Maths for Economics Second Edition Geoff Renshaw with contributions from Norman Ireland OXPORD UNIVERSITY PRESS
Brief contents Detailed contents About the author About the book How to use the book Chapter map Guided tour of the textbook features Guided tour of the Online Resource Centre Acknowledgements ix xiv xv xvii xviii xx xxii xxiv Part One Foundations 1 Arithmetic 3 2 Algebra 43 3 Linear equations 63 4 Quadratic equations 109 5 Some further equations and techniques 134 Part Two Optimization with one independent variable 6 Derivatives and differentiation 165 7 Derivatives in action 184 8 Economic applications of functions and derivatives 213 9 Elasticity 256 Part Three Mathematics of finance and growth i 10 Compound growth and present discounted value 297 11 The exponential function and logarithms 328 12 Continuous growth and the natural exponential function 342 13 Derivatives of exponential and logarithmic functions and their applications 368 Part Four Optimization with two or more independent variables 14 Functions of two or more independent variables 389 15 Maximum and minimum values, the total differential, and applications 441 16 Constrained maximum and minimum values 479 17 Returns to scale and homogeneous functions; partial elasticities; growth accounting; logarithmic scales 519
Pa it Five Some further topics CO o en 18 Integration 19 Matrix algebra 20 Difference and differential equations W21 Extensions and future directions (on the Online Resource Centre) 551 577 597 Appendix: Answers to chapter 1 self-test Glossary Index 623 624 632 O
Detailed contents About the author About the book How to use the book Chapter map Guided tour of the textbook features Guided tour of the Online Resource Centre Acknowledgements ijjgfl Part One Foundations 1 Arithmetic 1.1 1.2 Addition and subtraction with positive and negative numbers 1.3 Multiplication and division with positive 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 and negative numbers Brackets and when we need them Factorization Fractions Addition and subtraction of fractions Multiplication and division of fractions Decimal numbers Adding, subtracting, multiplying, and dividing decimal numbers Fractions, proportions, and ratios Percentages Index numbers Powers and roots Standard index form Some additional symbols SELF-TEST EXERCISE Algebra Rules of algebra Addition and subtraction of algebraic expressions Multiplication and division of algebraic expressions Brackets and when we need them Fractions Addition and subtraction of fractions Multiplication and division of fractions Powers and roots Extending the idea of powers Negative and fractional powers The sign of a" Necessary and sufficient conditions APPENDIX: The Greek alphabet xiv XV xvii xviii XX xxii xxiv 3 3 7 10 13 14 16 20 24 26 27 28 33 35 40 41 42 43 43 44 44 45 47 49 50 52 55 56 57 59 60 62 3 Linear equations 63 3.1 63 3.2 How we can manipulate equations 64 3.3 Variables and parameters 69 3.4 Linear and non-linear equations 69 3.5 Linear functions 72 3.6 Graphs of linear functions 73 3.7 The slope and intercept of a linear function 75 3.8 Graphical solution of linear equations 80 3.9 Simultaneous linear equations 81 3.10 Graphical solution of simultaneous linear equations 84 3.11 Existence of a solution to a pair of linear simultaneous equations 87 3.12 Three linear equations with three unknowns 90 3.13 3.14 3.15 3.16 3.17 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 5 5.1 5.2 5.3 5.4 5.5 5.6 Economic applications Demand and supply for a good The inverse demand and supply functions Comparative statics Macroeconomic equilibrium Quadratic equations Quadratic expressions Factorizing quadratic expressions Quadratic equations The formula for solving any quadratic equation Cases where a quadratic expression cannot be factorized The case of the perfect square Quadratic functions The inverse quadratic function Graphical solution of quadratic equations Simultaneous quadratic equations Graphical solution of simultaneous quadratic equations Economic application 1: supply and demand Economic application 2: costs and revenue Some further equations and techniques The cubic function Graphical solution of cubic equations Application of the cubic function in economics The rectangular hyperbola Limits and continuity 91 91 94 97 102 109 109 110 112 114 116 117 118 120 122 123 126 127 128 131 134 134 135 138 141 142 143
o a LU ilu O 5.7 Application of the rectangular hyperbola in economics 146 5.8 The circle and the ellipse 149 5.9 Application of circle and ellipse in economics 151 5.10 Inequalities 152 5.11 Examples of inequality problems 156 5.12 Applications of inequalities in economics 159 Part Two Optimization with one independent variable 6 Derivatives and differentiation 165 6.1 165 6.2 The difference quotient 166 6.3 Calculating the difference quotient 167 6.4 The slope of a curved line 168 6.5 Finding the slope of the tangent 170 6.6 Generalization to any function of x 172 6.7 Rules for evaluating the derivative of a function 173 6.8 Summary of rules of differentiation 182 7 Derivatives in action 184 7.1 184 7.2 Increasing and decreasing functions 185 7.3 Optimization: finding maximum and minimum values 187 7.4 A maximum value of a function 187 7.5 The derivative as a function of x 188 7.6 A minimum value of a function 189 7.7 The second derivative 191 7.8 A rule for maximum and minimum values 191 7.9 Worked examples of maximum and minimum values 192 7.10 Points of inflection 195 7.11 A rule for points of inflection 198 7.12 More about points of inflection 199 7.13 Convex and concave functions 206 7.14 An alternative notation for derivatives 209 7.15 The differential and linear approximation 210 8 Economic applications of functions and derivatives 213 8.1 213 8.2 The firm's total cost function 214 8.3 The firm's average cost function 216 8.4 Marginal cost 218 8.5 The relationship between marginal and average cost 220 8.6 Worked examples of cost functions 222 8.7 Demand, total revenue, and marginal revenue 229 8.8 The market demand function 229 8.9 Total revenue with monopoly 231 8.10 Marginal revenue with monopoly 232 8.11 Demand, total and marginal revenue functions with monopoly 234 8.12 Demand, total and marginal revenue with perfect competition 235 8.13 Worked examples on demand, marginal and total revenue 236 8.14 Profit maximization 239 8.15 Profit maximization with monopoly 240 8.16 Profit maximization using marginal cost and marginal revenue 242 8.17 Profit maximization with perfect competition 244 8.18 Comparing the equilibria under monopoly and perfect competition 246 8.19 Two common fallacies concerning profit maximization 248 8.20 The second order condition for profit maximization 248 APPENDIX 8.1: The relationship between total cost, average cost, and marginal cost 253 APPENDIX 8.2: The relationship between price, total revenue, and marginal revenue 254 9 Elasticity 256 9.1 256 9.2 Absolute, proportionate, and percentage changes 257 9.3 The arc elasticity of supply 259 9.4 Elastic and inelastic supply 260 9.5 Elasticity as a rate of proportionate change 260 9.6 Diagrammatic treatment 261 9.7 Shortcomings of arc elasticity 263 9.8 The point elasticity of supply 263 9.9 Reconciling the arc and point supply elasticities 265 9.10 Worked examples on supply elasticity 265 9.11 The arc elasticity of demand 268 9.12 Elastic and inelastic demand 270 9.13 An alternative definition of demand elasticity 272 9.14 The point elasticity of demand 273 9.15 Reconciling the arc and point demand elasticities 274 9.16 Worked examples on demand elasticity 275 9.17 Marginal revenue and the elasticity of demand 279 9.18 The elasticity of demand under perfect competition 282 9.19 Worked examples on demand elasticity and marginal revenue 284 9.20 Other elasticities in economics 288 9.21 The firm's total cost function 288 9.22 The aggregate consumption function 290 9.23 Generalizing the concept of elasticity 292
PartThree Mathematics of finance and growth 10 Compound growth and present discounted value 297 10.1 297 10.2 Arithmetic and geometric series 298 10.3 An economic application 300 10.4 Simple and compound interest 304 10.5 Applications of the compound growth formula 307 10.6 Discrete versus continuous growth 309 10.7 When interest is added more than once per year 309 10.8 Present discounted value 314 10.9 Present value and economic behaviour 316 10.10 Present value of a series of future receipts 316 10.11 Present value of an infinite series 319 10.12 Market value of a perpetual bond 320 10.13 Calculating loan repayments 322 11 The exponential function and logarithms 328 11.1 328 11.2 The exponential function y= 10* 330 11.3 The function inverse to/= 10 x 331 11.4 Properties of logarithms 333 11.5 Using your calculator to find common logarithms 333 11.6 The graph of/= log 10 x 334 11.7 Rules for manipulating logs 335 11.8 Using logs to solve problems 337 11.9 Some more exponential functions 338 12 Continuous growth and the natural exponential function 342 12.1 342 12.2 Limitations of discrete compound growth 343 12.3 Continuous growth: the simplest case 343 12.4 Continuous growth: the general case 346 12.5 The graph of y = ae re 347 12.6 Natural logarithms 349 12.7 Rules for manipulating natural logs 351 12.8 Natural exponentials and logs on your calculator 351 12.9 Continuous growth applications 353 12.10 Continuous discounting and present value 358 12.11 Graphs with semi-log scale 361 13 Derivatives of exponential and logarithmic functions and their applications 368 13.1 368 13.2 The derivative of the natural exponential function 369 13.3 The derivative of the natural logarithmic function 370 13.4 The rate of proportionate change, or rate of growth 13.5 Discrete growth 13.6 Continuous growth 13.7 Instantaneous and nominal growth rates compared 13.8 Semi-log graphs and the growth rate again 13.9 An important special case 13.10 Logarithmic scales and elasticity Part Four Optimization with two or more independent variables 14 Functions of two or more independent variables 389 14.1 389 14.2 Functions with two independent variables 390 14.3 Examples of functions with two independent variables 393 14.4 Partial derivatives 398 14.5 Evaluation of first order partial derivatives 401 14.6 Second order partial derivatives 403 14.7 Economic applications 1: the production function 411 14.8 The shape of the production function 411 14.9 The Cobb-Douglas production function 420 14.10 Alternatives to the Cobb-Douglas form 425 14.11 Economic applications 2: the utility function 428 14.12 The shape of the utility function 429 14.13 The Cobb-Douglas utility function 434 APPENDIX 14.1: A variant of the partial derivatives of the Cobb-Douglas function 439 15 Maximum and minimum values, the 371 371 374 377 378 379 381 total differential, and applications 441 15.1 441 15.2 Maximum and minimum values 442 15.3 Saddle points 448 15.4 The total differential of z = f(x, y) 452 15.5 Differentiating a function of a function 457 15.6 Marginal revenue as a total derivative 458 15.7 Differentiating an implicit function 460 15.8 Finding the slope of an iso-zsection 463 15.9 A shift from one iso-z section to another 463 15.10 Economic applications 1: the production function 465 15.11 Isoquants of the Cobb-Douglas production function 468 15.12 Economic applications 2: the utility function 470 15.13 The Cobb-Douglas utility function 472 15.14 Economic application 3: macroeconomic equilibrium 473 Lib I > m a o IT 2 CO
o a3a 15.15 The Keynesian multiplier 473 15.16 The IS curve and its slope 474 15.17 Comparative statics: shifts in the IS curve 475 16 Constrained maximum and minimum values 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 The problem, with a graphical solution Solution by implicit differentiation Solution by direct substitution The Lagrange multiplier method Economic applications 1: cost minimization Economic applications 2: profit maximization A worked example Some problems with profit maximization Profit maximization by a monopolist Economic applications 3: utility maximization by the consumer 16.12 -Deriving the consumer's demand functions 17 Returns to scale and homogeneous functions; partial elasticities; growth accounting; logarithmic scales 17.1 17.2 The production function and returns to scale 17.3 Homogeneous functions 17.4 Properties of homogeneous functions 17.5 Partial elasticities 17.6 Partial elasticities of demand 17.7 The proportionate differential of a function 17.8 Growth accounting 17.9 Elasticity and logs 17.10 Partial elasticities and logarithmic scales 17.11 The proportionate differential and logs 17.12 Log linearity with several variables Part Five Some further topics 18 Integration 18.1 479 479 480 482 485 486 490 496 501 502 508 510 512 519 519 520 522 525 531 532 534 537 539 540 542 544 551 551 ^ 18.2 The definite integral 552 18.3 The indefinite integral 554 18.4 Rules for finding the indefinite integral 555 18.5 Finding a definite integral 562 18.6 Economic applications 1: deriving the total cost function from the marginal cost function 18.7 18.8 18.9 18.10 19 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 19.10 19.11 19.12 19.13 19.14 20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 Economic applications 2: deriving total revenue from the marginal revenue function Economic applications 3: consumers' surplus Economic applications 4: producers' surplus Economic applications 5: present value of a continuous stream of income Matrix algebra Definitions and notation Transpose of a matrix Addition/subtraction of two matrices Multiplication of two matrices Vector multiplication Scalar multiplication Matrix algebra as a compact notation The determinant of a square matrix The inverse of a square matrix Using matrix inversion to solve linear simultaneous equations Cramer's rule A macroeconomic application Conclusions Difference and differential equations Difference equations Qualitative analysis The cobweb model of supply and demand Conclusions on the cobweb model Differential equations Qualitative analysis Dynamic stability of a market Conclusions on market stabilitv 565 567 569 570 572 577 577 578 579 579 580 582 583 583 584 587 589 590 592 594 597 597 598 601 605 610 612 615 616 620
W21 Extensions and future directions (on the Online Resource Centre) APPENDIX 21.3: The firm's maximum profit function with two products 21.1 APPENDIX 21.4: Removing the imaginary number 21.2 Functions and analysis 21.3 Comparative statics 21.4 Second order difference equations APPENDIX 21.1: Proof of Taylor's theorem Appendix: Answers to chapter 1 self-test 623 APPENDIX 21.2: Using Taylor's formula to relate Glossary 624 production function forms Index 632