You are previewing Mathematics for Economics and Finance.
O'Reilly logo
Mathematics for Economics and Finance

Book Description

Mathematics has become indispensable in the modelling of economics, finance, business and management. Without expecting any particular background of the reader, this book covers the following mathematical topics, with frequent reference to applications in economics and finance: functions, graphs and equations, recurrences (difference equations), differentiation, exponentials and logarithms, optimisation, partial differentiation, optimisation in several variables, vectors and matrices, linear equations, Lagrange multipliers, integration, first-order and second-order differential equations. The stress is on the relation of maths to economics, and this is illustrated with copious examples and exercises to foster depth of understanding. Each chapter has three parts: the main text, a section of further worked examples and a summary of the chapter together with a selection of problems for the reader to attempt. For students of economics, mathematics, or both, this book provides an introduction to mathematical methods in economics and finance that will be welcomed for its clarity and breadth.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. 1. Mathematical models in economics
    1. 1.1 Introduction
    2. 1.2 A model of the market
    3. 1.3 Market equilibrium
    4. 1.4 Excise tax
    5. 1.5 Comments
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  7. 2. Mathematical terms and notations
    1. 2.1 Sets
    2. 2.2 Functions
    3. 2.3 Composite functions
    4. 2.4 Graphs and equations
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  8. 3. Sequences, recurrences, limits
    1. 3.1 Sequences
    2. 3.2 The first-order recurrence
    3. 3.3 Limits
    4. 3.4 Special cases
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  9. 4. The elements of finance
    1. 4.1 Interest and capital growth
    2. 4.2 Income generation
    3. 4.3 The interval of compounding
    4. Worked examples
    5. Main topics/Key terms, notations and formulae
    6. Exercises
  10. 5. The cobweb model
    1. 5.1 How stable is market equilibrium?
    2. 5.2 An example
    3. 5.3 The general linear case
    4. 5.4 Economic interpretation
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  11. 6. Introduction to calculus
    1. 6.1 The rate of change of a function
    2. 6.2 Rules for finding the derivative
    3. 6.3 Marginal cost as a derivative
    4. 6.4 The derivative of a composite function
    5. 6.5 The derivative of an inverse function
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  12. 7. Some special functions
    1. 7.1 Powers
    2. 7.2 The exponential function and its properties
    3. 7.3 Continuous compounding of interest
    4. 7.4 The logarithm function
    5. 7.5 Trigonometrical functions
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  13. 8. Introduction to optimisation
    1. 8.1 Profit maximisation
    2. 8.2 Critical points
    3. 8.3 Optimisation in an interval
    4. 8.4 Infinite intervals
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  14. 9. The derivative in economics—I
    1. 9.1 Elasticity of demand
    2. 9.2 Profit maximisation again
    3. 9.3 Competition versus monopoly
    4. Worked examples
    5. Main topics/Key terms, notations and formulae
    6. Exercises
  15. 10. The derivative in economics—II
    1. 10.1 The efficient small firm
    2. 10.2 Startup and breakeven points
    3. Worked examples
    4. Main topics/Key terms, notations and formulae
    5. Exercises
  16. 11. Partial derivatives
    1. 11.1 Functions of several variables
    2. 11.2 Partial derivatives
    3. 11.3 The chain rule
    4. Worked examples
    5. Main topics/Key terms, notations and formulae
    6. Exercises
  17. 12. Applications of partial derivatives
    1. 12.1 Functions defined implicitly
    2. 12.2 The derivative of an implicit function
    3. 12.3 Contours and isoquants
    4. 12.4 Scale effects and homogeneous functions
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  18. 13. Optimisation in two variables
    1. 13.1 Profit maximisation again
    2. 13.2 How prices are related to quantities
    3. 13.3 Critical points
    4. 13.4 Maxima, minima, and saddle points
    5. 13.5 Classification of critical points – introduction
    6. 13.6 The classification of critical points in general
    7. Worked examples
    8. Main topics/Key terms, notations and formulae
    9. Exercises
  19. 14. Vectors, preferences and convexity
    1. 14.1 Vectors and bundles
    2. 14.2 Prices and budgets
    3. 14.3 Preferences, utility, and indifference curves
    4. 14.4 Linear and convex combinations
    5. 14.5 Choosing optimal bundles
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  20. 15. Matrix algebra
    1. 15.1 What is a matrix?
    2. 15.2 Matrix multiplication
    3. 15.3 How to make money with matrices
    4. Worked examples
    5. Main topics/Key terms, notations and formulae
    6. Exercises
  21. 16. Linear equations—I
    1. 16.1 A two-industry ‘economy’
    2. 16.2 Linear equations in matrix form
    3. 16.3 Solutions of linear equations by row operations
    4. 16.4 The echelon form in general
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  22. 17. Linear equations—II
    1. 17.1 Consistent and inconsistent systems
    2. 17.2 The rank of a consistent system
    3. 17.3 The general solution in vector notation
    4. 17.4 Arbitrage portfolios and state prices
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  23. 18. Inverse matrices
    1. 18.1 The square linear system
    2. 18.2 The inverse of a square matrix
    3. 18.3 Calculation of the inverse
    4. 18.4 The inverse of a 2×2 matrix
    5. 18.5 IS-LM analysis
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  24. 19. The input–output model
    1. 19.1 An economy with many industries
    2. 19.2 The technology matrix
    3. 19.3 Why is there a solution?
    4. Worked examples
    5. Main topics/Key terms, notations and formulae
    6. Exercises
  25. 20. Determinants
    1. 20.1 Determinants
    2. 20.2 The determinant as a test for invertibility
    3. 20.3 Cramer’s rule
    4. Worked examples
    5. Main topics/Key terms, notations and formulae
    6. Exercises
  26. 21. Constrained optimisation
    1. 21.1 The elementary theory of the firm
    2. 21.2 The method of Lagrange multipliers
    3. 21.3 The cost function
    4. 21.4 The efficient small firm again
    5. 21.5 The Cobb–Douglas firm
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  27. 22. Lagrangeans and the consumer
    1. 22.1 Lagrangeans: a more general formulation
    2. 22.2 The elementary theory of the consumer
    3. 22.3 The price ratio and the tangency condition
    4. 22.4 The consumer’s demand functions
    5. 22.5 The indirect utility function
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  28. 23. Second-order recurrence equations
    1. 23.1 A simplified national economy
    2. 23.2 Dynamics of the economy
    3. 23.3 Linear homogeneous recurrences
    4. 23.4 Non-homogeneous recurrences
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  29. 24. Macroeconomic applications
    1. 24.1 Recurrence equations in practice
    2. 24.2 Oscillatory solutions
    3. 24.3 Business cycles
    4. 24.4 Improved models of the economy
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  30. 25. Areas and integrals
    1. 25.1 The consumer surplus
    2. 25.2 The concept of area
    3. 25.3 Anti-derivatives and integrals
    4. 25.4 Definite integrals
    5. 25.4 Standard integrals
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  31. 26. Techniques of integration
    1. 26.1 Integration by substitution
    2. 26.2 Definite integrals by substitution
    3. 26.3 Integration by parts
    4. 26.4 Partial fractions
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  32. 27. First-order differential equations
    1. 27.1 Continuous-time models
    2. 27.2 Some types of differential equations
    3. 27.3 Separable differential equations
    4. 27.4 A continuous-time model of price adjustment
    5. Worked examples
    6. Main topics/Key terms, notations and formulae
    7. Exercises
  33. 28. Second-order differential equations
    1. 28.1 Market trends and consumer demand
    2. 28.2 Linear equations with constant coefficients
    3. 28.3 Solution of homogeneous equations
    4. 28.4 Non-homogeneous equations
    5. 28.5 Behaviour of solutions
    6. Worked examples
    7. Main topics/Key terms, notations and formulae
    8. Exercises
  34. Solutions to selected exercises
  35. Index