of matrix algebra are explained in chapter 8.
Unit analysis is briefly explained in appendix A; elementary multiplier analysis and equilibrium national income determination are summarized in appendix B; basic quantitative tools for management decisions are the focus of appendix C; a brief summary of some basic geometrical formulas and facts is given in appendix D; while selected review questions and math tables appear in appendices E and F, respectively.
I must record my grateful thanks for all the help and comments given, in particular to: South Carolina State University (Institutional Grants Program) for providing funds to cover typing and initial printing costs of the first edition; past and present School of Business deans and Department and Agribusiness and Economics chairpersons and departmental faculty for their support, enthusiasm and/or comments; all the students who expressed the need for such a handbook to improve math skills; and Ann's Secretarial Service for typing the manuscripts for the first through the third editions.
Stephen S. Kyereme
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