Foremost I wish to thanks Allah SWT for giving me the strength and commitment to complete writing this basic mathematical economic with application. My greeting and reward for the chosen Prophet Mohammed SAW, his family and followers with the opening surah Al Fatihah. Although I do not have the proper qualification and training on this quantitative subject I always like mathematics, especially when explaining the economic phenomena during studying the subject at the graduate level. I have written this text with the objective of letting the young generation to derive benefits out of its practical application in solving the real world problems. I believe that the theoretical economic subjects like microeconomics and macroeconomics can be understood more precisely with the use of mathematics. The teaching of economic theory always begins with the graphs which is hard to grasp and understand. For a better understanding of economic theory I believe that the graph presentation should be followed with the application of simple algebra to enlighten the simple economic models. In its application later the students should be illustrated with the practical statistical estimations. This current text is the result of teaching a one semester course in mathematical economics assigned to me during my undergraduate lectures. I have not totally completed writing this whole subject during the same semester. Its completion came the following alternative semester one year later when I have tried to complete the unfinished work on specific chapters namely the matrix system and a chapter on the application of optimal control theory to make this text relevant for research. I have to admit that the writing of this text is much drawn from examples and influence of the well-known mathematic textbook the Schaum's Outline Series "Introduction to Mathematical Economics" by Edward T. Dowling (Third Edition 2012) which I believe had helped innumerous number of non-mathematic major throughout the globe for its extremely easy explanations on the application of various mathematical methods. I have utilized and relied on the original Schaum's Series mathematic textbook during my Doctor of Philosophy degree in Natural Resource Economics. This academic training commenced at Washington State University's College of Agriculture in 1979 to 1983 under the Fulbright program. The other source of mathematical knowledge utilized for this text is the agricultural production economics which I found useful and applicable in farm management that can be practically useful in solving real world optimization problem. I own much of the knowledge from Professor Douglas Young. The mathematical exposure to optimal control system is primarily acquired through the process of writing and understanding the dynamic system of fisheries economics during exploring reading materials on literature review for thesis writing. For this development I personally would like to record my sincere appreciation to my supervisors Professor Walter R. Butcher and Professor Lee Roy Blakeslee. All of these professors are by now retired from the University. I am leaving this textbook, which is far from complete and perfect, for the coming generation of applied economists and anyone who would like to understand how economists can be a player in a the national game to recommend solutions to problems related to the practical agricultural and resource economics.

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MATHEMATICAL ECONOMICS

WITH APPLICATION

Nik Hashim Nik Mustapha

Program of Economics

School of Social and Economic Development

University Malaysia Terengganu

21030 Kuala Terengganu

CONTENTS

CHAPTER 1

INTRODUCTION TO MATHEMATICAL ECONOMICS 1

1.1 The relevance of Mathematical Economics 1

1.2 Empirical Aspect of Mathematical Economics 2

1.3 Relations and Functions 3

1.4 Mathematical Functions Used in Economics 4

1.5 Regression Estimate of Actual versus Classified Data 4

CHAPTER 2

REVISION OF BASIC MATHEMATICAL METHODS 6

2.1 Exponents 6

2.2 Polynomials 7

2.3 Linear and Quadratic Equations 8

2.4 Solving Simultaneous Equations 11

2.5 Functions 12

2.6 Graphs, Slopes and Intercepts 12

2.7 Identifying Functions and Equations 13

CHAPTER 3

EQUATIONS AND GRAPHS

IN MICROECONOMICS 14

3.1 Demand and Supply 14

3.2 Production Theory 17

3.2.1 Production Function with One Input 17

3.2.2 Isocost Line 18

3.2.3 Isoquant Curve 19

3.3 Production Possibility Curve 21

3.4 Estimate of PPC from Empirical Data 24

CHAPTER 4

UNDERSTANDING EQUATIONS AND GRAPHS

IN MACROECONOMICS 26

4.1 Basic Macroeconomic Model 26

4.2 Equilibrium Output for a Close Economy 27

4.2.1 Autonomous Spending and the Multiplier 27

4.2.2 Autonomous Consumption Spending 28

4.3 Understanding IS and LM curves 31

4.3.1 Commodity Market Equilibrium 31

4.3.2 Financial Market Equilibrium 32

4.4 Equilibrium in Commodity and Financial Market 33

CHAPTER 5

DIFFERENTIATION, EXTREMA AND OPTIMIZATION

IN SINGLE INPUT-OUTPUT RELATIONSHIP 37

5.1 Limits 37

5.2 Rules of Differentiation 38

5.2.1 Derivative of a Constant 38

5.2.2 Derivative of a Linear Function 38

5.2.3 Derivative of Power Function 39

5.2.4 Derivation of Product functions 39

5.2.5 Derivation of the Quotient Function 40

5.2.6 Derivative of Generalized Power Function 40

5.2.7 Other Differentiation Examples 41

5.3 Relative Extrema 42

5.4 Marginal Concepts in Economics 43

5.5 Optimization 49

CHAPTER 6

DIFFERENTIATION, EXTREMA AND OPTIMIZATION OF

SEVERAL OUTPUT-INPUT RELATIONSHIPS 52

6.1 Partial Derivatives of Several Variables 52

6.1.1 Product Rule 52

6.1.2 Quotient Rule 53

6.1.3 Generalized Power Function Rule 53

6.2 Second-Order Partial Derivatives 54

6.3 Constrained Optimization with Lagrange Multipliers 59

6.3.1 Constrained Maximization Problem 60

6.3.2 Constrained Minimization Problem 61

6.4 Graphical Solution using Economic Principle 62

6.5 Constrained Maximization with CES Production Function 65

CHAPTER 7

USE OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS

IN ECONOMICS 67

7.1 Exponential Functions 67

7.2 Logarithmic Functions 68

7.3 Natural Logarithmic Functions 70

7.4 Application of Exponential and Logarithmic Functions 71

7.5 Differentiation of Exponential and Logarithmic Functions 74

7.6 Optimum Forest Rotation 76

CHAPTER 8

INTEGRAL CALCULUS AND ITS USE IN ECONOMICS 78

8.1 Indefinite Integral 78

8.2 Definite Integral 82

8.3 Properties of Definite Integrals 83

8.4 Consumers' Surplus 84

8.5 Producers' Surplus 86

CHAPTER 9

MATRIX ALGEBRA AND ITS APPLICATION IN ECONOMICS 88

9.1 Definitions of a Matrix 88

9.2 Addition of Matrices 88

9.3 Subtraction of Matrices 89

9.3 Matrix Multiplication 90

9.5 Applications of Matrix System 91

9.6 Using System of Linear Equation in Economics 92

CHAPTER 10

SPECIAL TOPICS IN MATRIX FOR ECONOMISTS AND

THE RELATED DISCIPLINES 98

10.1 Jacobian Determinant 98

10.2 Hessian Matrix 99

10.3 Application of Hessian Matrix 101

10.3.1 Use of Hessian on Unconstrained Profit Maximization 101

10.3.2 Use of Hessian on Constrained Profit Maximization 102

10.3.3 Use of Hessian on Constrained Cost Minimization 104

10.4 Input-Output Analysis for a Simple Economy 105

10.4.1 Technical Coefficient Matrix 106

10.4.2 Solution to total output Using Matrix 108

CHAPTER 11

OPTIMAL CONTROL THEORY AND ITS APPLICATION

IN ECONOMICS 112

11.1 Solving Optimal Control Problem 113

11.2 Intertemporal Resource Use and Discount Rate 114

11.3 Intergenerational Distribution, Sustainability and Discount Rate 117

11.4 Intertemporal Fisheries Based on Clark's Model 119

11.5 Intertemporal Fisheries Management with Application 121

11.6 Simulation of Essential Fisheries Policy Variables 122

REFERENCES 127

INDEX

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