Maths for Economics. Geoff Renshaw with contributions from Norman Ireland OXPORD UNIVERSITY PRESS. Second Edition

Transcription

1 Maths for Economics Second Edition Geoff Renshaw with contributions from Norman Ireland OXPORD UNIVERSITY PRESS

2 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 Some further equations and techniques 134 Part Two Optimization with one independent variable 6 Derivatives and differentiation Derivatives in action Economic applications of functions and derivatives Elasticity 256 Part Three Mathematics of finance and growth i 10 Compound growth and present discounted value The exponential function and logarithms Continuous growth and the natural exponential function 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 Maximum and minimum values, the total differential, and applications Constrained maximum and minimum values Returns to scale and homogeneous functions; partial elasticities; growth accounting; logarithmic scales 519

3 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) Appendix: Answers to chapter 1 self-test Glossary Index O

4 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 Addition and subtraction with positive and negative numbers 1.3 Multiplication and division with positive 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 Linear equations How we can manipulate equations Variables and parameters Linear and non-linear equations Linear functions Graphs of linear functions The slope and intercept of a linear function Graphical solution of linear equations Simultaneous linear equations Graphical solution of simultaneous linear equations Existence of a solution to a pair of linear simultaneous equations Three linear equations with three unknowns 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

5 o a LU ilu O 5.7 Application of the rectangular hyperbola in economics The circle and the ellipse Application of circle and ellipse in economics Inequalities Examples of inequality problems Applications of inequalities in economics 159 Part Two Optimization with one independent variable 6 Derivatives and differentiation The difference quotient Calculating the difference quotient The slope of a curved line Finding the slope of the tangent Generalization to any function of x Rules for evaluating the derivative of a function Summary of rules of differentiation Derivatives in action Increasing and decreasing functions Optimization: finding maximum and minimum values A maximum value of a function The derivative as a function of x A minimum value of a function The second derivative A rule for maximum and minimum values Worked examples of maximum and minimum values Points of inflection A rule for points of inflection More about points of inflection Convex and concave functions An alternative notation for derivatives The differential and linear approximation Economic applications of functions and derivatives The firm's total cost function The firm's average cost function Marginal cost The relationship between marginal and average cost Worked examples of cost functions Demand, total revenue, and marginal revenue The market demand function Total revenue with monopoly Marginal revenue with monopoly Demand, total and marginal revenue functions with monopoly Demand, total and marginal revenue with perfect competition Worked examples on demand, marginal and total revenue Profit maximization Profit maximization with monopoly Profit maximization using marginal cost and marginal revenue Profit maximization with perfect competition Comparing the equilibria under monopoly and perfect competition Two common fallacies concerning profit maximization 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 Elasticity Absolute, proportionate, and percentage changes The arc elasticity of supply Elastic and inelastic supply Elasticity as a rate of proportionate change Diagrammatic treatment Shortcomings of arc elasticity The point elasticity of supply Reconciling the arc and point supply elasticities Worked examples on supply elasticity The arc elasticity of demand Elastic and inelastic demand An alternative definition of demand elasticity The point elasticity of demand Reconciling the arc and point demand elasticities Worked examples on demand elasticity Marginal revenue and the elasticity of demand The elasticity of demand under perfect competition Worked examples on demand elasticity and marginal revenue Other elasticities in economics The firm's total cost function The aggregate consumption function Generalizing the concept of elasticity 292

6 PartThree Mathematics of finance and growth 10 Compound growth and present discounted value Arithmetic and geometric series An economic application Simple and compound interest Applications of the compound growth formula Discrete versus continuous growth When interest is added more than once per year Present discounted value Present value and economic behaviour Present value of a series of future receipts Present value of an infinite series Market value of a perpetual bond Calculating loan repayments The exponential function and logarithms The exponential function y= 10* The function inverse to/= 10 x Properties of logarithms Using your calculator to find common logarithms The graph of/= log 10 x Rules for manipulating logs Using logs to solve problems Some more exponential functions Continuous growth and the natural exponential function Limitations of discrete compound growth Continuous growth: the simplest case Continuous growth: the general case The graph of y = ae re Natural logarithms Rules for manipulating natural logs Natural exponentials and logs on your calculator Continuous growth applications Continuous discounting and present value Graphs with semi-log scale Derivatives of exponential and logarithmic functions and their applications The derivative of the natural exponential function The derivative of the natural logarithmic function 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 Logarithmic scales and elasticity Part Four Optimization with two or more independent variables 14 Functions of two or more independent variables Functions with two independent variables Examples of functions with two independent variables Partial derivatives Evaluation of first order partial derivatives Second order partial derivatives Economic applications 1: the production function The shape of the production function The Cobb-Douglas production function Alternatives to the Cobb-Douglas form Economic applications 2: the utility function The shape of the utility function The Cobb-Douglas utility function 434 APPENDIX 14.1: A variant of the partial derivatives of the Cobb-Douglas function Maximum and minimum values, the total differential, and applications Maximum and minimum values Saddle points The total differential of z = f(x, y) Differentiating a function of a function Marginal revenue as a total derivative Differentiating an implicit function Finding the slope of an iso-zsection A shift from one iso-z section to another Economic applications 1: the production function Isoquants of the Cobb-Douglas production function Economic applications 2: the utility function The Cobb-Douglas utility function Economic application 3: macroeconomic equilibrium 473 Lib I > m a o IT 2 CO

7 o a3a The Keynesian multiplier The IS curve and its slope Comparative statics: shifts in the IS curve Constrained maximum and minimum values 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 Deriving the consumer's demand functions 17 Returns to scale and homogeneous functions; partial elasticities; growth accounting; logarithmic scales 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 Partial elasticities and logarithmic scales The proportionate differential and logs Log linearity with several variables Part Five Some further topics 18 Integration ^ 18.2 The definite integral The indefinite integral Rules for finding the indefinite integral Finding a definite integral Economic applications 1: deriving the total cost function from the marginal cost function 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

8 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