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Lecture Notes1Mathematical EcnomicsGuoqiang TIANDepartment of EconomicsTexas A&M UniversityCollege Station, Texas 77843([email protected])This version: August 20191The most materials of this lecture notes are drawn from Chiang’s classic textbookFundamental Methods of Mathematical Economics, which are used for my teaching and convenience of my students in class. Please not put them online or pass to any others.

Contents1234The Nature of Mathematical Economics11.1Economics and Mathematical Economics . . . . . . . . . . .11.2Advantages of Mathematical Approach . . . . . . . . . . . .3Economic Models52.1Ingredients of a Mathematical Model . . . . . . . . . . . . . .52.2The Real-Number System . . . . . . . . . . . . . . . . . . . .52.3The Concept of Sets . . . . . . . . . . . . . . . . . . . . . . . .62.4Relations and Functions . . . . . . . . . . . . . . . . . . . . .92.5Types of Function . . . . . . . . . . . . . . . . . . . . . . . . .112.6Functions of Two or More Independent Variables . . . . . . .122.7Levels of Generality . . . . . . . . . . . . . . . . . . . . . . . .13Equilibrium Analysis in Economics153.1The Meaning of Equilibrium . . . . . . . . . . . . . . . . . . .153.2Partial Market Equilibrium - A Linear Model . . . . . . . . .163.3Partial Market Equilibrium - A Nonlinear Model . . . . . . .183.4General Market Equilibrium . . . . . . . . . . . . . . . . . . .193.5Equilibrium in National-Income Analysis . . . . . . . . . . .23Linear Models and Matrix Algebra254.126Matrix and Vectors . . . . . . . . . . . . . . . . . . . . . . . .i

CONTENTSii4.2Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . .294.3Linear Dependance of Vectors . . . . . . . . . . . . . . . . . .324.4Commutative, Associative, and Distributive Laws . . . . . .334.5Identity Matrices and Null Matrices . . . . . . . . . . . . . .344.6Transposes and Inverses . . . . . . . . . . . . . . . . . . . . .355 Linear Models and Matrix Algebra (Continued)395.1Conditions for Nonsingularity of a Matrix . . . . . . . . . . .395.2Test of Nonsingularity by Use of Determinant . . . . . . . . .415.3Basic Properties of Determinants . . . . . . . . . . . . . . . .465.4Finding the Inverse Matrix . . . . . . . . . . . . . . . . . . . .515.5Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . .565.6Application to Market and National-Income Models . . . . .615.7Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . .645.8Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . .675.9Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .716 Comparative Statics and the Concept of Derivative776.1The Nature of Comparative Statics . . . . . . . . . . . . . . .776.2Rate of Change and the Derivative . . . . . . . . . . . . . . .786.3The Derivative and the Slope of a Curve . . . . . . . . . . . .806.4The Concept of Limit . . . . . . . . . . . . . . . . . . . . . . .806.5Inequality and Absolute Values . . . . . . . . . . . . . . . . .846.6Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . .856.7Continuity and Differentiability of a Function . . . . . . . . .867 Rules of Differentiation and Their Use in Comparative Statics7.1Rules of Differentiation for a Function of One Variable . . . .7.2Rules of Differentiation Involving Two or More Functionsof the Same Variable . . . . . . . . . . . . . . . . . . . . . . .919194

CONTENTS7.3Rules of Differentiation Involving Functions of Different Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89iii997.4Integration (The Case of One Variable) . . . . . . . . . . . . . 1037.5Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . . 1057.6Applications to Comparative-Static Analysis . . . . . . . . . 1077.7Note on Jacobian Determinants . . . . . . . . . . . . . . . . . 110Comparative-Static Analysis of General-Functions1138.1Differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1158.2Total Differentials . . . . . . . . . . . . . . . . . . . . . . . . . 1178.3Rule of Differentials . . . . . . . . . . . . . . . . . . . . . . . . 1188.4Total Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 1208.5Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . 1238.6Comparative Statics of General-Function Models . . . . . . . 1298.7Matrix Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 130Optimization: Maxima and Minima of a Function of One Variable1339.1Optimal Values and Extreme Values . . . . . . . . . . . . . . 1349.2Existence of Extremum for Continuous Function . . . . . . . 1359.3First-Derivative Test for Relative Maximum and Minimum . 1369.4Second and Higher Derivatives . . . . . . . . . . . . . . . . . 1399.5Second-Derivative Test . . . . . . . . . . . . . . . . . . . . . . 1419.6Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.7Nth-Derivative Test . . . . . . . . . . . . . . . . . . . . . . . . 14510 Exponential and Logarithmic Functions14710.1 The Nature of Exponential Functions . . . . . . . . . . . . . . 14710.2 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 14810.3 Derivatives of Exponential and Logarithmic Functions . . . 149

CONTENTSiv11 Optimization: Maxima and Minima of a Function of Two or MoreVariables15311.1 The Differential Version of Optimization Condition . . . . . 15311.2 Extreme Values of a Function of Two Variables . . . . . . . . 15411.3 Objective Functions with More than Two Variables . . . . . . 16111.4 Second-Order Conditions in Relation to Concavity and Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16311.5 Economic Applications . . . . . . . . . . . . . . . . . . . . . . 16712 Optimization with Equality Constraints17112.1 Effects of a Constraint . . . . . . . . . . . . . . . . . . . . . . 17212.2 Finding the Stationary Values . . . . . . . . . . . . . . . . . . 17312.3 Second-Order Condition . . . . . . . . . . . . . . . . . . . . . 17812.4 General Setup of the Problem . . . . . . . . . . . . . . . . . . 18012.5 Quasiconcavity and Quasiconvexity . . . . . . . . . . . . . . 18312.6 Utility Maximization and Consumer Demand . . . . . . . . . 18913 Optimization with Inequality Constraints19313.1 Non-Linear Programming . . . . . . . . . . . . . . . . . . . . 19313.2 Kuhn-Tucker Conditions . . . . . . . . . . . . . . . . . . . . . 19713.3 Economic Applications . . . . . . . . . . . . . . . . . . . . . . 20114 Differential Equations20714.1 Existence and Uniqueness Theorem of Solutions for Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . 20914.2 Some Common Ordinary Differential Equations with Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21014.3 Higher Order Linear Equations with Constant Coefficients . 21414.4 System of Ordinary Differential Equations . . . . . . . . . . . 218

CONTENTSv14.5 Simultaneous Differential Equations and Stability of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22314.6 The Stability of Dynamical System . . . . . . . . . . . . . . . 22415 Difference Equations22715.1 First-order Difference Equations . . . . . . . . . . . . . . . . 22915.2 Second-order Difference Equation . . . . . . . . . . . . . . . 23215.3 Difference Equations of Order n . . . . . . . . . . . . . . . . 23315.4 The Stability of nth-Order Difference Equations . . . . . . . 23515.5 Difference Equations with Constant Coefficients . . . . . . . 236

Chapter 1The Nature of MathematicalEconomicsThe purpose of this course is to introduce the most fundamental aspectsof the mathematical methods such as those matrix algebra, mathematicalanalysis, and optimization theory.1.1Economics and Mathematical EconomicsEconomics is a social science that studies how to make decisions in faceof scarce resources. Specifically, it studies individuals’ economic behaviorand phenomena as well as how individuals, such as consumers, households, firms, organizations and government agencies, make trade-off choices that allocate limited resources among competing uses.Mathematical economics is an approach to economic analysis, in whichthe economists make use of mathematical symbols in the statement of theproblem and also draw upon known mathematical theorems to aid in reasoning.Since mathematical economics is merely an approach to economic anal1

2CHAPTER 1. THE NATURE OF MATHEMATICAL ECONOMICSysis, it should not and does not differ from the nonmathematical approachto economic analysis in any fundamental way. The difference betweenthese two approaches is that in the former, the assumptions and conclusions are stated in mathematical symbols rather than words and in theequations rather than sentences so that the interdependent relationshipamong economic variables and resulting conclusions are more rigorousand concise by using mathematical models and mathematical statistics/econometric methods.I propose the "three-dimension and six-nature" methodology in studying and solving realistic social economic problems as well as arising yourmanagemental skills and leadership well.Three dimensions: theoretical logic, practical knowledge, and historical perspective;Six Characteristics/natures: scientific rigorous, preciseness, reality, pertinence, foresight and ideological.Only by accomplishing the above-mentioned "three-dimensional" and"six characteristics", can we put forward logical, factual and historical insights, high perspicacity views and policy recommendations for the economic development of a nation or and the world, as well as theoreticalinnovations and works that can be tested by data, practice and history.Only through the three dimensions, can your assessments and proposalsbe guaranteed to have these six natures and become a good economist.The study of economic social issues cannot simply involve real worldin its experiment, so it not only requires theoretical analysis based on inherent logical inference and, more often than not, vertical and horizontalcomparisons from the larger perspective of history so as to draw experience and lessons, but also needs tools of statistics and econometrics to doempirical quantitative analysis or test, the three of which are all indispens-

1.2. ADVANTAGES OF MATHEMATICAL APPROACH3able. When conducting economic analysis or giving policy suggestions inthe realm of modern economics, the theoretical analysis often combinestheory, history, and statistics, presenting not only theoretical analysis ofinherent logic and comparative analysis from the historical perspectivebut also empirical and quantitative analysis with statistical tools for examination and investigation. Indeed, in the final analysis, all knowledgeis history, all science is logics, and all judgment is statistics.As such, it is not surprising that mathematics and mathematical statistics/econometrics are used as the basic and most important analytical tools in every field of economics. For those who study economics and conductresearch, it is necessary to grasp enough knowledge of mathematics andmathematical statistics. Therefore, it is of great necessity to master sufficient mathematical knowledge if you want to learn economics well, conduct economic research and become a good economist.1.2Advantages of Mathematical ApproachMathematical approach has the following advantages:(1) It makes the language more precise and the statement of assumptions more clear, which can deduce many unnecessarydebates resulting from inaccurate verbal language.(2) It makes the analytical logic more rigorous and clearly states the boundary, applicable scope and conditions for aconclusion to hold. Otherwise, the abuse of a theory mayoccur.(3) Mathematics can help obtain the results that cannot be easily attained through intuition.(4) It helps improve and extend the existing economic theories.

4CHAPTER 1. THE NATURE OF MATHEMATICAL ECONOMICSIt is, however, noteworthy a good master of mathematics cannot guar-antee to be a good economist. It also requires fully understanding the analytical framework and research methodologies of economics, and havinga good intuition and insight of real economic environments and economic issues. The study of economics not only calls for the understanding ofsome terms, concepts and results from the perspective of mathematics (including geometry), but more importantly, even when those are given bymathematical language or geometric figure, we need to get to their economic meaning and the underlying profound economic thoughts and ideals. Thus we should avoid being confused by the mathematical formulasor symbols in the study of economics.All in all, to become a good economist, you need to be of original, creative and academic way of thinking.

Chapter 2Economic Models2.1Ingredients of a Mathematical ModelA economic model is merely a theoretical framework, and there is no inherent reason why it must mathematical. If the model is mathematical,however, it will usually consist of a set of equations designed to describethe structure of the model. By relating a number of variables to one another in certain ways, these equations give mathematical form to the set ofanalytical assumptions adopted. Then, through application of the relevantmathematical operations to these equations, we may seek to derive a setof conclusions which logically follow from those assumptions.2.2The Real-Number SystemWhole numbers such as 1, 2, · · · are called positive numbers; these are thenumbers most frequently used in counting. Their negative counterparts 1, 2, 3, · · · are called negative integers. The number 0 (zero), on theother hand, is neither positive nor negative, and it is in that sense unique.Let us lump all the positive and negative integers and the number zero5

CHAPTER 2. ECONOMIC MODELS6into a single category, referring to them collectively as the set of all integers.Integers of course, do not exhaust all the possible numbers, for we havefractions, such as 23 , 54 ,