INSEAD Library

Includes index

Digitized

Elements of Mathematics for Economics and Finance Contents 1. Essential Skills.............................................................................................. 1.1 Introduction ...................................................................................... 1.2 Numbers ............................................................................................ 1.2.1 Addition and Subtraction ....................................................... 1.2.2 Multiplication and Division ..................................................... 1 1 2 3 3 1.2.3 Evaluation of Arithmetical Expressions................................... 4 1.3 Fractions ........................................................................................... 5 1.3.1 Multiplication and Division ..................................................... 7 1.4 Decimal Representation of Numbers .................................................. 8 1.4.1 Standard Form..................................................................... 10 1.5 Percentages....................................................................................... 10 1.6 Powers and Indices ........................................................................... 12 1.7 Simplifying Algebraic Expressions .................................................... 16 1.7.1 Multiplying Brackets............................................................... 16 1.7.2 Factorization .......................................................................... 18 2. Linear Equations ............................................................................................ 23 2.1 Introduction .................................................................................... 23 2.2 Solution of Linear Equations ............................................................ 24 2.3 Solution of Simultaneous Linear Equations..................................... 2.4 Graphs of Linear Equations.............................................................. 2.4.1 Slope of a Straight Line .......................................................... 2.5 Budget Lines ..................................................................................... 2.6 Supply and Demand Analysis .......................................................... 2.6.1 Multicommodity Markets........................................................ 27 30 34 37 40 44 3. Quadratic Equations........................................................................................ 49 3.1 Introduction ..................................................................................... 49 3.2 Graphs of Quadratic Functions........................................................ 50 3.3 Quadratic Equations......................................................................... 56 3.4 Applications to Economics.................................................................. 61 4. Functions of a Single Variable ..................................................................... 69 4.1 Introduction ..................................................................................... 69 4.2 Limits .................................................................................................. 72 4.3 Polynomial Functions ........................................................................ 72 4.4 Reciprocal Functions......................................................................... 75 4.5 Inverse Functions.............................................................................. 81 5. The Exponential and Logarithmic Functions ............................................. 87 5.1 Introduction ..................................................................................... 87 5.2 Exponential Functions ..................................................................... 88 5.3 Logarithmic Functions ...................................................................... 90 5.4 Returns to Scale of Production Functions......................................... 95 5.4.1 Cobb-Douglas Production Functions ..................................... 97 5.5 Compounding of Interest ................................................................... 98 5.6 Applications of the Exponential Function in Economic Modelling ............................................................................ 102 6. Differentiation .............................................................................................. 6.1 Introduction ..................................................................................... 6.2 Rules of Differentiation........................................................................ 6.2.1 Constant Functions .............................................................. 6.2.2 Linear Functions ................................................................... 6.2.3 Power Functions .................................................................... 6.2.4 Sums and Differences of Functions........................................ 6.2.5 Product of Functions ............................................................. 6.2.6 Quotient of Functions............................................................ 109 109 113 113 114 114 114 116 117 6.2.7 The Chain Rule....................................................................... 117 6.3 Exponential and Logarithmic Functions .......................................... 119 6.4 Marginal Functions in Economics..................................................... 6.4.1 Marginal Revenue and Marginal Cost .................................... 6.4.2 Marginal Propensities ............................................................. 6.5 Approximation to Marginal Functions............................................... 6.6 Higher Order Derivatives .................................................................... 6.7 Production Functions ....................................................................... 121 121 123 125 127 129 7. Maxima and Minima ....................................................................................... 137 7.1 Introduction ................................................................................... 137 7.2 Local Properties of Functions ........................................................... 138 7.2.1 Increasing and Decreasing Functions .................................. 7.2.2 Concave and Convex Functions ............................................ 7.3 Local or Relative Extrema.................................................................. 7.4 Global or Absolute Extrema .............................................................. 7.5 Points of Inflection ............................................................................. 7.6 Optimization of Production Functions.............................................. 138 138 139 144 145 146 7.7 Optimization of Profit Functions ...................................................... 151 7.8 Other Examples................................................................................ 154 8. Partial Differentiation .................................................................................. 8.1 Introduction ................................................................................... 8.2 Functions of Two or More Variables ................................................. 8.3 Partial Derivatives.............................................................................. 8.4 Higher Order Partial Derivatives ...................................................... 159 159 160 160 163 8.5 Partial Rate of Change...................................................................... 165 8.6 The Chain Rule and Total Derivatives .............................................. 168 8.7 Some Applications of Partial Derivatives .......................................... 171 8.7.1 Implicit Differentiation ........................................................... 8.7.2 Elasticity of Demand ............................................................. 8.7.3 Utility ...................................................................................... 8.7.4 Production ............................................................................ 171 173 176 179 8.7.5 Graphical Representations.................................................... 181 9. Optimization ................................................................................................. 185 9.1 Introduction ................................................................................... 185 9.2 Unconstrained Optimization ........................................................... 186 9.3 Constrained Optimization ............................................................... 193 9.3.1 Substitution Method............................................................. 193 9.3.2 Lagrange Multipliers............................................................... 197 9.3.3 The Lagrange Multiplier A: An Interpretation........................ 201 9.4 Iso Curves..........................................................................................204 10. Matrices and Determinants....................................................................... 209 10.1 Introduction ................................................................................. 209 10.2 Matrix Operations........................................................................... 209 10.2.1 Scalar Multiplication.......................................................... 211 10.2.2 Matrix Addition.................................................................... 212 10.2.3 Matrix Multiplication ......................................................... 212 10.3 Solutions of Linear Systems of Equations ..................................... 220 10.4 Cramer's Rule ................................................................................ 222 10.5 More Determinants ........................................................................ 223 10.6 Special Cases ................................................................................. 230 11. Integration .................................................................................................. 233 11.1 Introduction ................................................................................. 233 11.2 Rules of Integration......................................................................... 236 11.3 Definite Integrals ............................................................................. 11.4 Definite Integration: Area and Summation..................................... 11.5 Producer's Surplus ...................................................................... 11.6 Consumer's Surplus..................................................................... 241 243 250 251 12. Linear Difference Equations ..................................................................... 261 12.1 Introduction ................................................................................. 261 12.2 Difference Equations ....................................................................... 261 12.3 First Order Linear Difference Equations ....................................... 12.4 Stability.......................................................................................... 12.5 The Cobweb Model............................................................................. 12.6 Second Order Linear Difference Equations .................................... 264 267 270 273 12.6.1 Complementary Solutions ................................................. 274 12.6.2 Particular Solutions ......................................................... 277 12.6.3 Stability ............................................................................. 282 13. Differential Equations ............................................................................... 287 13.1 Introduction ................................................................................. 287 13.2 First Order Linear Differential Equations ..................................... 288 13.2.1 Stability ............................................................................. 13.3 Nonlinear First Order Differential Equations ................................ 13.3.1 Separation of Variables...................................................... 13.4 Second Order Linear Differential Equations ................................. 13.4.1 The Homogeneous Case .................................................... 13.4.2 The General Case ............................................................... 13.4.3 Stability ............................................................................. 292 292 294 296 297 300 302 A. Differentials .................................................................................................. 305 Index..................................................................................................................................... 309