Elementary Mathematics for Economics

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

Nancy

M.A., M.Phil. Formerly Head, P.G. Dept. of Mathematics Doaba College, JALANDHAR

B. Tech.

VISHAL PUBLISHING CO. Future for WINNERS

JALANDHAR — DELHI

CONTENTS 1. LINEAR EQUATIONS 1–10 1. Introduction 1.1. Special Products 1.2. Definition of an Equation 1.3. Identity and Equations 1.4. Linear Equations Exercise–1 2. Economic application of linear equations in one variable Exercise–2 3. Economic Applications Exercise–3 Questions VSA and MCQ 2. SYSTEM OF EQUATIONS 11–24 1. Introduction 1.1. Simulataneous Linear Equations 1.2. Methods of Solving Simultaneous Linear Equations Exercise–1 2. Business application of Linear Equations in Two Variables Exercise–2 3. Market Equilibrium when demand and supply of two commodities are given Exercise–3 4. Economic Applications of Linear Equations 4.1. Effect of Taxes and subsides in Equilibrium Price and Quantity Exercise–4 Questions VSA and MCQ 3. QUADRATIC EQUATIONS 25–42 1. Defination 1.1. To solve the Standard Quadratic Equation Exercise–1 2. Equation reducible to Quadratic Equation Exercise–2 3. Equation of the form ax + b/x = c, where x is an expression containing the variable, may be solved by putting x = y. Exercise–3 4. Irrational Equations. An equation in which the unknown quantity occurs under a radical is called an irrational equation Exercise–4

5. Equation which can be put in the form ax2 + bx + p ax 2  bx  c + k=0 be solved by putting

ax 2  bx  c  y .

Exercise–5 6. Equation of type ax 2  bx  k  ax 2  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 Questions VSA and MCQ 4. FUNCTION, LIMIT AND CONTINUITY OF FUNCTIONS 43–76 1. Introduction 1.1. 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 function at a point Exercise – 1 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

Type 3. Use of Binomial Theorem for any index. Type 4. Rationlazing Method Type 5. Evaluation of limit when x  ¥ Exercise – 2 3. Some Important Limit Exercise–3 4. Continuity 4.1. Continuity Definitions 2 4.2. Type of Discontinuity of a Function Illustrative Examples Exercise–4 Questions VSA and MCQ 5. SETS 77–105 1. Introduction 1.1. Definition 1.2. Representation of Sets 1.3. Some Standard Sets Exercise–1 2. Types of Sets 2.1. Empty Set 2.2. Finite and Infinite Sets 2.3. Equal Sets Exercise – 2 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 Exercise–3 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 4.10. Symmetric Difference of two Sets 4.11. Complement of a Set 4.12. Complement Laws Exercise – 4 5. Number of Elements in a Set 6. Economic Application of Sets

Exercise – 5 Question VSA and MCQ 6. MATRICES 106–135 1. Introduction 1.1. Matrix 1.2. Types of Matrices Exercise – 1 2. Sum of Matrices 2.1. Properties of Addition of Matrices 2.2. Scalar Multiple of a Matrix 2.3. Properties of Scalar Multiplication Exercise – 2 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 Exercise – 3 4. Transpose of Matrix 4.1. Properties of Transpose of Matrices 4.2. Special Types of Matrices Exercise – 4 Questions VSA and MCQ 7. DETERMINANTS 136–174 1. Determinants 1.2. Determinant of a Matrix of order 3 × 3 1.3. Singular Matrix 1.4. Minor 1.5. Cofactors Exercise – 1 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 - 2 3. Solution of a System of Linear Equations 3.1. Homegeneous System of Linear Equations Exercise - 3 Questions VSA and MCQ

8. ADJOINT AND INVERSE OF A MATRIX 175–195 1.1. 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 |  0. Exercise – 1 2. 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 Exercise – 2 3. Rank of a Matrix 3.1. Steps to Determine the Rank of a Matrix Exercise – 3 Questions VSA and MCQ 9. SO LUTIO NS O F SIM ULTANEO US LINEAR EQUATIONS 196–210 1. Introduction 1.1. 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 Exercise – 1

Questions VSA and MCQ 10. NATIONAL INCOME MODEL 211–220

1. 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 Questions VSA and MCQ 11. STRUCTURE OF INPUT OUTPUT TABLE 221–237 1. Introduction 1.1. Characteristics of Input-Output Analysis

1.2. 1.3. 1.4. 1.5. 1.6. 1.7.

Assumptions of Input-Output Analysis Types of Input-output Models Main Concept of Input-output Model Input-output Analysis Techniques Technological Coefficient Matrix 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 Questions VSA and MCQ 12. DERIVATIVE 238–289 1. Introduction 1.1. Definition 1.2. Another Definition 1.3. Differentiation by delta method Exercise – 1 2. Derivation of some standard functions Exercise – 2 3. Differentiation of product of two functons Exercise – 3 4. Differentiation of quotient of two functions Exercise – 4 5. Differentiation of a function of a function : The chain rule Exercise – 5 6. If y = un, where u is function of x, then dy du = nun–1 . dx dx Exercise – 6 7. (a) If y = loga u, where u = f(x), then dy 1 du = loga e . dx u dx Exercise – 7 8. (a) If y = au, where u is function of x, then dy du = au log a , where a is constant. dx dx Exercise – 8 9. Differentiation of Implicit Function Exercise – 9 10. Differentiation of Parametric Functions Exercise – 10 11. Differentiation of a function w. r. to another function Exercise – 11

6. To show that

12. Logarithmic Differentiation Exercise – 12 13. Higher Derivatives Exercise – 13 Questions VSA and MCQ

n  [ f ( x)] f ( x ) dx 

Exercise – 6

13. PARTIAL AND TOTAL DIFFERENTIATION 290–311 1. Function of Two Variables 1.1. Partial Derivative Exercise – 1 2. Higher Order Partial Derivatives 2.1. Change of order of differentiation Exercise – 2 3. Homogeneous Functions 3.1. Linear Homogeneous function Exercise – 3 4. Properties of homogeneous functions 4.1. First property of homogeneous function 4.2. Second property of homogeneous function 4.3. Euler’s Theorem (Property III) Exercise – 4 5. Total Differential 5.1. Method to Determine Total Differential 5.2. Total Derivative Questions VSA and MCQ 14. INTEGRATION (WITH ECONOMIC APPLICATIONS) 312–360 1. Introduction 1.1. Constant of Integration 1.2. Basic Rules of Integration Exercise – 1 2. Integration by substitution Exercise – 2 3. To evaluate integrals of type

[ f ( x )]n  1  c, n   1. n 1

f ( x)

 ax  b dx .

Exercise – 3 4. Definite Integral Exercise – 4 5. To integrate an expression which involves linear , Method is, put linear = y. Exercise – 5

7. To show that



f ( x ) dx = log | f (x) | +c. f ( x)

Exercise – 7 8. To show that

a

f (x )

f ( x ) dx 

a f ( x) c, log a

where a is constant. Exercise – 8 9. Method of Partial Fraction Exercise – 9 10. Case II. Partial Fraction Exercise – 10 11. Case III. When the denominator contains linear repeated factors Exercise – 11 12. Type IV. When denominator contains a quadratic factor of the type x2 + a. Exercise – 12 13. Integration by parts 13.1. TYPE I. When integral of one of the function is not known. Exercise – 13 14. Type II. The single function whose integral is not known can also be integrated by integration by parts. Exercise – 14 15. Type III. When integral of both the functions is known, then we take polynomial in x as first function. Exercise – 15

16. Type IV.

 [ f ( x )  f ( x )

n

]e x dx  f ( x )e x

Exercise – 16 17. Application of Integration in Economics : Marginal cost. total cost. Exercise – 17 Questions