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Brief contents Detailed contents ix About the author xiii About the book xv How to use the book xvii Chapter map xviii Guided tour of the textbook features xx Guided tour of the Online Resource Centre xxii Acknowledgements xxiii 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 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; logarithmic scales; growth accounting 519 Part Five Some further topics 18 Integration 551 19 Matrix algebra 577 20 Difference and differential equations 597 W21 Extensions and future directions (on the Online Resource Centre) 000 Appendix: Answers to chapter 1 self-test 623 Glossary 624 Index 632 Detailed contents About the author xiii About the book xv How to use the book xvii Chapter map xviii Guided tour of the textbook features xx Guided tour of the Online Resource Centre xxii Acknowledgements xxiii Part 1 Foundations 1 Arithmetic 3 1.1 Introduction 3 1.2 Addition and subtraction with positive and negative numbers 4 1.3 Multiplication and division with positive and negative numbers 7 1.4 Brackets and when we need them 10 1.5 Factorization 13 1.6 Fractions 14 1.7 Addition and subtraction of fractions 16 1.8 Multiplication and division of fractions 20 1.9 Decimal numbers 24 1.10 Adding, subtracting, multiplying, and dividing decimal numbers 26 1.11 Fractions, proportions, and ratios 27 1.12 Percentages 28 1.13 Index numbers 33 1.14 Powers and roots 35 1.15 Standard index form 40 1.16 Some additional symbols 41 Self-test exercises 42 2 Algebra 43 2.1 Introduction 43 2.2 Rules of algebra 44 2.3 Addition and subtraction of algebraic expressions 44 2.4 Multiplication and division of algebraic expressions 45 2.5 Brackets and when we need them 47 2.6 Fractions 49 2.7 Addition and subtraction of fractions 50 2.8 Multiplication and division of fractions 52 2.9 Powers and roots 55 2.10 Extending the idea of powers 56 2.11 Negative and fractional powers 57 2.12 The sign of an 59 2.13 Necessary and sufficient conditions 60 Appendix: The Greek alphabet 62 3 Linear equations 63 3.1 Introduction 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 Economic applications 91 3.14 Demand and supply for a good 91 3.15 The inverse demand and supply functions 94 3.16 Comparative statics 97 3.17 Macroeconomic equilibrium 102 4 Quadratic equations 109 4.1 Introduction 109 4.2 Quadratic expressions 110 4.3 Factorizing quadratic expressions 112 4.4 Quadratic equations 114 4.5 The formula for solving any quadratic equation 116 4.6 Cases where a quadratic expression cannot be factorized 117 4.7 The case of the perfect square 118 4.8 Quadratic functions 120 4.9 The inverse quadratic function 122 4.10 Graphical solution of quadratic equations 123 4.11 Simultaneous quadratic equations 126 4.12 Graphical solution of simultaneous quadratic equations 127 4.13 Economic application 1: supply and demand 128 4.14 Economic application 2: costs and revenue 131 5 Some further equations and techniques 134 5.1 Introduction 134 5.2 The cubic function 135 5.3 Graphical solution of cubic equations 138 5.4 Application of the cubic function in economics 141 5.5 The rectangular hyperbola 142 5.6 Limits and continuity 143 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 2 Optimization with one independent variable 6 Derivatives and differentiation 165 6.1 Introduction 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 Introduction 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 Introduction 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 Introduction 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 Part 3 Mathematics of finance and growth 10 Compound growth and present discounted value 297 10.1 Introduction 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 Introduction 328 11.2 The exponential function y = 10x 330 11.3 The function inverse to y = 10x 331 11.4 Properties of logarithms 333 11.5 Using your calculator to find common logarithms 333 11.6 The graph of y = log10x 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 Introduction 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 = aerx 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 Introduction 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 371 13.5 Discrete growth 371 13.6 Continuous growth 374 13.7 Instantaneous and nominal growth rates compared 377 13.8 Semi-log graphs and the growth rate again 378 13.9 An important special case 379 13.10 Logarithmic scales and elasticity 381 Part 4 Optimization with two or more independent variables 14 Functions of two or more independent variables 389 14.1 Introduction 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 total differential, and applications 441 15.1 Introduction 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-z section 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 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 479 16.1 Introduction 479 16.2 The problem, with a graphical solution 480 16.3 Solution by implicit differentiation 482 16.4 Solution by direct substitution 485 16.5 The Lagrange multiplier method 486 16.6 Economic applications 1: cost minimization by the firm 490 16.7 Economic applications 2: profit maximization 496 16.8 A worked example 501 16.9 Some problems with profit maximization 502 16.10 Profit maximization by a monopolist 508 16.11 Economic applications 3: utility maximization by the consumer 510 16.12 Deriving the consumer's demand functions 512 17 Returns to scale and homogeneous functions; partial elasticities; growth accounting; logarithmic scales 519 17.1 Introduction 519 17.2 The production function and returns to scale 520 17.3 Homogeneous functions 522 17.4 Properties of homogeneous functions 525 17.5 Partial elasticities 531 17.6 Partial elasticities of demand 532 17.7 The proportionate differential of a function 534 17.8 Growth accounting 537 17.9 Elasticity and logs 539 17.10 Partial elasticities and logarithmic scales 540 17.11 The proportionate differential and logs 542 17.12 Log linearity with several variables 544 Part 5 Some further topics 18 Integration 551 18.1 Introduction 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 565 18.7 Economic applications 2: deriving total revenue from the marginal revenue function 567 18.8 Economic applications 3: consumers' surplus 569 18.9 Economic applications 4: producers' surplus 570 18.10 Economic applications 5: present value of a continuous stream of income 572 19 Matrix algebra 577 19.1 Introduction 577 19.2 Definitions and notation 578 19.3 Transpose of a matrix 579 19.4 Addition/subtraction of two matrices 579 19.5 Multiplication of two matrices 580 19.6 Vector multiplication 582 19.7 Scalar multiplication 583 19.8 Matrix algebra as a compact notation 583 19.9 The determinant of a square matrix 584 19.10 The inverse of a square matrix 587 19.11 Using matrix inversion to solve linear simultaneous equations 589 19.12 Cramer's rule 590 19.13 A macroeconomic application 592 19.14 Conclusions 594 20 Difference and differential equations 597 20.1 Introduction 597 20.2 Difference equations 598 20.3 Qualitative analysis 601 20.4 The cobweb model of supply and demand 605 20.5 Conclusions on the cobweb model 610 20.6 Differential equations 612 20.7 Qualitative analysis 615 20.8 Dynamic stability of a market 616 20.9 Conclusions on market stability 620 W21 Extensions and future directions (on the Online Resource Centre) 000 Appendix: Answers to chapter 1 self-test 623 Glossary 624 Index 632

Library of Congress Subject Headings for this publication:

Economics, Mathematical.