Mathematical Ecnomics

1 Lecture Notes1 Mathematical EcnomicsGuoqiang TIAND epartment of EconomicsTexas A&M UniversityCollege Station, Texas version: August 20181 The lecture notes are only for the purpose of my teaching and convenience of my students in class,please not put them online or pass to any The Nature of Mathematical Economics and Mathematical Economics .. Advantages of Mathematical Approach .. 22 Economic Ingredients of a Mathematical Model .. The Real-Number System .. The Concept of Sets.

2 Relations and Functions .. Types of Function .. Functions of Two or More Independent Variables .. Levels of Generality .. 103 Equilibrium Analysis in The Meaning of Equilibrium .. Partial Market Equilibrium - A Linear Model .. Partial Market Equilibrium - A Nonlinear Model .. General Market Equilibrium .. Equilibrium in National-Income Analysis .. 194 Linear Models and Matrix Matrix and Vectors .. Matrix Operations .. Linear Dependance of Vectors.

3 Commutative, Associative, and Distributive Laws .. Identity Matrices and Null Matrices .. Transposes and Inverses .. 295 Linear Models and Matrix algebra (Continued) Conditions for Nonsingularity of a Matrix .. Test of Nonsingularity by Use of Determinant .. Basic Properties of Determinants .. Finding the Inverse Matrix .. Cramer s Rule .. Application to Market and National-Income Models .. Quadratic Forms .. Eigenvalues and Eigenvectors .. Vector Spaces.

4 566 Comparative Statics and the Concept of The Nature of Comparative Statics .. Rate of Change and the Derivative .. The Derivative and the Slope of a Curve .. The Concept of Limit .. Inequality and Absolute Values .. Limit Theorems .. Continuity and Differentiability of a Function .. 707 Rules of Differentiation and Their Use in Comparative Rules of Differentiation for a Function of One Variable .. Rules of Differentiation Involving Two or More Functions of the Same Vari-able.

5 Rules of Differentiation Involving Functions of Different Variables .. Integration (The Case of One Variable) .. Partial Differentiation .. Applications to Comparative-Static Analysis .. Note on Jacobian Determinants .. 89ii8 Comparative-static Analysis of Differentials .. Total Differentials .. Rule of Differentials .. Total Derivatives .. Implicit Function Theorem .. Comparative Statics of General-Function Models .. Matrix Derivatives .. 1059 Derivatives of exponential and Logarithmic The Nature of exponential Functions.

6 Logarithmic Functions .. Derivatives of exponential and Logarithmic Functions .. 10910 Optimization: Maxima and Minima of a Function of One Variable Optimal Values and Extreme Values .. General Result on Maximum and Minimum .. First-Derivative Test for Relative Maximum and Minimum .. Second and Higher Derivatives .. Second-Derivative Test .. Taylor Series .. Nth-Derivative Test .. 12011 Optimization: Maxima and Minima of a Function of Two or More The Differential Version of Optimization Condition.

7 Extreme Values of a Function of Two Variables .. Objective Functions with More than Two Variables .. Second-Order Conditions in Relation to Concavity and Convexity .. Economic Applications .. 13312 Optimization with Equality Effects of a Constraint .. Finding the Stationary Values .. Second-Order Condition .. General Setup of the Problem .. Quasiconcavity and Quasiconvexity .. Utility Maximization and Consumer Demand .. 14913 Optimization with Inequality Non-Linear Programming.

8 Kuhn-Tucker Conditions .. Economic Applications .. 16014 Linear The Setup of the Problem .. The Simplex Method .. Duality .. 17415 Continuous Dynamics: Differential Differential Equations of the First Order .. Linear Differential Equations of a Higher Order with Constant Coefficients Systems of the First Order Linear Differential Equations .. Economic Application: General Equilibrium .. Simultaneous Differential Equations. Types of Equilibria .. 19416 Discrete Dynamics: Difference First-order Linear Difference Equations.

9 Second-Order Linear Difference Equations .. The General Case of Ordern.. Economic Application:A dynamic model of economic growth .. 20317 Introduction to Dynamic The First-Order Conditions .. Present-Value and Current-Value Hamiltonians .. Dynamic Problems with Inequality Constraints .. Economics Application:The Ramsey Model .. 208ivChapter 1 The Nature of MathematicalEconomicsThe purpose of this course is to introduce the most fundamental aspects of the mathe-matical methods such as those matrix algebra , Mathematical analysis, and Economics and Mathematical EconomicsEconomicsis a social science that studies how to make decisions in face of scarce , it studies individuals economic behavior and phenomena as well as howindividual agents, such as consumers, households, firms, organizations and governmentagencies.

10 Make trade-off choices that allocate limited resources among competing economicsis an approach to economic analysis, in which the e-conomists make use of Mathematical symbols in the statement of the problem and alsodraw upon known Mathematical theorems to aid in Mathematical economics is merely an approach to economic analysis, it shouldnot and does not differ from the nonmathematical approach to economic analysis in anyfundamental way. The difference between these two approaches is that in the former,the assumptions and conclusions are stated in Mathematical symbols rather than wordsand in the equations rather than sentences so that the interdependent relationship amongeconomic variables and resulting conclusions are more rigorous and concise by using math-1ematical models and Mathematical statistics/econometric study of economic social issues cannot simply involve real world in its experiment.