Essential Mathematics for Economic Analysis

Citation

Sydsæter, K., Hammond, P., Strøm, A., & Carvajal, A. (2021). Essential Mathematics for Economic Analysis (6th ed.). Pearson.

Chapter Summary

The sixth edition has twenty chapters in five parts, followed by an appendix on geometry, the Greek alphabet, a bibliography and solutions to the exercises. Pearson presents it as an extensive introduction to the mathematical tools needed to study economics at an intermediate level, progressing from elementary topics to more advanced ones. Every chapter mixes worked examples and exercises and ends with review exercises. This edition restructures the book: matrix algebra now comes before multivariable calculus, a new chapter on concave and convex functions of one variable and a new chapter on multiple integrals have been added, and the last four chapters are devoted entirely to optimisation, with constrained optimisation split into two chapters.

Part I, Chapters 1–5: Preliminaries

Chapter 1 (Essentials of Logic and Set Theory) introduces sets, logic, mathematical proofs and induction. Chapter 2 (Algebra) reviews real numbers, powers, fractions, inequalities, absolute values, sign diagrams, summation notation and the binomial formula. Chapter 3 (Solving Equations) covers equations with parameters, quadratic and some nonlinear equations, implication arrows and two linear equations in two unknowns. Chapter 4 (Functions of One Variable) treats linear models, quadratic, polynomial, power, exponential and logarithmic functions, and Chapter 5 (Properties of Functions) covers shifting graphs, inverse functions, graphs of equations, distance in the plane and general functions.

Part II, Chapters 6–11: Single-Variable Calculus

Chapter 6 (Differentiation) builds the derivative from slopes and tangents, with economic applications, a brief introduction to limits and the rules of differentiation. Chapter 7 (Derivatives in Use) covers implicit differentiation, the inverse function theorem, linear and polynomial approximations, Taylor's formula, elasticities, continuity, the intermediate value theorem, sequences and L'Hôpital's rule. The new Chapter 8 (Concave and Convex Functions) works from intuition to definitions, general properties, first- and second-derivative tests and inflection points; Pearson notes that it includes supergradients and subgradients. Chapter 9 (Optimization) treats extreme points, the extreme and mean value theorems and economic examples. Chapter 10 (Integration) covers indefinite and definite integrals, economic applications, integration by parts and substitution, and infinite intervals. Chapter 11 (Topics in Finance and Dynamics) covers effective rates, continuous compounding, present value, geometric series, mortgage repayments, the internal rate of return and first glimpses of difference and differential equations.

Part III, Chapters 12–13: Multi-Variable Algebra

Chapter 12 (Matrix Algebra) introduces matrices and vectors, systems of linear equations, matrix operations, the transpose, Gaussian elimination and the geometry of vectors, lines and planes. Chapter 13 (Determinants, Inverses, and Quadratic Forms) covers determinants, cofactor expansion, the inverse, Cramer's rule, the Leontief model, and, newly extended in this edition, eigenvalues, diagonalisation and quadratic forms.

Part IV, Chapters 14–16: Multi-Variable Calculus

Chapter 14 (Multivariable Functions) introduces functions of two and more variables, partial derivatives, convex sets, concave and convex functions, economic applications and partial elasticities. Chapter 15 (Partial Derivatives in Use) covers chain rules, implicit differentiation along a level curve, the elasticity of substitution, homogeneous and homothetic functions, linear approximations, differentials and differentiating systems of equations. The new Chapter 16 (Multiple Integrals) treats double integrals and integration over many variables.

Part V, Chapters 17–20: Multi-Variable Optimization

Chapter 17 (Unconstrained Optimization) moves from two choice variables to many, with necessary and sufficient conditions, linear models with quadratic objectives, the extreme value theorem, comparative statics and the envelope theorem. Chapter 18 (Equality Constraints) develops the Lagrange multiplier method, its interpretation, why it works, sufficient conditions and comparative statics. Chapter 19 (Linear Programming) uses a graphical approach and then duality theory and complementary slackness. Chapter 20 (Nonlinear Programming) handles inequality and nonnegativity constraints.

Key Concepts

Part I: Preliminaries

• Mathematical proof and induction: the forms of argument introduced at the outset and used throughout.
• Sign diagrams and summation notation: tools for analysing inequalities and handling sums.

Part II: Single-Variable Calculus

• Elasticity: the relative rate of change of one variable with respect to another, central to economic applications.
• Concave and convex functions: functions whose shape guarantees that stationary points are maxima or minima.
• Present value and internal rate of return: the valuation tools of the finance chapter.

Parts III–IV: Matrices and Multivariable Calculus

• Gaussian elimination and Cramer's rule: methods for solving linear systems.
• Quadratic forms and eigenvalues: tools for classifying second-order conditions.
• Homogeneous and homothetic functions: properties of production and utility functions.

Part V: Multivariable Optimization

• Envelope theorem: how the optimal value of a problem changes with its parameters.
• Lagrange multiplier: the method for equality-constrained optimisation, and its interpretation as a shadow price.
• Duality and complementary slackness: the links between a linear programme and its dual.

Critical Analysis

Strengths

• Broad and systematic. Twenty chapters run from set theory and logic to nonlinear programming, covering what Pearson describes as the mathematical tools needed to study economics at an intermediate level. Proofs and induction are introduced in Chapter 1, so students meet mathematical argument as well as techniques.
• Economic applications embedded in the mathematics. Sections on economic applications or examples appear in the chapters on differentiation, derivatives, optimisation, integration and multivariable functions, alongside elasticities, the Leontief model, the elasticity of substitution and comparative statics.
• Plenty of practice with solutions. Pearson stresses the numerous worked examples and exercises, the review exercises at the end of each chapter and the solutions to exercises in the book and online, which make it suitable for self-study. MyLab Economics is available for courses that use online assignments.
• Coherent sixth-edition structure. Moving matrix algebra ahead of multivariable calculus lets quadratic forms and eigenvalues be used in second-order conditions, and the four closing chapters give optimisation a part of its own.

Limitations

• Fast review for weak starters. Part I opens with logic and proofs and reviews algebra, equations and functions in four further chapters; students whose school algebra is rusty may need a gentler book first.
• Little on dynamics. Difference and differential equations receive only a few sections at the end of Chapter 11; dynamic optimisation is not covered.
• Long. Twenty chapters are more than a single course can cover; instructors need to select.

Where It Fits in the Course

Within this course list, Sydsæter and colleagues provide the main general-purpose textbook in mathematics for economists. Lis and Rosser's Basic Mathematics for Economists starts further back, rebuilding arithmetic and algebra and spending more space on financial mathematics; it is the better first book for students with weak school mathematics, and this one is the natural step after it. Ian Jacques's Mathematics for Economics and Business is another text aimed at business and economics students.

Chiang and Wainwright's Fundamental Methods of Mathematical Economics overlaps heavily and goes further into economic models and dynamics; students can use it as an alternative explanation of the same tools or as a next step for comparative statics and dynamic analysis. Brock and Malliaris's Differential Equations, Stability, and Chaos in Dynamic Economics is a graduate-level text on the dynamics that Chapter 11 only previews.

How to Use This Book

The book is aimed at first- and second-year economics students and at anyone preparing for intermediate microeconomics and macroeconomics. Basic school algebra and functions are enough to start, since Part I reviews them.

• First term: Part I (skimming familiar algebra) and Part II, with Chapters 6–9 as the core and Chapters 10–11 as the integration and finance component.
• Second term: Parts III–V. Chapters 12–15 and 17–18 are the core for intermediate economics; Chapter 16, Chapter 19 and Chapter 20 can be chosen according to the syllabus.
• Self-study: attempt the exercises before checking the worked solutions, and use the review exercises at the end of each chapter as a test before moving on.

Students who find Part I too fast should work through the early chapters of Lis and Rosser first; students preparing for graduate study should pay particular attention to Chapter 1 on proofs, Chapter 8 on concavity, and Chapters 17–20 on optimisation.