ELEMENTARY MATHEMATICS FOR ECONOMICS

Transcription

1 ELEMENTARY MATHEMATICS FOR ECONOMICS Catering the need of Second year B.A./B.Sc. Students of Economics (Major) Third Semester of Guwahati and other Indian Universities. 2nd Semester R.C. Joshi M.A., M.Phil. Formerly Head, P.G. Dept. of Mathematics Doaba College, JALANDHAR Nancy B. Tech. Future for WINNERS VISHAL PUBLISHING CO. JALANDHAR DELHI

2 CONTENTS 1. LINEAR EQUATIONS Special Products 1.2. Definition of an Equation 1.3. Identity and Equations 1.4. Linear Equations 2. Economic application of linear equations in one variable 3. Economic Applications 2. SYSTEM OF EQUATIONS Simulataneous Linear Equations 1.2. Methods of Solving Simultaneous Linear Equations 2. Business application of Linear Equations in Two Variables 3. Market Equilibrium when demand and supply of two commodities are given 4. Economic Applications of Linear Equations 4.1. Effect of Taxes and subsides in Equilibrium Price and Quantity 3. QUADRATIC EQUATIONS Defination 1.1. To solve the Standard Quadratic Equation 2. Equation reducible to Quadratic Equation 3. Equation of the form ax + b/x = c, where x is an expression containing the variable, may be solved by putting x = y. 4. Irrational Equations. An equation in which the unknown quantity occurs under a radical is called an irrational equation 5. Equation which can be put in the form ax 2 + bx + p ax 2 bx c + k=0 be solved by putting ax 2 bx c y. Exercise 5 6. Equation of type 2 2 ax bx k ax bx k = p, can be solved by putting A = ax 2 bx k, B = ax 2 bx k. Exercise 6 7. Reciprocal Equations Exercise 7 8. Simultaneous Quadratic Equations Exercise 8 9. Application in Economics Exercise 9 4. FUNCTION, LIMIT AND CONTINUITY OF FUNCTIONS Definition of Function 1.2. Image and pre-image 1.3. Domain 1.4. Real Valued Function 1.5. Types of Functions 1.6. Linear Homogeneous Function 1.7. Functions in Economics 1. Demand Function 2. Supply Function 3. Total Cost Function 4. Revenue Function 5. Profit Function 6. Consumption Function 7. Production Function 1.8. Value of a at a point 2. Limit 2.1. Left Limit 2.2. Definition : Left Hand Limit 2.3. Theorem on Limits 2.4. Methods of Finding Limit of a Function Type 1. Method of Factors Type 2. Method of Substitution

3 Type 3. Use of Binomial Theorem for any index. Type 4. Rationlazing Method Type 5. Evaluation of limit when x 3. Some Important Limit 4. Continuity 4.1. Continuity Definitions Type of Discontinuity of a Function Illustrative Examples 5. SETS Definition 1.2. Representation of Sets 1.3. Some Standard Sets 2. Types of Sets 2.1. Empty Set 2.2. Finite and Infinite Sets 2.3. Equal Sets 3. Subset 3.1. Proper Subset 3.2. Singleton Set or Unit Set 3.3. Power Set 3.4. Comparable Sets 3.5. Universal Sets 4. Venn Diagrams 4.1. Operations on Sets 4.2. Union of Sets Illustrative Examples 4.3. Definition 4.4. Some Properties of the Operation of Union 4.5. Intersection of Sets 4.6. Definition 4.7. Disjoint Sets 4.8. Some Properties of Operation of Intersection 4.9. Difference of Sets Symmetric Difference of two Sets Complement of a Set Complement Laws 5. Number of Elements in a Set 6. Economic Application of Sets Question VSA and MCQ 6. MATRICES Matrix 1.2. Types of Matrices 2. Sum of Matrices 2.1. Properties of Addition of Matrices 2.2. Scalar Multiple of a Matrix 2.3. Properties of Scalar Multiplication 3. Product of Two Matrices 3.1. Zero Matrix as the Product of Two non Zero Matrices 3.2. Theorem 3.3. Distributive Law 3.4. Associative Law of Matrix Multiplication 3.5. Positive Integral Powers of a Square Matrix A 3.6. Matrix Polynomial 4. Transpose of Matrix 4.1. Properties of Transpose of Matrices 4.2. Special Types of Matrices 7. DETERMINANTS Determinants 1.2. Determinant of a Matrix of order Singular Matrix 1.4. Minor 1.5. Cofactors 2. Properties of Determinants 2.1. To Evaluate Determinant of Square Matrices 2.2. Type I 2.3. Type II 2.4. Type III 2.5. Type IV 2.6. Type V Exercise Solution of a System of Linear Equations 3.1. Homegeneous System of Linear Equations Exercise - 3

4 8. ADJOINT AND INVERSE OF A MATRIX Theorem 1.2. (a) The Inverse of a Matrix (b) Singular and Non-Singular Matrix 1.3. The Necessary and Sufficent Condition for a Square Matrix to Possess its Inverse is That A Elementary Transformation 2.1. Symbols for Elementary Transformation 2.2. Equivalent Matrices 2.3. Elementary Matrices 2.4. Theorem 2.5. Inverse of a Matrix by Elementary Transformation 2.6. Method to Compute the Inverse Illustration Examples 3. Rank of a Matrix 3.1. Steps to Determine the Rank of a Matrix 9. SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS To solve simultaneous linear equations with the help of inverse of a matrix 1.2. Criterion of Consistency 1.3. Type-I 1.4. Type-II 1.5. Type-III 10. NATIONAL INCOME MODEL National Income Model 1.1. Solving National Income Model Using Inverse Method or Matrix Method 1.2. Partial Equilibrium Market Model 1.3. Application of partial equilibrium market model Exercise 11. STRUCTURE OF INPUT OUTPUT TABLE Characteristics of Input-Output Analysis 1.2. Assumptions of Input-Output Analysis 1.3. Types of Input-output Models 1.4. Main Concept of Input-output Model 1.5. Input-output Analysis Techniques 1.6. Technological Coefficient Matrix 1.7. Steps to determine Gross Level of Output and Labour Requirements 1.8. The Hawkins-Simon Conditions or Viability Conditions of the Input-output Model Three Sector Economy Exercise 12. DERIVATIVE Definition 1.2. Another Definition 1.3. Differentiation by delta method 2. Derivation of some standard s 3. Differentiation of product of two functons 4. Differentiation of quotient of two s 5. Differentiation of a of a : The chain rule 6. If y = u n, where u is of x, then = nun 1 du dx. dx Exercise 6 7. (a) If y = log a u, where u = f(x), then dx = 1 u log a e du dx. Exercise 7 8. (a) If y = a u, where u is of x, then dx = au log a du, where a is constant. dx Exercise 8 9. Differentiation of Implicit Function Exercise Differentiation of Parametric Functions Differentiation of a w. r. to another 1

5 12. Logarithmic Differentiation Higher Derivatives PARTIAL AND TOTAL DIFFERENTIATION Function of Two Variables 1.1. Partial Derivative 2. Higher Order Partial Derivatives 2.1. Change of order of differentiation 3. Homogeneous Functions 3.1. Linear Homogeneous 4. Properties of homogeneous s 4.1. First property of homogeneous 4.2. Second property of homogeneous 4.3. Euler s Theorem (Property III) 5. Total Differential 5.1. Method to Determine Total Differential 5.2. Total Derivative 14. INTEGRATION (WITH ECONOMIC APPLICATIONS) Constant of Integration 1.2. Basic Rules of Integration 2. Integration by substitution 3. To evaluate integrals of type dx. ax b 4. Definite Integral 5. To integrate an expression which involves linear, Method is, put linear = y. 6. To show that n 1 n [ ] [ ] f ( x) dx c, n 1. n 1 Exercise 6 7. To show that Exercise 7 8. To show that f ( x) dx = log f (x) +c. a a f ( x) dx c, log a where a is constant. Exercise 8 9. Method of Partial Fraction Exercise Case II. Partial Fraction Case III. When the denominator contains linear repeated factors Type IV. When denominator contains a quadratic factor of the type x 2 + a Integration by parts TYPE I. When integral of one of the is not known Type II. The single whose integral is not known can also be integrated by integration by parts Type III. When integral of both the s is known, then we take polynomial in x as first Type IV. n x x [ f ( x) ] e dx e Application of Integration in Economics : Marginal cost. total cost. 7 Questions